Topology Optimisation of Structures Exposed to Large ...

Topology Optimisation of Structures Exposed to Large (non-linear) Deformations

Jesper Christensen September 2015

A thesis submitted in partial fulfilment of Coventry University’s requirements for the Degree of Doctor of Philosophy

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Acknowledgements I would like to thank my academic advisors Dr. Christophe Bastien and Professor Mike V. Blundell for their commitment, support, patience, expertise and interest in my work; for which I am very grateful. I would like to thank my wife Rachael, my parents Børge and Jane, my brother Lars, his wife Anja as well as my parents-in-law Kim and Jane for all of their support, patience and understanding before and during the development of my work. I would like to express my gratitude to the Faculty of Engineering and Computing at Coventry University and the many members of staff who have taken an interest and help support my work. Finally I would like to thank all contributors to the Low Carbon Vehicle Technology Project as well as the Towards Affordable Recyclable Future Low Carbon Vehicle (TARF-LCV) for providing info and background information and opportunities to promote and develop my research activities.

Acknowledgements

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Table of Contents

1.

Critical Review Document

2.

Contribution 1: Low Carbon Vehicle Technology Project

3.

Contribution 2: Effects of Roof Crush Loading Scenario Upon Body in White Using Topology Optimization

4.

7.

97

121

Contribution 4: The Feasibility of ESLM for BIW Roof Structure Development and Optimisation

6.

29

Contribution 3: Buckling Considerations and Cross-Sectional Geometry Development for Topology Optimised Body in White

5.

1

145

Contribution 5: Non-Linear Topology Optimisation Review, Development and Case Studies

171

Contribution 6: BEETS Software

314

Table of Contents

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Critical Review Document

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Acronyms BC

Boundary Condition

BEETS

Bi-directional Evolutionary Entropy Tabu search Simulated annealing

BESO

Bi-directional Evolutionary Structural Optimisation

BIW

Body In White

ESLM

Equivalent Static Load Method

FE / FEM / FEA

Finite Element / FE Model /FE Analysis

HEV

Hybrid Electric Vehicle

SIMP

Solid Isotropic Material with Penalisation

VDM

Variable Density Method

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Table of Contents Table of Contents .................................................................................................................................... 3 1. Introduction......................................................................................................................................... 4 2. Autobiographical Context for the Portfolio of Evidence ..................................................................... 5 3. Chronological Description of Portfolio Development ......................................................................... 7 3.1 Contribution 1 ............................................................................................................................... 7 3.2 Contribution 2 ............................................................................................................................... 8 3.3 Contribution 3 ............................................................................................................................... 9 3.4 Contribution 4 ............................................................................................................................... 9 3.5 Contributions 5 and 6 .................................................................................................................... 9 4. Evaluative Description of Originality ................................................................................................. 11 4.1 Basis for Selection of Outputs for Portfolio of Evidence ............................................................. 11 4.2 Contribution 1 ............................................................................................................................. 11 4.3 Contribution 2 ............................................................................................................................. 13 4.4 Contribution 3 ............................................................................................................................. 13 4.5 Contribution 4 ............................................................................................................................. 14 4.6 Contributions 5 and 6 ................................................................................................................. 14 5. Evaluative Review of Contribution to Subject Area .......................................................................... 16 6. Synthesis and Evaluation of Links between Outputs and Portfolio .................................................. 18 7. Critical Reflection of Candidate’s Development as a Research Practitioner .................................... 20 8. Contributions of Others .................................................................................................................... 22 9. Conclusions and Future Work ........................................................................................................... 23 References............................................................................................................................................. 27

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1. Introduction The purpose of this document is to describe the context for the overall PhD portfolio submitted, which has been developed over a 5 year period. Before the details are discussed it is important to list the proposed portfolio title as well as the aims and objectives of the research: Proposed title for the PhD by Portfolio Topology optimisation of structures exposed to large (non-linear) deformations. Research aim 

To investigate if topology optimisation can be used for the development of mechanical structures exposed to large (non-linear) deformations.

Research objectives 1. Analyse and critically evaluate the potential for using state of the art commercially available Finite Element software (and associated topology optimisation algorithms) for topology optimisation of structures exposed to large-deformations. 2. Based on 1 (where feasible) suggest, develop and critically appraise opportunities, methodologies and tools for enhancing the accuracy and precision of current state of the art topology optimisation algorithms for non-linear applications. 3.

Based on the outcomes of 1 and 2 define / refine and integrate a topology optimisation algorithm / methodology with enhanced levels of accuracy for structures exposed to large (non-linear) deformations.

4. Critically analyse and assess the outcomes of the tool developed in 3 to competing algorithms and “sound engineering judgement” using case-studies and objectively evaluate the potential for further development/refinement of the proposed algorithm / methodology.

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2. Autobiographical Context for the Portfolio of Evidence The PhD candidate of this Portfolio commenced his Bachelor of Science (BSc) studies in Industrial Engineering at Aalborg University, Denmark, in 2004 and was awarded the BSc in June of 2007. Following this, the candidate enrolled on a Master of Science (MSc) degree at Aalborg University in September of 2007 and graduating in January 2010 specialising in Design of Mechanical Systems. During the last 18 months of study the candidate attended 4 modules at Coventry University (CU), which reinforced links with the automotive industry, culminating with a 6 month placement at Jaguar Land Rover (JLR) where the candidate investigated the effects of seat foam material properties upon automotive crash safety. Following this the candidate completed his individual thesis (6 month full time) investigating the potential for modelling airbag tear seams through plate theory. The combination of these 3 events coupled with previous courses of optimization principles, Finite Element (FE) modelling and Structured Programming (SP) inspired a deep interest in the development and application of structural optimisation algorithms to non-linear (large deformation) problems. In this context the applicant successfully applied for a Research Associate (RA) role at Coventry University 2 months after graduating. The full-time RA role was dedicated to the Low Carbon Vehicle Technology Project (LCVTP), (LCVTP, nd) a multi-partner project funded by the European Regional Development Fund (ERDF), Advantage West Midlands (AWM) and a number of industrial partners including Tata Motors European Technology Centre and JLR. The candidate worked full time on this project for approximately 12 months specifically focussing on lightweighting of automotive crash structures. The outcome of this work, documented by contribution 1 to this portfolio see section 1, could in essence be considered to be the full time equivalent of the first year of a “conventional” PhD candidate. Based on the achievements of the LCVTP work the candidate was promoted to a permanent lecturing position at CU, with the title of “lecturer in Stress Analysis. Although this meant that the candidate would no longer be able to continue his research on a full time basis the development of the PhD portfolio did continue as the teaching activities were in relevant and associated fields; this ultimately led to the publications of contributions 2, 3 and 4 of this portfolio in October 2011, May 2013 and December 2013 respectively. Due to an increase in and variation of responsibilities of the candidate the work of the portfolio did continue at varying intensity, as the candidate was firstly promoted to the role of Senior Lecturer in Stress Analysis and subsequently to Senior Research Fellow. Although the candidate has maintained and developed a continuous portfolio of research relevant to this PhD portfolio throughout his career there has also been emphasis

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on other areas of research and development; a natural consequence of being an early career academic. The candidate has however made specific targeted efforts to maintain focus on his true passion; the development and implementation of structural optimisation algorithms. The most noteworthy achievements being presentations in the USA, Germany, at the House of Commons in Westminster and a publishing contract with Elsevier, scheduled for publication in December 2015.

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3. Chronological Description of Portfolio Development As indicated in section 2, the portfolio has been developed over a period of approximately 5 years. The portfolio contains a total of 6 contributions: 1. One html document containing models and reports from an ERDF / AWM / Industrial funded project entitled LCVTP. 2. One double-blind peer reviewed journal paper entitled “Effects of roof crush loading scenario upon body in white using topology optimisation” published in the International Journal of Crashworthiness in October 2011 (Christensen et. al., 2011). 3. One double-blind peer reviewed journal paper entitled “Buckling Considerations and CrossSectional Geometry Development for Topology Optimised Body In White” published in the International Journal of Crashworthiness in May 2013 (Christensen et. al., 2013). 4. One double-blind peer reviewed journal paper entitled “Feasibility of ESLM for BIW Roof Structure Development and Optimisation” published in the Journal of Mathematical Research and Applications in December 2013 (Christensen et. al., 2013a). 5. One monologue presenting the definition, programming and initial case studies of a non-linear FE based topology optimisation algorithm entitled Bi-directional Evolutionary Entropy Tabu search Simulated annealing (BEETS). 6. One software program, BEETS, for non-linear FE based topology optimisation as documented by contribution 5. The following sub-sections will detail the chronological description of the 6 contributions to the portfolio.

3.1 Contribution 1 The starting point was the LCVTP project; wherein the work was associated with Work-Package (WP) 7: lightweighting. The LCVTP project was extensive and contained a total of 15 WPs under simultaneous development; with an overall project objective of mapping out technical directions for future low carbon vehicles. Although the project did contain a demonstrator vehicle; a converted Jaguar XJ, the focus of the project was predominately on the development of techniques, tools and methodologies enabling future low carbon vehicles. This was to be achieved through a wealth of activities the most relevant ones in this context are materials, aerodynamic performance, powertrain selection and development in addition to lightweight

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design (WP 7). Due to the overall purpose of the project it was necessary to monitor and consider the project advances in the fields of material development, aerodynamic performance and powertrain selection as all of these would of course have an effect upon the lightweight design. As WP 7 was scheduled to run concurrently with the other aforementioned WPs it was evident that some fundamental assumptions had to be made in order to commence WP7. The ultimate scope of the PhD candidates’ work within WP7 was to: “Define an optimisation strategy / methodology for developing lightweight automotive vehicle structures capable of meeting current (2010) and future crash performance standards” Based on discussions with relevant project partners and WP owners the following list of assumptions was made: 1. The material properties were assumed to be isotropic. 2. Two competing drive train scenarios were considered: a. A fully electric vehicle. b. A range extended electric vehicle. 3. A case study would be based around the Tata Beacon concept vehicle. The work rapidly started to focus on topology optimisation technologies as it would be an ideal starting point for creating a truly lightweight concept vehicle. The initial idea was to investigate and critically analyse the opportunities for utilising existing (2010) state-of-the-art Finite Element Analysis (FEA) and optimisation principles to create a lightweight vehicle based upon an exterior styling envelope, i.e. a draft CAD design. The ideology was that implementing topology optimisation at such an early stage of the design cycle would enable the engineer to determine “ideal” locations for key electric powertrain components with an overall view of minimising the mass of the vehicle. The output of the work, as detailed in contribution 1, provided a draft methodology for this process including a substantial sensitivity study aimed at determining the robustness of the approach. The work also included a number of extended studies for example investigating manufacturing feasibility and full crash structure development; although these have not been submitted for consideration in this PhD portfolio.

3.2 Contribution 2 Contribution 2 is a direct extension of the LCVTP work documented by contribution 1, and followed approximately 6 months after the LCVTP program terminated. The work of contribution 2 also emphasises on sensitivity studies; it did however focus on a different

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aspect of vehicle crash structures; namely the roof structure. There were several reasons for this. Firstly, the roof crush is a very severe scenario dominated by buckling and structural instability. At the time there was also a lot of focus upon roof crush modelling and test procedures in general, including several publications which proposed changes to the Federal Motor Vehicle Safety Standards (FMVSS) standard for roof crush; FMVSS 216. The work reviewed the potential impacts of the proposed changes to the FMVSS 216 standard upon hybrid vehicle roof design, using topology optimization, and found the most severe scenario, i.e. combination of pitch and roll angles, matching that proposed in other papers investigating the physical test procedures, for example (Parent et. al., 2010).

3.3 Contribution 3 Contribution 3 was developed as a direct and natural extension of contribution 2, and was published approximately 18 months after contribution 2. The reason for the natural link between contributions 2 and 3 is that although contribution 2 answered questions relating to the probable effects of the proposed changes to FMVSS 216 it also raised some severe shortcomings of the approach, as shall be further discussed in sections 4.3 and 4.4. The shortcomings of the approach were know in advance, and the work leading up to the publication of contribution 3 was aimed at proving that “taken for granted” assumptions about the approach were not true in this context.

3.4 Contribution 4 The background work for Contribution 4 is based on some of the ideas developed during the LCVTP project, and was initiated at around the same time as contribution 3, this is the reason why contribution 4 was published approximately six months after contribution 3. Contribution 4 investigates a supposedly significantly different approach to non-linear topology optimisation; namely the use of equivalent static loading, as shall be further discussed in section 4.5.

3.5 Contributions 5 and 6 The underlying concepts leading to contributions 5 and 6 were initially developed during the LCVTP project and were slowly matured from that point onwards; i.e. from 2011. The further development of the ideas and principles were not fully addressed until the start of 2014, which included revisiting a number of previous papers (included in contribution 5), as well as

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reviewing contemporary developments of topology optimisation algorithms from 2011 onwards. As indicated in the preceding sections the contributions in this portfolio are thereby numbered chronologically in the order they were completed. It is worth noting that contributions 5 and 6 were developed simultaneously.

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4. Evaluative Description of Originality The purpose of this section is to review and evaluate the originality of each individual contribution to the proposed PhD portfolio. With the research aims and objectives listed in section 1 in mind, the next sub-section will explain the basis for which the contributions to the portfolio have been selected.

4.1 Basis for Selection of Outputs for Portfolio of Evidence The outputs have been carefully chosen to represent the overall evolution of the work firstly showing that “taken for granted” assumptions about linear static optimisation are not entirely supported by contemporary evidence, contributions 1, 2 and 4. This is followed by the development of additional tools for improving the optimisation outcomes, contribution 3. Finally, primarily based on the first 4 contributions a new and novel optimisation algorithm has been developed utilising aspects of multiple algorithms in addition to mathematical principles from other disciplines and finally integrated into a software tool, contribution 6. Contribution 5 contains the documentation for the software, as well as a series of case-studies critically assessing the software and overall methodology. Complementary to the work directly included in the portfolio an additional 4 journal papers and 6 conference papers have been published with the candidate as the main author. Additionally the candidate has given national and international presentations based on this work including the Set for Britain (SFB) 2013 competition in the House of Commons and to the German Association for Computational Mechanics in Dresden, Germany. Furthermore, the candidate has signed a publishing contract with Elsevier publishers for a text book entitled “Non-linear Optimization of Vehicle Safety Structures” as a co-author. Finally, the work has been extended through at least 5 individual Master’s thesis’, and more recently it has led to a spin-off PhD on Hybrid Optimisation partly funded by Coventry University and MIRA (£150k) in addition to being the backbone of the recently EPSRC/TSB awarded Virtual Exhaust Prototype (VExPro) project with a total project value of £780k, for which the candidate is the Principal Investigator (PI).

4.2 Contribution 1 The originality in terms of contribution to knowledge for contribution 1 is to be found in the application of the modelling and the depth to which the investigations are conducted. A general assumption is that linear topology optimisation techniques cannot be used for

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automotive crash structure development. The work undertaken in contribution 1 challenges this conception by critically reviewing the fundamental FE modelling techniques with a particular emphasis on model setup including mesh size and Boundary Conditions (BC) review. At the time very little detailed and in depth material was published about the use of topology optimisation for crash structure development. One of the most relevant and detailed investigations was the Future Steel Vehicle project (FSV, 2011) which demonstrated the use of topology optimisation as a tool in the design process. In terms of topology optimisation application and analysis the level of detail and depth of work is however considerably higher in contribution 1 than that of the FSV project. In particular contribution 1 provided original contributions to knowledge by determining the following: 

Commercially available linear static topology optimisation can advantageously be used for development of automotive crash structures based upon exterior styling envelopes.



A specific FE modelling methodology was developed to ensure the dynamic nature of front, rear, side impact, pole and roof crush accident scenarios can be appropriately considered.



The importance of the use of Inertia Relief (IR) boundary conditions as opposed to Single Point Constraints (SPC) was thoroughly investigated.



Sensitivity studies were completed in order to map and critically evaluate the general sensitivity of topology optimisation results to mass distribution and load application angles.



A general list of requirements for topology optimisation algorithms dealing with (nonlinear) structures exposed to large deformations and dynamic loading was defined, and current state-of-the-art algorithms were reviewed and their suitability for further development was identified.



A topology optimisation algorithm capable of conducting optimisation of non-linear structures was proposed. This method is based on the principles of linear static topology optimisation, thus “inheriting” the majority of advantages thereof. The proposed method shares some commonality with the Equivalent Static Load Method (ESLM) which has subsequently been introduced into multiple commercial FE suites such as Altair HyperWorks. This methodology is however not the one implemented in contributions 5 and 6 as shall be subsequently explained.

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4.3 Contribution 2 The originality in terms of contribution to knowledge for contribution 2 is to be found in the unique application focusing on roof crush modelling including the associated post-processing and identification of roof topology trends as a function of variations in the applied load angles. Although the paper utilises an unconventional approach, i.e. linear static topology optimisation, to assess the effects of various roof crush scenarios the determination of the worst case scenario corresponds to that identified by other papers (focusing on physical roof crush experimentation and accurate FE modelling). The paper did however also state that the mass of the roof structure does not change significantly as a function of load application angles; contradictory to the majority of other published material. Although this unconventional method was used contribution 2 was the first paper to demonstrate the potential use of this methodology in this particular context. The paper also highlighted some shortcomings of the approach taken; predominately the inability to incorporate buckling and localised crushing effects as parameters in the topology optimisation. The paper does make suggestions as to how this may be corrected; e.g. by the adaptation of homogenisation based optimisation, as also discussed in contribution 1.

4.4 Contribution 3 There are a number of original contributions to knowledge in contribution 3. Firstly the paper overall demonstrates that the proposed methodology, i.e. use of topology, shape and size optimisation can indeed be used to design a structure capable of meeting the structural performance criteria of a crash scenario. Although the resulting topologies are significantly different than “conventional” automotive roof structures it should be noted that the vehicle layout is significantly different, i.e. a Hybrid Electric Vehicle (HEV), see contribution 1, and that the roof crush loading conditions are also significantly different as discussed in contribution 2. The paper also demonstrates how buckling can be incorporated as a parameter in the optimisation process by using size optimisation subsequent to the topology optimisation. This is done by estimating local second moment of area values assuming a tubular cross-section. Following the size optimisation a “conventional” vehicle roof structure cross-section is used as the starting point of a shape optimisation step with an aim of “matching” the second moments of area found during the size optimisation step in order to produce a more feasible crosssection subject to maximum exterior dimension constraints.

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4.5 Contribution 4 The overall original contribution to knowledge of contribution 4 can be defined as a critical and detailed review of the Equivalent Static Load Method (ESLM) to vehicle crash structure topology and highly non-linear / dynamic problems in general. Since the time when contribution 1 was developed a number of “new” optimisation algorithms had been implemented into commercial FE software; the most prominent one being ESLM. As there are many parallels between ESLM and the methodology applied in contributions 1, 2 and 3 it was a relevant and important step to review the ESLM approach. Contribution 4 firstly compared the theoretical aspects of ESLM to the “conventional” Variable Density Method (VDM) and Solid Isotropic Material with Penalisation (SIMP) interpolation scheme, including potential optimisation parameters. Subsequently a large case study using identical FE models to those used in contributions 1, 2 and 3, was completed in order to benchmark the ESLM approach against the linear static VDM SIMP optimisation algorithm of HyperWorks. It was found that the ESLM approach had significant stability issues with the non-linear / large deformation nature of the models. A parameter study was therefore undertaken in an attempt to improve the stability of the models. This was somewhat successful; however a significant number of the ESLM models retained significant stability issues. At this point it was double checked that the level of non-linearity and dynamics of the models did not violate the stipulations for ESLM; via manuals and direct contact with the software vendors. When comparing the “successful” ESLM models to those using the VDM SIMP optimisation algorithm it was deemed that the substantial increase in CPU time associated with the ESLM approach did not provide a comparable increase of topology refinement justifying the substantial increase in CPU time and requirements.

4.6 Contributions 5 and 6 Based on the knowledge obtained in contributions 1 through 4 the aim of contributions 5 and 6 was to develop an optimisation algorithm specifically for structures exposed to large (nonlinear) deformations and dynamic loading. The original contribution to knowledge of contribution 5 is the definition, development, implementation and initial assessment of a topology optimisation algorithm capable of appropriately catering for (non-linear) deformations and highly dynamic loading. The original contribution to knowledge in contribution 6 is the software tool itself.

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As previously mentioned at this stage there had been significant developments in terms of topology optimisation availability in commercial FE codes since contribution 1 was developed. In addition to this the “original” literature review included in contribution 1 was conducted in a slightly more “narrow and industrial” context due to the environment in which it was completed than may be anticipated in a “true” research environment. As a consequence of these two facts it was decided to thoroughly revise and extend the literature review in contribution 5. The purpose was to propose and define an optimisation algorithm capable of appropriately catering for (non-linear) deformations and highly dynamic loading using Finite Element Analysis (FEA), whilst meeting the “non-linear topology optimisation criteria” list set forth in contribution 1. The outcome was that Bi-directional Evolutionary Structural Optimisation (BESO) was the most suitable candidate for this purpose. The algorithm does however have some severe limitations; which were addressed through the implementation of principles from other algorithms and mathematical principles relating to statistics and uncertainty. The ultimate outcome of this extended literature review was the definition of a flow chart for an optimisation algorithm entitled Bi-directional Evolutionary Entropy Tabu search Simulated annealing (BEETS). The BEETS algorithm was subsequently programmed into a PowerShell script, contribution 6, enabling it to utilise commercial FE software for structural analysis. At present this is limited to LS-Dyna keyword 971 input decks. Finally a series of case studies were completed in order to test the validity, relevance and extended potential of the algorithm for non-linear and dynamic structural optimisation problems. At this stage it should be highlighted that the purpose of this work was not to fully validate the BEETS algorithm; rather to initially investigate its potential for further development. It is concluded that the BEETS algorithm does demonstrate potential for further development, although a few aspects of the algorithm must be additionally reviewed before any extended development is undertaken.

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5. Evaluative Review of Contribution to Subject Area In the authors opinion the portfolio has successfully contributed to the subject area in a variety of different ways and with varying impact. The variation in the type of impact and the level of impact is not surprising given the varying nature of the work, from application and modelling techniques to theoretical developments and software implementation. The specific impact of contribution 1 is difficult to directly measure; this is predominately because the majority of work completed was bound by confidentiality in terms of practical applications (direct case studies relating to the Tata Beacon). The impact of the work could however be indirectly measured in terms of invited presentations and queries relating to the work. In this context a number of invited presentations to senior industrial experts were completed. These included invited presentations of the work and methodology to Craig Bonham, Chief Engineer Body & Exteriors at Tata Motors European Technical Centre (TMETC) in February 2011, Tayeb Zeguer, CAE Manager at Jaguar Land Rover in March 2011, and a presentation to Damian Dry, Principle Engineer Body & Exterior BIW Feasibility, SAIC in December 2011. The work also led to invited presentations at the 4th colloquium for the German Association of Computational Mechanics (GACM) in Dresden, Germany, in August 2011 as well as an invited presentation at the Altair Americas HyperWorks user Technology Conference (HTC) in Detroit, MI, USA in May 2012 including an associated video interview. Contributions 2 and 3 were both published in the International Journal of Crashworthiness which has an impact factor of 0.88. Although this may seem a relatively low impact factor the journal itself is very relevant to the work undertaken. Unfortunately it has not been possible to review the number of official citations of these articles based upon the Taylor and Francis publishers’ data. According to (Taylor & Francis, 2015) contribution 2 has had 138 views and contribution 3 has had 124 views (Taylor & Francis, 2015a), indicating a moderate amount of interest in the work. The moderate amount of interest in these particular publications was expected due to the controversy of the applied principles therein. Contribution 4 was published in the Journal of Mathematical Research and Applications (JMRA), the impact factor of which is unknown. JMRA is an open access journal which under normal circumstances charges a fee for publication; it should be noted that the fee was waived for contribution 4, i.e. no fee was paid for this publication. It has not been possible to obtain neither the number of citations nor the number of views for this particular article. Although the direct impact and reach of contributions 2, 3 and 4 are not well documented it should be

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noted that these 3 papers was used as examples of work in connection with applying for a book publishing contract with Elsevier publishers. The book entitled “Non-linear Optimization of Vehicle Safety Structures” is scheduled for publication in December 2015 (Elsevier, 2015), and will include aspects of the work presented in contributions all contributions to this PhD portfolio. The approval process for the book was subject to reviews of 4 external subjects experts who are unknown by name, but were listed with the following titles: 

Reviewer 1: Professor XXX, Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI USA.



Reviewer 2: Dr XXX, School of Mechanical Engineering, University of Leeds.



Reviewer 3: XXX, Professor of Solid Mechanics, University of Leeds.



Reviewer 4: Professor XXX, Technical University of Denmark.

As previously stated the publishing contract was successfully awarded with the book currently undergoing final reviews as of May 2015. Although the original contribution to knowledge of contributions 5 and 6 were clear to see the impact of these is low at present. The primary reason for this is simply that this work has not yet been published. There are a number of reasons behind this, primarily because the author intends to further develop and test the methodology before any proposed publications are submitted for review including validation studies. The first instance of public dissemination of the work included in contributions 5 and 6 will be the aforementioned text book.

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6. Synthesis and Evaluation of Links between Outputs and Portfolio All 6 outputs are linked as a continuous chronological evolvement of research into non-linear topology optimisation, as indicated through previous sections. The below paragraphs will further explain the direct links between all 6 contributions. 

Contribution 1, a HTML file, which documents the outcomes of the candidates’ involvement in the ERDF funded LCVTP project. This includes a novel and substantial investigation into the advantages, disadvantages, pitfalls and limitations of applying linear static topology optimisation techniques for developing structures exposed to large deformations, in this case automotive crash-structures. This includes aspects such as sensitivity to variations of input parameters, e.g. load application angles, material characteristics, boundary conditions etc. The work is based around Hybrid Electric Vehicle (HEV) architectures and was initially based upon approximately 332 FE models. The contribution also contains critical appraisal reports of the findings in addition to a literature review which ultimately outlines suggestions for further development of an algorithm for non-linear topology optimisation.



Contribution 2 is a published double-blind peer reviewed journal paper, which utilises the lessons learnt in contribution 1 to investigate the potential effects of proposed changes to simulation and testing standards (FMVSS 216) of automotive roof crush scenarios upon Body In White (BIW) / automotive roof structure design for HEVs. The paper not only focuses upon the variation of load cases but also draws conclusions on general “topology trends” in addition to critically appraising the suitability of the method applied.



Contribution 3 is also a published double-blind peer reviewed journal paper published in the same journal as contribution 2. Contribution 3 is in fact a direct continuation of contribution 2, as it picks up some of the areas for further research, analysis and development identified in contribution 2. More specifically contribution 3 demonstrates the feasibility / accuracy of the topologies obtained in 2 and proves, for the first time, that the resulting structures are in fact able to meet the structural requirements set forth in 2, thus proving that “taken for granted” assumptions about the limitations of linear static optimisation can be disproven; knowing the limitations is

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essential to the successful application of linear static topology optimisation to nonlinear problems. 

Contribution 4 is a published double-blind peer reviewed journal paper critically assessing and comparing the techniques developed in contributions 1, 2 and 3 to the competing (but very similar) ESLM which at the time had been released in commercial (FE) software. The paper highlights the issues with ESLM, and thoroughly discusses why the limitations of linear static optimisation also applies to ESLM, thus concluding that this approach is not a “stronger candidate” for non-linear topology optimisation when compared to the methodology developed and refined in contributions 1, 2 and 3.



Contribution 5 is a monologue containing three major parts. The first of which is a detailed literature review of optimisation algorithms and critically reviewing recent developments thereof. The overall aim of this part was to define a novel new optimisation algorithm fulfilling the requirements set forth based on contributions 1 4; with an ultimate aim of resolving the majority of limitations associated with the methodology used in contributions 1 - 4. The second part of the monologue defines and explains how the optimisation algorithm from the first part was integrated into software, contribution 6. The final part of the monologue contains a series of casestudies comparing results obtained using the developed software / algorithm to competing algorithms in order to critically assess the potential of the algorithm for non-linear topology optimisation.



Contribution 6 is the novel and new algorithm embedded in software documented by contribution 5.

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7. Critical Reflection of Candidate’s Development as a Research Practitioner At the early stages of the development of the PhD portfolio the candidate was highly influenced by his previous work as an engine fitter as well as his undergraduate and postgraduate engineering studies. This meant a strong predisposal towards quantitative research methods, which is also prominent throughout the majority of contributions included in the portfolio. Contribution 1 is a prime example of this where emphasis is on the percentage mass reduction of each model. Due to the exploratory nature of the research and the associated difficulties with conclusively demonstrating the benefits, relevance and indeed accuracy of the methods developed and applied, i.e. adaptation of linear static optimisation and the initial development of the BEETS method, has however “forced” the candidate to include aspects of qualitative research in combination with the quantitative approach. Although outside the “comfort zone” of the candidate, who prefers mathematical proof, this was necessary when considering time and capacity limitations. However even without these limitations the required intricacy of FE based topology optimisation algorithms for structures exposed to large (non-linear) deformations and dynamic loading, for example through “the curse of dimensionality”, see contribution 1, suggests that a fully qualitative based proof of the accuracy, validity and relevance of such an algorithm may be impossible to obtain. The first instance where this became apparent to the candidate was through the latter stages of developing contribution 1, forcing the candidate to “change” to a qualitative approach in order to further the research; the outcome of this being the “list of requirements” for a non-linear topology optimisation algorithm originally listed in contribution 1 and repeated in contribution 5. This mixed approach of qualitative and quantitative research methods and argumentation was adapted through the remainder of the portfolio development. The publication process of contribution 2 proved to be another significant landmark in developing the candidates mixed research approach; as one of the blind peer reviewers was very critical of the approach; citing the “unconventional” nature of the methodology as the main reason for recommending that the article not be published without a major rewrite. The author did however dispute this recommendation using a series of quantitative and above all qualitative measures. Eventually the editor chose to ignore the recommendation of that particular peer reviewer based on the evidence and argumentation set forth by the author, and thus the journal paper was published. As previously stated this was a significant turning point; reinforcing the potential powerful

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argumentations and evidence that can be found via qualitative research methods even in a mechanical engineering context. Contributions 3 and 4 were primarily based on quantitative research methods; whereas contributions 5 and 6 contained a mixed approach of qualitative and quantitative research methods. In summary the development of the portfolio has significantly changed the candidates’ perspective of the importance, efficiency and potential benefits of using a mixed approach of qualitative and quantitative research methods; however due to his mechanical engineering background the candidates’ preference remains to apply quantitative research methods where feasible.

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8. Contributions of Others The first 4 contributions in this portfolio all have named co-authors; the following will describe the specific contributions of those individuals: 

Christophe Bastien and Mike Blundell. Both Christophe and Mike have acted as line managers of the PhD candidate, and has supported the candidate in an identical way to that of PhD supervisors for a conventional PhD candidate. For example by reviewing ideas, results, overall research direction and internal review of papers prior to submission for external publication.



Pierre Andre Batt. Pierre Andre was a student intern at Coventry University from January to June of 2012. Pierre Andre worked under the supervision of Jesper Christensen and Christophe Bastien developing a shape optimisation tool featured in section 3 of contribution 3. It should however be noted that the actual application featured in contribution 3 was completed by the candidate, including writing the entire journal paper.

Contributions 5 and 6 were entirely and exclusively developed by the candidate Jesper Christensen.

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9. Conclusions and Future Work In the authors opinion this portfolio covers a substantial amount of work equivalent to that of a conventional PhD. The work contains a mixture of in-depth critical reviews, derivations, development, programming and verification / validation work both from theoretical as well as practical viewpoints. All 6 contributions have been carefully selected to clearly and concisely demonstrate the natural progression of the candidate as a research practitioner as well as the gradual and “natural” development of a number of significant contributions to knowledge. The work contained within the portfolio of evidence constitutes a continuous, novel and coherent contribution to the development of non-linear topology optimisation as well as non-linear optimisation in general; specifically the work within the portfolio addresses the following areas of “significant and original contribution to knowledge” as stipulated by the University regulations (CU, nd): o

Develop a new model, paradigm or conceptual framework and test it in application. (Contributions 5 and 6)

o

Successfully challenge an existing model or paradigm and show how it can be improved or why it should be discarded (in certain circumstances). (Contributions 1 - 4)

o

Show that “taken for granted” truths or assumptions are not substantiated by contemporary evidence. (Contributions 1 - 3)

o

Develop an existing methodology, form of enquiry or tool set for data collection, analysis, display or interpretation and show how its use in application proved to be superior in some circumstances compared to other tools etc. (Contributions 1 - 3 and 5 - 6)

o

Show limitations and errors in existing dominant methodologies, forms of enquiry or use of existing tools or analytical methods and the consequences for interpretation of previous structures. (Contributions 1 - 4)

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o

Add progressively to understanding of an issue, part of a field of a complex problem (e.g. multidisciplinary one), social or natural phenomenon or professional practice by a series of linked in-depth studies or experiments. (Contributions 1-5)

In the authors view the work included in the portfolio can in general be divided into two main aspects: A1. Development of advanced application methodologies and critical appraisal of linear static optimisation methods, contributions 1 - 4. A2. Definition, development, implementation, programming, application and critical evaluation of a non-linear dynamic topology optimisation algorithm, contributions 5 and 6. Using these “divisions of work” definitions as well as the areas of “significant and original contribution to knowledge” stipulated above general conclusions of the work can be drawn; starting with A1. In the authors opinion the work included in A1 has successfully challenged the existing “model” for FE based topology optimisation; the VDM SIMP approach. Contributions 1, 2 and 3 “demonstrated how this approach could be improved” by developing a specific methodology for appropriate creation of models to represent dynamic and non-linear load crash scenarios using a linear static solver. This included detailed review and critical appraisal of aspects such as BCs as well as creation and definition of equivalent static loads. This methodology was applied and refined using a large number of case studies focusing on lightweight Hybrid Electrical Vehicle (HEV) Body In White (BIW) / crash structure design and optimisation. Contributions 2 and 3 added further to this by demonstrating that the “final design” was actually capable of meeting the structural performance criteria stipulated for at the topology optimisation stage, thus showing that “taken for granted” assumptions about linear static optimisation were not substantiated for this particular application. In the authors experience the “taken for granted assumptions” about linear static optimisation varies significantly throughout the engineering community. Automotive engineers are generally highly sceptic about this approach when applied to structures exposed to large deformations and dynamic loading. This attitude does however seem to have changed somewhat with the introduction of ESLM. However, as shown in contribution 4, and also revised in contribution 5 the introduction of ESLM retains some of the significant issues relating to linear static FE based

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optimisation; predominately the continued use of Young’s modulus to dictate the material stiffness, regardless of stress and strain levels, most likely leading to an underestimation of the structural compliance when exposed to large deformations. Contribution 4 also demonstrated some of the “practical” pitfalls, i.e. limitations of the approach, including excessive CPU requirements with no significant improvement in terms of accuracy, precision or confidence of results thus concluding that the ESLM approach contributes very little in terms of advancing state-of-the-art topology optimisation algorithms for structures exposed to large (non-linear) deformations and dynamic loading, concluding that this approach should be discarded. Aspect A2 utilises the knowledge obtained through A1, in particular with respect to generating a “list of requirements” for a topology optimisation algorithm for structures exposed to large (non-linear) deformations and dynamic loading, as included in contributions 1 and refined in contribution 5. This list was subsequently used in contributions 5 and 6 to develop a “new model or conceptual framework” which was initially tested through fundamental case studies. The new optimisation algorithm was denoted BEETS; it was concluded that the cases studies using the algorithm demonstrate potential for further development although there are a few “teething” problems partially related to the algorithm and partially related to the method of implementation which must be investigated and resolved before additional development of the algorithm is undertaken. The author recognises that the portfolio contains no physical experimentation results which may prove critical for further development, validation and correlation of any algorithm developed. It is important to emphasise that any optimisation algorithm will only be as accurate as the information that is fed into it. FE based optimisation algorithms will therefore only be as accurate as the FE analysis that provides the input information. It is the authors’ opinion that the portfolio of evidence, i.e. the 6 contributions has “progressively added to the understanding of the complex issue of topology optimisation of mechanical structures exposed to large (non-linear) deformations and dynamic loading” thus fulfilling this criteria for a significant and original contribution to knowledge. As specified in section 4 the 6 contributions to this portfolio of evidence was carefully selected from a significantly larger bulk of work, which in combination with the work included in the portfolio has led to a number of additional achievements and activities including: a) A signed publishing contract with Elsevier; to be published in December 2015.

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b) A £780k TSB/EPSRC funded VExPro project, partnered with Unipart (EPSRC, 2015). c) A fully funded PhD student working on Hybrid Optimisation in collaboration with MIRA. d) A self-funded PhD student working on Holistic Optimisation. e) Further integration into the CU “Transport” Faculty Research Centre. Point e) above relates to the fact that the candidate has been selected to be part of a recently formed Faculty Research Centre (FRC) at Coventry University focusing on the overall theme of Transport and Mobility. It is the candidates’ intention to seek further funding for the continued development and critical evaluation / validation of the BEETS algorithm, in order to better assess its potential for topology optimisation of mechanical structures exposed to large (nonlinear) deformations and dynamic loading.

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References Christensen, J., Bastien, C., Blundell, M. V. (2011) “Effects of Roof Crush Loading Scenarios Upon Body in White using Topology Optimisation”. International Journal of Crashworthiness 17 (1), 29-38. Christensen, J., Bastien, C., Blundell, M. V., Batt, P.A. (2013) “Buckling Considerations and Cross-Sectional Geometry Development for Topology Optimised Body In White” International Journal of Crashworthiness 18 (4), 319-330. Christensen, J., Bastien, C., Blundell, M. V. (2013a) “The Feasibility of ESLM for BIW Roof Structure Development and Optimisation”. Journal of Mathematical Research and Applications 1 (2), 34-47. CU (nd) “Coventry University, PhD by Portfolio information pack” (online) available from: (20 May 2015) Elsevier (2015) “Elsevier store” (online) available from: < http://store.elsevier.com/Non-linear-Optimization-of-Vehicle-SafetyStructures/Jesper-Christensen/isbn-9780124172975/> (19 May 2015) EPSRC (2015) “Engineering and Physical Sciences Research Council” (online) available from: < http://gow.epsrc.ac.uk/NGBOViewGrant.aspx?GrantRef=EP/M506771/1> (20 May 2015) Low Carbon Vehicle Technology Project - LCVTP (nd) “Low Carbon Vehicle Technology Project” (online) available from: < http://www2.warwick.ac.uk/fac/sci/wmg/research/lcvtp/> (18 May 2015) Parent, D.P., Kerrigan, J.R. and Crandall, J.R. (2010) “Comprehensive computational rollover sensitivity study part 1: Influence of vehicle pre-crash parameters on crash kinematics and roof crush, ICRASH 2010 Conference, Washington, DC, USA, 22–24 September 2010. Taylor & Francis (2015) “International Journal of Crashworthiness” (online) available from: < http://www.tandfonline.com/doi/abs/10.1080/13588265.2011.625640> (19 May 2015)

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Taylor & Francis (2015a) “International Journal of Crashworthiness” (online) available from: < http://www.tandfonline.com/doi/abs/10.1080/13588265.2013.792442> (19 May 2015)

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CONTRIBUTION

1

HTML document Applied Research Documentation from the Low Carbon Vehicle Technology Project The printed is the non-linear topology optimisation report. All reports, associated models and full documentation of the contribution can be found on the enclosed CD

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Non-Linear Finite Element based Topology optimisation for LCVTP

Jesper Christensen Christophe Bastien

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November 2011

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Table of Contents Acronyms..................................................................................................................................... 33 1. Introduction............................................................................................................................. 35 1.1 Summary of LIFE TOPOPT.................................................................................................. 35 1.2 LIFE TOPOPT limitations .................................................................................................... 38 2. NOLIFE TOPOPT ....................................................................................................................... 42 2.1 Definitions and limitations ................................................................................................ 42 2.2 General topology optimisation theory and principles ...................................................... 43 2.3 The curse of dimensionality .............................................................................................. 45 3. Mathematical Based Optimisation Algorithms ....................................................................... 48 3.1 The Variable Density Method (VDM) ................................................................................ 48 3.2 Gradient based algorithms ................................................................................................ 57 3.2.1 The SIMP interpolation scheme ................................................................................. 58 3.2.2 The SRV constraint ..................................................................................................... 67 3.2.3 Design Sensitivity Analysis (DSA)................................................................................ 69 3.2.4 Other algorithms ........................................................................................................ 71 3.2.5 Feasibility of employing gradient based algorithms for NOLIFE TOPOPT .................. 72 3.3 Homogenisation Based Optimisation (HBO) ..................................................................... 76 3.3.1 Basic principles of HBO algorithms ............................................................................ 76 3.3.2 Feasibility of employing HBO algorithms for NOLIFE TOPOPT ................................... 80 4. Evolutionary Based Optimisation Algorithms ......................................................................... 82 4.1 General theory .................................................................................................................. 82 4.1.1 Evolutionary Structural Optimisation, ESO ................................................................ 82 4.1.2 Additive Evolutionary Structural Optimisation, AESO ................................................ 84 4.1.3 B-directional Evolutionary Structural Optimisation, BESO ........................................ 85 4.2 Feasibility of employing evolutionary algorithms for NOLIFE TOPOPT ............................. 86 5. NOLIFE TOPOPT Algorithm Selection ...................................................................................... 87 5.1 Key points of consideration for selection process ............................................................ 87 5.1.1 Linear and non-linear material behaviour ................................................................. 87 5.1.2 Combined topology-, shape- and size-optimisation .................................................. 87 5.1.3 Mesh dependency ...................................................................................................... 88 5.1.4 CPU cost and performance ........................................................................................ 88

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5.1.5 Feasibility of topology optimisation results ............................................................... 88 5.1.6 Convexity, concavity and global versus local extrema ............................................... 89 5.1.7 Constraints implementation ...................................................................................... 89 5.1.8 Material model ........................................................................................................... 89 5.1.9 Material microstructure and future potential ........................................................... 90 5.1.10 Current limitations ................................................................................................... 90 5.2 Selection of NOLIFE TOPTOP algorithm and future steps ................................................. 91 List of references ......................................................................................................................... 94

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Acronyms BC

Boundary Condition(s)

BIW

Body In White

Cijkl

Constitutive tensor (elasticity tensor)

{Ey}

Elasticity vector (design variable)

Eymin

Minimum allowable value for any entry of {Ey}

ε

Nominal (engineering) strain

εE

Nominal (engineering) elastic strain

εP

Nominal (engineering) plastic strain

{F}

External load (force) vector

F(x), F(ρ)

Objective function

FEA

Finite Element Analysis

gi(x), gi(ρ)

Constraint function

HEV

Hybrid Electrical Vehicle

IR

Inertia Relief

[K]

Global stiffness matrix

[K]-1

Inverse global stiffness matrix

LCVTP

Low Carbon Vehicle Technology Project

LIFE TOPOPT

LInear Finite Element based TOPology OPTimisation

MLS

Moving Least Squares (approximation technique / curve fitting)

n

Integer number

NCAP

New Car Assessment Program

NOLIFE TOPOPT

Nonlinear Finite Element based TOPology OPTimisation

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NURBS

Non Uniform Rational B Spline

PI

Performance Index

RR

Rejection Ratio

Sijkl

Compliance tensor

SIMP


σYield

Nominal Yield stress value

σUTS

Nominal Ultimate Tensile Strength

SPC

Single Point Constraint(s)

SRV

Sum of Reciprocal Values

  y

Design function (density as a function of (FE) element y)

{U}

Global displacement vector

W  x

Strain energy density function



And (mathematical symbol)



Column vector

Continuous interval or numerical value

 

Inverse matrix



Matrix

 

Row vector

1

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1. Introduction The purpose of this document is to document Coventry University's research activities into the field of Nonlinear Finite Element (NOLIFE) based TOPology OPTimisation (TOPOPT) carried out as part of Workstream 7.6 of the Low Carbon Vehicle Technology Project (LCVTP). This study originates from the outcomes of the previously conducted LInear Finite Element (LIFE) based TOPOPT study which intended to determine the ideal loadpaths for a Body In White (BIW) subjected to NCAP equivalent static load cases (crash scenarios). The LIFE study also included determining the ideal positions (with respect to minimising BIW mass) of key components for a Hybrid Electrical Vehicle (HEV); these included battery pack and range extender. In addition, the effects of Single Point Constraints (SPC) and Inertia Relief (IR) Boundary Conditions (BC) were also initially investigated; the final models all utilised IR boundary conditions. Furthermore, the LIFE TOPOPT study also included sensitivity analysis with respect to variation of the load (force) application angles. Finally, the possibilities of utilising the structural integrity of electrical component housing; e.g. the battery box, to meet the structural (crash) performance requirements of the BIW; partially under the assumption of zero failure rate electronics [1], [2] was investigated. The following section will briefly summarise the setup and overall outcome of the LIFE TOPOPT. Please note that full documentation for the LIFE study is available in separate documents.

1.1 Summary of LIFE TOPOPT In order to appreciate the limitations of the obtained results from the LIFE TOPOPT it is important to firstly define the applied loadcases which are illustrated in Figure 1.

Figure 1, illustration of loading scenarios.

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The magnitude of the equivalent linear static forces applied to the individual loadcases illustrated in Figure 1 are listed in Table 1. Table 1, applied forces. Applied force magnitude,

#

Load case

Applied force

1

Pole impact

300 kN

300 kN

2

Side barrier impact

300 kN

300 kN

3

Roof crush (A-pillar)

2.5  g  EVM

29.5 kN

4

Low speed rear impact

150 kN

150 kN

5

High speed rear impact

60  g  EVM

707 kN

6

Front impact(ODB)

60  g  EVM

707 kN

EVM = 1200 kg

In Table 1 g is the gravitational acceleration (9.82 m/s2) and EVM is the Estimated Vehicle Mass which has been set to 1200 kg. The force magnitudes listed in Table 1 are distributed over areas with the dimensions specified in Table 2.

#

Contribution 1

Table 2, initial barrier dimensions. Barrier dimensions (mm) Load case (width  height)

1

Pole impact

Ø300

2

Side barrier impact

1560  475

3

Roof crush (A-pillar)

250  250

4

Low speed rear impact

250  250

5

High speed rear impact

1375  250

6

Front impact(ODB)

1000  650

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The material used for the LIFE TOPOPT was a mild grade steel, the properties of which are listed in Table 3. Table 3, material properties. Young’s’ Modulus (MPa) 210000 Poisson’s’ ratio

0.3

Density (kg/m2)

7850

Please note that the material properties listed in Table 3 represent those of a linear elastic isotropic material; the significant importance of this will become increasingly underlined in section 1.2. For clarification; the objective of the LIFE TOP OPT was to minimise the BIW mass; subjected to the constraints of specified (maximum) displacement criteria. The LIFE TOPOPT study involved defining and solving approximately 400 different TOP OPT models. The final model (BIW) which was chosen as the basis for the continued work of other research activities within workstream 7 such as the manufacturing, battery box interaction and crash structure development was model 65. The LIFE TOPOPT results of model 65 are illustrated in Figure 2.

Figure 2, ISO metric view of model 65 from LIFE TOPOPT.

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Figure 3 illustrates a cross-sectional view of the "floor" of model 65 from the LIFE TOPOPT.

Figure 3, cross sectional view of model 65 from LIFE TOPOPT. The topology illustrated in Figure 3 can readily be used as a basis for discussing the limitations of LIFE TOP OPT; this will be the focus of attention in the following section.

1.2 LIFE TOPOPT limitations In order to fully appreciate the limitations of LIFE TOPOPT it is important to initially define the meaning of LIFE (LInear Finite Element) based topology optimisation. This simply means that the topology optimisation algorithm is based upon FE analysis that assumes linear elastic material behaviour as well as linear geometry behaviour. Furthermore, the utilised material behaviour has been defined as isotropic; this means that Hooke's generalised law, i.e. equation (1), is applicable.

 ij  Cijkl   kl

(1)

 ij  Sijkl   kl

Equation (1) is written in classic tensor notation; using the summation index. C ijkl represents the materials constitutive tensor ( material elasticity tensor) and Sijkl represents the material compliance tensor. In other words a linear relationship between stress  ij and strain  ij is assumed in LIFE TOPOPT. This effectively eliminates any dynamic effects (from loading), as any variation of e.g. force magnitude between time = t1and time = t2 will simply be equivalent to scaling the resulting strains with a factor equal to that of the difference in force magnitude between t1and t2. Consequently any dynamic loading can be converted to equivalent static

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loading and implicit Finite Element Analysis (FEA) can be employed. This means that the governing equilibrium equation between the external forces {F} and the resulting displacements {U} can be solved by inverting the stiffness matrix [K], expression (2).

F    K   U  (2)

 K   F   U  1

By utilising expression (2) a stable FEA solution can be obtained by means of implicit (linear) FEA. If the influence of equation (1) and expression (2) are taken into consideration when observing the LIFE TOPOPT results displayed in Figure 2 and Figure 3 the limitations of LIFE TOPOPT become increasingly clear. The topology of Figure 2 and in particular Figure 3 clearly displays a widespread use of triangles (triangulation). This makes perfect sense given equation (1) and (2) combined with the fact that the stiffest geometry in static solid mechanical terms is indeed the triangle. Therefore an optimum structure (in terms of high stiffness to mass ratio) based upon linear static and linear elastic material modelling will naturally attempt to utilise triangulation. However, an actual crash event is inherently a dynamic scenario (loadcase). When taking a more holistic approach to the task of designing a BIW it soon becomes evident that the dynamic effects from crash scenarios are imperative to estimating the severity of occupant injuries during such scenarios. In order to minimise the potential occupant injuries it is very important to have controlled (plastic) deformation of e.g. the front crash structure (denoted 1 in Figure 3), in order to dissipate parts of the “crash energy”. At this point the widespread triangulation becomes a problem; simply because of the triangles relatively high resistance to buckling which undoubtedly will affect the crushability of e.g. the front crash structure, consequently leading to a negative influence upon the crash performance of the BIW. The question of whether the above mentioned controlled deformations need to be elastic and plastic or solely elastic is of course partially material dependent. The vital crash structure, e.g. the front crash structure, of a typical modern day passenger vehicle primarily consists of high strength steel, i.e. a ductile material with relatively high ultimate tensile strength (  UTS ). The reason for this, along with the significance of utilising plastic deformations, can be appreciated by observing Figure 4 which represents a typical stress (  ) strain (  ) curve, i.e. a Prandtl diagram, of a structural steel.

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Figure 4, Prandtl diagram of a typical structural steel. In Figure 4

 E is the elastic strain,  p is the plastic strain and ε is the total (mechanical) strain;

any thermal strain is ignored. Please note that although Figure 4 represents tensile strain it can also in principle be used for compressive strains. Figure 4 can thus be used to further substantiate the need for including the non-linear material behaviour into the topology optimisation as follows. The green area in Figure 4 represents the elastic strain energy UE whilst the red area represents the (potential) plastic strain energy UP; the total strain energy U = UE + UP is equal to the anticipated magnitude of the aforementioned dissipated energy from the crash (impact). By observing Figure 4 it is obvious that the majority of the total strain energy U originates from the plastic strain energy (UP) contribution thereby underlining the importance of including the plastic region of the Prandtl diagram into the topology optimisation. In the plastic region the material behaves in a non-linear manner, consequently, Hooke’s generalised law, i.e. equation (1) is invalid in this region, and the solution strategy of equation (2) becomes unfeasible. Figure 4 can also be used to appreciate the reasoning for utilising high strength steel for crash structures. Simply by increasing the magnitude of σUTS and / or σYield the respective areas underneath the graph will increase thus increasing the potential strain energy, i.e. energy dissipation during crash scenarios.

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To summarise LIFE TOPOPT only considers the material behaviour in the green area in Figure 4, whereas NOLIFE TOPOPT would consider the material behaviour in the red area of Figure 4. Please note that the magnitude of the red area represents the maximum potential UP. In actual design situations this area is likely to slightly decrease as it is not customary to design to strain levels in excess of the respective

 UTS value. It could be argued that the dynamic effects of

the crash scenario has been partially taken into consideration by utilising “equivalent static loading” in addition to the choice of employing IR constraints as opposed to SPC in connection with the LIFE TOPOPT. However, as found during the LIFE TOPOPT study these effects are indeed localised to the areas in the immediate vicinity of the original SPC, and will not be further discussed in this report. The above discussion underlines the two main reasons for the vast potential of introducing NOLIFE TOPOPT into the BIW design process. To summarise these are: 1. The necessity to include dynamic effects. 2. The necessity to include non-linear material behaviour. If the two points above are successfully included into the topology optimisation; the resulting topology is very likely to provide a significantly improved starting point for the continued BIW design process thereby leading to a decrease in overall BIW design time and an increase in the ratio of BIW crash performance relative to BIW mass. With the main reasons for introducing NOLIFE TOPOPT established the next chapter will introduce the basic principles of NOLIFE TOPOPT and initially highlight some of the advantages and disadvantages of some of the potential optimisation algorithms to be utilised.

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2. NOLIFE TOPOPT The purpose of this chapter is to firstly define NOLIFE TOPOPT in the context of this report; and secondly to present some of the possible theories and algorithms that can potentially be used to solve the governing task.

2.1 Definitions and limitations The purpose of this section is simply to define NOLIFE TOPOPT; the assumptions, limitations and overall scope of the work documented by this report. The definition of Non-Linear Finite Element based TOPology OPTimisation (NOLIFE TOPOPT) solely consists of non-linear material behaviour. The primary assumptions relating to the behaviour of the material utilised for the remainder of this report are: 1. Isotropic material behaviour. 2. Negligible strain rate effects. 3. Negligible change in material characteristics as a function of temperature change. 4. No viscoelastic effects. 5. Only linear stress-strain behaviour as a function of geometry. Based upon the definition and assumptions listed above a “typical” material that would satisfy these criteria could be the plastic region of the majority of structural steels. These are often used for primary crash structures within the automotive industry for an operating temperature T as listed in the interval (3) below. However employing such a material in connection with topology optimisation will include both linear and non-linear material behaviour as indicated in Figure 4; this must also be taken into account during the following discussions.

50C  T  450C

(3)

The above assumptions does thus eliminate the usage of e.g. composite materials as these are in general not isotropic; however this does not mean that non-isotropic materials will be completely omitted from the remainder of the report; they will however primarily be indirectly considered during the subsequent discussions. Future BIW designs will very likely include multiple materials, e.g. steel and aluminium in combination with e.g. plastics and composite materials therefore it will become increasingly important to consider these in future topology optimisation algorithms; however this exceeds the scope of this project.

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In addition other considerations such as multicriteria and multiphysics or multidisciplinary optimisation will moreover not be directly considered during the subsequent analyses and discussions; as the primary aim of this report is to illuminate the possibilities of implementing non-linear material behaviour in connection with topology optimisation. With the above definitions, assumptions and limitations presented the next section will provide a general introduction into the general theories, principles and techniques of topology optimisation.

2.2 General topology optimisation theory and principles The purpose of this section is to give a general introduction into the general theory and principles of topology optimisation. In general, topology optimisation can be defined as: Topology optimisation seeks to find the optimum distribution of material within a given design volume. This is performed with respect to achieving a predefined objective. The optimisation process can be controlled by defining one or several constraints, relating to e.g. displacement or stress values. Topology optimisation has evolved rapidly in light of the vast development of FEA. For 3D topology optimisation in connection with Finite Element Analysis (FEA), the objective can e.g. be achieved by varying the mass density of the individual elements within the design volume. For 2D this is equivalent to varying the thickness of the elements. For 1D elements the objective is typically achieved by varying the cross sectional dimension(s). The outcome of a typical FEA based topology optimisation is thus an indication of relative material (mass) density throughout the design volume. This outcome does therefore not contain high detailed information with respect to e.g. panel thickness' or cross sectional geometry in general. If desired, this information can subsequently be obtained by the application of e.g. shape-, size- and topography- optimisations. An overall flow chart of the general FE based topology optimisation is illustrated in Figure 5.

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Figure 5, flow chart of general topology optimisation. The flow illustrated in Figure 5 can briefly be summarised as: 1. Setup FE model of design volume, including loadcase(s), objective and constraint(s). 2. Conduct FE analysis 3. Evaluate the objective 4. Evaluate the constraints. 5. If the constraints are not violated check if the objective has converged. 6. If the constraints are violated, check if the maximum number of iterations is obtained. 7. If the maximum number of iterations is not obtained; adapt the FE model and return to step 2. By utilising a commercial available FE solver the actual steps of the optimisation algorithm can be defined as those enclosed in black boxes in Figure 5 above. This effectively means that 5 steps needs to be defined in order to construct the optimisation algorithm. Three of these five steps are relatively straightforward to define; for example step 4 and step 6 can be obtained by means of a simple Boolean function. Checking the convergence of the algorithm, i.e. step 5, is also relatively simple to implement once the objective has been evaluated. By comparing the value of the objective of the current iteration to that of the previous one (or multiple) a decision can once again be made based upon a Boolean function. This could e.g. be based upon the percent wise change of the objective between iterations.

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Another option could be to base the decision upon the slope of a graph of the objective vs. iteration such as the one illustrated in Figure 6.

Figure 6, objective value vs. iteration number. Leaving step 2 (FEA) aside, the remaining and initially more cumbersome and time consuming steps are steps 2 "Evaluate objective", and step 7, "Adapt FE model". These two steps are very closely linked, and heavily dependent upon the chosen optimisation algorithm. Potential ways of defining and implementing these will be thoroughly discussed in the following sections. Before that is done it is essential to further elaborate and present some of the general theory of topology optimisation, and introduce the different types of algorithms and approaches this will be the focus of attention in the following section.

2.3 The curse of dimensionality The general topology optimisation problem based on Finite Element calculations can essentially be defined as scaling a property, e.g. the mass of the individual elements by a factor of either 1 (for full mass) or 0 (for no mass), this type of problem is often referred to as a 0-1 problem. The selection process of how these element masses are scaled can be defined in a vast number of different ways. Consider the Finite Element mesh illustrated in Figure 7, where the blue dots represent the nodes and the black lines represent the elements. Please note that neither constraints nor applied loads are illustrated as these are irrelevant in this context.

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Figure 7, FE mesh example, iteration i. The numbers displayed in Figure 7 indicate the outcome of the current optimisation iteration i.e. the mass scale factors. For a small number of elements such as that illustrated in Figure 7 this may seem as a simple problem with relatively few solutions, due to the fact that the scaling factor is discrete, as it can only assume the value 0 or 1. However, the number of possible solutions rapidly becomes very large. This can be appreciated by considering the truss structure in Figure 8, which consists of 11 "beams".

Figure 8, truss structure. The task of minimising the mass and maximising the stiffness of the truss structure in Figure 8 can simply be treated as a discrete optimisation problem where 0 means that a specific beam should be removed and 1 means that the beam in question should remain. The simple optimisation problem in Figure 8 has a total of 2048 possible solutions, equation (4)

211  2048

(4)

If 1 beam in Figure 8 represents 1 element in the FE mesh it is obvious that the number of possible solutions rapidly becomes infinite; e.g. 25 elements leads to approximately 33.5 million possible solutions while 30 elements leads to in excess of 1 billion possible solutions.

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As soon as the number of elements exceed 1,000 the number of possible solutions effectively becomes infinite. At this stage aspects such as symmetry can be taken into consideration. Nevertheless; if 3 plane symmetry is utilised for a 3D component the total number of elements before an infinite number of solutions can be found is a mere 8,000. Medium to large FE models often have in excess of 100,000 elements which underlines the magnitude of the optimisation task. This means that for the vast majority of models it will be unfeasible albeit impossible to evaluate all possible solutions; which must be taken into account when defining the optimisation algorithm. The primary purpose of the optimisation algorithm is essentially to define how the discrete scale factors are decided, i.e. assigning 0 or 1 values to elements. This process can, as previously mentioned, be implemented in an abundance of different algorithms. In general, these algorithms are divided into two different types, which are: 1. Mathematical based algorithms. 2. Evolutionary / heuristic algorithms. The overall approach, fundamental principles and implementation of the two types of algorithms differ significantly. The first type, the mathematically based algorithms focus upon mathematically formulating and evaluating the objectives by means of objective functions , which are indeed functions of the design variables, i.e. the aforementioned scale factors. This means that the mathematical approaches are often relatively complex; however they often ensure stable and controllable optimisation runs. The evolutionary based algorithms take on a more direct approach to optimisation. These are, as listed above often based upon heuristics and are usually "easier" implemented than the mathematical algorithms. Typical issues with evolutionary methods are related to stability, robustness and controllability of the optimisation process. The fundamental principles, typical implementation, advantages, disadvantages and overall suitability of the individual algorithms for employment in NOLIFE TOPOPT will be thoroughly discussed in the following sections; starting with the mathematical based algorithms.

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3. Mathematical Based Optimisation Algorithms As stated in the previous chapter, mathematical based optimisation algorithms includes mathematical definition of a single objective function (5).

F  x

(5)

In (5) x can represent any quantity, for linear static topology optimisation x often represents Young's modulus, E or the mass density, ρ, i.e. the objective function is often as defined in (6). Please note that the objective function F(x) can in general contain more than 1 design variable, i.e. x1, x2 etc. For the purpose of the current discussion only 1 design variable will be included.

F  x  F   

(6)

In expression (6) and the previous chapter the mass density was used as an example of a design variable, this is because the majority of algorithms within the field of topology optimisation utilise a Material Distribution Concept (DMC), i.e. the material distribution is the main parameter (variable) during the optimisation. Two main variations of the DMC is generally used; these are: 

Variable Density Method (VDM)



Homogenisation Based Optimisation (HBO)

Very generally the difference between VDM and HBO can be described as the latter utilises the material microstructure during the optimisation whereas the former does not. The VDM approach is the one used in most present day commercially available FE optimisation software, such as Altair HyperWorks. In the following section the VDM method will be discussed. The HBO approach will be discussed in section 3.3, further information regarding HBO can also be found in [3] and [4].

3.1 The Variable Density Method (VDM) For topology optimisation the function F in equation (6) often represents stiffness or compliance (strain energy). In these cases it may seem odd to define e.g. the strain energy as a function of the mass density, as there is no direct link between these two with respect to mechanics of materials. Nevertheless, ρ is used as the design variable in most applications of topology optimisation.

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The reason for this is essentially related to human intuition, as most persons find it "easier" to interpret a plot of density, as opposed to a plot of stiffness. Initially the relationship between density and (material) stiffness is defined as equation (7). 0 Cijkl  y     y   Cijkl

(7)

where :

  y   0,1 In equation (7) Cijkl(y) represents the material properties of the individual element y. For a linear elastic isotropic material Cijkl only contains two independent variables Young's modulus E, and Poisson's ratio ν. For the purpose of conducting linear static (isotropic) topology optimisation Poisson's ratio generally assumes a constant value, as it is sufficient to vary E. 0 Cijkl represents the "original" (and constant) material properties.

Finally, the relative element/material "density" ρ(y) represents the design function, i.e. the function that determines the scaling factor attributed to the individual element y. For the discrete formulation ρ(y) can therefore only assume the values 0 and 1. Thus equation (7) states that a linear relationship exists between ρ and the constitutive tensor Cijkl. Due to the linear relationship between ρ and Cijkl (E) the aforementioned respective plots of "stiffness" and "density" will be identical, provided that the 0-1 discretisation is used. Equation (7) is only valid in the linear elastic region of the Prandtl diagram (stress-strain curve) as Hooke's law is only valid in that region. This limitation is ignored for the time being, as the initial purpose is to discuss the governing optimisation principles. It should however be noted that in general Cijkl is applicable to linear isotropic and linear non-isotropic materials. In general the constitutive tensor is not used to describe non-linear material behaviour, instead this is often mathematically described by a strain energy function. In addition to the objective function (6) one or several constraint functions, or constraints also need to be defined, (8).

gi     Vali

gn     Valn

(8)

Vali and Valn in (8) represents the limiting values of the respective constraints. Please note that the individual constraint functions in (8) can consist of equalities as well as inequalities.

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The number n of constraints (or constraint functions) in (5) is in principle infinite. In practice n is however most often a finite number, typically in the range of 1 - 20. These constraints often relate to displacement values, stress levels or a maximum mass value of the component. If n is set to zero then no constraints will exist, and the topology optimisation will most likely remove all material from the model, leaving a "blank" design volume; which obviously makes no sense with respect to the "practical" overall aim of e.g. minimising the mass of a given component subjected to external loading. This type of optimisation is known as unconstrained optimisation. Due to the NCAP requirements for the BIW unconstrained optimisation is of no interest in this context. Only constrained optimisation, i.e. optimisation where maximum allowable values (e.g. displacement) are active are relevant. This seemingly trivial observation does underline an important aspect of the mathematical based optimisation algorithms; namely that they strictly work based on mathematical equations. Consequently extra care must be taken when defining the objective and associated constraint functions, in order to ensure a feasible, i.e. "practical" outcome of the topology optimisation algorithm is obtained. This obviously relates to the example listed above, but also to more general aspects such as one of the main reasons for conducting this research, namely the fundamental differences between static and dynamically loaded structures. With the objective function F(ρ) and the constraints gi(ρ) defined, the optimisation task can e.g. be defined as expression (9).

minimise F    (9)

subject to : g1     Val1 ; g 2     Val2

; g3     Val3

In (9) the objective function may represent the compliance (strain energy); therefore the optimisation task is to minimise the objective function. The task could have also been to e.g. maximise the stiffness. For a linear elastic body not subjected to initial strains nor stresses; the linear strain energy, U, can be defined as equation (10).

U









1 1 T T      dV       E     dV  2 2

(10)

In (10) dV indicates that the function must be integrated over the volume of the body (component) in order to obtain the combined strain energy. By combining equation (7) and equation (10) a fully defined objective function can thus be found, equation (11).

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



1 T     E       dV  2

F  U  

(11)

It is important to note that equation (11) is a linear function, if all of the constraint functions are also linear the optimisation problem can be described as linear. This is in principle independent of whether linear of non-linear FE is utilised. Providing that all functions in (9) are linear the optimisation problem will per definition be linear. If any of the functions in (9) are non-linear then the optimisation problem is inherently non-linear. This means that there is a fundamental difference between non-linear FE based topology optimisation and non-linear optimisation in general. The effects of this relates to how the specific optimisation problem is formulated. With equation (9) and equation (11) the optimisation problem becomes a "simple" mathematical problem of minimising a function subject to given constraints. As it was previously discussed, it is not feasible to calculate all possible permutations; simply because the number of these rapidly becomes infinite as the number of elements increase. By virtue of equation (9) the problem can thus seemingly be solved by means of differentiation, as the extrema (maximum and minimum points) of a continuous function can be found via differentiation, (12).

F     F      F   

(12)

Thus the extrema of F(ρ) can be found by solving equation (13).

F     0

(13)

Thereby the optimisation problem of (9) can theoretically be solved numerically by e.g. utilising a simple Newton-Raphson algorithm. Subsequently the constraints can be used to eliminate any violating solutions and validate the remaining one(s). All algorithms utilising the steps represented by equation (12) and equation (13) is generally referred to as gradient based optimisation algorithms. Unfortunately equation (13) does not immediately solve the overall problem at hand, due to the assumptions related to the gradient based approach of (12) and (13). These are: 1. The function must be continuous. 2. The function must be convex or concave to ensure that global extrema are found.

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In order to better appreciate what this means, please see Figure 9.

Figure 9, example of a continuous convex function. Figure 9 represents a continuous convex function, with the global minimum point indicated, when dealing with this type of problem it is often referred to as convex (or concave) optimisation. Please note that Figure 9 essentially represents a 1D optimisation problem, i.e. a single variable. For multivariable optimisation (not to be confused with multicriteria optimisation) such as the topology optimisation of a BIW, Figure 9 would have as many axes as parameters, i.e. elements. The fact that the function in Figure 9 is continuous simply means that F(x) is defined for every value of x. From observing the graph in Figure 9 it is straightforwardly clear where the global minimum of the function is. This can, as previously explained, be found by means of differentiation, because the extrema are located at the only position where the slope of the tangent is equal to zero. The gradient based approach will thus also reveal the two global maximum points of the function in Figure 9; however these are easily separated from the minimum simply by observing the F(x) values. The gradient based approach applied to a convex (or concave) function such as the one in Figure 9 will only yield the three extrema. In the case of Figure 9 these are the two global maximum points and the singular global minimum point. Thus the optimal solution to (9) is relatively easily obtained, even for an infinitely large number of possible permutations i.e. elements. However, if the function is non-convex (or non-concave) the global extrema become increasingly difficult to distinguish from the local ones. In order to appreciate this, please see Figure 10.

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Figure 10, example of non-convex continuous function. In Figure 10, the global minimum is once again distinguishable, but the gradient based approach will also yield the additional local extrema. These additional points could also be "filtered" by observing the F(x) values. Figure 10 does however only display two additional extrema which are easily filtered. A non-convex function does in principle add an infinite number of extrema complicating the mathematical problem. Returning to the first assumption of the gradient based approach, it is immediately obvious that the objective function F(ρ) is not a continuous function as it was defined in section 2.3 that the "scale factors", i.e. ρ for the individual element can either assume the value of 0 or 1. Consequently F(ρ) is only defined as F(0) and F(1). Making the objective function discrete, i.e. not continuous. Therefore the differential of F(ρ) only exist as defined in expression (14).

F      F   0  F  1

(14)

As the objective function is not continuous it is not possible to utilise the gradient based approach of (12) and (13). It is however highly desirable to be able to utilise this approach; this can be made possible by allowing that ρ(y) in equation (7) to assume continuous values between 0 and 1, this is in mathematical terms known as "relaxation" [5]. The introduction of relaxation may initially seem to be a disadvantage as the number of possible solutions thus becomes infinitely larger. However, as previously discussed, the possible number solutions to the discrete (initial) optimisation problem (such as the one in Figure 8) becomes infinite anyway, when the number of elements exceeds approximately 8000.

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Thus, from a mathematical viewpoint it can be argued that the increase in the number of possible solutions is of less importance, when compared to the advantages of the gradient based solution methodology. It was previously discussed that extra care must be taken with respect to "practical" aspects of the optimised design. An obvious example of this is the introduction of the continuous ρ(y) function; which undoubtedly will lead to solutions suggesting element densities of e.g. 0.5. To better comprehend this issue, consider the FE model illustrated in Figure 11.

Figure 11, example design volume for topology optimisation. In Figure 11 the red elements indicate a ρ(y) value of 1. The model contains 45 elements; it is constrained at all nodes (blue) on the left hand side and subjected to forces on all nodes of the right hand side. Suppose the objective of the topology optimisation is to minimise the mass of the model; whilst the constraints consist of a maximum displacement magnitude of the nodes where the load is applied, in addition to a maximum stress value applied to all elements in the model. A possible solution to this topology optimisation problem could be the one illustrated in Figure 12.

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Figure 12, potential solution to topology optimisation problem. The "white" elements in Figure 12 denote elements where ρ(y) is equal to zero. The topology illustrated in Figure 12 is thus a mathematically viable solution; however there are some severe limitations from an engineering point of view. First of all the structure in Figure 12 would be very difficult, if not impossible to manufacture. Furthermore, the "real world" version of the topology in Figure 12 would be inherently unstable. This is not the case of the FE model, simply because of the FE discretisation itself. In this context all forces, displacements etc. are transferred via the nodes. Hence, the compliance of the topology in Figure 12 is almost as low as that off Figure 11, however the "mass" (i.e. red elements) of Figure 11 is 45 whereas it is only 23 for Figure 12. The problem illustrated by Figure 12 is a very common problem of topology optimisation, and is often referred to as "checkerboard effect". The checkerboard effect does in principle arise from the introduction of a continuous ρ(y) function. This fact can be appreciated by acknowledging that the displacements, stiffness values etc. in a FE model are all calculated for the individual nodes, not the individual elements. Figure 13 illustrates a close-up of 4 elements in Figure 12.

Figure 13, close-up of elements from topology in Figure 12.

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Elements 1 and 4 in Figure 13 have a ρ(y) value of 1, whereas elements 2 and 3 have a ρ(y) value of 0. The green node belongs to all 4 elements, therefore the ρ(y) value in this node will be equal to the average from the 4 elements, i.e. a value of 0.5. Another example of the complications of introducing a continuous ρ(y) function can be seen in Figure 14, which also represents a possible solution to the model in Figure 11.

Figure 14, 2nd potential solution to topology optimisation problem. The pink elements in Figure 14 are representative of a ρ(y) value of e.g. 0.5. Figure 14 contains a total of 45 elements thus leading to a "mass" of Figure 13 equal to approximately 23, i.e. similar to that of Figure 12. However, the feasibility of the proposed topology of Figure 14 is equally "unfortunate" as that of Figure 12. The solution based upon Figure 14 is simply to replace the material with one that has a mass density ρ and a Young’s' modulus (E) scaled by 0.5 to that of the "original" material in the model. This obviously defeats the purpose of topology optimisation. Thereby it can be concluded that topology optimisation with continuous scale factors is not always likely to return the "optimum" solution from an engineering viewpoint. The introduction of a continuous ρ(y) function is convenient from a mathematical solution viewpoint; however it conveys additional "practicality" problems such as the checkerboard effect. The checkerboard effect can be relieved by using higher order elements, i.e. elements with e.g. parabolic shape functions, this does however imply a substantial increase in CPU time. Typical values indicate an increase in CPU time by a factor of 16 or above. This renders this particular solution to the checkerboard effect infeasible; as this increase in CPU time is unacceptable for larger models. Therefore it becomes necessary to introduce another method of "steering" the

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optimisation towards a 0-1 solution. Some of the most widely used methodologies currently used to achieve this will be discussed in the following section. Other methods, such as the usage of continuous material distribution has recently been proposed, suggesting that the mesh dependency and checkerboard effects can be completely avoided [6]. This is the one of the main advantages of HBO optimisation which will be discussed in section 3.3. Before the HBO methodology is presented, the next sections will be aimed at discussing the VDM approach to topology optimisation, including a popular technique used to minimise the checkerboard effect.

3.2 Gradient based algorithms The purpose of this section is to briefly introduce the governing theory of some of the mathematical gradient based optimisation algorithms, i.e. the algorithms utilising expressions (12) and (13). The vast majority of mathematically based optimisation algorithms are gradient based, or alternatively partially utilise gradient based techniques. In general, these types of algorithms can be applied to any optimisation problem as long as the objective can be defined mathematically. An example of the objective function was previously defined in equation (11). According to the "correct" mathematical definition expressions (12) and (13) can subsequently be used to solve the optimisation problem. In the case of FE based topology optimisation the approach is however not as straightforward as that. This is because equation (11) contains the





strain vector ε which is found via FEA, i.e. ε  E  . This means that a FE analysis must be conducted for each iteration of the optimisation, somewhat "deviating" from the mathematical gradient based approach to optimisation. This type of algorithm contains considerable more "finesses" than described in this paper, these primarily relate to the specifics of the selection technique which "selects" which elements to change / adjust between individual iterations of the optimisation. This is a central part of the detailed understanding of the gradient based optimisation algorithms, however it is not essential in understanding / appreciating the overall / general methodology of the algorithms, which is the main purpose of this paper. Therefore the specifics relating to the "selection process" has been omitted. Many articles exist on the topic, e.g. [7] and [8]. It is useful to note that the "selection process" is represented by "Adapt FE model" in Figure 5. For mathematical algorithms this typically involves Design Sensitivity Analysis (DSA), which will be discussed in section 3.2.3.

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3.2.1 The SIMP interpolation scheme The Solid Isotropic Material with Penalisation (SIMP) method is an interpolation scheme aimed at "steering" the topology optimisation towards the 0-1 solution by "penalising" intermediate values of the design variable(s) e.g. ρ in expression (9). Thereby the number of unfeasible designs created by e.g. the checkerboard effect can be minimised. This is done by defining a mathematical "power law" linking ρ and E. These were previously linked in expression (7). The SIMP interpolation scheme utilises the "power law for representation of elasticity properties", as proposed by [7], expression (15). 0 Cijkl  y     y   Cijkl p

where :

(15)

p  1    y   0 :1 At first glance expression (15) is very similar to expression (7). There are however subtle differences which hold significant influence. Firstly it should be noted that ρ(y), i.e. the design function, can now assume all values in the continuous interval between 0 and 1. Secondly expression (15) contains the power factor p which theoretically can assume all values above or equal to 1.0. If p is indeed set to 1.0 expression (15) does essentially become equal to expression (7) overlooking the fact that the former is defined for a continuous ρ(y) interval whereas the latter is not. Figure 15 is a graphical illustration of the influence of the penalisation factor, p, of the SIMP method as defined in expression (15). 1

p=1

Cijkl(y)

0.8

p=2

0.6

p=3 0.4

p=6

0.2

p = 10

0 0

0.2

0.4

0.6

0.8

1

ρ(y) Figure 15, graphical illustration of effects of penalisation factor, p.

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From Figure 15 it can be seen that if p equals 1 there is in essence no penalisation for intermediate ρ(y) values, as the relationship between ρ and Cijkl is linear. As p increases the penalisation for intermediate densities becomes increasingly severe, thus "forcing" the optimisation towards a 0-1 solution. For high p values, e.g. 6 or 8 it is obvious that any element with a ρ(y) value less than approximately 0.6 essentially has zero stiffness, which is in line with the overall purpose of the method. However the graph also displays a very steep slope between ρ(y) values in the interval between 0.8 to 1. This is highly likely to lead to severe instability problems, primarily from an optimisation viewpoint, but also potentially from an FEA view point. The underlying issue is that a relatively minor variation of ρ(y) can lead to a significantly large change in stiffness; thus making the model very (if not extremely) sensitive to minor variations of ρ(y). The model illustrated in Figure 16 can be used to demonstrate the influence of the penalisation factor, p upon the topology optimisation results.

Figure 16, illustration of FE model. The model illustrated in Figure 16 is exposed to a load of 1000N applied at a single node, and is rigidly clamped partially along the left hand side. The FE model utilised 2D, i.e. shell, elements and was discretised using 5000 quad (square) elements with linear shape functions. The objective function was to minimise the mass, only a single constraint was applied. This constraint denoted the maximum displacement magnitude of the node where the external force, i.e. the 1000N, was applied. Figure 17 shows the effects of varying the penalisation factor p, in equation (15), between 1.0 and 6.0.

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Figure 17, topology optimisation results, varying penalisation factor, p - 5000 quad elements. The results illustrated in Figure 17 indicates that the value of the penalisation factor may hold an influence upon the topology optimisation results, it is evident that "higher" values of p may lead to a significant change of the resulting topology. Figure 17 may also be used to appreciate the "practical" effects of the penalisation factor, as the topology optimisation is "forced" towards a 0-1 solution. This can be substantiated by noting that "p = 6" in Figure 17 primarily consists of "red" elements, i.e. ρ(y) = 1., whereas "p=1" contains significantly more "yellow", "green" and "blue" elements indicating intermediate ρ(y) values. Figure 18 represents results from topology optimisations of the same basic problem as Figure 17 does, i.e. the one illustrated by Figure 16. The only difference between Figure 17 and Figure 18 is that the latter was re-meshed using 4450 triangular (2D) elements with linear shape functions, i.e. the results of Figure 17 and Figure 18 represents models with approximately the same number of elements.

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Figure 18, topology optimisation results, varying penalisation factor, p - 4450 triangular elements. By comparing the results of Figure 18 to those of Figure 17 it can be seen that the choice of element type also has an effect upon the results, however this is not likely to be directly linked to the SIMP interpolation scheme, but rather an effect of general FE discretisation and modelling techniques. Furthermore, it can be seen that the majority of load paths in Figure 17 are also present in Figure 18, the most significant differences occur in the "p = 6" plots, which can be attributed to the previously discussed "instability" issues that can be appreciated by viewing the trends of the graphs in Figure 15. It could be argued that the above comparison is biased, as the element size in Figure 18 is a factor of 1.5 times larger than that of Figure 17. In addition, the re-meshing also implies that the node locations of the respective models has changed, possibly "skewing" the resulting topologies. Therefore, a 3rd series of models has been solved, these are based upon the same mesh used for the results in Figure 17, however the quad elements have been split, resulting in

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10000 triangular elements with identical nodal locations for these models (Figure 17 and Figure 19), excluding the added nodes in the latter.

Figure 19, topology optimisation results, varying penalisation factor, p - 10000 triangular elements. By comparing Figure 19 to Figure 18 it can be seen that similarities between the respective topologies exist. However, significant variations also exist, thereby highlighting the mesh dependency issue. Furthermore, if Figure 17 is compared to Figure 19 it can also be seen that the respective topologies are similar in general. Further examination clearly revel differences between the respective results, underlining the previous statement that the outcome of a topology optimisation merely serves as a suggestion as to the location of the loadpaths. The "p = 6" results of Figure 17, 18 and 19 are all significantly different, thus substantiating the instability issue relating to "high magnitude" p values in connection with the SIMP interpolation scheme, Figure 15.

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The above findings thus indicate that the selection of the p value is not always straightforward as no single answer exists to this problem. Guidelines such as the ones proposed by [7] does however exist, expression (16) .

4   2 2D: p  max  ; 0 0   1  1   

(16)

  1  0  3  1  0   3D: p  max 15  ;   0  7  5  2  1  2 0      In expression (16) ν0 represents Poisson's ratio of the linear elastic isotropic material in

question. According to (16), the minimum (integer) value of the penalisation factor p should be equal to or larger than 2 for shell (2D) and 3 for solid (3D) elements, given a ν0 equal to 0.3. It is therefore seemingly beneficial to choose a relatively high value of p in order to further penalise the intermediate ρ(y) values, enforcing the 0-1 solution, thereby ensuring a more refined topology. However, as previously discussed, there are also significant issues associated with choosing a large p value. This can also be appreciated by studying the graph for p equal to 10 in Figure 15. With the general principle of the SIMP interpolation scheme presented the focus can now turn to the implementation of SIMP into FE analysis. The SIMP interpolation scheme is used in many state of the art FE codes and software suites, an example of which is Altair HyperWorks. The general strengths of the SIMP method lies within the efficiency, versatility and primarily stability of the scheme. This is particularly exploited in relation to linear FE based analysis. The most efficient (and stable) way to conduct linear static FE analysis is to invert the global stiffness matrix [K] and subsequently employ matrix and vector multiplication to solve the overall governing equation. The overall steps of this approach are demonstrated in (17).

F   K  U 

K 

1

 K   F  U  1

(17)

The stiffness matrix [K] contains data from the constitutive tensor Cijkl, and is hence a function of material properties (as well as geometry). The resulting global displacement vector {U} in (17) is rapidly obtained once [K]-1 has been calculated. The most CPU intensive step of the process in (17) is to invert the stiffness matrix [K] which theoretically needs to be repeated if the geometry or material of the model changes. Consequently the stiffness matrix should in principle be reassembled and re-inverted after each individual iteration of the topology optimisation. However, the SIMP scheme is often

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implemented in a way that eliminates the need to repeat these steps. This is partially due to the isotropic and indeed linear nature of the material. Due to this behaviour of the material, combined with the previous statement that Poisson's ratio, ν, remains constant throughout the optimisation, the inverted stiffness matrix [K]-1 can be defined as equation (18).

K 

1

 E y    K  E y 

1

(18)

In equation (18) [K]-1 is the "original" inverted stiffness matrix of (17); {Ey} is an "Elasticity

 

1

vector" containing Young's modulus for the individual (y) element and  K E y  contains



algebraic expressions including Young's modulus for the individual (y) element. Thereby an "updated" inverted stiffness matrix can swiftly be constructed for iteration n+1 of the topology optimisation by utilising the {Ey} vector obtained from iteration n. A similar approach to updating the (inverted) stiffness matrix could be taken for a non-isotropic material; however it is not possible for non-linear materials, as a simple scaling function such as (18) would not suffice, this is related to the fact that non-linear material behaviour is most often mathematically formulated by the usage of e.g. strain energy functions. Thus it can be avoided to re-assemble and invert the stiffness matrix due to changes in material properties during the (linear FE based) topology optimisation. However, if elements are removed from the design volume, i.e. if changes of geometry occur, the stiffness matrix will once again require reassembly and inverting. This is however not the case for most SIMP based optimisation algorithms, as elements are in general not deleted, consequently there is no need to re-assemble the stiffness matrix, and the entire procedure of (17) only requires to be conducted once; and it is therefore only required to redefine [K]-1 in accordance with (18) between iterations of the topology optimisation. Thereby the most CPU intensive step of linear FEA is not repeated, and the CPU time required for the overall topology optimisation process is significantly reduced. There is however complications associated to this approach; namely that the resulting model will contain artificial stiffness, as the "geometry", i.e. nodes, will remain within the model throughout the entire topology optimisation process. This is however only deemed as a minor issue, as the vast majority of stiffness (in most cases) originates from the material properties, not the geometry itself. Thus the mathematical optimisation algorithm utilising the SIMP interpolation scheme becomes very effective as the CPU intensive step of inverting the stiffness matrix does not require repetition. In addition the FE analysis is generally very stable

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and robust. Finally, the SIMP based optimisation algorithm is in general also very stable, as no elements are deleted. Coincidentally the algorithm is also very "flexible", if required, as the relative stiffness of any element can be extremely low in one iteration and very high in the next one; simply by varying {Ey} in (18). At this point it should be noted that the stiffness value, i.e. the Ey value in (18) of any element is never set to zero for the purpose of conducting topology optimisation in connection with the SIMP method. This is simply because if any Ey value is set equal to zero [K] effectively becomes singular. Consequently no inverted matrix exists; (17) cannot be used and [K] will need to be reassembled. Alternatively Ey values may be set very low to represent a "0 element", therefore reassembly is not required. This leads to the interval of ρ(y) (in equation (15)) to be as defined in equation (19).

0    y  1

(19)

The lack of the need to re-assemble the stiffness matrix can be further exploited, as it has previously been in the initial topology optimisation stages of the LCVTP, by employing Inertia Relief constraints to the optimisation thereby further decreasing the necessary CPU time if multiple load cases are including in the topology optimisation model. One of the major problems associated with the 0-1 problem, which is not directly solved by the introduction of SIMP is the lack of existence of solutions to the problem. This fact is well document by e.g. [9], [10] and [11]. The significance of this may be cumbersome to understand. However, an appreciation of this may be obtained by the following. Firstly it is important to acknowledge that the 0-1 optimisation problem of e.g. the BIW is in principle a vast series of topology optimisation problems, i.e. an optimisation problem to choose either 0 or 1 for each element. In other words if an FE model contains 100,000 elements then the optimisation may be defined as 100,000 individual topology optimisation problems that combined make up the design volume, e.g. the BIW. However these individual problems are necessarily related. Therefore the overall objective function which regards the problem as being global was introduced. This objective function is thus a generalised function, which is also often referred to as a distributed problem in mathematical terms. Solutions to this type of problem may in many cases not exist; this is what is what is discussed in more detail in [9], [10] and [11] . As a direct result of this the topology optimisation becomes mesh dependent. This fact can be appreciated by observing Figure 20, which represents three "identical" topology optimisation

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problems, where the difference between them is mesh size. The models were solved using HyperWorks which utilises linear static topology optimisation and the SIMP scheme. The penalisation factor in Figure 20 (i.e. p in equation (15)) was set to 2.0.

Figure 20, example of mesh dependency. Figure 20 thus clearly demonstrates the above claim that the topology optimisation process is mesh dependent; this fact is also substantiated by Figure 17 and Figure 18. The mesh dependency will not be further discussed here, however modelling techniques, including mesh sizes will be thoroughly discussed in chapter 5. It should be noted that the mesh dependency of topology optimisation is a widespread problem, which theoretically is applicable to all FE based optimisation algorithms due to the discretisation used for FEA. Furthermore Figure 20 also reveals that even though the SIMP interpolation scheme penalises the intermediate ρ(y) values it does not eliminate the previously mentioned checkerboard effect. It can also be seen that the mesh size (i.e. average element size) has a noticeable influence upon the "severity" of the checkerboard effect. Therefore extra care must always be taken when meshing the design volume prior to conducting a topology optimisation. Thereby the general presentation of the SIMP interpolation scheme is complete. The following section will discuss the governing theory of the SRV constraint.

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3.2.2 The SRV constraint The Summation of Reciprocal Values (SRV) constraint is another methodology that can be used to "steer" the continuous 0-1 optimisation problem towards integer values. The following description of the SRV constraint is primarily based upon [12]. The SRV constraint can serve as an alternative as well as an addition to the SIMP interpolation scheme as presented in the previous subsection. This is simply because the SRV is a constraint, which is imposed on the optimisation problem, in the same manner as e.g. a limiting displacement value would be. The initial governing theory relating to the definition of an objective function, using continuous ρ(y) values etc. as explained in the previous sections is thus also applicable to the SRV constraint. The only potential difference is that the SIMP methodology is optional, alternatively the penalisation factor p in equation (15) may be set to 1.0, either way no penalisation of intermediate ρ(y) values is imposed. For the purpose of clarity, identical notation to that used in the previous subsection will be utilised throughout this subsection. The starting point is equation (15), more specifically ρ(y) which represents the individual design variable attributed to the individual element. The basic form of the SRV constraint thus takes the form of equation (20). m

1

y 1

y

SRV   where :

(20)

 y    y m in :1 In equation (20) m represent the number of elements in the FE model, whilst ρ(y)min defines the minimum value of ρ(y). In order for equation (20) to be valid for all elements it follows that ρ(y)min must be larger than zero, so in order to successfully apply the SRV constraint equation (15) needs to be redefined as equation (21). 0 Cijkl  y     y   Cijkl p

where :

(21)

p  1  0    y  1 By introducing equation (21) it follows that the implementation of the SRV constraint means that the topology optimisation problem is in essence no longer a 0-1 problem. However, this

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needn't have a significant influence upon the overall solution. This is because even though there is a finite lower value for ρ(y) in equations (20) and (21), this can be set very low, as the only stipulation is that this value is non-zero. Furthermore, the same criterion was previously discussed in connection with the implementation of the SIMP scheme in the preceding subsection. By specifying that the values of ρ(y) to be equal to ρ(y)ymin or 1 the vector is still discrete. In addition, if the number of unity values for ρ(y) in the model is defined as n, the SRVdiscrete constraint can be defined as equation (22).

SRVDiscrete 

 m  n n  m  n   n 1   y min   y m in

(22)

It can be mathematically proven that for any non-discrete vector {ρ(y)} equation (23) is valid [13].

SRV   y   SRVDiscrete

(23)

The optimum { ρ(y)}vector is discrete (as the purpose is to solve a 0-1 problem), it therefore follows that SRVopt is identical to SRVDiscrete, equation (24).

SRVOpt  SRVDiscrete

(24)

Equation (23) in combination with equation (24) thus states that SRVOpt must be a local (or global) maximum, as SRV Opt must be discrete. This can be defined as the topology optimisation problem of expression (25) [12].

min F    y   m

where

   y   n  m  n    y  y 1

1 mn  n    y min y 1   y  m

min

(25)

0    y  1

In expression (25) the equality constraint represents the total number of elements, n + m, and the inequality constraints represents the SRV constraint. Only one option exists to satisfy the latter, and that is to satisfy the former, i.e. the equality constraint. The only means of achieving

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that is to have ρ(y) values of either ρ(y)min or 1. Thereby the objective function may only contain discrete values. The compatibility between the SIMP interpolation scheme and the SRV constraint are subtle. The primary point is the fact that both approaches require that ρ(y) assumes non-zero values, as can be seen in (19) and (25) for SIMP and SRV respectively. As previously stated, the SIMP and SRV approaches may be used individually or in combination. Further examples of the implementation including case studies utilising the SRV and the SIMP-SRV approach respectively can be found in [12], [13], [14] and [15]. Other alternatives to the SIMP approach also exists, such as the SINH method, which utilises the hyperbolic sine function (SINH) to dictate the relationship between e.g. ρ(y) and the compliance tensor Cijkl. However, the SINH method essentially follows the same steps as the SIMP interpolation scheme, thus making many of the "issues" with SIMP and SINH identical [16] with respect to application to nonlinear topology optimisation. The next subsection will briefly present the fundamental principles of Design Sensitivity Analysis (DSA), which is not generally used as "stand alone" algorithms in connection with topology optimisation. Instead, it is typically used as an "intermediate step" in the overall optimisation algorithms for instance in connection with the SIMP interpolation scheme, as previously stated. 3.2.3 Design Sensitivity Analysis (DSA) The purpose of DSA, a.k.a. Sensitivity Analysis (SA), is simply to obtain an overview of the sensitivities of the individual parameters of the objective function. This knowledge can then be used to "increase the efficiency" of the optimisation by initially changing the magnitudes of the most sensitive parameters and subsequently "fine tune" the optimisation by varying the "less sensitive" parameters. DSA can not only be used in connection with topology optimisation, but is also applicable for shape, size and any other form of optimisation, including nonstructural optimisation. The sensitivity of the individual parameters are found by means of differentiation, making the

 

DSA or SA a gradient based approach. For example, if the objective function F ρ y in equation

 

(11) is used, the gradient vector F ρ y can be expressed as expression (26).

  F   y  F   y  F   y    , , 2  1 

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F   y    , n  

(26)

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In expression (26), y represents the element number, and n represents the total number of elements in the model. The gradient vector of expression (26) can be obtained by three different approaches:

 

1. Analytical, this requires that F ρ y is fully defined analytically. This is the most robust and efficient option.

 

2. Overall Finite Difference, if F ρ y is purely numerically defined.

 

3. Semi-analytical, if F ρ y is partially defined analytically.

 

In the particular case of FE based topology optimisation, F ρ y is generally fully analytically defined, such as equation (11), thereby the analytical option can be utilised. However, the strain vector ε in equation (11) will be a function of Young's modulus E, and thereby also ρ.

 

Unsurprisingly, this means that the objective function F ρ y "becomes" dependent upon the FE analysis, seeking to obtain equilibrium for equation (2), i.e.:

F   K  U 

 

Please note that F ρ y represents the objective function, whilst F in equation (2) represents the external load vector. By combining equation (2) with expression (26) and using the chain rule of differentiation, an expression for the sensitivity with respect to design variable ρ y (i.e. the density of element y) can be found, expression (27).

 F   y



K   y

 U    K  

 U   y (27)

K   y

 U  

 F   U   K   y  y

The lower equation in expression (27) thus represents the displacement sensitivities of the model, with respect to the relative mass density of element y.At this point is should be noted that the CPU cost for solving equation (27) is relatively low, and is directly proportional to the number of nodes / elements in the FE model. In the case of linear topology optimisation utilising e.g. the SIMP interpolation scheme, equation (27) will only have to be derived once, as the stiffness matrix [K] will remain Contribution 1

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"constant", i.e. an algebraic function of ρ as discussed in section 3.2.1. This means that in a particular application, and in line with the SIMP and SRV approaches, a large part of the efficiency of DSA can be attributed to the fact that the [K] matrix only requires assembly once. Thereby, DSA or SA theoretically becomes "less efficient" if the approach is adopted for NOLIFE TOPOPT, however it does not completely eliminate the potential usage of DSA / SA in this connection. Further information regarding DSA can be found in [17]. The following subsection will briefly discuss some of the other gradient based algorithms available for structural optimisation. 3.2.4 Other algorithms The purpose of this subsection is to briefly introduce some of the "remaining" gradient based optimisation algorithms. In general, these algorithms are not specifically used for topology optimisation, but rather structural optimisation in general, e.g. shape optimisation. However, as previously discussed, the idealistic optimisation procedure for any structural design would include both topology, shape, size and even topography into a single optimisation algorithm, thereby obtaining a truly optimum structure. The non-linear topology discussed in this report merely represents the initial steps towards achieving this ultimate aim. In order to progress towards this ultimate aim it is therefore very beneficial to gain insight and obtain a basic understanding of these algorithms, in order to consider potential compatibility issues between these. Furthermore, some of the algorithms discussed below theoretically have the potential to be utilised in connection with topology optimisation. Practical aspects do however severely limit these potentials, as shall be discussed. Design Of Experiments (DOE) is in principle very similar to DSA. The purpose of the two is in principle also identical, namely to determine the most influential parameters upon the objective function, i.e. sensitivity. DOE also attempts to construct an approximate model that can be used as a surrogate model, i.e. a meta-model, in order to "substitute" the CPU intensive FE solver with less CPU intensive "experiments". This approach will in most cases significantly reduce the CPU cost, thus increasing the effectiveness of the optimisation. The major disadvantage and ultimately the pitfall of DOE in relation to NOLIFE TOPOPT is the robustness of the method, which undoubtedly will become an issue in light of the complexity and large number of parameters used for topology optimisation, which is highly likely to cause the results to diverge as opposed to converge. These issues can however potentially be avoided by the usage of full factorial DOE, this will however incur an enormous amount of numerical

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experiments, consequently the reduction of CPU cost will become insignificant. More detailed information of DOE can e.g. be found in [18]. Response Surface Method (RSM) may be used as a statistical approach to optimisation. This approach also seeks to determine a relationship between the input variables and the response of e.g. the objective function. By the knowledge obtained by conducting a series of these numerical experiences a response surface is constructed and approximated typically by a 2nd order polynomial. Once this has been defined the optimisation can relatively rapidly be conducted. Another variation of the RSM method is denoted Sequential Response Surface Method (SRSM), where the response surface is reconstructed sequentially, e.g. after a set number of iterations. The major disadvantages of RSM and SRSM is the accuracy of the 2nd order polynomial in connection with the vast number of parameters in a typical topology optimisation. For instance a model with 5000 elements will require 5000 polynomials to build a combined "response surface" for all of these parameters. At this point the previously discussed curse of dimensionality becomes an issue. For 5000 elements a minimum of 12,507,501 experiments is required. In practice RSM or SRSM methods are limited to a maximum of 7 or 8 variables due to the curse of dimensionality. This makes them well suited for e.g. shape and size optimisation, but unsuited for topology optimisation. In addition, the numerical experiments used to build the response surface must be carefully planned in order to obtain useful information. This is why RSM is often used in combination with DOE or DSA. An example of this is the LSTC software LS-OPT and Altair HyperStudy. Both of which can be used for size and shape optimisation. Additional information regarding RSM can be found in [19]. The above is merely a short presentation of some of the optimisation algorithms and techniques available, many more such as Genetic Algorithms (GA), LeapFrog OPtimisation (LFOP) and Adaptive Simulated Annealing (ASA) also exist. Furthermore, non-gradient based algorithms such as Simulated Biological Growth (SBG), Particle Swarm Optimisation (PSO) and Metamorphic Development (MD) also exist. Neither of these will however be further considered, primarily because these are deemed unfeasible for topology optimisation. With the above discussion, the next subsection will briefly summarise the outcomes and key findings of gradient based optimisation, presented throughout this section. 3.2.5 Feasibility of employing gradient based algorithms for NOLIFE TOPOPT The purpose of this subsection is to briefly summarise the findings of the preceding subsections, which presented and discussed the general theory associated with gradient based

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topology optimisation algorithms, this included the SIMP interpolation scheme and the SRV constraint. Figure 5 of section 2.2 illustrates the overall flow of an FE based (topology) optimisation. This can be further expanded for a mathematical approach, Figure 21.

Figure 21, overall flow diagram mathematically based topology optimisation algorithm. The difference between Figure 5 and Figure 21 is the addition of the step "Define objective function and constraints" in the latter. This step may seem trivial, but it is very central in the difference between linear and non-linear optimisation. This is because the objective function (and / or the constraints) become non-linear. Once this step is completed the general steps of the linear and the non-linear optimisation algorithm are in theory identical. This means that the "loops" in Figure 21 may be used to represent linear as well as non-linear optimisation algorithms. The details of the "FEA" and the "Adapt FE model" steps in Figure 21 will however most likely be significantly different between the linear and the non-linear algorithms, as briefly discussed in subsection 3.2.3. Furthermore, there is the practical issue of the increased CPU cost to consider in connection with non-linear FEA in general. In light of Moore's law which states that the CPU performance is increasing exponentially, the CPU cost may become much less significant in the near future. Under this assumption the most significant step / issue with non-linear algorithms becomes the "Adapt FE model" step in Figure 21. As discussed above, this step can be conducted by the use of e.g. the SIMP interpolation and / or the SRV constraint. However, these are not directly applicable to non-linear material

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behaviour, simply because both utilise the compliance tensor, Cijkl, which contains the (linear) elastic material parameters. However, this does not completely eliminate the usage of e.g. SIMP in connection with non-linear optimisation. The relevance of this statement can be appreciated by considering "the degree of non-linearity". In other words, SIMP and (or) SRV may be useable in connection with non-linear topology optimisation, depending on the material model being used. Many different material models exist to describe non-linear behaviour. One approach is to introduce a strain energy density function W(x) that relates the strain energy of element x to the deformation gradient, i.e. the relative level of deformation, once the yield stress has been exceeded. The stresses σ in the body can subsequently be found by derivation of W(x) with respect to the strains ε . An example of this type of material behaviour can be seen in Figure 22.

Figure 22, non-linear material behaviour. Figure 22 thus presents a "typical" Prandtl diagram for a structural steel. It is clear that e.g. the SIMP interpolation scheme cannot be utilised in connection with topology optimisation in the non-linear area of the curve. However, non-linear material behaviour can also be described by means of piecewise linearity. For instance the curve in Figure 22 can be approximated by the additional curves illustrated in Figure 23.

Figure 23, linear approximation of non-linear material behaviour.

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By using the various E values as indicated in Figure 23 it is potentially possible to utilise e.g. SIMP or SRV and DSA in connection with non-linear optimisation, by approximating the nonlinear behaviour with piecewise linearity. Subsequently the optimisation could be conducted by a series of two (or more) "SIMP (or SRV) driven optimisations" where the Ey values, equation (18), are adjusted in accordance with the current stress (or strain) values (Figure 23). The feasibility of such an approach will as previously mentioned be dependent upon the "degree of non-linearity of the material", i.e. the "shape" of the non-linear section of the curve. In this view the red curve in Figure 22 (Figure 23) may be perceived as idealistic for the suggested methodology. Nevertheless, the potential for this approach remains intact. If successful in applying this methodology the main benefits such as the stability, robustness and low CPU cost of e.g. the SIMP algorithm may be "regained". From the above discussions it can thus be determined that the material model chosen for the optimisation may hold a significant influence upon the "applicable" methodologies for the topology optimisation. Other material models which could be utilised to describe the nonlinear behaviour is e.g. a polynomial or even NURBS (Non Uniform Rational B Spline). In any case the discussion above has highlighted the necessity for careful selection of material model in connection with the non-linear topology optimisation. Regardless of which material model is chosen it is still necessary to ensure that the two "basic" requirements for gradient based optimisation are met: 

Convexity / concavity of objective function



Continuity of objective function

If these requirements are met, the approach explained above can be directly utilised. If it cannot be established that the function is e.g. convex, then the potential effects of this must be closely considered. Furthermore, the below issues must also be meticulously considered in the process of selecting / defining an optimisation algorithm for NOLIFE TOPOPT: 

Mesh dependency



Material modelling

The above four bullet points can thus be used as a very broad conclusion for the initial presentations and discussions of the gradient based optimisation algorithms of the previous sections. The discussions have clearly raised some significant issues with the presented algorithms in connection with NOLIFE TOPPOPT. However, it must also be noted that these discussions have not ultimately eliminated the possibility of utilising gradient based algorithms in connection with NOLIFE TOPOPT. The main part of this section has assumed that the

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Variable Density Method (VDM) is utilised. The following section will present and discuss a different type of gradient based optimisation algorithm, HBO.

3.3 Homogenisation Based Optimisation (HBO) In the preceding section different approaches to topology optimisation using the Variable Density Method (VDM) was presented and discussed. For linear elasticity the VDM approach assumes an explicit relationship between Young's modulus (E) and the relative mass density (ρ) of the material. The topology can subsequently be obtained by varying ρ, and thereby E to obtain a topology and the associated changes of structural performance of the component in question. The VDM approach thus seeks to solve the topology optimisation problem via the macroscopic material properties. This approach works well for linear elastic problems, however issues relating to the effectiveness in relation to NOLIFE TOPOPT were disclosed. Furthermore, the FE based VDM approach is also prone to the problem known as the checkerboard effect, leading to unfeasible designs, Figure 12. Finally, the issue of mesh dependency was demonstrated by Figure 20. As previously stated, an alternative to the VDM approach is denoted Homogenisation Based Optimisation (HBO), which will be the focus of attention in this section. 3.3.1 Basic principles of HBO algorithms The VDM approach is by far the most widely used in modern day commercial FE software, despite the previously discussed issues. Possible reasons for this could be the implementation with FE combined with the proven robustness, efficiency and the "relative simplicity" of the algorithms. HBO algorithms use a combination of macroscopic and microscopic consideration to perform topology optimisation. The (theoretical) primary advantages of HBO as opposed to VDM are: 1. HBO can simultaneously conduct topology, shape and size optimisation. 2. HBO is mesh independent. 3. HBO completely avoids the checkerboard effect. If the three advantages stated above can indeed be obtained, the potential for HBO based algorithms will be significantly larger than VDM based algorithms. However, given the ultimate aim other aspects such as implementation with non-linear material behaviour, CPU cost, robustness, diversity and accuracy must all be carefully considered before such a statement can be fully justified. In order to further address these aspects it is necessary to present the general theory of HBO. In mathematical terms homogenisation is the study of partial

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differential equations with rapidly oscillating coefficients. In order to relate this to continuum mechanics consider the following. All materials are inhomogeneous (heterogeneous) at some scale, e.g. the microstructure of steel varies significantly as a function of location. Nevertheless for practical applications it is most convenient to assume e.g. that structural steel has isotropic, i.e. homogeneous, properties such as Young's modulus, Poisson's ratio and mass density. For most (engineering) applications this assumption is reasonable and provides a satisfactorily level of accuracy, whilst significantly reducing the complexity of the design problem. This approach is e.g. one of the basic assumptions of the overall LCVTP BIW optimisation study. The above example demonstrates the general principle of homogenisation. In order to demonstrate this in a mathematical sense suppose that equation (28) describes Young's modulus  E  of a material as a function of location u  x  , where the  x  describes the location.

E

   x  u  A  x     x 

(28)

The value of E at a small distance  from x will thus (rapidly) vary according to the magnitude

x  contribution, leading to a significant variation of E across the material /  

of the A 

component in question. In order to "avoid" this variation equation (28) is re-defined as equation (29) according to the homogenisation theory.

E

  * u  A  x  x 

(29)

Please note that E in equation (28) and (29) is merely an example, and could be substituted by any other parameter. In equation (29) A* is a constant representing the "effective property" of E, thus making equation (29) "less fluctuating", which may e.g. be applied in a local region of the material / component. The latter part of the last sentence is key to understanding the general principle of HBO. The above explanation is merely a brief introduction into the general homogenisation theory. In order to fully understand the principles and finer detail it is imperative that the reader as a minimum possesses extensive knowledge about continuum mechanics and differential equations. Therefore, a more in depth explanation of fundamental homogenisation theory exceeds the scope of this paper. Furthermore, while the in depth Contribution 1

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understanding is desirable it is not crucial to the continued discussion of the potential application to topology optimisation. The transition between equation (28) and equation (29) can thus be used to represent the general homogenisation theory. The "behaviour" (response) of equation (28) and equation (29) can thus be described as two "extremes", as the former dictates a rapidly oscillating function, whilst the latter dictates a significantly less rapidly oscillating function. In HBO based algorithms the material behaviour is typically described as intermediate between these two extremes, and varying periodically. In other words, e.g. E will periodically obtain its constant value (A*) at given points in the material / component and vary (equation (29)) in between these points. A graphical illustration of this can be seen in Figure 24, where "E" denote the "locations" of A*( i.e. E), the remaining points are thus defined by equation (29), notice the periodicity illustrated in the figure.

Figure 24, illustration of HBO principle. As Figure 24 indicates, E has thereby become a "continuous variable" throughout the material (or component) in question. In principle, this means that the topology optimisation problem can be described without the need for discretisation, consequently the checkerboard effect and mesh dependency will not occur. At this point it is however important to remember that in connection with topology optimisation it is often highly desirable to apply constraints relating to e.g. maximum displacement or stress values. These can in principle be obtained by the usage of mesh-free methods, i.e. without the usage of FE [20]. This approach is however often very cumbersome, if not impossible to define without the usage of discretisation. Nevertheless, advances are made in this context, and the potential for future usage remains

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[21]. This will however not be further discussed in this report, and the focus will return to FE based optimisation. Consequently, the structural analysis associated with the optimisation will be mesh dependent. The impending task is thus to define the periodic material properties (Figure 24), in relation to FE analysis. This could e.g. be done by introducing continuous FE approximation of design variables, as previously discussed. The following methodology of implementing HBO is primarily based upon the approach of [6]. In order to include the microstructure in the FE analysis a unity honeycomb base cell could be employed, Figure 25.

Figure 25, honeycomb base cell for HBO. By utilising the honeycomb base structure illustrated in Figure 25, the material microstructure is considered during FEA. Furthermore, the approach also allows for isotropic behaviour, as the honeycomb base cell will mimic isotropic behaviour on a macroscopic scale [6][23]. In order to utilise the honeycomb base cell in connection with HBO and FEA the "normal" interpolated shape functions used to construct the stiffness matrix [K] are substituted with Moving Least Squares approximation (MLS) shape functions. This method is in essence a curve fitting technique and can briefly be defined as consisting of three primary components: 1. A weighting function. 2. A basis function. 3. A set of coefficients. Combined the three components listed above works similar to e.g. a B-spline, i.e. a weighted average "decides" how the function acts in any given interval, this will be further explained in an example below. Additional info on MLS can e.g. be found in [23]. By introducing MLS the design variable (e.g. the material density) becomes a continuous function over the design domain (in general), and the honeycomb base cell (Figure 25) may be used as a design variable, equation (30). m

r e    y ry

(30)

y 1

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Equation (30) represents an example of the MLS approach to HBO. The honeycomb base cell size r e (Figure 25) at any location e in the design domain can be interpolated via equation (30). m in the equation represents the total number of design variables (e.g. the total number of elements in the FE model), whilst  y represents the interpolation shape functions, i.e. the ones which replaced the "original" FE shape functions. From equation (30) the density at location e can be derived as equation (31). For details regarding this derivation please see [6]. m

e  y y

(31)

y 1

Equation (31) can thus be used in a similar manner to the general implementations discussed in section 3.1 and 3.2, however the material microstructure has been taken into account in equation (31). The above presentations and discussions thus represent a very brief overview of the general principles and possibilities of HBO based algorithms. The following subsection will summarise the findings of this section. 3.3.2 Feasibility of employing HBO algorithms for NOLIFE TOPOPT The following discussions are based upon the findings of the preceding subsection. In order to conduct a more in depth discussion of HBO algorithms it is, as previously mentioned, imperative that the reader possesses substantial knowledge of continuum mechanics and partial differential equations, thereby exceeding the scope of this report. HBO based algorithms hold very desirable potential with respect to NOLIFE TOPOPT, and also for the next steps in the development of future topology optimisation algorithms, such as the (efficient) inclusion of e.g. composite, i.e. non-isotropic materials. The overall potential advantages of HBO algorithms discussed in this report may be summarised as: 1. Mesh independency. 2. Absence of checkerboard effect. 3. Inclusion of material microstructure. 4. Ability to cater for non-isotopic materials. At first glance the HBO algorithms therefore hold many significant benefits when compared to the VDM based algorithms discussed in section 3.1 and 3.2. However, many of these benefits become less valuable due to the inability of implementing these into actual code, especially in relation to structural analysis, where FEA is still utilised. The HBO algorithms thus

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hold significant potential for NOLIFE TOPOPT, but in order to fully enable this potential it seems that mesh-free analysis [20], [21] plays a vital role. Nevertheless, HBO algorithms have recently been subjected to extended research and case studies (e.g. [4] and [6]), making these an interesting, and viable alternative to VDM based algorithms. Thereby the discussions and presentations relating to mathematical based optimisation algorithms are complete. The next chapter will focus on the evolutionary based optimisation algorithms.

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4. Evolutionary Based Optimisation Algorithms The purpose of this chapter is to present an alternative type of algorithm to the mathematically based ones presented and discussed in the previous chapter. This alternative type is generally referred to as evolutionary algorithms, also generally referred to as Evolutionary Structural Optimisation (ESO) algorithms.

4.1 General theory The general methodology of the ESO algorithms is essentially very simple, and is based loosely upon the theory of evolution, i.e. the general principle of "survival of the fittest". In other words, this type of algorithm ensures that the "fittest elements" survive by gradually removing "inefficient" material from the structure in question. The first book on ESO algorithms was published by Y. M. Xie and G. P. Steven in 1997 [24] making this approach relatively new. The following sections are based upon [24]. 4.1.1 Evolutionary Structural Optimisation, ESO The ESO algorithms solely rely on FE analysis to obtain the data necessary for conducting the topology optimisation. Thereby the general flowchart of an evolutionary based algorithm can be defined as illustrated in Figure 26.

Figure 26, flowchart of evolutionary algorithm.

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As illustrated in Figure 26, the flowchart of the evolutionary based algorithms is slightly different than that that of the mathematical based algorithms, Figure 21. The governing theory behind the evolutionary based algorithms is in general less complicated than that of the mathematical algorithms. An Evolutionary Structural Optimisation (ESO) algorithm may work as described by the following 7 steps. 1. Define an FE model 2. Define initial Rejection Ratio (RR) and Performance index (PI). The Rejection Ratio is the methodology used in ESO to determine which elements are "inefficient". This could e.g. be based on strain or stress values, for instance the Rejection Ratio could be defined as equation (32).

RR  In equation (32)

y  max

(32)

 i is the stress of the individual element, y, and σ max is the maximum

allowed stress, for linear applications this could e.g. be the yield stress value. Typical values of RR range between 1% and 25%. The Performance Index (PI) is a recent addition to ESO, and was initially suggested by M. Querin [25]. The ideology behind the PI is in essence the same as the objective function of the mathematically based algorithms, namely to provide a single number "ranking" the performance of the topology in that specific iteration, enabling the "optimum" structure / topology to be identified via this single number. Further info including case studies can be found in [25] and [27]. 3. Conduct an FE analysis (linear or non-linear). 4.

Adapt FE model. This step simply consists of deleting those elements which RR is below that specified in equation (32).

5. Evaluate the Performance Index and check if a significant change has occurred since the last iteration, this step is very similar to that used in the mathematical based algorithms as previously presented. 6. If the PI has not converged, check if the RR has reached a steady state, i.e. if no elements (or very few) has been deleted in the current iteration, compared to the preceding one(s). 7. If the RR has reached a steady state increase it according to equation (33).

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RRupdated  RR  ER

(33)

In equation (33) RRupdated will replace the current RR value by adding the Evolutionary Ratio (ER). The ER is also a predefined value to be selected from model to model. A typical value of ER is 5%. The above steps are repeated until the Performance Index has converged. The overall steps of the ESO and the mathematically based algorithms are thus very similar. However, the way in which these steps are conducted is quite different. The most important thing to note is that ESO does not reduce the stiffness (or density) of any element, it straightforwardly deletes the "inefficient" ones. This strategy has its advantages as well as its disadvantages. The advantage lies in the fact that the geometry is updated according to the optimisation, i.e. no "artificial" stiffness remains in the model as it does in e.g. the SIMP scheme (where E of an element is lowered while the "geometrical stiffness" remains). The disadvantages of deleting the elements primarily relate to instability issues, and the "irreversibility" of the process. If an element is deleted it cannot be "recreated" at a later stage in the process if required. This is possible using SIMP, as the "scaling" factors of [K] can simply be adjusted. Furthermore, ESO algorithms need to reassemble [K] for every iteration, as the topology (geometry) changes. As previously mentioned, this is not (initially) required for SIMP. However, when the focus turns to NOLIFE TOPOPT this will have to be re-evaluated. Variations of the ESO algorithm have been proposed, these have primarily been developed to overcome some of the above issues relating to the algorithm. The two main variations of ESO are denoted: 

Additive Evolutionary Structural Optimisation (AESO).



Bi-directional Evolutionary Structural Optimisation (BESO).

4.1.2 Additive Evolutionary Structural Optimisation, AESO The AESO algorithm works very similar to the ESO algorithm, expect instead of deleting elements it adds them. This means that the optimal structure is not found from "a full design volume" rather an empty one. This can be better understood by observing Figure 27 which may be used to illustrate the starting point for an ESO and an AESO algorithm respectively.

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Figure 27, illustration of ESO and AESO starting points. Please note that Figure 27 is merely an illustration, a "minimum" number of elements is required in order to carry out the initial FE analysis. Further information relating to AESO can be found in e.g. [28]. Due to the overall similarities and the "identical" approach of ESO and AESO the two types of algorithm experience the same difficulties, namely that of the inability to "reverse" the steps, i.e. recreating(ESO) or deleting (AESO) elements. On this basis the Bidirectional Evolutionary Structural Optimisation (BESO) algorithm was developed. BESO essentially "combines" ESO and AESO so that the algorithm is capable of creating as well as deleting elements, to a certain extent. 4.1.3 B-directional Evolutionary Structural Optimisation, BESO In theory the BESO algorithm works well and does resolve some of the obvious disadvantages of the ESO and AESO algorithms. However, some significant practical issues severely restrain the AESO algorithm. These primarily relate to the CPU cost of the algorithm which in general is high, as a full FE analysis is required for each iteration. Furthermore, research papers have severely questioned the validity of ESO topology optimisation results [29], [30] and [31]. In addition to the above discussion it must also be noted that all (presented) ESO algorithms are based upon FEA, making the previously presented issue of mesh dependency a factor in evaluating the feasibility of ESO, AESO and BESO algorithms for NOLIFE TOPOPT. This has been a very brief introduction of evolutionary based topology optimisation algorithms. A more in depth understanding of the basic principles and implementation with FE can be found in [24], [25] and [26]. The next subsection of this chapter will summarise the findings presented in this chapter and overall evaluate the feasibility of employing evolutionary algorithms for NOLIFE TOPOPT.

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4.2 Feasibility of employing evolutionary algorithms for NOLIFE TOPOPT In principle the evolutionary based algorithms presented above need only very minor adaptations to enable the usage of these in connection with NOLIFE TOPOPT. Indeed, initial case studies using ESO algorithms for non-linear optimisation has been conducted [32]. The relative simplicity of the evolutionary based algorithms is also an appealing factor during the implementation phase and possible subsequent changes to the methodology. One of the largest drawbacks of this type of algorithms is undoubtedly the high CPU cost associated with the need for a full FE analysis for each iteration. However, as previously discussed this factor may become less significant in the view of modern day CPU development. Another significant drawback of the evolutionary algorithm is the briefly discussed validity and stability issues associated with the methodology. As the general principle of the algorithms is relatively simple the theoretical foundation of ESO is subjected too much criticism. One of the primary concerns is the inability to substantiate that the results obtained by ESO are in fact optimum solutions. The general conclusion regarding the usage of evolutionary based algorithms for NOLIFE TOPOPT must therefore be that these algorithms hold significant potential, however there are significant limitations which require further evaluation.

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5. NOLIFE TOPOPT Algorithm Selection The purpose of this chapter is firstly to summarise the outcomes of the discussions conducted in the preceding chapters and secondly based on these define the continued steps of the NOLIFE TOPOPT algorithm. The discussions of the previous chapters can be summarised into 10 key topics which must all be considered during the subsequent selection process.

5.1 Key points of consideration for selection process The 10 key points of consideration that are to form the basis of the NOLIFE TOPOPT algorithm are listed and discussed in the following subsections. 5.1.1 Linear and non-linear material behaviour As previously stated, the intended purpose of the NOLIFE TOPOPT algorithm is to incorporate non-linear material behaviour into a topology optimisation algorithm. However, it does not automatically follow that linear material behaviour is therefore disregarded. Despite the development of e.g. composite materials it is highly likely that future BIW designs will continue to contain a considerable amount of structural steel. In order to create a versatile algorithm it is therefore highly desirable to construct an algorithm that is capable of considering linear material behaviour in addition to non-linear material behaviour. This can be substantiated by the fact that it is desirable to ensure that parts (i.e. areas / loadpaths) of e.g. the BIW structure does not exceed the yield stress limit. If these areas were optimised using a nonlinear algorithm they would potentially be "under optimised" as the required structural performance of the component(s) will be "exaggerated", thereby (unnecessarily) adding mass to the load path(s) in question. Given the previously discussed "lack of cross sectional load path details" often associated with topology optimisation (section 2.2), the above may seem indifferent. However, as discussed in section 3.3, the potential for simultaneously conducting topology-, shape- and sizeoptimisation exists, underlining the importance of the above discussion. 5.1.2 Combined topology-, shape- and size-optimisation This issue is not central to the optimisation algorithm selection process, rather it should be consider as a potential "added bonus". Theoretically the combination of topology -, shape- and size-optimisation into a single (simultaneous) algorithm is highly desirable. However, it must also be carefully considered that although the four may have the same objective, e.g. to minimise mass, they may be ineffective if applied simultaneously thus significantly increase the CPU cost and lead to potential instability / robustness issue. The only type of algorithm,

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presented here, that initially has the potential to simultaneously conduct the three types of optimisation is the HBO based one, section 3.3. Alternatively, the three types of optimisation may be conducted sequential to each other, e.g. in a modular based algorithm. 5.1.3 Mesh dependency The issues relating to the mesh dependency of FE based topology optimisation have been continuously highlighted throughout the foregoing discussions, emphasized by Figure 20. This will be an issue regardless of which algorithm is selected, providing it is FE based. The only exception presented in this report is the meshless approaches discussed in connection with the HBO algorithm of section 3.3. Thereby the mesh dependency becomes a central issue, not only with respect to the obtained results, but also with respect to the CPU cost, subsection 5.1.4. Based on this it is crucial to investigate different modelling / meshing techniques including the potential for utilising "simplified" models using e.g. spring and point masses to represent the structural properties of the battery mass. An example of this type of approach is the DRI/KRASH modelling technique originally developed to simulate aircraft crash scenarios [33]. At this point the following phrase for meshing / modelling should be considered: A model should be as simple as possible and as detailed as necessary 5.1.4 CPU cost and performance The CPU performance is directly linked to the topic of the preceding subsection, namely that of modelling technique. In section 3.2.5 the rapid development of CPU performance was briefly discussed based on this it is likely that the CPU cost will be a less influential factor in the near future. However, this does not mean that the optimisation algorithm shouldn't be as "efficient as possible". It is important to note that one of the "advantages" of utilising topology optimisation should be a reduced BIW design time, whilst actively designing to meet e.g. safety and crash performance criteria at a very early stage in the design process. This means that the CPU should be considered as a factor in the selection process, however it may not be perceived as a dominating factor. 5.1.5 Feasibility of topology optimisation results This topic is central in the algorithm selection process, simply because of the importance of a feasible design. In this context this is attributed to manufacturing constraints in particular. If it is possible to obtain a topology requiring a "minimum of interpretation" this would in theory be an ideal situation, because it holds a significant potential with respect to further decreasing

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the BIW design process time. Therefore it is important to consider the algorithms "ability" (or lack thereof) to produce feasible optimisation results / topology and avoid problems such as the checkerboard effect, section 3.1. However, at this point it must also be considered that a fully manufacturable design may not be "ideal" either. This relates to the vast number of possible solutions to a topology optimisation problem, section 2.3, meaning that the solution found may not be the global optimum, and in the case of ESO an optimum at all. 5.1.6 Convexity, concavity and global versus local extrema This topic is directly linked to the feasibility of topology optimisation results discussed above, and is thereby also a central topic in the algorithm selection process. The feasibility, combined with the possibility of simultaneously conducting topology-, shape- and size-optimisation, subsection 5.1.2, are linked with the level of confidence that a global extrema has been found, thereby ensuring that the proposed topology is indeed an optimum solution with respect to the objective. In the case of a convex or concave objective function this does not pose a major obstacle. However, the convexity of the objective cannot be guaranteed, and given the general degree of complexity of topology optimisation problems it is highly likely that the majority of tasks will include non-convex functions. This is also an issue with the evolutionary based algorithms as they, in essence, do not have any theoretical / mathematical validation or indication that the result is indeed an optimum, be it local or global, chapter 4. It is therefore essential to consider the ability of the algorithm to identify or indicate, with a certain level of confidence, that an optimum solution has indeed been found. 5.1.7 Constraints implementation This topic is naturally a very central issue that any structural optimisation algorithm must address. The concept of unconstrained optimisation was previously discussed, chapter 2, and is of no interest in the context of FE based topology optimisation. Constraints implementation is thus a very significant factor, however, it is included in all the algorithms presented throughout this report, meaning that all contain the potential for constraining the optimisation. The task of evaluating the algorithms potential for constraining the optimisation results thus becomes a consideration of how the constraints are implemented into the algorithm. 5.1.8 Material model The material may at first glance not seem to play an essential role in the selection process. However, as previously discussed in subsection 3.2.5, it may be used to include or exclude

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certain algorithms e.g. by using linear approximation of non-linear material behaviour. This means that extensively developed linear topology optimisation algorithms may be partially utilised, and the proven robustness, versatility and efficiency of these may be utilised. In addition, this poses a simple method of combining linear and "non-linear" material behaviour as discussed in subsection 5.1.1. 5.1.9 Material microstructure and future potential Throughout this study it has been continuously stressed that only isotropic materials would be utilised. However, future BIW designs are likely to include multiple materials, including isotropic, e.g. structural steel, in combination with e.g. plastics or composite materials which in general are non-isotropic. This means that even though this study is limited to isotropic materials the potential for considering the material microstructure or as a bare minimum the non-isotropic macroscopic behaviour of the material should be considered, as this may be of significant importance in the very near future. 5.1.10 Current limitations In addition to the above considerations with emphasis on the future potential, i.e. the potential for future development of the algorithm, it is also important to review the previously defined limitations, i.e. the assumptions listed in section 2.1, these are repeated below for convenience. 1. No Strain rate effects. 2. Negligible change in material characteristics as a function of temperature change. 3. No viscoelastic effects. 4. Only linear stress-strain behaviour as a function of geometry. In light of the conclusions in the preceding subsections of this chapter, the "validity" of some of the majority of the limitations listed above becomes questionable. This primarily relates to the absence of strain rate effects and the negligible change of characteristics as a function of temperature. Nevertheless, these may not necessarily require to be directly included into the optimisation algorithm itself, rather as an overall sensitivity study similar to those previously conducted for the BIW topology conducted as part of WS7.6 of the LCVTP and documented in the associated *.html document. Further limitations were specified throughout chapter 2, the most significant ones are listed below.

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1. Multiple material optimisation. This is different from that discussed in subsection 5.1.9, where different materials where pre-specified, in this context it means that the material selection becomes a parameter in the optimisation process. 2. Multidisciplinary optimisation. This means that the topology optimisation becomes part of a larger series of optimisations typically "stretching" across several engineering disciplines. Consequently the optimisation results "feed into each other". An example of this could be shape optimisation based upon aerodynamics and topology optimisation based upon FEA. Please note that the objective of both optimisations may be identical, e.g. to minimise the BIW volume, and thereby mass. 3. Multicriteria optimisation. This may be used in combination with multidisciplinary optimisation, but can also be solely based upon e.g. FE based topology optimisation. In a multicriteria optimisation two (or more) objective (functions) would be defined, however the design variable(s) would in most cases be identical, i.e. to maximise stiffness and minimise mass by varying the relative mass density, ρ. The above limitation should all be considered during the optimisation algorithm selection process, however at this point only the potential of utilising these are taken into account, as the primary aim remains the same, namely to define a non-linear topology optimisation algorithm for an isotropic material subject to the assumptions listed above. With this the summation of the general outcomes of the discussions throughout this report is complete. The next section will discuss and evaluate the previously presented algorithms against the 11 key points of consideration listed above. Based upon this the future steps in the derivation and implementation of the algorithm are presented.

5.2 Selection of NOLIFE TOPTOP algorithm and future steps Based upon the discussions throughout this report as summarised above, the individual algorithms have been meticulously evaluated in relation to NOLIFE TOPOPT. The potential of employing the mathematical gradient (VDM) based optimisation algorithms is in essence fully dependent upon the definition of material modelling. If the piecewise linear approximation is deemed acceptable, the widely used SIMP interpolation scheme could e.g. be utilised, thereby (partially) exploiting the proven robustness versatility and efficiency, including the relatively low CPU cost However, this does also imply that the "dynamic effects" are simplified, potentially having a significant effect upon the topology optimisation results, as

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found during the previously conducted LIFE TOPOPT study, chapter 1. The SIMP based approach does in theory not eliminate any of the issues raised above, e.g. material microstructure or multidisciplinary optimisation. The simultaneous execution of topology-, shape- and size-optimisation is however not initially possible, nevertheless the modular approach suggested in subsection 5.1.2 holds significant potential. The HBO based approach does theoretically hold the largest potential for NOLIFE TOPOT of the ones presented. This primarily relates to the "3 in 1" optimisation of topology, shape and size, the mesh independency and avoidance of checkerboard effect. However, these are heavily dependent upon the continued development of meshless methods. Nevertheless, these may still be utilised in connection with FEA, thus "reinstating" the mesh dependency and possibly the checkerboard effect. The evolutionary based optimisation algorithms are very attractive due to their relative simplicity and logical approach. However, as previously stated these have recently been subjected to harsh criticism primarily related to the feasibility of the obtained topologies. Furthermore, the lack of theoretical rigour makes it difficult to claim that the obtained topology is indeed an optimum. Finally, the versatility of the approach may be limited, as e.g. optimising the first natural frequency of a structure is problematic [34], in addition to the generally high CPU cost of the approach. The selection of the NOLIFE TOPOPT algorithm is therefore neither an easy nor straightforward task, as all algorithms hold potential. However, these potentials are all dependent upon given limitations. Based on this it is deemed that the potential of all three types of algorithms listed above require further study in order to enable a better appreciation of the limitations versus the advantages of the individual algorithms. This will include practical case studies from which actual topology optimisation results can be compared. The future steps of the selection process will therefore initially contain three separate assignments, as dictated below.

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1. A series of SIMP based topology optimisations, piecewise linear material behaviour is assumed. 2. HBO based optimisation algorithm, based upon FEA, meshless methods will not be further considered. 3. BESO based optimisation algorithm. The ideal scenario would include comparing case studies of identical (FE) models solved via each of the three algorithms listed. The extended evaluation of all three the HBO and BESO based optimisation algorithms above would incur specific and dedicated programming in e.g. C++ or MatLab in order to conduct the case studies, as no commercial software is available. However, at this point in time this large task is beyond the remaining scope of the LCVTP. Commercial software, such as the GENESIS software provided by GRM consulting LTD [35] offer a similar approach to the one suggested in step 1 above, namely equivalent static loading. In order to maximise the outcome of the LCVTP it has therefore been chosen to focus on the continued steps of the BIW development, i.e. the development of the actual crash structures from the topology optimisation, as this will provide a much clearer indication of the potential for using linear approximation of material behaviour to model non-linear crash scenarios. This is the focus of attention in some of the associated reports which may be accessed via the associated html document.

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CONTRIBUTION

2

Double blind peer review journal paper published in the International Journal of Crashworthiness on th October 13 2011 [Format has been edited to comply with Coventry University regulations]

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This article was downloaded by: [Jesper Christensen] On: 13 October 2011, At: 06:25 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

International Journal of Crashworthiness Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tcrs20

Effects of roof crush loading scenario upon body in white using topology optimisation a Faculty

J. Christensen a , C. Bastien a & M. V. Blundell a of Engineering and Computing, Coventry University, Coventry, UK Available online: 13 Oct 2011

To cite this article: J. Christensen, C. Bastien & M. V. Blundell (2011): Effects of roof crush loading scenario upon body in white using topology optimisation, International Journal of Crashworthiness, DOI:10.1080/13588265.2011.625640 To link to this article: http://dx.doi.org/10.1080/13588265.2011.625640 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-andconditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. International Journal of Crashworthiness iFirst 2011, 1–10

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Effects of roof crush loading scenario upon body in white using topology optimisation J. Christensen∗, C. Bastien and M.V. Blundell Faculty of Engineering and Computing, Coventry University, Coventry, UK (Received 6 May 2011; final version received 19 September 2011)

This paper investigates the effects of variations in modelling of roof crush loading scenarios upon topology and mass of a body in white (BIW) for a hybrid electric vehicle (HEV). These variations incorporated the proposed changes to the Federal Motor Vehicle Safety Standards (FMVSS) 216 standard. The base model used for the investigation in this paper was based upon a series of optimisation studies. The overall purpose was to minimise the BIW mass of an HEV subjected to multiple crash scenarios including high-speed front impact, offset deformable barrier (ODB), side impact, pole impact, high-speed rear impact and low-speed rear impact in addition to a roof crush scenario. For the purpose of achieving this goal, finite element (FE) topology optimisation was employed. Owing to the limitations of present-day FE optimisation software, all models utilised linear static load cases. In addition, all models made use of inertia relief (IR) boundary conditions. With the above approach, the BIW topology was investigated. Keywords: finite element topology optimisation; FMVSS 216; roof crush; body in white (BIW); lightweight vehicle architecture; hybrid electric vehicle; inertia relief

1. Introduction 1.1. Topology optimisation In 2003, it was estimated that approximately 10,000 fatalities occurred on US roads due to rollovers [21]. This highlights the severity of the crashes associated with rollovers; the influence of the roof strength is discussed in [10]. In the light of this, recent changes to the Federal Motor VehicleSafety Standards (FMVSS) 216 standard have been proposed. The study in this paper aims to investigate the effects of these proposed changes upon the body in white (BIW) for a hybrid electric vehicle (HEV) by means of topology optimisation. Topology optimisation seeks to find the optimum distribution of material within a given design volume. This is performed with respect to achieving a predefined objective, which, in this particular case, was to minimise the BIW mass. The optimisation process can be controlled by defining one or several constraints, relating to, for example, displacement values. For 3D

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topology optimisation in connection with finite element analysis (FEA), the objective may be achieved by varying the mass density of the individual elements within the design volume (i.e. BIW). The outcome of an FEA-based topology optimisation is thus an indication of relative material (mass) density throughout the design volume. This outcome therefore does not contain highly detailed information with respect to, for example, panel thicknesses or crosssectional geometry in general [14]. If desired, this information can subsequently be obtained by the application of, for example, shape, size and topography optimisations, which in return can feed back into the overall BIW design process [4,5]. The above methodology thus suggests an alternative approach to the ‘typical’ BIW design process utilised by original equipment manufacturers (OEMs). The validity of this approach has been discussed by the authors of this paper in [8]. In addition, a similar approach has been taken in the Future Steel Vehicle project (FSV; http://www.futuresteelvehicle.org/), which designed a crashworthy (NCAP compliant) vehicle, initially based upon topology optimisation. The design volume utilised for the study in the current paper can be seen in Figure 1. The maximum external dimensions of the design volume were (X, Y, Z) 3865 mm × 1850 mm × 1530 mm. The design volume illustrated in Figure 1 was discretised utilising solid (3D) tetra elements with linear shape functions and an average element size of 25.0 mm, leading to the creation of approximately 103,000 nodes and 527,000 elements.

Figure 1, the design volume used for topology optimisation.

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1.2. Material model The material model used for the optimisation study was defined as linear elastic with the material characteristics of an isotropic mild grade steel. The values used are defined in Table 1. Table 1, material characteristics used for optimisation models. Parameter Young’s modulus (E) Poisson’s ratio (v) Volumetric mass density (ρ)

Value

SI unit

210,000

MPa

0.3

N/A

7,850

kg/m3

Please note that due to the limitations of present day commercial optimisation software, nonlinear material behaviour cannot be accommodated within the topology optimisation procedure [3, 9, 13, 16, 25, 26].

1.3. Applied loading and boundary conditions Despite the fact that the primary purpose of this study was to investigate the effects of variations in the force application angles for the roof crush scenario, additional load cases were also included in the topology optimisation set-up. A total of six load cases were initially applied. These were the following: 1. pole impact 2. side impact barrier 3. roof crush 4. low-speed rear impact (centred) 5. high-speed rear impact 6. high-speed front impact; offset deformable barrier (ODB). As previously stated, this paper will only be concerned with the effects of the roof crush loading scenario; however, in the continued BIW design process, the effects of the remaining load cases listed above must also be further investigated [12, 15, 23, 24]. An illustration of the locations and directions of the individual load cases listed above can be seen in Figure 2. In Figure 2, the terms encapsulated in ‘()’ denote the direction of the applied forces. For example, for load case number 1, the (Y) in Figure 2 denotes that the force was applied in the positive Y-direction relative to the global coordinate system illustrated in the figure.

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Figure 2 Applied load cases for topology optimisation. Owing to the nature of the models, equivalent linear static forces were defined in order to simulate the dynamic impact loads. Previously conducted topology optimisation studies had revealed that the finite element (FE) models in general were sensitive to the external force application angles, i.e. relatively minor changes in these had considerable effects upon the resulting BIW topology. In order to take this into account, additional eight load cases were added to the pre-existing six load cases listed above. Four of these additional eight were essentially replicas of the high-speed front impact, i.e. load case (6), whilst the remaining four were replicas of the high-speed rear impact, i.e. load case (5) above. The respective load case magnitudes and application points were identical for all the load cases in question. The difference consisted of variations of the force application angle as illustrated in Figure 3.

Figure 3, Force application angles for high-speed front and rear impact loading scenarios. The additional load cases illustrated in Figure 3 were defined as follows: 1. high-speed front crash +5◦ 2. high-speed front crash –5◦ 3. high-speed front crash +10◦ 4. high-speed front crash +5◦ 5. high-speed rear crash +5◦

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6. high-speed rear crash –5◦ 7. high-speed rear crash +10◦ 8. high-speed rear crash +5◦ In addition to the external forces illustrated in Figures 2 and 3, the HEV components which had to be included within the BIW consisted of a 150 kg battery pack and a 110 kg range extender/fuel tank. In the case of time dependent loading, these masses would inherently lead to inertial effects of considerable magnitude, relative to the external loads illustrated in Figures 2 and 3. These inertial effects would have to be reacted by the BIW structure, and were thus most likely to have a considerable influence upon the overall BIW topology, i.e. the outcome of the topology optimisation process. However, as the nature of the models was linear static, these inertial effects could not be taken directly into account. Subject to the choice of the type of boundary condition, this issue could simply be resolved by introducing a force applied to, for example, the battery pack’s centre of mass whilst acting in the opposite direction of the respective external force, as illustrated in Figures 2 and 3. Adopting the above approach to incorporate the inertial effects implied that there is no direct coupling between the external forces, the boundary conditions and the point masses (e.g. battery pack). This is the case when single point constraints (SPC) are used, i.e. when the degrees of freedom (DOFs) of individual nodes are constrained In order to achieve force equilibrium of the structure. In the case of SPC, the external force will ‘simply’ transfer from the point(s) of application to the constrained node(s) (SPC). The results of a topology optimisation would therefore remain unaffected by any point masses within the design volume when not considering the gravitational acceleration. Therefore, if applying SPC it is necessary to include forces to represent the inertial effects of the two HEV components. Even though arguments could be made to justify the usage of the above approach to incorporate the inertial effects, further issues could be raised with respect to the application of SPC. The primary concern was the fact that all externally applied forces would ultimately be reacted in the specific locations where the nodal DOFs were constrained. In the case of the BIW modelling, an obvious choice of location to apply the boundary conditions would be the centre of the wheels. However, by doing so, it immediately follows that all external loading will ultimately be reacted at these points, which in the case of real world crash scenarios is highly unlikely. Instead, the external forces were more likely to be (fully) reacted by local deformations and accelerations (stress waves) throughout the

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structure. The basis of this problem is inherently linked to the simplifications of implicit (linear static) versus explicit (non-linear dynamic) crash modelling. However, as previously defined, the limitations of present-day commercially available FE optimisation software dismiss the usage of dynamic (time dependent loading) topology optimisation. An alternative to applying SPC is the usage of inertia relief (IR). IR can be applied to linear static load cases, but it does not include the necessity to constrain the DOF of any nodes in order to obtain force equilibrium of the individual load scenarios. Instead, IR works by balancing the external loads, translational and rotational accelerations within the actual structure, giving rise to body forces that when combined react with the external loads and thus equilibrium is achieved. More specifically, this is done by adding an additional displacement-dependent load to the stiffness matrix [k] when solving Equation (1).

 k  F   kIR   u   0

0   u  kadd 

(1)

In Equation (1), [kIR] is the stiffness matrix used for IR, [k] is the original stiffness matrix and [kadd] represents the additional terms in the stiffness matrix. Owing to the reasoning discussed above, all models in the topology optimisation study were solved using IR boundary conditions; additional information relating to the implementation of IR in FE models can be found in [2].

1.4. Stiffness and mass density As the impending optimisation was to be performed in a linear static manner, the relationship between the stiffness matrix [k] or [kIR] and the volumetric mass density (ρ) needed to be defined. This was done by utilising the ‘power law for representation of elasticity properties’ as Equation (2) [1].

k      p k 

(2)

In Equation (2), [k] (ρ) is the penalised stiffness matrix, and p is the penalisation factor, which is used to determine the ‘type’ of relationship between [k] and ρ. As long as p is equal to 1.0, the two were directly proportional, as illustrated in Figure 4.

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Figure 4, relationship between [k] and ρ.

This relationship can be adjusted by varying p with the effects as indicated in Figure 4. The reason for adjusting this relationship is to typically penalise intermediate density values in order to avoid ‘vague’ definitions of topology; this is also sometimes referred to as ‘chequerboard effect’ [3]. However, initial analyses revealed that this was not a widespread problem for the models of this study. Therefore, in the remainder of this paper, the value of p will be 1.0, i.e. a linear relationship between the stiffness matrix [k] and the mass density ρ will exist.

1.5. Roof crush scenario The primary purpose of this paper was to investigate the changes in BIW topology when subjected to variations of the pitch and roll angles associated with the roof crush scenario illustrated in Figure 2. The definition of pitch and roll angles relative to the FMVSS 216 standard can be seen in Figure 5.

Figure 5, pitch and roll angle according to the FMVSS 216 standard. The pitch and roll angles that were utilised throughout the study in this paper are defined in Table 2, additional information can be found in [6]. The values listed in Table 2 represent the only differences between the 11 FE models used for the present study. The proposed changes to the FMVSS 216 standard, as discussed in [6, 11, 20, 22], also include changing the

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magnitude of the force that the roof structure must withstand to 3.0 times the vehicles’ unloaded weight, i.e. excluding fuel, passengers, etc. Table 2, pitch and roll angle values for the topology optimisation. Model #

Pitch angle φ (◦)

Roll angle θ (◦)

1

0

0

2

5

25

3

5

40

4

10

40

5

15

40

6

5

45

7

10

45

8

15

45

9

5

50

10

10

50

11

15

50

As the estimated unloaded mass of the vehicle used for this study was 1,500 kg, the force value used for the roof crush scenario was set to 45,000 N. This magnitude of force has been used for the roof crush scenario of all models utilised in the present study.

2. Topology optimisation results The purpose of this section is to present and highlight the results of the topology optimisation study.

2.1 Post-processing of results As previously stated, all 11 models solved contained approximately 523,000 nodes and 14 separate load cases, and utilised IR as the boundary conditions. The average CPU time of the 11 models was approximately 6,562 s or 109 min, which could be considered to be negligible relative to the overall CPU time required to solve (dynamic) crash models at a later stage in the design process. The primary objective of the topology optimisation was to extract the idealised load paths of the BIW, thereby minimising the BIW mass. The post-processing of the topology optimisation results could therefore be divided into two major parts, namely the mass reduction value and the overall topology. These two parts were initially evaluated for each

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individual model contained within the study, and ultimately combined in order to evaluate the overall outcome of the study. The two individual parts of the post-processing will be the focus of attention in the next two subsections, followed by a combined discussion of the two parts.

2.2 Mass reduction values The mass reduction values associated with the topology optimisation study were calculated as the difference between the mass value of the initial iteration, i.e. iteration 0, and the mass value of the final iteration, provided that represented the lowest mass value. The mass reduction value was mathematically formulated as Equation (3).

 Massiteration 0  Mass reduction  %   100    100  Mass final iteration  

(3)

The mass reduction values of the individual models defined in Table 2 are listed in Table 3. Model #


Roll angle θ (◦)

Mass reduction value (%)

1

0

0

90.4

2

5

25

90.6

3

5

40

90.6

4

10

40

90.6

5

15

40

90.6

6

5

45

90.7

7

10

45

90.9

8

15

45

90.6

9

5

50

90.5

10

10

50

90.7

11

15

50

90.7

A more comprehensive overview of the mass reduction values may be obtained by means of Figure 6, which is a graphical illustration of the results listed in Table 3. The numbers on top of the individual columns in Figure 6 are the model numbers as defined in Table 2.

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Figure 6, mass reduction values from topology optimisation. By observing Figure 6, it can be seen that the model that displayed the largest mass reduction value constitutes model 7, i.e. a pitch angle of 10 and a roll angle of 45, whilst the lowest mass reduction value was obtained from model 1, i.e. for 0 pitch and roll angles. The difference in mass reduction values between the two was a mere 0.5%, thereby representing the largest difference in mass reduction value obtained during the study. Subsequent studies based upon the topology optimisation results of this paper have estimated that the final BIW mass (ready for manufacturing) will be less than 200 kg, when the material properties listed in Table 1 were utilised. The actual difference in mass between models 7 and 1 thereby became approximately 1 kg. The second largest mass reduction values were found in models 6, 10 and 11, which all resulted in a value of 90.7%. The difference in mass between these three models and model 7 (the largest mass reduction value) thus became 0.2% or an estimated 0.4 kg. The above results did therefore indicate that the differences of BIW mass as a function of varying the angles associated with the roof crush loading scenario were minor. The maximum mass difference estimated to be 1 kg, found via the study, could nevertheless also be interpreted as being of significant magnitude. Such an interpretation must however also be considered in context with the overall design process, as the topology optimisations of this paper are intended to represent the initial steps of the overall design process associated with the production of a BIW. The topology, cross-sectional areas and overall geometry were only coarsely defined at this very early stage in the design process. Subsequent to the topology optimisation, other aspects such as manufacturing, dynamic loading, buckling, local crushing, noise vibration and harshness (NVH) and fatigue/durability must also be taken into account. When the BIW design process enters these phases, the level of detail in the BIW design is

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significantly increased compared with the topology optimisation stage, thus making an estimated mass difference of 1.0 kg at this initial stage less significant than in the later stages. In the light of the above explanation of the ‘relative simplicity’ of the models at this very early stage in the design process, it must be stressed that the above mass reduction values are merely estimations. A significant part of the final BIW mass, i.e. when the BIW is ready for production, is defined via the engineering interpretation of the topology optimisation results. This will include FEA with considerably higher levels of details of the design than presented in this paper. This will result in more accurate predictions of the structural (crash) performance of the roof, thereby enabling a deeper understanding of the specific geometrical details of the roof and consequently enabling increasingly precise estimations of the total BIW mass. In fact, several studies have shown that a 10 pitch angle combined with a 45 roll angle constitutes ‘the worst case’ roof crush scenario, resulting in ‘poor’ roof crush performance, for example, resulting in deep intrusion into the passenger cell, thus increasing the risk of severe occupant injuries [6, 11, 19]. This may seem to contradict the preliminary topology optimisation results of this paper, as model 7 (10 pitch angle and 45 roll angle) represented the highest mass reduction. However, as explained above, the maximum difference was only found to be 0.5%. Furthermore, the relative simplicity of the models at this very early stage in the design process clearly plays a significant role in this context. Finally, the conclusions of this paper cannot be solely based upon the mass reduction values, but must also include the actual topology obtained via the conducted optimisation runs. The conclusion, solely based upon the mass reduction values, must therefore be that the variation of the pitch and roll angles associated with the roof crush loading scenario only incurred minor changes to the BIW mass.

2.3 Overall topology For the purpose of post-processing the resulting topologies, the global topology of each of the 11 individual models was observed. Based upon these initial observations, it was concluded that the primary differences in topology were found in the roof area. The definition of the roof or roof area can be understood by observing Figure 7, where the area is highlighted (and denoted as 1). This initial finding was in agreement with the linear nature of the models, in addition to the fact that the only difference between the 11 models was the force application Angles were associated with the roof crush loading scenario.

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Figure 7, definition of the roof area. When the resulting topologies of the roof areas obtained by solving the 11 models were postprocessed, significant variations were however found. By observing the topological trends of the roof area for the 11 models, they were divided into six different groupings, based upon similarities between the individual model topologies. The six groupings are listed in Table 4. Table 3, grouping of models based upon the roof topology. Grouping #

Model #


Roll angle θ (◦)

6

5

45

9

5

50

10

10

50

3

5

40

7

10

45

8

15

45

1

0

0

2

5

25

IV

4

10

40

V

11

15

50

VI

5

15

40

I

II

III

2.3.1 Grouping I Models 6, 9 and 10 were classed as belonging to group I. All three of these models utilised pitch angles (φ) of 5◦ and 10◦, and roll angles (θ) of 45◦ and 50◦. An example of the general roof topology obtained from these three models is illustrated in Figure 8, which displays the

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roof topology of model 6 viewed in the XY-plane. For clarity, Figure 8 also indicates the locations of the ‘windscreen’, ‘passenger cell’ and ‘rear end of the vehicle.

Figure 8, roof topology of model 6 at φ = 5◦ and θ = 45◦. The topology displayed in Figure 8 could be characterised as highly unconventional when compared with more ‘traditional’ roof bow structures often found in modern-day (fossilfuelled) vehicles. The topology also indicates that a considerable amount of material, i.e. mass, must be used for the roof, which may have significant effects upon the vehicle dynamics [18]. The topology illustrated in Figure 8 displays a widespread usage of triangles, which is also compliant with the linear nature of the models used. In addition, the models contained within group I displayed a clear tendency to utilise two significantly curved load paths indicated as ‘A’ in Figure 8. 2.3.2 Grouping II Models 3, 7 and 8 were classed as belonging to group II. All of these models utilised pitch angles (φ) of 5◦, 10◦ or +15◦. The roll angles (θ) were 40◦ or 45◦. An example of the general roof topology obtained from these three models is illustrated in Figure 9, which displays the roof topology of model 7.

Figure 9, roof topology of model 7 at φ = 10◦ and θ = 45◦.

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According to, for example, Chirwa and Peng, Grzebieta et al. and Parent et al. [6, 11, 19], the roof crush scenario of model 7 (10 pitch angle and 45 roll angle) constitutes the ‘worst-case’ loading scenario. However, according to the mass reduction values of this study, model 7 constitutes the largest mass reduction value, with a maximum difference of 0.5%. The significance of the magnitude of this value must however, as previously explained, be evaluated in combination with the relative simplicity of the models at this very early stage of the design process. Indeed, it is important to remember that topology optimisation uses ‘relative mass densities’, i.e. these vary throughout the resulting geometry (topology) and are therefore a significant factor in the overall mass reduction value. In plots, such as Figure 9, the lower value ‘relative mass densities’ are indicated by ‘darker colours’. This indicates that ‘relatively less’ mass is required in these specific areas. However, it must be stressed that these indications are based upon linear static FEA. This however does not mean that these areas are insignificant with respect to the structural (crash) performance of the BIW. They may, however, be used to explain the differences in mass reduction values between models. By comparing Figure 8 with Figure 9, it can be seen that the former contains more ‘lighter coloured areas’, whereas the latter contains more ‘darker coloured areas’. This indicates that the ‘relative mass density’ in Figure 9, in general, is less than that of Figure 8, thereby contributing to the difference in mass reduction. As previously explained, this will become increasingly significant, and most likely subject to change, during the highly detailed FEA and final physical test validation of the structural performance of the roof [7, 20]. In line with the findings related to grouping I, the topology found in grouping II also displayed a complicated roof topology/geometry, including the widespread usage of triangles. The two load paths denoted as ‘A’ in Figure 8 do not seem to exist in Figure 9. These have seemingly been ‘replaced’ by the two load paths denoted as ‘B’ in Figure 9; however, these latter two (‘B’) were also distinguishable in Figure 8. This meant that even though the two roof topologies defined as groups I (Figure 8) and II (Figure 9) at first glance appeared to be very dissimilar, common topological trends could still be identified between the two groupings. 2.3.3 Grouping III Models 1 and 2 were classed as belonging to group III. These models utilised pitch angles (φ) of 0◦ and 5◦, in addition to roll angles (θ) of 0◦ and 25◦. An example of the general roof topology obtained from these two models is illustrated in Figure 10, which displays the roof topology of model 1.

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Figure 10, roof topology of model 1 at φ = 0◦ and θ = 0◦. When comparing Figures 8 and 9 with Figure 10, it soon became clear that the topology displayed in Figure 10 was less complicated than those displayed by the other two. Given the linear nature of the models and the 0◦ pitch and roll angles used for model 1, this made sense, as the loading associated with the roof crush became perpendicular to the plane of the roof, i.e. the XY-plane as defined by Figure 10. When comparing Figure 9 with Figure 10, it could be seen that the two load paths denoted as ‘B’ existed in both topologies, which as previously concluded also existed in Figure 8. In addition, the topology illustrated in Figure 10 also displayed the ‘triangulation’ as Figures 8 and 9 did. It is also worth noticing that the two models belonging to group III represent the current FMVSS 216 (φ = 5◦, θ = 25◦) and EuroNCAP (_φ = 0◦, θ = 0◦) test specifications. In other words, the load cases specified by these two led to considerably less complicated roof topologies when compared with the remaining roof crush load cases of this study, which represented some of the proposed changes to the current FMVSS 216 standard. Despite the fact that the results from model 1 (Figure 10) represented the least complicated geometry of the study, it also represented the lowest mass reduction value found in the study, indicating that low complexity of roof topology/ geometry and low mass reduction values were inversely linked. However, the significance of this statement must be accompanied by the discussion in Section 2.2, which concluded that the maximum difference in mass between all 11 models was found to be 0.5% or an estimated 1.0 kg for a final BIW mass of 200 kg. 2.3.4 Groupings IV, V and VI These groups only contained a single model each. The individual differences between the models in these three groupings mainly consisted of subtle differences towards the rear end of the structure (Figure 8). Figure 11 illustrates the results from grouping VI.

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Figure 11, , roof topology of model 5 at φ = 15◦ and θ = 40◦. In line with the previously presented results, all three models in grouping IV, V and VI also contain the load paths denoted as ‘B’ in Figure 11. At this stage, all the ‘general’ topologies relating to the groupings defined in Table 4 have been presented and individually compared. The next step involved summarising the findings obtained from these comparisons whilst highlighting some of the topological trends that were found to be consistent / distinguishable throughout the study. This will be the focus of attention in the following subsection.

2.4 Summation of topological tendencies The purpose of this subsection is to highlight and underline the general trends that were found by post-processing the results obtained by solving the 11 models defined for this topology optimisation study. The discussions of this section will primarily be based upon the topologies displayed in Figures 8-11. The main similarity between the roof topologies presented and discussed throughout the previous section was the load paths denoted as ‘B’ in, for example, Figure 11. These particular load paths were distinguishable in all the topologies presented, thus underlining the significance of these. The point denoted as ‘C’ in Figure 12 represents the areas where the ‘B’ load paths intersect.

Figure 12, similarities between roof topology results.

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The significance of this point may not be immediately clear however by studying all roof topologies obtained, a tendency started to immerge. This tendency suggested that point ‘C’ displaced along the centre line of the design volume, i.e. the X-axis, whilst the variations of roof topology complied with the location of this point. In certain models, this point seemed to be replaced by two points, ‘C1’ and ‘C2’, leading to a ‘duplication’ of the ‘B’ load paths as illustrated in Figure 13.

Figure 13, variation of point C in models 5 and 1. The load paths denoted as ‘B’ were however not the only ones ‘repeated’ in the topologies presented. Further load paths were also present in all topologies, thereby underlining their particular relevance. The load paths in question were particularly apparent towards the front end of the design volume, specifically within the area denoted as ‘D’ in Figure 12. Finally, the load paths denoted as ‘A’ in Figure 8 were also distinguishable in the results of grouping V, meaning that these load paths were found in a total of four models, i.e. in excess of one third of the models contained within the study. The presence/importance of these load paths thereby must be taken into consideration when concluding the overall topological trends found during this optimisation study. On the basis of the discussions and illustrations of this paper, the overall topological trends found by means of this optimisation study were summarised. This ‘summation of topology’ is illustrated in Figure 14. The denotations used in Figure 14 correspond to those used throughout the discussions of this paper. The curved load paths denoted as ‘A’ in Figure 14 are presented as dotted lines because the presence of these were not found to be consistent throughout all models. In most models, the ‘B’ load paths joined each other at a single point denoted as ‘C’ in Figure 14; however, in some models, ‘C’ was replaced by two points (Figure 13).

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Figure 14, summation of the found roof topology tendencies. The implications of whether ‘C’ was a single point or two points also influenced the length, size and indeed existence of the two load paths illustrated between ‘C’ and the two points ‘E’ in Figure 14. Finally, the topology towards the front end of the design volume, i.e. in the vicinity of ‘D’ in Figure 14, was found to be very consistent throughout the optimisation study.

3. Conclusion and validity of results This study has investigated the possible effects of the proposed changes to the FMVSS 216 standard upon the roof topology of a BIW intended for an HEV vehicle, based upon linear static topology optimisation results. The methodology utilised in this study is thus significantly different from the ‘typical’ BIW design process used by OEMs. However, ongoing research by the authors of this paper indicates that relevant and useful information for the BIW architecture can be extracted via topology optimisation [8]. This claim is substantiated by the findings of the FSV project (http://www.futuresteelvehicle.org/). It should be noted that the authors of this paper did not participate in the FSV project. The present study included a total of 11 combined variations of the pitch and roll angles associated with the roof crush loading scenario. These combinations also included the current EuroNCAP values as well as the values specified in the current FMVSS 216 standard. The study found that the estimated mass value of the BIW for the HEV did not vary significantly as a function of the pitch and roll angle variations. This may initially seem controversial, as other papers investigating the effects of proposed FMVSS 216 changes, such as [17], found that the magnitude of force distribution within the roof changes significantly as a function of the pitch and roll angles. However, at this point, it is important to remember that the topology optimisation extracts the most ‘efficient’ load paths according to the applied load cases. As the magnitudes of the applied forces have remained constant throughout the study, the effects of the change in angles can be accommodated by changing the load paths, with only minor changes in BIW mass. This statement is consistent with the results found during the study.

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Furthermore, Mao et al. [17] also discuss the effects and importance of buckling and localised crushing of the roof pillars as significant factors in the overall crash performance of the roof, thus making these potential parameters of the topology optimisation. However, it is not feasible to implement the above as parameters into the optimisation models. Therefore, it is important to recognise the ‘relative simplicity’ of the models in question. As previously stated, the outcome of FE-based topology optimisation does not contain detailed information relating to, for example, cross-sectional geometry, which is required in order to draw accurate conclusions on the presence or absence of buckling and localised crushing of the roof pillars. In order to utilise buckling and localised crushing (with a satisfactory level of accuracy) as parameters in connection with topology optimisation, alternative optimisation algorithms, such as the homogenisation method, need to be employed [3]. This is not currently included in commercially available FE software. The implementation of the two parameters above is not likely to lead to significant changes in the results of the present study. This is primarily linked to the objective of the optimisation, which was to minimise the BIW mass whilst constrained by maximum displacement criteria. It therefore follows that the optimisation will define/retain the load paths where they are most ‘efficient’. Consequently, the optimisation will attempt to maximise the forces in the individual load paths, inadvertently increasing the possibility of buckling. Initial post-processing of the global BIW topologies determined that the pitch and roll angle variations did not incur significant changes to the global BIW topology. However, it was found that the angle variations did imply significant changes to the roof topology. Initially, the results of the 11 models led to six individual groupings, suggesting that only minor similarities existed. None of the 11 topologies bore a close resemblance to the conventional roof bow structures typically found in modern-day fossil fuelled vehicles. Additional post-processing of the topologies did however expose some general topological trends that were identifiable across these six groupings. These trends were ultimately combined to enable the construction of a general roof topology representing the outcomes of the topology optimisation study. This combined topology is represented in Figure 14. The overall conclusion of this study, which utilised linear (implicit) topology optimisation, was therefore that it is unlikely that the proposed changes to the FMVSS 216 standard will lead to an increase in BIW mass; however, it is very likely that it will lead to significant changes of the BIW roof area topology for an HEV, subject to the BIW design

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process described in this paper. Further steps of the BIW design process, including dynamic (explicit) FEA containing increased levels of, for example, cross-sectional geometry, are however required in order to further substantiate the above proposed tendency.

Acknowledgements The authors of this paper would like to thank Mr Mike Dickison, Mr Richard Nicholson (both of Coventry University),Mr Andrew Gittens of MIRA Ltd., Tata Motors European Technical Centre (TMETC), Jaguar Land Rover (JLR), Warwick Manufacturing Group (WMG), AdvantageWestMidlands (AWM), the European Regional Development Fund (ERDF) and other contributors to the Low Carbon Vehicle Technology Project (LCVTP) for supplying data and guidance to assist in the making of this paper.

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References [1] Altair Engineering, Inc., Altair Hyperworks Manual, Altair Engineering, Inc., Troy, MI, 2010. [2] R. Barnett, T.W. Widrick, and D.R. Ludwiczak, Closed-Form Static Analysis With Inertia Relief and Displacement-Dependent Loads Using a MSC/NASTRAN DMAP Alter, NASA, Washington, DC, 2005. Available at http:// ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/199500132331995113233.pdf. [3] M.P. Bendsoe and O. Sigmund, Topology Optimization,Theory Methods and Applications, Springer-Verlag, Berlin, 2004. [4] K.U. Bletzinger and K. Maute, Towards generalized shapeand topology optimization, Eng. Optimiz. 29 (1997), pp. 201–216. [5] M.Bremicker,M.Chirehdast,N.Kikuchi, and P.Y. Papalambros, Integrated topology and shape optimization in structural design, Mech. Base. Des. Struct. Mach. 19 (1991), pp. 551–587. [6] E.C. Chirwa and Q. Peng, Modelling of roof crush using the newly updated FMVSS 216, ICRASH 2010, Washington, DC, 22–24 September 2010. [7] E.C. Chirwa, R.R. Stephenson, S.A. Batzer, and R.H. Grzebieta, Review of the Jordan Rollover System (JRS) vis-à-vis other dynamic crash test devices, Int. J. Crashworthiness 15 (2010), pp. 553–569. [8] J. Christensen, C. Bastien, M.V. Blundell, A. Gittens, and O. Tomlin, Lightweight hybrid electrical vehicle structural topology optimisation investigation focusing on rashworthiness, Int. J. Veh. Struct. Syst. 3 (2011), pp. 113–122. [9] F. Duddeck, Multidisciplinary Optimization of Car Bodies, Springer, London, 2007. Available at http://www.springerlink.com/content/2741p87038110l70/. [10] D. Friedman and C.E. Nash, Measuring rollover roof strength for occupant protection, Int. J. Crashworthiness 8 (2003), pp. 97–105. [11] R.H. Grzebieta, A.S. McIntosh, and M. Bambach, How stronger roofs prevent diving injuries in rollover crashes, ICRASH 2010, Washington, DC, 22–24 September 2010. [12] M.F. Horstemeyer, X.C. Ren, H. Fang, E. Acar, and P.T. Wang, A comparative study of design optimisation methodologiesfor side-impact crashworthiness, using injury-based versus energy-based criterion, Int. J. Crashworthiness 14 (2009), pp. 125–138. [13] N. Jones, Dynamic material properties and inelastic failure in structural crashworthiness, Int. J. Veh. Struct. Syst. 6 (2009), pp. 7–18. [14] J. Klose, A note on the core topology used in optimization, Optimization 23 (1992), pp. 27– 40.

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[15] X. Liao, Q. Li, X. Yang, W. Li, and W. Zhang, A two-stage multi-objective optimisation of vehicle crashworthiness under frontal impact, Int. J. Crashworthiness 13 (2008), pp. 279– 288. [16] Z. Liu and J.G. Korvink, Adaptive moving mesh level set method for structure topology optimization, Eng. Optimiz. 40 (2008), pp. 529–558. [17] M. Mao, E.C. Chirwa, and W. Wang, Assessment of vehicle roof crush test protocols using FE models: Inverted droptests versus updated FMVSS no. 216, Int. J. Crashworthiness 11 (2006), pp. 49–63. [18] M. Okçuo˘glu, K. Friedman, D. Mihora, J. Hutchinson, J. Wiedmann, K. Fink, S. Reid, and S. Chan, Considerationof vehicle handling and stability with improved roof strength, Int. J. Crashworthiness 12 (2007), pp. 541–547. [19] D.P. Parent, J.R.Kerrigan, and J.R. Crandall, Comprehensive computational rollover sensitivity study part 1: Influence of vehicle pre-crash parameters on crash kinematics and roof crush, ICRASH 2010, Washington, DC, 22–24 September 2010. [20] R. Rhoads Stepenson, The case for a dynamic rollover test, ICRASH 2010, Washington, DC, 22–24 September 2010. [21] S.A. Richardson, G. Rechnitzer, R.H. Grzebieta, and E. Hoareau, An advanced methodology for estimating vehicle rollover propensity, Int. J. Crashworthiness 8 (2003), pp. 63–72. [22] F. Tahan, K. Digges, and P. Mohan, Sensitivity study of vehicle rollovers to various initial conditions—Finite element model based analysis, ICRASH 2010, Washington, DC, 22–24 September 2010. [23] B. Zhang, J. Yang, and Z. Zhong, Optimisation of vehicle side interior panels for occupant safety in side impact, Int. J. Crashworthiness 15 (2010), pp. 617–623. [24] Y. Zhao,G.F.Rosala, I.F. Campean, and A.J. Day, A response surface approach to front-car optimisation for minimising pedestrian head injury levels, Int. J. Crashworthiness 15 (2010), pp. 143–150. [25] H. Zimmer, M. Prabhuwaingankar, and F. Duddeck, Topology and geometry based structure optimization using implicit parametric models and LS-OPT, 2009. Available at http://www.dynalook.com/european-conf-2009/F-II-02.pdf/view. [26] K.T. Zuo, L. Chen, Y.Q. Zhang, and J. Yang, A hybrid topology optimization algorithm for structural design, Eng. Optimiz. 37 (2005), pp. 849–866.

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CONTRIBUTION

3

Double blind peer review journal paper published in the International Journal of Crashworthiness on st May 1 2013 [Format has been edited to comply with Coventry University regulations]

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This article was downloaded by: [Jesper Christensen] On: 01 May 2013, At: 07:45 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

International Journal of Crashworthiness Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tcrs20

Buckling considerations and cross-sectional geometry development for topology optimised body in white J. Christensen a , C. Bastien a , M. V. Blundell a & P. A. Batt a of Engineering and Computing , Coventry University , Coventry , UK Published online: 01 May 2013

a Faculty

To cite this article: J. Christensen , C. Bastien , M. V. Blundell & P. A. Batt (2013): Buckling considerations and cros sectional geometry development for topology optimised body in white, International Journal of Crashworthiness, DOI:10.1080/13588265.2013.792442 To link to this article: http://dx.doi.org/10.1080/13588265.2013.792442 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-andconditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents Will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

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Buckling considerations and cross-sectional geometry development for topology optimised body in white J. Christensen∗, C. Bastien, M.V. Blundell and P.A. Batt Faculty of Engineering and Computing, Coventry University, Coventry, UK (Received 11 January 2013; final version received 2 April 2013)

This paper will investigate how current state-of-the-art structural optimisation algorithms, with an emphasis on topology optimisation, can be used to rapidly develop lightweight body in white (BIW) concept designs, based on a computer aided design envelope. The optimisation models included in the paper will primarily focus on crashworthiness and roof crush scenarios as specified in the Federal Motor Vehicle Safety Standards (FMVSS) 216 standard. This paper is a continuation of a previously published paper, which investigated the potential effects of recently proposed changes to FMVSS 216 upon BIW mass and architecture using topology optimisation. The paper will investigate the possibilities of including buckling considerations of roof members directly into current state-of-the-art topology optimisation algorithms. This paper will also demonstrate the potential for developing a detailed BIW design including crosssectional properties based on a styling envelope. Keywords: finite element topology optimisation; FMVSS 216; roof crush; body in white (BIW); lightweight vehicle architecture; hybrid electric vehicle; inertia relief

1. Introduction In 2003, it was estimated that approximately 10,000 fatalities occurred on US roads due to rollovers [12]. This highlights the severity of the crashes associated with rollovers; the influence of the roof strength is discussed in [10]. In light of this, recent changes to the FMVSS 216 standard has been proposed. The potential effects of these changes upon body in white (BIW) architecture and mass have previously been investigated using topology optimisation, inertia relief and a hybrid electrical vehicle case study [5]. The only variation between the individual models of the study in [5] was the (pitch and roll) angles associated with the roof crush scenario. The pitch angle φ and the roll angle θ are illustrated in Figure 1.

Figure 1, pitch and roll angle according to FMVSS 216.

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Furthermore, the applied loading magnitude was increased to three times the unloaded ehicle weight (45 kN), i.e. excluding fuel, passengers etc., as suggested in [3,11,13,14]. The study in [5] included 11 models and found that the estimated mass value of the roof, denoted as 1 in Figure 2, did not vary significantly as a function of the pitch and roll angle variations.

Figure 2, roof area definition. This may initially seem controversial, as other papers investigating the effects of the proposed FMVSS 216 changes, such as [4], found that the magnitude of force distribution within the roof changes significantly as a function of the pitch and roll angles. However, at this point, it is important to remember that topology optimisation extracts the most ‘efficient’ load paths according to the applied load cases based on linear static finite element (FE) modelling. As the magnitudes of the applied forces did remain constant throughout the study [5], the effects of the change of angles could be accommodated by changing the load paths, with only minor changes to overall BIW mass. The effects and importance of buckling and localised crushing of the roof pillars as significant factors in the overall crash performance of the roof have been discussed in [4], thereby making these potential parameters of topology optimisation. Using commercially available topology optimisation software, it is not feasible to directly implement the above parameters as variables into optimisation models. Firstly, buckling cannot be calculated for solid (three dimensional (3D)) elements. Secondly, it is important to recognise the ‘relative simplicity’ of the models in question. In general, the outcome of FE-based topology optimisation does not contain detailed information relating to e.g. cross-sectional geometry, which is required in order to draw accurate estimations on the likelihood of buckling and/or localised crushing of the roof pillars occurring. To include buckling and localised crushing (with a satisfactorily level of accuracy), as parameters in connection with topology optimisation, alternative optimisation algorithms, such as the homogenisation method, could possibly be employed [2]. However, these algorithms have proven extremely difficult, if not impossible to apply to anything but very simple geometries, let alone a complex’ Contribution 3

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design volume, such as the one illustrated in Figure 2 (including a large number of elements and design space based on complex Boolean operations). The implementation of buckling and localised crushing as parameters in topology optimisation is not certain to lead to significant changes in the results of the FMVSS 216 study of [5]. This is primarily linked to the objective of the optimisation, which was to minimise the BIW mass whilst constrained by maximum displacement criteria. It therefore follows that the optimisation will define/retain the load paths where they are most ‘efficient’. Consequently, the optimisation will attempt to maximise the forces in the individual load paths, inadvertently increasing the likelihood of buckling. This fact raises an interesting question with respect to obtaining the lightest roof structure. To prevent buckling of ‘long beams’, i.e. the roof members, obtained using commercial software, as illustrated in Figure 3 (displaying relative element density), the cross sections of these will subsequently have to be dimensioned accordingly during the optimisation process.

Figure 3, example of roof topology; relative element density. This does in essence mean that (for a constant value of Young’s modulus) the second moment of area I, needs to be defined. As there is a clear limit to the outer dimension of these beams, due to packaging space, i.e. design envelope, the only remaining option is to increase the material gauge, inadvertently leading to an increase in mass, disregarding a possible change of material. On the other hand, if the aspects and limitations stated above were considered and actively used during the topology optimisation process, the general outcome (topology) may be significantly different. This is based on the fact that ‘shorter beams’ will be less likely to experience buckling, hence I may be decreased. This is, however, likely to considerably increase the complexity of the roof structure, possibly invoking manufacturing constraints. Finally, it is far from certain that the overall mass of the ‘short beam’ structure is less than that of the ‘long beam’ structure, this warrants further analysis and is the main focus of attention

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of this paper. The problem will be approached using existing topology optimisation algorithms as used in [5]. Figure 4 illustrates the proposed overall algorithm for BIW development using structural optimisation.

Figure 4, overall proposed optimisation methodology. The main focus of this paper will be step 3 of Figure 4. Before this is completed step 2 as was originally completed in [5] will be revisited; I value estimations. There are some immediate complications of how the I-value estimation could be made an active parameter within the topology optimisation algorithm, let alone as a constraint. This will be the focus of attention in the following section.

2. Implementation of I-value To fully implement an accurate I-value within topology optimisation a meta-modelling technique similar to that used in the shape and size optimisation of [9] could be adopted. However, this approach was deemed infeasible in this connection, particularly as topology optimisation determines the general load paths without providing high levelled details of cross-sectional properties. Therefore, it was much more feasible to include an approximation of the local I-value within the topology optimisation algorithm. To do so, the following points of consideration needed to be addressed: (1) Establish how the local I-values could be estimated. (2) Determine how the I-values could be used to evaluate the probability of the occurrence of buckling. (3) Define how to implement step 2 into the topology optimisation algorithm. To determine a feasible methodology for estimating the appropriate I-values for the ‘local roof beams’, a case study was performed. The intention was to use size optimisation of a fixed cross-sectional geometry, i.e. no shape optimisation, and include the buckling factor as a

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constraint. To do so, wireframe models of representative topologies of four models from the results of the topology optimisation study [5] were used. The four selected models are listed in Table 1. Table 1, selected models for case study. Model #


Roll angle θ (◦)

1

0

0

4

10

40

6

5

45

7

10

45

It is important to note that the selected models in Table 1 contain the current EuroNCAP setup (φ = θ = 0◦); this grouping also included the current FMVSS 216 set-up [5]. Model 7, also listed in Table 1, represented the highest mass reduction of the original FMVSS 216 topology optimisation study, although the maximum mass difference between all models of this study was found to be less than 0.5% [5]. It is also worth noting that other studies, such as [1,3,13], have found that the combination of a 10◦ pitch angle φ, and a 45◦ roll angle θ constitutes the ‘worst case loading scenario’. To evaluate the likelihood of buckling eigenvalue based algorithms are commonly used in FE. However, due to the usage of inertia relief in connection with the topology optimisation, it is not feasible to use an eigenvalue-based approachas the two methods are incompatible. Therefore, it was chosen to use Euler’s buckling formula to estimate the critical buckling load, Fcrit, Equation (1),

Fcrit 

2 EI k  L2

(1)

where E isYoung’smodulus, I is the second moment of area, k is the slenderness ratio and L is the length of the beam member. For the remainder of this paper, k will be set to 1. The length L of each beam member was predetermined (and fixed) based on the outcome of the topology optimisation results. L was determined on a member basis (based on a constant mesh size), as opposed to an element basis, as illustrated in Figure 5.

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Figure 5, example of wireframe model. For the purpose of conducting the size optimisation, in order to estimate the I-values, a tubular cross section of unity thickness was selected for convenience, as this left only one variable parameter: the diameter d. It has to be noted that the general cross-sectional geometry (shape) was irrelevant, as the impending task was to estimate appropriate I-values for buckling estimations. With the above definitions, the buckling factor, Bf, could be estimated using Equation (2):

Bf 

Element axial force 1 Fcrit

(2)

The ‘element axial force’, used in Equation (2), is determined by the maximum axial force occurring within any element of the member in question. As long as Bf remains less than 1, linear buckling is not likely to occur. The above methodology was relatively simple to include as a constraint within a sizing optimisation, using commercial software. The results from the four models listed in Table 1, are listed in Table 2, and illustrated in Figure 6.

Table 2, results from I-value estimation (sizing optimisation). Model #


Roll angle θ (◦)

Final estimated mass (kg)

1

0

0

42.1

4

10

40

41.0

6

5

45

39.7

7

10

45

44.4

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Figure 6, results from diameter sizing.

The estimated masses listed in Table 2 include A-,B-, C and D-pillars as well as the main roof structure, excluding roof sheets, as illustrated in Figure 6. The values are not to be interpreted as the final (ready for production) masses, as shape-optimisation has not been completed as this stage. In fact another sizing optimisation will need to be completed at a later stage in order to further optimise the structure. It is important to remember that the overall purpose of the above was to investigate the first step of the potential for including buckling considerations within topology optimisation, i.e. step 2 of Figure 4. Furthermore, the wireframe models included in the study (Tables 1 and 2) were kept as close to the ‘original’ as possible, in particular this meant the inclusion of a significant number of ‘curved beams’, as indicated by Figure 5 and Figure 6. Revising the topology prior to ‘I-value determination’ may have a significant effect on the estimated mass results in Table 2. Nevertheless, the mass results listed in Table 2 may be compared relative to each other in order to better understand the ‘relative influence’ of the pitch and roll angles upon the roof structure. On this basis, it can be seen that the current EuroNCAP test, φ = φ = 0◦, is of ‘medium severity’, based on the final mass estimation. Even more interesting model 7 (φ = 10◦ , φ = 45◦) represents a significantly higher estimated mass value with a difference of approximately 11% relative to model 6 (φ = 5◦ , θ = 45◦) highlighting the necessity for Contribution 3

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subsequent sensitivity studies as discussed in [5]. This is in line with the findings of [1,3,13], who established that the ‘worst crash loading scenario’ occurs when φ = 10◦ and θ = 45◦, thus substantiating the appropriateness of the results obtained by means of topology optimisation and the subsequent I-value estimation by scaling the diameters. This does, however, not solve the previously proposed problem of a ‘long beam structure’ versus a ‘short beam structure’, as the I-value is only considered subsequent to the completion of the topology optimisation. The next steps of how this could be achieved will be the focus of attention in the following section.

2.1. Implementation of I-value estimation within topology optimisation Although the results from the size optimisation estimating the local I-values suggestively produce reasonable approximations they cannot straightforwardly be implemented within a topology optimisation algorithm, such as the one employed in [5]. Using a mathematically based topology optimisation algorithm and the solid isotropic material with penalisation (SIMP) interpolation scheme two primary problems needed to be addressed; the identification of ‘beams’, i.e. load paths, and how to include the I-value estimations as active parameters within the algorithm. At this point, it is important to highlight that it was necessary to use static modelling as 3D (solid) elements were used for the topology optimisation [5]. Therefore, it was not feasible to consider using one or several (explicit) dynamic crash analyses. This is due to the fact that the central processing unit time would dramatically increase as a function of the small timestep (∆t); a consequence of the small element size required for the 3D elements. To increase ∆t element mass could be added; this would, however, need to be of significant magnitude making the FE model incompatible with the real world physics of the crash event. Consequently linear static (implicit) finite element analysis (FEA) would be the starting point for the I-value estimations. With the starting point defined, the next part-task for the I-value estimations could be approached. This was the issue of how to ‘define/identify’ the load paths within a specific iteration of the topology optimisation process, in order to model these as ‘beams’ for estimation of appropriate Ivalues. This was, however, a very complex task, which may be envisaged by observing the example in Figure 7, illustrating relative element density.

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Figure 7, example of topology iteration; relative element density. The example in Figure 7 clearly demonstrates the complexity of (explicitly and uniquely) identifying the primary structural load paths within any iteration. Considering current state-ofthe-art modelling/optimisation algorithms, it is believed that this proposed technique has the highest potential to resolve this issue by means of logical mathematical expressions embedded within the FE solver, i.e. observing and using Boolean expressions based on the current (iteration) values of the relative element densities. If this problem was to be solved, it would also aid in the often lengthy task of post-processing the optimisation results, by offering a certain level of ‘automation’ with the aim of obtaining a computer-aided design (CAD) model to be used for further analysis, or indeed additional structural optimisation. Implementation of this feature would, however, require disassembly of the commercial FE solver, which is prohibited and therefore not possible at this stage. Following the identification of load paths the next step would be to include the I-value estimations as active constraints within the topology optimisation.

2.2. Active estimation of I-values Assuming that the above problem was solved, the next step would then be to implement the Ivalue estimations within the topology optimisation algorithm. This would initially not pose a significantly challenging problem, as this could simply be obtained by means of a constraint imposed within the topology optimisation problem definition, in the exact same manner as e.g. a displacement constraint. In this case, the methodology proposed in the previous section could be straightforwardly employed. However, this would theoretically neither provide a significant engineering nor scientific advancement as the end results would most likely not differ considerably from those illustrated in Figure 6 or those listed in Table 2. The reason for this is simply that such an approach would merely include the local I-values as ‘inactive constraints’, and not actively guiding/influencing the topology optimisation process

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or results. If the I-value estimations were to be actively included, the task at hand would become significantly more complex, as the ‘fundamental’ decision-making process of the topology optimisation algorithm would have to be redefined [1,2]. The fact that the I-value estimations include the length of the beam, which would not be defined at that stage of the topology optimisation process, adds further complexity to the problem, as previously discussed and illustrated by Figure 7. The two aspects discussed above thus make it extremely difficult to implement the I-value estimations, and indirectly the buckling considerations, as an active parameter within a topology optimisation algorithm. Evaluating whether or not solving the extremely complex tasks discussed above would be worthwhile taking on was in essence a question of estimating the potential effects of ‘long fat beams’ versus ‘short thin beams’ as previously discussed. In this context, it was deemed that the influence upon the final roof design would most likely be negligible, partially due to manufacturing limitations in terms of cost versus the potential mass reduction in percent. In addition, it needed to be considered that the topology optimisation was completed using linear static FEA, as discussed in [5-9,15]. Furthermore, the Eulerian buckling formula, Equation (1), assumes linear material behaviour. Thereby the method of superposition was in principle applicable, as the penalisation factor p equalled 1 in [5]. Consequently it was unlikely that any significant difference in results would be found between completing the I-value estimations subsequent to the topology optimisation, or including these directly in the topology optimisation step, provided that the latter were not actively guiding the optimisation process, further analyses are, however, required to support this claim. It therefore became clear that if the buckling phenomena were to be ‘fully implemented’ into a topology optimisation algorithm, using 3D elements, the fundamental principles of the algorithm would have to be revised, as well as disassembly of the FE solver. In light of the above, it was deemed acceptable to complete the I-value estimations subsequent to the topology optimisation as previously completed and listed in Table 2 and illustrated by Figure 6.

2.3. Substantiation of results Prior to generating (manufacturable) local cross sections, the results obtained via the topology and subsequent size optimisation were substantiated via dynamic (explicit) nonlinear FE modelling. This was conducted using the information from the topology optimisation as well as the I-value estimations, to create (two-dimensional (2D)) shell models, as illustrated in Figure 6. The four models were separately meshed, using an element size of 4 mm, leading to the creation of approximately 360,000 elements per model. The models used the LS-Dyna

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‘MAT 003: Plastic Kinematic’ material card with the parameters listed in Table 3. Table 3, material parameters for explicit analysis. Parameter

Value

SI Unit

210,000

MPa

0.3

N/A

Volumetric mass density(ρ)

7,850

kg/m3

Plastic tangent hardening modulus (ETAN)

1,000

N/A

Young’s modulus (E) Poisson’s ratio (ν)

A rigid plate was subsequently inserted above the A-pillars in all four models, with the angles defined in Table 1. During the analysis, these rigid plates were subjected to a prescribed displacement of 250 mm perpendicular to the surface of the plate, i.e. moving in the yz-plane, Figure 8, with a velocity of 5mph [3], thus simulating a roof crush scenario. The model set-up is illustrated in Figure 8.

Figure 8, representation of FE model. The contact force between the rigid plate and the roof structure was measured during the analyses. This could subsequently be compared to the applied force of 45 kN used for the topology and size optimisations [5], in order to evaluate the performance of the roof structure relative to the original requirements. Figure 9 displays the contact force (between the rigid plate and the roof structure) as a function of time. Figure 9 substantiates the validity of the approach, as the peak contact force of all four individual models exceed 45 kN.

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Figure 9, results of explicit FE analysis, force vs. time. Figure 10 illustrates the deformed roof (model 7) at t = 0.15 s, i.e. at a rigid plate displacement of 250 mm.

Figure 10, deformed roof structure, model 7, t = 0.15 s. The results in Figure 10 indicate that the roof structure does not exhibit significant buckling. The figure also indicates that only localised deformation occurs, indirectly suggesting that the structure is ‘over dimensioned’. It must, however, be reiterated that the structure has been derived based on topology optimisation incorporating; front, rear, side and pole crash scenarios, which inevitably have influenced the topology of the structure. The results of the FE

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analyses thus substantiate that the obtained structures perform as anticipated, the next step is therefore to generate (detailed) local cross-sectional geometry.

3. Generation of local cross-sectional geometry To generate the detailed local cross sections of the roof, in preparation for manufacturing, an optimisation algorithm was created. This did also build on the outcome of the topology optimisation as well as the I-value estimations. The first step in this process was to define a roof segment model to allow for the cross sections of the roof to vary in size, and shape, throughout the structure, with the aim of minimising the roof mass.

3.1. Segment model As roof members are typically manufactured from (metal) sheets, it was decided that ‘conventional’ manufacturing processes would be used in connection with the roof design. The cross section of each roof member would consist of a series of segments. The cross section of a flat metal sheet would consist of one single segment, as illustrated in Figure 11.

Figure 11, cross section of metal sheet. The segment example in Figure 11 is located in the yz-plane, is defined by the two endpoints A and B with the coordinates (YA, ZA) and (YB, ZB), respectively. The coordinates of the centroid C of the segment are (YC, ZC). Finally, the thickness of the segment is denoted t and the angle between the segment and the y-axis is denoted α. The area of the segment, the angle α and the coordinates for the centroid C can all be obtained providing A, B and t are known parameters. Using the parallel axis theorem, the expressions for the second moment of area around the centroid C with respect to the global coordinate system axes can be defined as expression (3):

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I CY 

I CY 1  I CZ 1 I CY 1  I CZ 1 t 3  l AB  cos  2  I CY 1  2 2 12

(3)

l   t I I I I  CY 1 CZ 1  CY 1 CZ 1 cos  2  I CY 1  AB 2 2 12 3

I CZ

In expression (3) lAB is the length of the line segment between point A and point B.

3.2. Six segment model The cross section of a modern day vehicle does not only consist of a single segment. Typically six segments are used as illustrated in Figure 12.

Figure 12, segments for cross section. These are defined by the letters A-G, excluding C which denotes the centroid of the entire cross section. The local (segment i) coordinate axes are denoted Yi and Zi, αi denotes the angle of rotation between the local and global coordinate system. It was assumed that t remains constant for all segments; as previously stated the roof members were to be manufactured using metal sheets of uniform thickness. The coordinates for the centroid C were obtained by Equation (4):

YC 

1 ACS

1 ZC  ACS

6

 AS  Y i 1

i

Ci

(4)

6

 AS  Z i 1

i

Ci

In Equation (4) ACS is the area of the entire cross sections, ASi is the area of the individual segment, Figure 11, whilst YCi and ZCi are the coordinates of the centroid for the individual segment. The second moments of area for the entire cross section can thus be obtained by firstly determining these for the individual segment i, with respect to the global coordinate

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system using the parallel axes theorem, expression (5):

 ICY i   ICiY i  SAi   dz  2  ICZ i   ICi Z i  SAi   dY  2

(5)

In expression (5) dY and dZ are the relative distances from the centroid of the individual segment to the centroid of the combined cross section. The second moments of area for the entire cross section can subsequently be obtained using expression (6): 6

I CY    I CY i i 1

(6)

6

I CZ    I CZ i i 1

Thereby the second moment of area for any cross section can be calculated regardless of the number of segments within the cross section. The methodology presented above was subsequently programmed into a spreadsheet, which could be used to immediately calculate the second moments of area for any cross section. The calculations of the spreadsheet were subsequently validated using commercial CAD and FE software as well as manual calculations. Following this, the spreadsheet was set up to be able to calculate and store the second moments of area for a series of cross sections in connection with each other, which could be used to calculate these for the entire roof structure.

3.3. Optimisation of cross sections With the ability to calculate the second moments of area for all roof members, these could be equated to the outcome of the individual I-value estimations discussed in Section 1, illustrated in Figure 6. It should also be noted that parts of the cross section will follow the outer profile, i.e. the A surface of the vehicle. This meant that the I-value could be used as a constraint in connection with shape and size optimisation with the objective of minimising the roof mass. Alternatively, the objective could also be to maximise the I-values of the cross-sections, as the anticipated change in roof mass was low. Nevertheless, the ultimate aim of the exercise was to define manufacturable cross-sections of the roof. The optimisation was to be conducted using commercial software, which aimed to use design of experiments (DOE), as described in [9] in order to build the response surface of the vehicle cross section. Due to the vast number of design variables and design variable levels, it was found to be advantageous to perform a direct optimisation eliminating the need to compute the response surface from experimental points in order to then extract the minimum section mass.

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Indeed, computing a response surface from more than 1,000 points was not an efficient method of optimisation, as it used a significant amount of disk space and caused an excessive memory shortfall when DOE response points had to be reloaded for the computation of the response surface. The direct optimisation method was mostly chosen because solving each optimisation in an Excel spreadsheet took just a few seconds with acceptable results, as will be discussed later. To conduct the optimisation, it was necessary to define the design variables including initial values and constraint magnitudes. The majority of these would have to be set on a local level, i.e. for each roof member (section); however, parameters such as the mass of the roof would be monitored on a global level. Figure 13 represents a ‘typical’ cross sectional roof profile and can be used as a starting point for an example to define design variables.

Figure 13, Initial design variables for shape optimisation. The coordinates of points A-G (excluding C) were potential design variables for the optimisation process. In this particular case, A, B and D were fixed as these in essence define the external profile, i.e. the A-surface of the vehicle. Thereby three design variables remained, namely the coordinates of points E, F and G. There were, however, some restrictions on how these would be able to change, which was primarily related to the resulting interior cabin space (head room). In addition to the actual coordinates of E, F and G, the gauge thickness of the individual segments (1 to 6) could also be design variables. However, as modern day roof structures usually use a minimum steel gauge thickness of 0.7 mm for reasons other than crash performance, it was decided to set the gauge thickness of the six segments to 0.7 mm. Figure 13 also displays four potential reinforcements (R1, R2, R3 and R4) which could be ‘inserted’ or ‘removed’ using simple Boolean operators. Furthermore, the gauge thickness of the four potential reinforcements, TR1, TR2, TR3 and TR4 were also included as design variables, which could either assume the discrete values of 1.0 mm or 0.7 mm. Finally, manufacturing constraints were taken into account during the optimisation process; it was

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assumed that the roof structure would be tack welded and use a sheet metal forming process, which meant that blank holders and blend surfaces would have to be used. The optimisation parameters are summarised in Table 4. Table 4, parameters for initial shape and size optimisation. Design variables

YE,ZE,YF,ZF,YG,ZG,R1,R2,R3,TR1,TR2,TR3,TR4

Optimisation responses

ACS,ICY,ICZ

Optimisation constraints

ICY=fixed; ICZ=fixed

Optimisation objective

Minimise ACS

Based on the information summarised by Table 4, the optimisation problem was set up using Microsoft Excel coupled with Altair HyperStudy to directly solve the optimisation problem. Using the roof rail as a case study, it was found that the most efficient solution, given the design variable in Table 4 used the reinforcement R1 with a gauge thickness (TR1) equal to 1.0 mm; this scenario met all (mathematical) optimisation constraints, including matching the Ivalue to that of the previous estimation. Despite this, there were concerns about the manufacturability of the proposed design; hence, it was decided to insert two additional points (H and J) to guide the profile of the reinforcement R1, the updated design variables are illustrated in Figure 14.

Figure 14, updated design variables for shape optimisation. With these changes the optimisation process was repeated and a manufacturable design was obtained. This consisted of three parts: the inner panel, the outer panel and the reinforcement, as illustrated by Figure 15.

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Figure 15, optimised cross section. To substantiate the feasibility of the proposed manufacturing methodology with respect to the design in Figure 15, the geometry was analysed using FE-based sheet metal forming analysis (Altair HyperForm). This analysis used 2D (shell) elements with an average mesh size of 5 mm; the material parameters used are defined in Table 5. Table 5, material parameters for sheet forming analysis. Parameter

Value

SI Unit

210,000

MPa

Poisson’s ratio (ν)

0.3

N/A

Yield stress value (σy)

315

MPa

0.125

N/A

Young’s modulus (E)

Friction coefficient (FC)

Figure 16 displays the relative thinning of the sheets after the forming process.

Figure 16, results from sheet metal forming analysis of optimised cross section.

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Figure 16 displays relatively minor variation in thickness for all three parts analysed. The part found to have the largest variation of gauge thickness, as a function of the forming process was the outer panel. In this case, the maximum increase (of thickness)was 0.429% and the minimum decrease was −0.235%, meaning a maximum thickness of 0.703 mm and a minimum thickness of 0.698 mm, based on an initial gauge thickness of 0.7 mm. All elements of the analyses were below the failure level of the forming limit diagram. The results of the sheet forming analyses thus substantiate that the parts can be manufactured without any significant variations of thickness, indicating that the anticipated structural performance of the previously conducted FE analyses can indeed be obtained.

4. Conclusion and next steps The starting point for this paper was complications uncovered during a previous study, which used topology optimisation to investigate the potential effects of proposed changes to FMVSS 216 upon BIW architecture [5]. The use of topology optimisation enables the possibility of significant reductions in BIW mass, particularly in connection with ‘alternatively’ fuelled vehicles, such as HEV. Although the outcome of the study suggested that currently available topology optimisation software can be used for BIW load path extraction, it was recognised that buckling effects, which is the primary failure mode for rollover events, were not taken into account. The study in this paper has investigated various options for considering buckling as part of the optimisation process. Initially, it was investigated how this could be included (as an active parameter) during the topology optimisation process. This approach was, however, deemed unfeasible using current commercially available (topology) optimisation algorithms, as the detailed information related to the geometry (of the roof sections) are not available during the topology optimisation. As buckling could not be included as an ‘active parameter’ within topology optimisation, the potential for implementing it as a ‘passive’ parameter was investigated. Recognising that topology optimisation uses linear static FEA (and a penalisation factor of 1 was used in [5]), the method of superposition was deemed applicable, meaning that the buckling calculations could be performed subsequently to the completion of the topology optimisation. Consequently, an ‘I-value’ estimation study was set up using the load paths from the topology optimisation and circular tubes with a gauge thickness of 1.0 mm to estimate the required I-values of each roof member in order to successfully withstand buckling failure in the event of a vehicle rollover. Dynamic (explicit) non-linear FE analysis was used to

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substantiate the structural performance of the obtained load path and I-values. The results of these analyses clearly indicated that the structures obtained were fully capable of withstanding the applied loading. Subsequently, a combined shape and size optimisation study was set up in order to derive manufacturable cross sections based on the topology optimisation and I-value estimations. Finally, the manufacturability of these cross sections were substantiated using FE-based sheet metal forming analysis to estimate the variations of gauge thickness, and indirectly the structural performance of the manufactured parts. In conclusion, this paper has demonstrated the potential for using current state-of-the-art commercially available optimisation software to obtain a manufacturable BIW able to meet crash performance requirements, based on a styling (CAD) surface. The paper also recognises that revised (topology) optimisation algorithms would be very beneficial in the effectiveness and validity of the overall methodology.

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References [1] Altair Engineering, Inc. Altair Hyperworks Manual, Altair Engineering, Inc., Troy, MI, 2010. [2] M.P. Bendsoe and O. Sigmund, Topology Optimization, Theory Methods and Applications, Berlin, 2004. [3] E.C. Chirwa, and Q. Peng, Modelling of roof crush usingthe newly updated FMVSS 216, ICRASH,Washington, DC, 2010. [4] E.C. Chirwa, R.R. Stephenson, S.A. Batzer, and R.H. Grzebieta, Review of the Jordan rollover system (JRS) vis-à-vis other dynamic crash test devices, Int. J. Crashworthiness 15 (5) (2010), pp. 553–569. [5] J. Christensen, C. Bastien, and M.V. Blundell, Effects of roof crush loading scenario upon body in white using topology optimisation, Int. J. Crashworthiness 12 (1) (2012), pp. 29–38. [6] J. Christensen, C. Bastien, M.V. Blundell, A. Gittens, and M. Dickison, Integration of electric motor and alternator insmart lightweight vehicles, Proceedings of the 4th International Conference on Mechanical Engineering and Mechanics, 2011, pp. 921–932. ISBN978-1933100-40-1. [7] J. Christensen, C. Bastien, M.V. Blundell, A. Gittens, and O. Tomlin, Lightweight hybrid electrical vehicle structural topology optimisation investigation focusing on crashworthiness, Int. J.Vehicle Struct. Syst. 3 (2) (2011), pp. 113–122.

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[8] J. Christensen, C. Bastien, M.V. Blundell, O. Grimes, A. Apella, G. Bareham, and K. O’Sullivan, Modelling of lightweight hybrid electric vehicle architectures in case of the newly updated FMVSS 216 roof crush scenarios, TheInternational Crashworthiness Conference, ICRASH, Milano, Italy, 2012. [9] J. Christensen, C. Bastien, M.V. Blundell, and J. Kurakins, Lightweight body in white design using topology-, shape and size optimisation, Electric Vehicle Symposium (EVS26), Los Angeles, California, 2012. [10] D. Friedman, and C.E. Nash, Measuring rollover roof strength for occupant protection, Int. J. Crashworthiness 8 (1) (2003), pp. 97–105. [11] R.H. Grzebieta, A.S. McIntosh, and M. Bambach, How stronger roofs prevent diving injuries in rollover crashes, ICRASH, Washington, DC, 2010. [12] S.A. Richardson, G. Rechnitzer, R.H. Grzebieta, and E. Hoareau, An advanced methodology for estimating vehicle rollover propensity, Int. J. Crashworthiness 8 (1) (2003), pp. 63–72. [13] R.R. Stephenson, The case for a dynamic rollover test, ICRASH, Washington, DC, 2010. [14] F. Tahan, K. Digges, and P. Mohan, Sensitivity study of vehicle rollovers to various initial conditions – finite element model based analysis, ICRASH, Washington, DC, 2010. [15] J. Trollope, C. Bastien, J. Christensen, C. Kingdom, and M.V. Blundell, Optimisation of a common vehicle architecture including electric vehicle, hybrid electric vehicle and internal combustion engines propulsion systems for durability and crash performance, The International Crashworthiness Conference (ICRASH), Milano, Italy, 2012.

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CONTRIBUTION

4

Double blind peer review journal paper published in the Journal of Mathematical Research and Applications (JMRA) on rd Dec 23 2013 [Format has been edited to comply with Coventry University regulations]

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Journal of Mathematical Research and Applications (JMRA)

Dec. 2013, Vol. 1 Iss. 2, PP. 34-47

The Feasibility of ESLM for BIW Roof Structure Development and Optimisation J. Christensen*1, C. Bastien2, M. V. Blundell3 Department of Mechanical and Automotive Engineering, Coventry University Priory Street, Coventry, CV1 5FB, United Kingdom *1 [email protected] ; [email protected] ; [email protected] Abstract- This paper is part of an extensive research effort exploring the opportunities and feasibility of using structural optimisation algorithms to develop crash structures and Body In White (BIW) architecture directly from a styling envelope. Previously published papers have explored the limitations of “conventional”, i.e. linear static, topology optimisation using the Variable Density Method (VDM) and the Solid Isotropic Material with Penalisation (SIMP) interpolation scheme to develop BIW architecture, including an investigation into the potential effects of recently proposed changes to FMVSS 216 upon roof structure design. The most significant limitation of the VDM and SIMP based optimisation algorithms, found in most commercial Finite Element (FE) software, is that they are limited to linear and isotropic material behaviour only. The most recent (generally adopted) method for topology optimisation of nonlinear material behaviour (and non-linear geometrical behaviour) is the Equivalent Static Load Method (ESLM). The purpose of this paper is to investigate the differences between “conventional”, i.e. linear static and isotropic, topology optimisation and ESLM. This will be achieved by comparing both the theoretical aspects as well as a case study of roof crush modelling and optimisation in accordance with FMVSS 216. Keywords: Non-linear Finite Element Topology Optimisation; FMVSS 216; Roof Crush; Body In White (BIW); Lightweight Vehicle Architecture; Equivalent Static Load Method (ESLM)

1. Introduction The potential of structural optimisation algorithms is widely accepted and recognised within all branches of engineering. Despite being a relatively new branch of engineering the use of Finite Element (FE) based structural optimisation algorithms is rapidly gaining momentum, particularly within the mechanical and automotive sectors. The potential for expanding the use of such algorithms within the automotive sector becomes even more significant in light of the development (and requirements) for alternatively fuelled vehicles, such as Hybrid Electrical Vehicles (HEV). In addition, the continued development of active safety technology and accident avoidance systems substantially enhances the opportunities for “alternative” vehicle layouts. This opportunity could for example be used to improve vehicle handling characteristics, maximising interior cabin space or superior visibility angles. This enhanced flexibility of vehicle layout could be simultaneously utilised for significantly reducing the overall Body In White (BIW) mass and improve crashworthiness. Most state of the art,

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commercially available, Topology Optimisation (TO) algorithms are based on the Variable Density Method (VDM) in combination with the Solid Isotropic Material with Penalisation (SIMP) interpolation scheme. The most significant limitation of this type of algorithm is the fact that they are confined to linear (and isotropic) material behaviour. This is indeed a significant hurdle for the increased use and development of TO particularly within the automotive sector, where BIW architecture and crash structures could potentially be rapidly and very costefficiently developed based on a styling envelope, i.e. a Computer Aided Design (CAD) model. The purpose of any automotive crash structure is in essence to absorb energy via deformation primarily bending and buckling. The importance of non-linear material behaviour in this context can be appreciated by observing Fig. 1, representing a “typical” stress-strain curve for steel.

Fig. 1, example of stress-strain curve. From Fig. 1 it is clear that the largest area underneath the curve, i.e. the energy absorbed, is in the non-linear (plastic region), highlighting the importance of non-linearity in the design and performance of crash structures. Due to the inherent non-linear nature of crash structures; both in terms of geometry and material behaviour, Fig. 1, current state of the art TO algorithms are not ideally suited for this application, as assessed in [1] & [2]. FE based algorithms “claiming” to be able to conduct non-linear TO do exist, an example of which is the Evolutionary Structural Optimisation (ESO) based algorithms [3]. This type of algorithm does in essence work on the principle of “survival of the fittest”, and generally requires 1 FE analysis per optimisation iteration, which can be extremely CPU intensive, particularly if the system being optimised is a vehicle crash structure. In addition, it is not widely accepted (and proven)

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that this type of algorithm converges towards an optimum solution [4-6]. Other algorithms not based on the VDM approach also exist. An example of which is the Homogenisation Based Optimisation (HBO), which could potentially perform topology, shape and size optimisation simultaneously. This does however heavily depend on the development of meshfree / meshless methods [7, 8], which are not presently able to accurately obtain stress and strain levels in combination with a complex geometry, such as an automotive crash structure. The present emphasis on development of commercial TO tools is therefore adaptations of the combined VDM and SIMP approach, primarily because these are well developed and proven for linear static optimisation, in addition to already being well integrated into commercial FE software. Equivalent Static Load Method (ESLM) is now available in several commercial FE based software suites which amongst others include Altair HyperWorks and VR & D Genesis. The ESLM approach does in essence use the “conventional” VDM and SIMP setup as successfully adapted for linear static (and isotropic) TO. The primary difference lies in the generation of loadsteps as well as an additional iteration loop; the overall flow chart of the ESLM optimisation process is illustrated in Fig. 2.

Fig. 2, ESLM flowchart. Fig. 2 contains four primary stages, which will be subsequently explained, in addition to two iteration loops: · Internal Iterations (II); these are the optimisation iterations contained within Step 3, Fig. 2. · External Iterations (EI); these are the iterations relating Step 1 to Step 3, Fig. 2. The two iteration loops above are important, as only the Internal Iterations (II) are utilised in conventional linear static topology optimisation; the II and EI acronyms will be utilised throughout the remainder of this paper. Step 1 in Fig. 2 consists of an FE analysis, exactly the same as a “conventional”, i.e. linear static, topology optimisation algorithm. The difference is that the FE analysis in connection with ESLM is not confined to linear static modelling, but may Contribution 4

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be dynamic and non-linear. For the subsequent case studies of this paper three types of ESLM based analysis were available, all of which have been analysed: 

Geometric non-linear (implicit static)



Geometric non-linear (implicit dynamic)



Geometric non-linear (explicit dynamic)

Following the FE analyses Step 2 creates individual (linear static) load steps(s) in order to obtain a representative response field to the analysis completed in Step 1. The number of load steps created is automatically set equivalent to the number of timesteps. Once these Equivalent Static Load Steps (ESLS) have been created Step 4 conducts a “conventional” linear static topology optimisation based on the outcomes of Steps 1 and 2, whilst using the VDM and SIMP approach, i.e. completing the Internal Iterations (II). Pending convergence- and/or termination criteria the algorithm will either return to Step 1 and repeat the steps with the updated geometry via the External Iteration (EI) loop, or terminate the analysis in Step 4. It has been documented in [9] that the ESLM method has been developed for “Nonlinear responses from implicit static analysis, implicit dynamic analysis and explicit dynamic analysis”. It is generally recognised that the ESLM approach is best suited for low levels of non-linearity, for example small displacement non-linear analysis. The ESLM approach should therefore be well suited for development of automotive roof structures, which generally experience a modest level of non-linearity in connection with roll over events. A previously published paper, [1], investigated the likely effects of recently proposed changes to FMVSS 216 upon roof structure geometry by means of linear static topology optimisation. Where relevant, the outcomes of this previously published paper will be used in the current paper, with an overall aim of drawing comparisons, highlighting significant differences, as well as evaluating the overall performance of ESLM versus linear static TO based upon an automotive roof crush scenario. This will be completed in order to evaluate the feasibility, importance and potential for using ESLM for non-linear TO in general. The overall study will consist of three main phases documented in this paper: 1. Initial study; this will contain the description of initial ESLM model setup and results. 2. Parameter sensitivity study; which will focus on the influence of optimisation control parameters in connection with ESLM as well as linear static TO models. 3. Revised study; this will combine the lessons learnt from Stages 1 and 2 above in order to refine the ESLM models before the final comparisons and associated conclusions draw the study to a close.

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The individual parts described above will be the focus of attention in the following three sections of this paper.

2. Initial Study The combined case studies to be presented in this paper included a total of 131 TO models. The general FE analysis setup is illustrated in Fig. 3.

Fig. 3, Illustration of general FE model. The general FE model illustrated in Fig. 3 had the approximate external dimensions (x,y,z); 3,000 mm; 1,700 mm; 550 mm; consisted of approximately 67,000 solid (tetra) elements with an average mesh size of 25.0 mm and utilised first order displacement functions. These parameters were all specified based on previously conducted studies [1, 2]. Using Single Point Constraints (SPCs) all six Degrees Of Freedom (DOF) were locked at the “base” of the roof pillars as indicated in Fig. 3. The external loading consisted of a uniformly distributed load with a combined magnitude of 45 kN spread across a total of 17 nodes; the application angle will be subsequently discussed. The topology optimisation was set up with the main parameters listed in Table 1: Table 1, optimisation parameters. Optimisation objective parameter

Description

Value

Objective

Minimise mass

N/A

Constraint

Maximum compliance

1,000,000 mm/N

Constraint

Maximum nodal displacement

50 mm

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2.1 Initial Model Setup With the above main parameters defined, the specific models to be included in the comparison case study were to be set up. As the purpose of the overall study was to draw direct comparisons between linear static topology optimisation and the ESLM approach an ideology of avoiding “over constraining” the optimisation algorithms was therefore adopted. This meant that as few optimisation control parameters as possible would be used; in order to get a more “realistic” overview of the results from the individual optimisation algorithms. Therefore only the “global checkerboard control” and the MINimum DIMension of member size (MINDIM) parameters were activated in order to reduce the likelihood of checkerboard effect occurring. Apart from these, the only other optimisation control parameter initially activated was the Penalisation Factor (PF) for the SIMP interpolation scheme. These were applied to both the linear static as well as the ESLM models. Previously conducted research had shown that a PF of 1.0 was appropriate for this application [1]. With the PF determined the remaining parameters to be varied between different models were chosen, these are listed and summarised in Table 2 as well as explained in the following. Table 2, parameters and possible values for initial model. Parameter #

Description

Possible values Linear static

1

Load Step Type (LST)

Geometric non-linear, implicit static Geometric non-linear, implicit dynamic Geometric non-linear, explicit dynamic

Maximum number of Internal 2

(MIE) 3

External Load Angle (ELA)

4

Material model

5

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1

iterations per External iteration

Non-linear Material Parameters (NMP)

10 Pitch angle: 0°, Roll angle: 0° Pitch angle: 10°, Roll angle: 45° Linear elastic Johnson-Cook Elastic Plastic Mild grade steel High strength steel

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1) Load Step Type (LST): The Load Step Type (LST) listed as Parameter 1 in Table 2 represents the possible options for linear and ESLM based optimisation in the software used, and refers to Step 1 of Fig. 1. It was necessary to include all three options for non-linear optimisation in order to obtain a thorough understanding of ESLM. The time step for the models was calculated using Oasys Primer FE software and set to: 2.7E-9 s, which remained constant throughout all models detailed within this paper. 2) Maximum Number of Internal iterations per External iteration (MIE): The Maximum number of Internal iterations per External iteration (MIE) defined as Parameter 2 in Table 2 refers to the number of Internal Iterations (II) in Step 3, Fig. 1, between the External Iterations (EI), linking Steps 1 and 3 in Fig. 1. The ideology behind choosing 1 II per EI was that this should theoretically provide the highest possible “control” within the ESLM optimisation process itself. This is because a (non-linear) FE analysis would be conducted; Step 1, Fig. 1, following each II optimisation ensuring updated Equivalent Static Load Step (ESLS) for every single EI, maximising the accuracy of the ESLS. This should however also theoretically provide the highest Central Processing Unit (CPU) requirements due to the potentially large number of (non-linear) FE analysis required for each model. Nevertheless, the CPU time and cost was a secondary aspect in this context. The reason for choosing 10 II per EI was to estimate the influence of this parameter, indirectly evaluating the potential trade-off between accuracy and CPU requirements. Initially it was not deemed suitable to further increase the number of II, as a very large value of this parameter would theoretically “revert” ESLM to linear static topology optimisation, as the number of EIs completed would most likely be minimised. 3) External Load Angle (ELA): The External Load Angles (ELAs) listed as Parameter 3 in Table 2 were defined based on previous research papers [1, 10]. The roll angle was defined as the angular rotation of the roof structure around the x-axis in Fig. 3; whereas the pitch angle was defined as the angular rotation around the y-axis in Fig. 3. Roll and pitch angles of 0° represent the current European New Car Assessment Program (NCAP) roof crush test scenario, whilst the combination of a 10° pitch angle and a 45° roll angle represent the worst case scenario for roof crush [10]. The lessons learnt in [2], demonstrated the necessity of incorporating sensitivity analysis for topology optimisation. This was however of minor concern in this study, as the primary

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objective was the comparison of optimisation algorithms. The loading for either of the two possible ELAs was assigned a time history curve as illustrated by Fig. 4.

Fig. 4, Force vs. time history. The loading was applied with a constant acceleration from 0 to 45 kN, as illustrated by Fig. 4, representing an impact velocity of 2.22 m/s (8 km/h) [10]. It was chosen to extend the termination time to 0.085 seconds, in order to ensure that a steady state was obtained in the model. It should be noted that the models did not contain any strain dependencies with respect to the material parameters. 4) Non-linear Material Parameters (NMP): The material models (Parameter 4) and the NMP (Parameter 5) used for the models of the initial studies, as listed in Table 2, were linear elastic and Johnson-Cook Elastic-Plastic. The latter represented an extension of the former, as the “only” difference occurs in the non-linear region of the material curve. The non-linear part of the Johnson-Cook Elastic-Plastic material model [9], is controlled by the yield stress value, a plasticity hardening parameter as well as a plasticity hardening exponent. The material parameters used for the study are listed in Table 3. Table 3, material parameters. Material model

Material description

(isotropic) Linear elastic

Mild grade steel (linear)

Johnson-

Mild grade steel

Cook

(linear elastic-plastic)

Johnson-

High strength steel

Cook

(linear elastic-plastic)

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Young’s

Yield

Plasticity

Plasticity

Modulus

stress

hardening

hardening

(MPa)

(MPa)

parameter

exponent

210,000

N/A

N/A

N/A

210,000

276

0.076

210,000

430

0.017

0.643

0.51

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Based on the above definitions and parameters an initial study containing 26 individual TO models was completed. This included 2 linear static models and 24 ESLM models, consisting of 8 geometric non-linear implicit static, 8 geometric nonlinear implicit dynamic and 8 geometric non-linear explicit dynamic models. The models were all solved using Symmetric Multicore Processing (SMP) on a High Performance Computer (HPC), utilising 10 Processors Per Node (PPN) on a single node of a Linux 64 bit based system.

2.2 Results of Initial Study Obtaining useful and relevant results from the 26 models of the initial study proved significantly difficult, apart from the two linear static optimisation models which had already been refined courtesy of [1] and [2]. Both the implicit and explicit ESLM models seemed to have substantial stability problems. The root of the problems was initially thought to be the parameters controlling the implicit and explicit time steps. These parameters were therefore reviewed; specifically the allowed maximum number of iterations to obtain equilibrium within each time step increased in an attempt to stabilise the models. With this update all 24 (ESLM) models were solved once again, the adjustment of implicit and explicit time step parameters did however prove not to have a noteworthy effect on the stability of the models. Therefore further investigation into the problem was undertaken. It was found that the initial FE analyses, i.e. Step 1 in Fig. 2, were completed successfully. Additional investigation uncovered that the problem was rooted in the checkerboard effect as can be perceived from Fig. 5, representing a “typical” outcome of the 24 initial ESLM models.

Fig. 5, checkerboard effect, penalisation factor = 1.0.

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The model illustrated by Fig. 5 represents the outcome of the II returned to the EI in order to update the ESLS, i.e. what was returned from Step 3 to Step 1, Fig. 2. Unsurprisingly the subsequent FE analyses failed to achieve equilibrium during the implicit (or explicit) time steps, which caused the models to error terminate. The maximum number of II (MIE), i.e. 1 or 10, Table 2, was not found to have a considerable effect on this “phenomenon”, which most often occurred between the first and second EI loop. The checkerboard effect as illustrated in Fig. 5 occurred despite the fact that the “built-in” global checkerboard control of the FE software was activated. In addition the MINimum DIMension of member size (MINDIM) optimisation control parameter had also been previously activated. This aims to penalise the formation of small members, but does however not necessarily prevent the formation of checkerboard patterns during the optimisation[9, 11]. In an attempt to resolve the checkerboard issue all 26 models from the initial study, including those utilising linear static optimisation, were updated with a Penalisation Factor (PF) of 3.0, which is the recommended maximum value for solid elements [9]. Initially the results obtained from the 26 models with a PF of 3.0 indicated that the changes had had the overall desired effect, as the average number of EI increased from 6 (PF = 1.0) to 10 (PF = 3.0). In addition, the average mass reduction, calculated as the ratio of the initial mass relative to the final mass, increased from 72% to 93%. Further post-processing of the models did however reveal that the feasibility of the obtained structures had not drastically improved. Fig. 6 illustrates an “identical” model to that displayed by Fig. 5, the only difference between the two is that Fig. 5 represents a PF of 1.0, whereas Fig. 6 represents a PF of 3.0.

Fig. 6, checkerboard effect, penalisation factor = 3.0.

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The model in Fig. 5 successfully completed 1 EI and obtained a relative mass reduction of 30%, the corresponding values for the model in Fig. 6 were 4 and 73% respectively. The above example provided a very good indication of the overall effects of the increased Penalisation Factor (PF). Although a significant number of models had indeed been stabilised by the increased PF further inspection of the individual models soon revealed that the initial issue of a large number of models only completing 1 EI was still considerable. Approximately 40% of the 24 ESLM models with a PF of 3.0 terminated during Step 1, Fig. 2, when initiating the 2nd EI. This was once again contributed to the checkerboard effect illustrated in Fig. 5. In an attempt to increase the stability of the models during these initial stages it was decided to further “experiment” with the PF, by repeating all 26 initial models using a PF of 0.5. This may initially seem controversial, however it should theoretically increase the stability of the optimisation models during the “early” EI, thus maximising the number of EI, with the ultimate aim of obtaining an enhanced definition of topology. The influence of lowering the PF upon the results can be appreciated by observing Fig. 7.

Fig. 7, effects of penalisation factor. As illustrated by Fig. 7, the effect of reducing the PF to 0.5 did in essence correspond to “reversing” the effects of the penalisation in the SIMP interpolation scheme, as it “encouraged” the use of elements with low relative element density and high relative element stiffness. Strictly speaking this was also the “equivalent” of reversing the scale for the relative element density, and would most likely have an effect on the estimated mass of the resulting topologies. This meant that the models utilising a PF of 0.5 could not be directly compared to Contribution 4

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the relative mass reduction values of the ones utilising PFs of 1.0 and 3.0. On the other hand, the models utilising a PF of 0.5 should enhance the stability of the ESLM optimisation models. With the above in mind, all 26 models from the initial study were recomputed using a PF of 0.5. Fig. 8 displays the results of the model equivalent to the ones displayed in Fig. 5 and Fig. 6, apart from the fact that Fig. 8 utilised a PF of 0.5.

Fig. 8, checkerboard effect, penalisation factor = 0.5. As indicated by Fig. 8, the results obtained by lowering the PF to 0.5 did not significantly change the resulting topology, as can be seen by comparing Fig. 8 to Fig. 6. The change of PF did however have the anticipated effect on the number of EI, which increased to an average of 19 per model. Nevertheless, the feasibility of the topologies obtained was, as previously discussed, questionable. The results of the “corresponding” linear static models to the ESLM models of Figs. 5, 6 and 8 are represented by Fig. 9.

Fig. 9, corresponding linear static model.

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The results of all 3 linear static models were identical to the naked eye, and obtained an identical relative mass reduction of 90.6 % (within 0.1%). This included the model with a PF of 0.5 thus substantiating the initial hypothesis that a PF of 0.5 could indeed be used. However, the overall results obtained from the initial 78 models (26 times 3) indicated that the basic model setup had to be reviewed in order to better comprehend the effects of ESLM.

2.3 Parameter Sensitivity Study When the initial study, as documented above, was set up the underlying ideology was to use as few optimisation control parameters as possible in order to better comprehend the “true” effects of ESLM. Nevertheless, the results from the 78 models of the initial study demonstrated that it was necessary to review this approach. Therefore, a parameter sensitivity study involving a total of 10 ESLM models, using the LST geometric non-linear implicit static (Table 2) only, was conducted. The main aim of the parameter sensitivity study was to enhance the stability of the models, as well as increase the feasibility of the topology results. The starting point of the study was the general FE model illustrated in Fig. 3. The individual subsections below highlight the main parameters addressed during the parameter sensitivity study. 1) Aggressiveness of Topology Optimisation Algorithm (ATOA): Analysing the results files which contained the detailed specification of the optimisation process, it soon became clear that the “aggressiveness” of the Topology Optimisation (TO) algorithm would have to be reduced. This entailed reducing the maximum allowable percentage of material (mass) to be removed in a single optimisation iteration from 50% to 10%. This was found to have a significant (positive) effect on the stability of the ESLM models. 2) MINimum DIMension of member size (MINDIM): The MINDIM parameter, as previously explained, attempts to prevent the formation of small “clusters” of materials, thus reducing the likelihood of widespread checkerboard effect, was increased from 3 times the average mesh size to 4 times the average mesh size. This was found to have a positive effect on the feasibility of the results obtained by ESLM. 3) Maximum number of Internal iterations per External iteration (MIE): During the setup of the initial study it was postulated that having only one single Internal Iteration (II), Step 3, Fig. 2, per External Iteration (EI) would provide the best possible control of the Equivalent Static Load (ESL) models as the load cases would be updated for each II of the TO. However, as illustrated by Fig. 5, the checkerboard effect was often widespread after a

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single II. By reducing the ATOA, as listed above, this tendency was likely to decrease. If the MIE was increased simultaneously, the likelihood of widespread checkerboard effect would be further reduced, as the material would be allowed to be increasingly stressed in specific locations prior to recalculating the ESL. Excessively increasing the MIE would mean that the ESLM models would “converge” towards a “standard” linear static topology optimisation as the influence of the non-linear (and potential dynamic) effects would be minimised due to the reduced ratio of EI to II. It could also be argued that the main underlying reason for the widespread checkerboard effect in Fig. 5 was the fact that the displacement as well as the stress levels were considerably low in the initial analysis, i.e. Step 1 of EI 1, Fig. 2. This meant that the initial analyses were in essence linear, as the magnitudes of the applied loading were not sufficiently large to induce non-linear responses. This was known prior to the initial study, as the thickness of the design volume was approximately 70 mm where the external loading was applied, i.e. in the z-direction, Fig. 3. However, as both the design volume and the applied loading were carefully selected it was not deemed feasible to adapt these, in order to evoke a non-linear response from the model during the initial iteration. This finding seemed to suggest that the ESL method had stability problems if no non-linear responses were obtained during Step 1, Fig. 2. A possible method of overcoming this problem would be to “refine” the design volume prior to the optimisation, this would however (partially) counteract the overall purpose of conducting a topology optimisation. In order to obtain a more in-depth understanding of the stability issue the model represented by Fig. 6 was reanalysed. This revealed that in Step 1, EI 1, Fig. 2, the maximum plastic strain value obtained was approximately 0.01. During the analysis in EI 7 the maximum plastic strain value obtained was 0.27 indicating that the structure had entered the non-linear region. All results obtained at this point clearly suggested that a MIE value of 1 or even 10 was not sufficient; the MIE value was therefore set to 100. 4) Relative Convergence Tolerance (RCT): The effects of this parameter were the last to be investigated during the parameter sensitivity study. The reasons behind investigation the RCT parameter was linked to the ATOA and MIE parameters discussed above. The RCT parameter controls when the II terminates, by comparing the relative convergence of the objective function between successive iterations, in other words it terminates the II when the optimisation becomes “ineffective”; the default RCT value was 0.5%. Increasing this value to 5% meant that the II would most likely terminate more

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rapidly, reducing the overall CPU time. This was found to have a positive effect on the overall stability of the ESLM models, in combination with the ATOA and MIE parameters discussed above.

3. Revised Study Based on the results obtained by the initial study and the information gathered during the parameter sensitivity study a revised study was defined. The general FE model illustrated in Fig. 3 was “reused”, including the optimisation constraints listed in Table 1. The variable optimisation parameters used for the study remained identical to the ones listed in Table 2, apart from Parameter 2, MIE, which was set to 100. In addition, a 6th parameter; the PF which could assume the values of 0.5, 1.0 and 3.0 as presented in Section 2.2, was set. Finally, the additional optimisation control parameters discussed in Section 2.2, and listed in Table 4, were implemented. With the revised values a total of 43 additional models were created. Table 4, optimisation control parameters for revised study. Description

Value

Aggressiveness of Topology Optimisation Algorithm (ATOA)

10%

MINimum DIMension of member size (MINDIM)

100

Relative Convergence Tolerance (RCT)

5%

3.1 Results from Revised Study The models from the revised study proved to be significantly more stable than the ones from the initial study, with approximately 70% of the models successfully completing in excess of 1 EI, clearly demonstrating that the application of the additional control parameters listed in Table 4 had the desired effects. In the following subsections the effects of individual parameters upon the topology optimisation results will be compared and discussed. The main topics covered in these sections will be the influences of Penalisation Factor (PF), Load Step Type (LST), Non-linear Material Parameters (NMP), External Load Angles (ELA) and scalability of ESLM.

3.2 Penalisation Factor (PF) The Penalisation Factor (PF) is always a parameter of interest when dealing with optimisation algorithms utilising the SIMP interpolation scheme. Although the main purpose of the parameter is to reduce the checkerboard effect, it may also have a significant effect upon the Contribution 4

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overall resulting topology [1, 2, 11]. The effects of this parameter were studied on a smaller scale in connection with the parameter sensitivity study presented in Section 2.2. An expansion of this was included within the revised study, which included a total of 13 models with a PF of 0.5, 16 models with a PF of 1.0 and 13 models with a PF of 3.0. However, only 13 PF 1.0 models were included in this comparison, the reason for the additional 3 PF 1.0 models will be presented and discussed in another subsection. Furthermore, the models being discussed in this section will be ESLM models only. Table 5 provides an overview of the results obtained via the three different PFs. Table 5, overview of results. Average value

Penalisation Factor (PF)

Mass reduction (%)

# of External Iterations

CPU time (s)

0.5

95.6

9

19,150

1.0

88.6

7

1,280,141

3.0

82.3

8

784,630

At this point it should be reiterated that the mass reduction listed in Table 5 for PF 0.5 was not straightforwardly comparable to the ones for 1.0 and 3.0 models, due to previously discussed reasons primarily relating to the ratio of relative element stiffness to relative element density as illustrated in Fig. 7. Initially disregarding PF 0.5 Table 5 revealed a minor difference in the average mass reduction between the PF 1.0 and PF 3.0, where the former displayed an approximate increase of 7% relative to the PF 3.0 models. This was obtained via an average of 7 EI for PF 1.0, as opposed to a comparable 8 EI for PF 3.0, indicating that a PF of 1.0 was more effective. The average CPU time between the two PF groupings was also different, with the PF 1.0 models using an average CPU time as 1.6 times that of the PF 3.0 models. The PF 0.5 grouping clearly represented the lowest CPU time, averaging less than 1% and 2.5% of the CPU time relative to the PF 1.0 and 3.0 groupings respectively. One of the primary reasons behind this significant difference originated from the explicit models, as shall be discussed later. In addition, the influence of the checkerboard effect was once again significant. Fig. 10 represents the results from the revised study of the model “identical” to the one illustrated in Fig. 5, i.e. with a PF of 1.0.

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Fig. 10, model with updated optimisation control parameters, penalisation factor = 1.0. Coincidentally the results to the “corresponding” PF 0.5 and 3.0 models, relative to Fig. 10, were now well represented by the topology illustrated in Fig. 5. This tendency was occasionally observed, however, no general tendencies could be directly deducted from it. The overall conclusion of the study was therefore that the influence of the PF could be very significant, however, it was not a “stand-alone” parameter, as found in the “initial study”; it was in fact very dependent upon other optimisation control parameters. With the current setup, the trend was however, that the models with a PF of 1.0 were the most stable and efficient (in terms of CPU time) in addition to providing the most feasible results, as illustrated by Fig. 10.

3.3 Load Step Type (LST) The Load Step Type (LST) was, as previously presented, divided into three possible categories: non-linear geometry (implicit static), non-linear geometry (implicit dynamic) and non-linear geometry (explicit static). The comparison to the linear static models will be subsequently completed. Fig. 11 represents the results of the non-linear geometry (explicit dynamic) LST with a PF of 1.0.

Fig. 11, non-linear geometry (explicit dynamic) LST. Contribution 4

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The model illustrated by Fig. 11 is thus “identical” to the model represented by Fig. 10, which represents the non-linear geometry (implicit static) model. The results of the “remaining” model with the non-linear geometry (implicit dynamic) LST are well represented by Fig. 6. Therefore by comparing Figs. 6, 10 and 11, an initial understanding of the influence of LST could be obtained. By comparing Fig. 10 to Fig. 11, it is obvious that the topology of the latter was better defined than the former, i.e. reduced checkerboard effect. This meant that the nonlinear geometry (explicit dynamic) LST models provided clearer definition than the non-linear geometry (implicit static) models did. This trend was found to be largely applicable throughout all models. Table 6 contains the average values obtained across all ESLM models when divided into groups according to LST applied. Table 6, overview of results based on LST. Average value

Load Step Type (LST) (Non-linear geometry)

Mass reduction (%)


CPU time (s)

Implicit static

75

8

4,061

Implicit dynamic

79

8

16,607

Explicit dynamic

91

6

2,936,834

Table 6 supports the trend identified from Figs. 10 and 11, as explicit dynamic models obtained an average mass reduction value considerably larger than those obtained via the implicit static and implicit dynamic models. Furthermore, the explicit dynamic models obtained this considerably larger mass reduction value using 6 EI on average, as opposed to 8 EI for either of the implicit LST. In other words, the explicit dynamic models seemed to be the most “efficient” per EI. At this point, it should however be noted that, as listed in Table 6, the implicit static LST models were completed using an average CPU time of 4,061 seconds, the implicit dynamic models were completed using 16,607 seconds and the explicit dynamic models were obtained using 2,936,834 seconds. In other words, the implicit static and implicit dynamic models used 0.1% and 0.5% of the CPU time used by the explicit dynamic models, respectively. Throughout the study not a single “set” of models, i.e. “identical” models apart from the LST were found, where all 3 models provided feasible/comparable results to an extent beyond that demonstrated by Figs. 6, 10 and 11 as discussed above. Generally the explicit dynamic models were found to be the most stable, but at the same time also the most CPU intensive. In general it was also found that the topologies obtained from comparable models using different LSTs did not display significant differences beyond those illustrated by Figs. 10 and 11.

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The overall conclusion upon the effects of the LST was therefore that the choice of LST should predominately be made as a function of the type of load scenario desired, as would be customary for a “conventional” standalone FE analysis, and not as a function of the optimisation itself.

3.4 Non-linear Material Parameters (NMP) The Non-linear Material Parameters (NMP) discussed in this section purely relates to those listed as Parameter 5 in Table 2, i.e. mild grade steel (linear elastic-plastic) or high strength steel (linear elastic-plastic), both using the Johnson-Cook material model [9]. The specific NMPs were listed in Table 3. Fig. 12 represents an example of the “typical” difference found between two “identical” models utilising different NMP.

Fig. 12, effects of Non-linear Material Parameters (NMP). As illustrated in Fig. 12 the models utilising the high strength steel values generally displayed more “distinct” topology results. The mild grade steel model in Fig. 12 was obtained after 7 EI, achieved a mass reduction of 93.3% and had a combined CPU time of 13,825 seconds. The corresponding values for the high strength steel model were 6 EI, a mass reduction of 92.8% and a CPU time of 9,252 seconds. The above results were obtained using implicit dynamic LST, and indicated that using a high strength steel model increases the “efficiency” of the optimisation, by obtaining a similar mass reduction using approximately 30% less CPU time. This was generally found to be the case using implicit dynamic LST. Dividing the models into NMP groupings and including all LST did however not support this trend, as documented by Table 7.

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Table 7, overview of results based on NMP. Average value

Non-linear Material Paramater (NMP)

Mass reduction (%)


CPU time (s)

Mild grade steel

80

8

244,756

High strength steel

7971

5

468,508

Table 7 clearly shows that the average CPU time consumed in connection with mild grade steel was only approximately 52% of that used for the high strength steel models. In addition, the mild grade steel models obtained an average mass reduction value of 80% using 8 EI as opposed to 71% using 5 EI for the high strength steel models. Based on the above no clear trend or tendency could be found with respect to the influence of the NMP. Trends in subsets of models, such as the one for the implicit dynamic models discussed above were however identified; indicating that the influence of the NMP was not a “stand-alone” parameter, but was heavily dependent upon other factors such as the LST used. At this point it should be noted that there are many factors which could potentially influence the above, one of which would be the degree of non-linearity of the individual models, i.e. maximum displacement values. These are likely to have an effect on the above results, but were not further investigated during the study documented in this paper.

3.5 External Load Angles (ELA) The reason for including the External Load Angles (ELAs) as a parameter within the study was in essence to obtain an indication of the sensitivity of the models, as this is of primary concern with respect to crash structures and to a lesser extent TO in general. Previously conducted studies [2] have demonstrated the importance of this. The underlying intention in this particular case was to establish whether any significant variations of the “level” of sensitivity existed between different LST sets. In other words, the intention was for example to determine whether the “non-linear geometry (explicit dynamic)” models were more sensitive to the ELA than the “non-linear geometry (implicit static)” models. Throughout the study a very clear tendency emerged, strongly suggesting that there were no distinct differences in the level of sensitivity between different groupings of LST. This was not surprising given the fact that all models utilised identical optimisation parameters, the minor differences that were found between sets of LST were most likely caused by the difference in “starting point”, i.e. the difference in ESLS obtained during Step 1, Fig. 2. Before summarising the findings of the

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ESLM study a study devoted to exploring the scalability of the ESL method will be presented in the following subsection.

3.6 Scalability of ESLM The purpose of this section is to document a subset of the case study devoted to the scalability of ESLM. The underlying reason for conducting this study was based on suspicions arising in connection with the investigation into the stability problems previously discussed. This study contained a total of 3 non-linear geometry (implicit static) models, accounting for the additional 3 models using this LST type, as previously mentioned. Initially all optimisation results were obtained using a High Performance Computer (HPC) using a single Node and 12 Processors Per Node (PPN). At first it was found that a significant number of models error terminated after a very short duration; in the region of 10 seconds of effective CPU time. The only error code returned from the FE solver was “Abnormal termination in non-linear geometry subcase”, indicating that the solver was unable to achieve equilibrium. It was suspected that the issue was related to the amount of memory available, the PPN was therefore reduced to 10 prior to the initial study commencing, which relived the problem to a certain extent. Nevertheless, it was decided to compare the results of two completely identical models; one solved using 1 PPN and another one solved using 10 PPN. This was found to have a very dramatic effect upon the results, as the model which utilised 1 PPN completed a total of 30 EI (which was the maximum allowed value), whereas the model with 10 PPN completed a total of 8 EI prior to terminating due to excessive checkerboard effect as illustrated in Fig. 5. The mass reduction between the two models was of course significant due to the significant difference in EI. Nevertheless, the 30 External Iterations (EIs) were completed using a total of 3,132 seconds as opposed to 1,562 seconds to complete the 8 EIs on 10 PPN. The FE software utilised Sequential Multicore Processing (SMP), all models were solved on the same Linux 64 bit based HPC. The results obtained in this brief study suggested that the models would be more stable if run on 1 PPN. However, it was decided not to repeat the study using 10 PPN. The main reason behind this decision was the significant increase in real time, as documented by Table 6; a single non-linear geometry (explicit dynamic) model used 2,936,834 seconds on average, the equivalent of 816 hours or 34 days to solve.

3.7 Overall Conclusion of Optimisation Study The above subsections have highlighted the majority of key findings in the revised study. It was clear that not all results provided conclusive tendencies, as there were many factors to be

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considered, many of which proved not to have significant effects individually but proved to be considerably influential in combination with other parameters. The most obvious example of this was the influence of the Non-linear Material Parameters (NMP) as previously discussed. The Penalisation Factor (PF), the Load Step Type (LST) and the NMP were predominantly found to be influential upon the overall “definition” of the resulting topology. In other words, these were not found to significantly change the topology in general, but were in essence found to reduce the checkerboard effect. This also had a direct effect upon the stability of the models, and consequently upon the number of EIs completed, CPU time used and mass reduction obtained. The theoretical ideology of a Maximum Internal iterations per External iteration (MIE) value of 1 initially adopted proved not to be practically feasible, due to significant checkerboard and associated instability problems experienced. It could be argued whether this specific problem was related to the ESLM algorithm or indeed to the specific software used. Prior to the models included in this study another study was conducted using the most recent version of the FE software employed. This did however prove not to be able to produce any useable results, after which the models were repeated using a Beta version of the software, indicating that the stability issues could potentially be software based. This suspicion was substantiated by the outcome of the ESLM scalability study, as previously discussed. Regardless of the root of the problem the fact remained that the stability of the majority of models was significantly increased by the introduction of additional optimisation control parameters as listed in Table 4. The overall study did only include one design volume and 2 different sets of External Load application Angles (ELAs) which could potentially also have a significant effect upon the stability of the models. The main reasoning behind this argument was that the initial analysis, i.e. Step 1, Fig. 2 did not display high stress nor strain levels, meaning that the structure was initially “under-loaded”, i.e. experienced linear behaviour only. This caused the models to exhibit significant checkerboard effects in the subsequent analyses, i.e. EI 2, as the relative material density, and therefore element stiffness, had been significantly reduced destabilising the model. Although this problem was overcome by increasing MIE and decreasing the Aggressiveness of the Topology Optimisation Algorithm (ATOA), it could still be argued that this “un-favoured” the ESL approach. In the same context the question of whether or not the ESL models should be exposed to a certain degree of nonlinearity, e.g. significant displacement values at Step 1, EI 1, Fig. 2, in order to be effective could also be asked. If this is the case the structure, i.e. design volume, would need to be relatively “well defined” prior to the optimisation commencing meaning that the fundamental advantages of conducting a

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topology optimisation study would be diminished. In order to fully answer this question additional studies using multiple software programmes would have to be completed.

3.8 ESLM Versus Linear Static Topology Optimisation With the overall conclusion of the ESLM based optimisation study completed it was now possible to compare these to “conventional” linear static optimisation. A total of 12 linear static optimisation models were included in the overall study, however, valuable knowledge was also drawn from [1] and [2] which combined contained in excess of 300 linear static TO models. The results obtained from all 12 models did not vary significantly, although minor differences did appear. The results obtained from linear static topology optimisation were thus well represented by Fig. 9, and so a direct comparison of the ESL models in Figs. 10, 11 and 12 would provide an indication of the difference in results between ESL and linear static optimisation. By doing so it immediately became clear that the singular load path extracted in Fig. 9 was also noticeably distinct in all ESL based results. In fact, it could be argued that Fig. 9 and Fig. 12 (high strength steel) display the most refined topology of the four results depicted. These are all very good indications that the primary (and optimal) load path should be located as indicated in Fig. 9; regardless of whether the dynamic effects are taken into account or not. The conclusion based on the results of the study containing 131 topology optimisation models must therefore be that the use of ESL does not provide any significant detailed information with respect to load path generation, particularly if the significant increase in CPU time is taken into account. The average CPU time for the linear static models was 1,768 seconds as opposed to 1,075,100 seconds across all ESL based models included within the study. In other words, the excessive increase in CPU time used for ESL based optimisation as opposed to linear static optimisation could not be justified by a significant change in optimisation results.

4. Conclusion and Next Steps Based on the model results and statistic information included within the study presented in this paper the use of ESL based optimisation for development of automotive crash structures cannot be justified. This is primarily related to the excessive increase in CPU time and the lack of increased information (detailed load path generation) obtained therefrom. In addition, the ESL models experienced significant stability issues which could be attributed to a number of factors, the majority of which were investigated and documented in this paper. It could also be argued that the stability issues may have been caused, or increased as a function of applied loading and/or the design volume, as the preliminary analysis indicated that the initial loading Contribution 4

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coupled with the specific design volume did not invoke a non-linear response from the models in question. According to [9] the ESL method “can be applied at the concept design phase as well as design fine tuning phase”. This means that the use of ESL throughout this study is well within the realm of the ESL method. Nevertheless, the “optimisation engine” of the two methods remains identical. Previously conducted research [1] and [2] did in essence utilise a “manual” version of ESL, as the dynamic crash loads were represented by “equivalent” static load cases defined by using “engineering intuition and experience”. This did however allow for the use of linear static modelling, with the additional benefit of significantly lower CPU time relative to the ESL models. The fact does however remain that linear static modelling does not provide feasible results for non-linear (and highly dynamic) crash scenarios, such as a head on vehicle collision. The results from this study clearly indicate that the use of ESL based models did not significantly enhance the feasibility of the topology results. The primary reason behind this is most likely that the two approaches to optimisation is in essence identical, as discussed in [12], which states that “we cannot use design variables or functions which cannot be defined in linear static response optimization”. It could nevertheless be explored whether ESL might be advantageously utilised in a later stage of the design process, once the design has initially “matured”, i.e. initial load paths have been extracted. Regardless of whether or not this approach was to be successful, the fact still remains that truly non-linear topology optimisation of crash structures requires substantial improvements to provide highly efficient and feasible results. Furthermore, the previously highlighted possible “limitations” of the design volume and applied loading adopted throughout this paper could be further investigated. This could for example be completed via a case study focusing on the development of an automotive front crash structure which would be exposed to very large non-linear deformation as well as dynamic loading. Such a study could follow the lines previously set out in [2], which utilised linear static topology optimisation. Nevertheless, based on the findings of this paper the main conclusion must be that ESL does not provide a significant improvement for non-linear topology optimisation of automotive crash structures.

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References [1] J. Christensen, C. Bastien, and M. V. Blundell, Effects of Roof Crush Loading Scenario Upon Body In White Using Topology Optimisation, International Journal of Crashworthiness, vol. 12, iss. 1, pp. 29-38, 2012. [2] J. Christensen, C. Bastien, M. V. Blundell, A. Gittens, and O. Tomlin, Lightweight Hybrid Electrical Vehicle Structural TopologyOptimisation Investigation Focusing on Crashworthiness, International Journal of Vehicle Structures and Systems, vol. 3, iss. 2, pp. 113-122, 2011, Online-ISSN: 0975-3540, Print ISSN: 0975-3060. [3] Y. M. Xie and G. P. Steven, Evolutionary Structural Optimization, Springer Verlag, London, UK, 1997. [4] M. Zhou and G. I. N. Rozvany, On the validity of ESO type methods in topology optimization, Structural and Multidisciplinary Optimization, vol. 21, iss. 1, pp. 80-83, Springer Verlag, Berlin, Germany 2001. [5] X. Hunag and Y. M. Xie, A new look at ESO and BESO optimization methods, Structural and Multidisciplinary Optimization, vol. 35, iss. 1, pp. 89-92, Springer Verlag, Berlin, Germany 2008. [6] X. Hunag and Y. M. Xie, A further review of ESO type methods for topology optimization, Structural and Multidisciplinary Optimization, vol. 41, iss. 5, pp. 671-683, Springer Verlag, Berlin, Germany 2010. [7] G. R. Liu and Y. T. Gu, An Introduction to Meshfree Methods and Their Programming, Springer Verlag, Berlin, Germany, 2005. [8] J. M. Ferreira, E. J. Kansa, G. E. Fasshauer and V. M. A. Leitão, Progress on Meshless Methods, Springer Verlag, Berlin, 2009, ISBN 978-1-4020-8821-6. [9] Altair Engineering, Inc, Altair HyperWorks manual, Troy, Michigan, USA, 2013. [10]E. C. Chirwa, Q. Peng, Modelling of roof crush using the newly updated FMVSS 216, ICRASH 2010, Washington DC, USA, 22-24 September 2010. [11]M. P. Bendsoe and O. Sigmund, Topology Optimization, Theory Methods and Applications, Berlin, Germany, 2004. [12]G-J. Park, Equivalent Static Loads Method for Non Linear Static Response Structural Optimization, LS-Dyna Forum, Bamberg, 2010, available from

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CONTRIBUTION

5

Monologue Completed December 2014

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Acronyms 

Lagrangian Multiplier, element sensitivity number

ACO

Ant Colony Optimisation

ANS

Average Neighbourhood Strain energy

AR / ARMAX

Admissions Ratio / AR Maximum



Lagrangian Multiplier

BC

Boundary Condition

BEETS

Bi-directional Evolutionary Entropy Tabu search Simulated Annealing

CA / CAA

Cellular AutomatA

CAN

Cellular AutomatoN

CNE

Current Number of non-void Elements

COOLIT

Cooling Interval

COP / DCOP / SCOP

Combinatorial Optimisation Problem / Deterministic COP/ Stochastic COP

COOLUSE

Boolean operator to activate / deactivate cooling function

D

Design variable

t

Time step size

DV

Design Volume

e

Base of natural logarithm

E

Young’s Modulus / Element strain energy

ER

Evolutionary Ratio

ESE

Element Strain Energy

ESLM

Equivalent Static Load Method

ESN

Element Sensitivity Number

ESO / AESO / BESO

Evolutionary Structural Optimisation / Additive ESO / Bi-directional ESO

f

Generic function

F

Externally applied force

FE / FEM / FEA

Finite Element / FE Model /FE Analysis

FL

Fuzzy Logic

G

Constraint

HCA

Hybrid Cellular Automata

L

Function

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log / log2

Natural logarithm (base 10) / base 2

Max_iter

Maximum Number of Iterations

MEP

Maximum Entropy Principle

MEPUSE

Boolean operator to activate / deactivate the use of MEP

MCS

Monte Carlo Simulation

MNEA

Maximum Number of Added Elements

NEA

Number of Elements Added

NESTL

Number of Elements Stored in the Tabu List

NITS

Number of Iterations per Tabu Search

NN

Neural Network

NNels

Number of Neighbouring Elements



Weighting function

P

Probability function

PSO / BPSO

Particle Swarm Optimisation / Binary PSO

Q/q

Local / global quantity or constraint

r

Radius / distance between element centres

RADIUS

Parameter to define distance for neighbourhood influence

RMS

Root Mean Square

S

Entropy

SA / QSA

Simulated Annealing / Quasi-SA

SCH

Stochastic Hill Climbing

SFCOOL

Scale Factor for cooling function

SIMP


t

Time (s)

T

Temperature

TABU_offset

Function variable to offset the start of TS

TABU_LIST

Boolean operator to activate / deactivate the Tabu list

TERMTOL

Termination tolerance

TLF

Tabu List Fraction

TS / TSI / TSF

Tabu Search / TS Iteration / TS Fraction

TV / TVF

Target Volume / TV Fraction

V

Volume

VE

Volume of individual Element

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VDM

Variable Density Method

VR

Variation Ratio

VRE

Volume of Removed Elements

VTS

Variations per Tabu Search

VTSE

Volume to Total Strain Energy ratio

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List of Figures Figure 1, BESO flow diagram. .................................................................................................... 187 Figure 2, A) full design volume B) elements void (deleted) C) elements added ....................... 190 Figure 3, proposed algorithm flowchart. .................................................................................. 223 Figure 4, detailed flowchart of step 1. ...................................................................................... 225 Figure 5, illustration of neighbourhood influence and strain energy calculation. .................... 226 Figure 6, detailed flowchart of step 5. ...................................................................................... 229 Figure 7, Tabu Search iteration implementation. ..................................................................... 233 Figure 8, BEETS optimisation algorithm flowchart.................................................................... 237 Figure 9, main folder contents. ................................................................................................. 247 Figure 10, example folder contents A) standard iteration, B) Tabu Search iteration. .............. 247 Figure 11, example of Main_INFO.txt file content.................................................................... 248 Figure 12, general FE model for case study 1 - bracket. ........................................................... 249 Figure 13, load vs time. ............................................................................................................. 250 Figure 14, sum of strain per iteration case study 1. .................................................................. 252 Figure 15, total volume per iteration case study 1. .................................................................. 252 Figure 16, total strain (Von Mises strain) case study 1, iteration 0, t = 1.1. ............................. 253 Figure 17, total strain (Von Mises strain) case study 1, iteration 18, t = 1.1. ........................... 254 Figure 18, total strain (Von Mises strain) case study 1, iteration 23, t = 1.1. ........................... 255 Figure 19, optimised bracket from BESO. ................................................................................. 255 Figure 20, optimised bracket from Optistruct using unconstrained optimisation. .................. 256 Figure 21, Optimised bracket from Optistruct using compliance constrained optimisation.... 257 Figure 22, general FE model for case study 2 - Warren bridge. ................................................ 257 Figure 23, sum of strain per iteration case study 2. .................................................................. 259 Figure 24, sum of volume per iteration case study 2. ............................................................... 259 Figure 25, case study 2 iteration 12 and iteration 75. .............................................................. 260 Figure 26, optimised Warren bridge from BESO. ...................................................................... 261 Figure 27, overlay of BEETS and BESO results. .......................................................................... 262 Figure 28, optimised Warren bridge from Optistruct .............................................................. 263 Figure 29, FE model for case study 3. ....................................................................................... 264 Figure 30, BEETS optimisation result from case study 3. .......................................................... 266 Figure 31, BESO optimisation result from case study 3. ........................................................... 266 Figure 32, Optistruct optimisation result from case study 3. ................................................... 267

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Figure 33, fundamental FE model for non-linear case studies. ................................................ 268 Figure 34, final deformed shape at iteration 0. ........................................................................ 270 Figure 35, BEETS optimisation results case study 4. ................................................................. 271 Figure 36, sum of strain per iteration case study 4. .................................................................. 272 Figure 37, sum of volume per iteration case study 4. ............................................................... 273 Figure 38, first 3 iterations BEETS case study 5......................................................................... 275 Figure 39, final structure from BEETS case study 5................................................................... 276 Figure 40, sum of volume per iteration case study 5. ............................................................... 276 Figure 41, sum of volume per iteration case study 6. ............................................................... 278 Figure 42, sum of volume per iteration case study 7. ............................................................... 280 Figure 43, optimised shapes case study 7. ................................................................................ 281 Figure 44, sum of volume per iteration case study 8. ............................................................... 284 Figure 45, sum of volume per iteration case study 9 M1 and M2. ........................................... 287 Figure 46, sum of strain per iteration case study 9 M1 and M2. .............................................. 288 Figure 47, resulting topologies case study 9 M1 and M2. ........................................................ 289 Figure 48, load vs. time M3 and M4 case study 9. .................................................................... 290 Figure 49, sum of strain per iteration case study 9 M3 and M4. .............................................. 291 Figure 50, resulting topologies case study 9 M3 and M4 ......................................................... 292 Figure 51, sum of volume per iteration case study 10 M1 and case study 7 M1. .................... 294 Figure 52, sum of strain per iteration case study 10 M1 and case study 7 M1. ....................... 295 Figure 53, resulting topologies case study 10 M1 and case study 7 M1................................... 296 Figure 54, sum of volume per iteration case study 10 M2. ...................................................... 298 Figure 55, sum of strain per iteration case study 10 M2. ......................................................... 298 Figure 56, resulting topology case study 10 M2. ...................................................................... 299

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List of Tables Table 1, advantages and disadvantages of CA / HCA algorithms. ............................................. 186 Table 2, advantages and disadvantages of BESO algorithm. .................................................... 194 Table 3, results from first numerical FE example. ..................................................................... 215 Table 4, results from second numerical FE example................................................................. 216 Table 5, results from third numerical FE example. ................................................................... 217 Table 6, results from the fourth numerical FE example............................................................ 217 Table 7, importance of resolving BESO disadvantages. ............................................................ 221 Table 8, overview of Powershell files ........................................................................................ 240 Table 9, BEETS userdefined inputs. ........................................................................................... 243 Table 10, material parameters for case study 1 - bracket. ....................................................... 250 Table 11, BEETS input parameters for case study 1 - bracket................................................... 251 Table 12, BEETS input parameters for case study 2 – Warren bridge. ..................................... 258 Table 13, BEETS input parameters for case study 3. ................................................................. 265 Table 14, material properties for non-linear case studies. ....................................................... 269 Table 15, BEETS input parameters for case study 5. ................................................................. 274 Table 16, BEETS input parameters for case study 6. ................................................................. 277 Table 17, BEETS input parameters for case study 7. ................................................................. 279 Table 18, BEETS input parameters for case study 8. ................................................................. 283 Table 19, BEETS input parameters for case study 9. ................................................................. 286 Table 20, BEETS input parameters for case study 10 M1. ........................................................ 293 Table 21, BEETS input parameters case study 10 M2. .............................................................. 297 Table 22, findings in response to "original" BESO disadvantages. ............................................ 300

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Table of Contents Acronyms................................................................................................................................... 172 List of Figures ............................................................................................................................ 175 List of Tables .............................................................................................................................. 177 1.

Introduction ...................................................................................................................... 180

2.

Review of topology optimisation algorithms .................................................................... 181 2.1 Mathematical Algorithms.............................................................................................. 181 2.2 Heuristic & Meta-Heuristic Algorithms ......................................................................... 183 2.2.1

Hybrid Cellular Automata .................................................................................. 183

2.2.2

Bi-directional Evolutionary Structural Optimisation ......................................... 187

2.2.3

Combinatory Optimisation Problems................................................................ 195

2.2.4

Ant Colony Optimisation ................................................................................... 196

2.2.5

Stochastic Hill Climbing ..................................................................................... 197

2.2.6

Tabu Search ....................................................................................................... 199

2.2.7

Simulated Annealing ......................................................................................... 200

2.2.8

Particle Swarm Optimisation ............................................................................. 203

2.2.9

Neural Networks ............................................................................................... 204

2.3 General Principles ......................................................................................................... 206

3.

2.3.1

Robust Optimisation.......................................................................................... 206

2.3.2

Fuzzy Logic ......................................................................................................... 207

2.3.3

Monte Carlo Simulation, Probability and Entropy ............................................ 207

Definition and Implementation of Non-linear Topology Optimisation Algorithm............ 219 3.1 Algorithm definition ...................................................................................................... 219 3.1.1

Calculate Sensitivity Numbers ........................................................................... 223

3.1.2

Average Sensitivity Numbers and Determine Target Volume .......................... 227

3.1.3

Construct New Design ....................................................................................... 229

3.1.4

Tabu Search Implementation ............................................................................ 232

3.1.5

Simulated Annealing Implementation .............................................................. 235

3.2 Algorithm implementation and Software Development .............................................. 238

4.

3.2.1

Selection of programming language and software platform ............................ 238

3.2.2

Overview of BEETS PowerShell Software .......................................................... 239

Case Studies ...................................................................................................................... 249 4.1 Linear topology optimisation case studies .................................................................... 249

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4.1.1

BEETS Case Study 1 ........................................................................................... 250

4.1.2

BESO Case Study 1 ............................................................................................. 255

4.1.3

HyperWorks Case Study 1 ................................................................................. 256

4.1.4

BEETS Case Study 2 ........................................................................................... 257

4.1.5

BESO case study 2 ............................................................................................. 261

4.1.6

HyperWorks case study 2 ................................................................................. 263

4.1.7

BEETS case study 3 ............................................................................................ 264

4.1.8

BESO case study 3 ............................................................................................. 266

4.1.9

HyperWorks case study 3 .................................................................................. 267

4.2 Non-linear topology optimisation case studies ............................................................ 268 4.2.1

BEETS case study 4 ............................................................................................ 269

4.2.2

BEETS case study 5: effects of ER and ARMAX magnitudes .............................. 274

4.2.3

BEETS case study 6: cooling function influence ................................................ 277

4.2.4

BEETS case study 7: neighbourhood influence ................................................. 279

4.2.5

BEETS case study 8: Tabu Search influence....................................................... 283

4.2.6

BEETS case study 9: effects of MEP ................................................................... 285

4.2.7

BEETS case study 10: combined effect of BEETS ............................................... 292

4.3. Conclusion of the potential of BEETS for non-linear topology optimisation ................ 299 5.

Conclusion ......................................................................................................................... 303

References................................................................................................................................. 305 Appendix A ................................................................................................................................ 311

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1.

Introduction

This monologue which constitutes Contribution 5 to the PhD portfolio aims to: 1. Review potential approaches for non-linear (material and geometry) topology optimisation of structures exposed to large deformations. 2. Present the theoretical aspects as well as documenting the development of the optimisation algorithm embedded within Contribution 6 to this PhD portfolio. 3. Present and discuss a series of case studies using the software which constitutes Contribution 6 to this PhD portfolio. The ultimate aim of chapter 2 is to define a strategy for the continued research into and definition of a non-linear topology optimisation algorithm for structures exposed to large deformations. The first chapter will therefore provide the overarching algorithm which will form the basis for the final, i.e. the sixth, contribution to this PhD portfolio, the optimisation software. Chapter 3 of this monologue will document the initial development and implementation of the algorithm from chapter 1 into the software, i.e. Contribution 6, including flowcharts, parameters, variables, user inputs, initial verification / validation of the software etc. Chapter 4 of this monologue will present a series of topology optimisation case studies used to compare the embedded algorithms to competing algorithms, including commercial software where applicable. The ultimate aim of chapter 4 is not to fully validate the software (contribution 6), but rather to analyse the performance of the algorithm and assess the feasibility for further development work.

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2.

Review of topology optimisation algorithms

Contribution 1 to this PhD portfolio contains a report entitled “Non-Linear Finite Element based Topology Optimisation for Low Carbon Vehicle Technology Project (LCVTP)”, which reviews the potential for the use of the Solid Isotropic Material with Penalisation (SIMP) interpolation scheme in combination with the Variable Density Method (VDM) as well Evolutionary Structural Optimisation (ESO) algorithms for non-linear optimisation. This report was completed in November 2011. As there has been some considerable developments in terms of commercially available (Finite Element based) optimisation software since then it was deemed appropriate to revisit and extend the literature review at this stage. Optimisation algorithms may be categorised in a number of different ways. In this monologue the focus will be aimed at two main categories: mathematical and heuristic / meta-heuristic algorithms as shall be subsequently presented. Moreover, optimisation algorithms may also be divided into approximation algorithms and exact algorithms, (Vazirani, 2001). Exact algorithms work on the principle that all permutations of a given optimisation problem be computed; the optimisation problem thus reduces to a “simple” selection problem. However, as discussed in Contribution 1, the curse of dimensionality rapidly eradicates the feasibility of employing exact algorithms for most topology optimisation applications. (Hosny, 2010) states: “The term approximation algorithm is often used to refer to an optimization algorithm which provides a solution that is guaranteed to be within a certain distance from the optimum solution every time it is run, with provable runtime bounds”

All algorithms discussed in this monologue may therefore generally be considered to belong to the approximate algorithm family, although certain limitations will apply, as shall be discussed throughout the remainder of this chapter.

2.1

Mathematical Algorithms

From a “practical” engineering viewpoint mathematical based algorithms are “ideal” for structural optimisation, as these generally have a clear and systematic structure, enabling a relatively straightforward implementation into numerical analysis software such as commercial FE programs. Gradient based methods such as the Optimality Criteria Method (OCM), Moving Asymptotes (MA) and Sequential Linear Programming (SLP) are most often employed. From a scientific viewpoint the perhaps most pivotal feature of mathematical algorithms is that they (generally) allow it to be proven (within certain limitations) that the optimisation is converging

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towards a (global) optimum solution, as opposed to a (local) “random” solution, provided a unique global optimum solution exists. This “proof” is of course dependent upon a number of parameters governing the specific optimisation problem. As mentioned above, Contribution 1 of this PhD portfolio contains a literature review of mathematical based algorithms including Homogenisation Based Optimisation (HBO). As concluded in Contribution 1, the use of approaches such as HBO has extremely limited applications within engineering in general, as it is not possible to define a continuous function to describe e.g. the stress state throughout a component. The opportunities of utilising such an approach thus addresses a much wider aspect of analytical and numerical modelling, and would (if solved) make the use of Finite Element obsolete for stress analysis. This approach will therefore not be further discussed. Contribution 1 also contains detailed discussions relating to the VDM method and the SIMP interpolation scheme, which are perhaps the most widely used and accepted approaches to topology optimisation of linear elastic and isotropic problems. Indeed many adaptations and variations of the SIMP interpolation scheme has been used for a wide variety of applications outside of the linear elastic and / or isotropic remits of the original formulation. (Sorensen and Lund, 2012) did for example attempt to utilise the scheme for optimisation of composite ply lay-up. The “Non-Linear Finite Element based Topology Optimisation for Low Carbon Vehicle Technology Project (LCVTP)” document also contains suggestions as to how the SIMP scheme could be “modified” to cater for (minor) levels of material non-linearity, however the general issues related to SIMP as highlighted throughout the document are still applicable. It can be justified that the overall topology optimisation study of Contribution 1 utilised a “manual” version of the Equivalent Static Load Method (ESLM) originally proposed by (Park, 2011). Since Contribution 1 was completed in November 2011, the ESLM method has become available in commercial software such as Altair HyperWorks. In light of this Contribution 4 contains a more detailed comparison of the underpinning differences between the “single” linear static topology optimisation algorithm and the ESLM approach, this includes a detailed case study. Contribution 4 concurs with the conclusions of Contribution 1, in concluding that the SIMP interpolation scheme is not suited for large deformation non-linear topology optimisation. In light of this information and the infeasibility of formulating continuous functions, e.g. stress functions, for most mechanical engineering structural analysis / optimisation problems the purely mathematical algorithms are dismissed as a potential approach for a non-linear topology optimisation algorithm in connection with this PhD portfolio. Instead the focus will now turn to heuristic algorithms.

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2.2

Heuristic & Meta-Heuristic Algorithms

If the mathematical optimisation algorithms presented in section 2.1 are considered capable of providing the “exact” (optimum) solution to a given optimisation problem, heuristic algorithms may (in the extreme) be considered to take the most “practical” approach to optimisation. Pure heuristic algorithms are most often experience based and utilise a rational and above all practical approach to optimisation and problem solving in general. The scenario of a structural engineer reviewing and modifying an initial design based upon personal experience and numerical analysis results may thus be defined as a heuristic “algorithm”. Hence pure heuristic algorithms have inherent uncertainties with e.g. repeatability (imagine that the structural engineer is replaced) and relevance of solution (how can it be justified that the solution is indeed an optimum?). Heuristic algorithms thus generally have many undesirable factors of uncertainty associated with them. Meta-heuristic algorithms are an adaption of heuristic algorithms and overall address many of the common issues associated with heuristic algorithms. Meta-heuristic algorithms most often adapt ideologies from other sciences such as biology and evolution, and combine these with physical experimentation results as well as experience. The following sub-sections will introduce some of the competing meta-heuristic algorithms. 2.2.1

Hybrid Cellular Automata

Hybrid Cellular Automata (HCA) is the optimisation algorithm incorporated in LSTC’s commercially available LS-TaSC optimisation software capable of handling large deformation non-linear problems, such as vehicle crashworthiness (LSTC, 2011). According to (LSTC, 2011 a) the algorithm is a heuristic one. However, according to (LSTC, 2011) the algorithm was first derived by (Tovar, 2004) which clearly utilises bone biology to define the Hybrid Cellular Automata algorithm, arguably making it a meta-heuristic algorithm. The HCA algorithm is an adaptation of the Cellular Automata (CAA) model originally developed by Stanislaw Ulman and John Von Neuman in the 1940s, (originally Cellular AutmatoN (CAN). The general principle behind CAA is as follows. Imagine a series of “individual cells” which may have finite states, e.g. ”1” or “0” as is the basic principle of FE based topology optimisation. Each individual cell is “related” to a set of cells denoted its “neighbourhood”. An initial state of each cell is set (at t=0), a new overall state (at t = 1) is then achieved using a fixed rule, for example the outcome of an FE analysis, the new state of each cell (at t = 1) is then obtained based upon its current state (at t = 0) and “the responses” from its neighbourhood. The neighbourhood may be

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defined in slightly different ways; one of the most often used neighbourhoods is the Von Neumann neighbourhood (Barati, 2014). Comparing the above description to that of the Evolutionary Structural Optimisation (ESO) algorithms presented in Contribution 1 the parallels are very distinct; as the ESO algorithms use a “snapshot” of the current state, i.e. FE analysis results to choose “initial states” and thereafter use “neighbourhoods” to determine the state of each element (cell). The basic topology optimisation problem in LS-TaSC is solved using a uniform internal energy density as the objective function, (Roux, 2011). The topology optimisation problem in LS-TaSC is formulated as equation (2.1) (Roux, 2011a):

  x i  j 1  N  subject to :    xi Vi  M *  i 1  C lj  C j  C uj , j  1, 2..., J   xmin  xi  1.0 

min   w jU j  xi   U *j  N

L

(2.1)

In (2.1) w is a weighting factor, U is the internal energy density of element I, Vi is the element volume, U*is the internal energy density set point and Cj is constraint j with lover (l) and upper (u) subscript boundaries, (Roux, 2011). The change in the individual design variable I at time t and the updated design variable at t+1 are defined in expression (2.2) (Roux, 2011a):

x  t i

K U it  U *  (2.2)

U* xit 1  xit  xit

In (2.2) K is a scale factor. According to (Roux, 2011), LS-TaSC is controlled by 3 user-defined parameters: Mass Fraction (MF), Convergence Tolerance Parameter (GTP) and Proximity Tolerance Parameter (PTL). LS-TaSC utilises a “full design volume” for a starting point, as does, see Contribution 1 “Non-Linear Finite Element based Topology Optimisation for Low Carbon Vehicle Technology Project (LCVTP)” Figure 27. In Contribution 1 “Non-Linear Finite Element based Topology Optimisation for Low Carbon Vehicle Technology Project (LCVTP)” the objective function and constraints for the Variable

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Density Method were formulated as equation (9) and (11). These are repeated in (2.3) and (2.4) for convenience:

minimise F    subject to :

(2.3)

g1     Val1 ; g 2     Val2 ; g3     Val3

F  U  





1 T     E       dV  2

(2.4)

Comparing (2.3) and (2.4) to (2.1) it is clear that there are distinct similarities between the two “algorithms”. The main difference lies in the fact that (2.1) utilises the strain energy density while (2.4) uses the (total) linear elastic strain energy. It should be noted that (2.4) is an example of how VDM works in general, the actual implementation will vary from software to software. (Altair, 2009) does however support that the above explanation is indeed the way that topology optimisation is implemented into Altair HyperWorks’ optimisation module Optistruct. (Altair, 2009) also confirms that Optistruct utilises a gradient based approach to perform the actual optimisation, as discussed in Contribution 1. Despite the similarities discussed above HCA and LS-TaSC does not utilise a gradient based approach, (LSTC, 2011 a): “This method updates the density of elements based on information from its neighbors. No gradient information was required”. This means that LS-TaSC / HCA performs the optimisation based on “current” information, and is not influenced by “time-history”, i.e. previous iterations are not considered as part of the optimisation process. In other words, the optimisation process has no “memory” as e.g. Optistruct does (Altair, 2009). The above statement thus reveals a distinct difference between CA and HCA, as the latter does not use “time-history”, i.e. gradient information, whereas the former (CA) does. From an overall perspective this means that HCA / LS-TaSC is in essence very similar to evolutionary algorithms which are discussed in the “Non-Linear Finite Element based Topology Optimisation for Low Carbon Vehicle Technology Project (LCVTP)” report contained within Contribution 1. This also suggests that LS-TaSC “shares” some of the inherent problems of evolutionary algorithms with respect to proven convergence towards a truly optimum solution

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as also discussed in Contribution 1. As previously discussed this is one of the major advantages of mathematical based algorithms, however when large levels of non-linearity is introduced the “predictive” (linear) nature of the systems and the associated benefits diminish as highlighted in Contribution 4. In light of the above discussions it was chosen to abandon the HCA and ESO based algorithms. Furthermore, the lack of “time-history” influence, e.g. gradient based information” is a potential area of concern, as it is very likely that both the HCA and ESO algorithms will rapidly move towards a specific solution (topology). For relatively “simple” topology optimisation problems, i.e. small design volumes, this may in fact be advantageous, but for larger more complex problems it indicates that neither algorithm is able to “diversify” its search pattern and avoid local extremum / optimum points. The general advantages and disadvantages of the HCA / CA based algorithms are listed in Table 1. Table 1, advantages and disadvantages of CA / HCA algorithms. Advantages Strong, direct and relatively simple implementation into FEA. Local and global constraint control. Direct link between objective function and FEA.

Disadvantages No time-history data considered. No convergence justification. Relatively high computational cost. Low dispersion of search within potential

Neighbourhood influence.

solution space, high probability of “converging” towards local optimum points.

In addition to the continuous comparison between CA and BESO throughout this section it is convenient to review the most recent developments of the BESO based algorithm in isolation. This is because BESO was found to have the highest potential for non-linear topology optimisation of the three “conventional” evolutionary based algorithms as discussed in “NonLinear Finite Element based Topology Optimisation for Low Carbon Vehicle Technology Project (LCVTP)”, Contribution 1.

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2.2.2

Bi-directional Evolutionary Structural Optimisation

An extensive search for publications and original research into the most recent developments of BESO, AESO and ESO for non-linear topology optimisation has returned very little documentation. ESO based optimisation can in general be classified as belonging to the “greater family” of optimisation algorithms known as Genetic Algorithms (GA). Expanding the literature review to include GA at this stage did provide additional publications but none directly related to non-linear topology optimisation. The most recent paper, (Huang and Xie, 2008), from the “fathers” of BESO, Y.M. Xie and X. Huang directly focusing on non-linear topology optimisation is from 2008, (Researchgate, 2014) , indicating that the development of BESO for non-linear topology optimisation has not been widely pursued recently. Given the general similarities between GA and BESO and that the literature review did not disclose any publications from other authors specifically relating to the development of GA or BESO for non-linear topology optimisation post 2008 it is presumed that (Huang and Xie, 2008) contains the most recent version of BESO for non-linear topology optimisation; the following subsection will therefore utilise the information of this publication, as this is representative of GA algorithms in general. The paper focuses on both geometrically as well as material non-linearity under displacement loading, and focuses on the robustness of the algorithm. Please note that robust optimisation will be further discussed in section 2.3.1. Based on information in (Huang and Xie, 2008) an overall flowchart for the revised BESO algorithm has been created, Figure 1.

Figure 1, BESO flow diagram.

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The paper highlights three main aspects which have been improved in the “updated” BESO algorithm as: a) Filtering scheme (2). b) Improved sensitivity analysis (1, 3). c) A new procedure for adding and removing elements (5). The brackets above relate to the algorithm steps illustrated in Figure 1. It can be noted that the “filtering scheme step” falls in between the two parts of the “sensitivity analysis”, and indeed the two are intertwined. The algorithm in (Huang and Xie, 2008) aims to maximise the structural stiffness of the structure with a constraint based on maximum allowed displacements. The objective function thus seeks to maximise the external work which is set equal to the total strain energy under the caveat of a quasi-static load condition. While this condition may not be of major concern to certain topology optimisation problems it is certainly an aspect which must be thoroughly considered if dealing with e.g. high velocity crashworthiness scenarios. This potential concern could possibly be circumnavigated by introducing a very low timestep in the FE analysis; this would however have a significant effect upon the overall CPU run time. The revised sensitivity analysis and filtering scheme aims to determine the influence upon the individual (FE) element upon the response of the objective function, i.e. the sensitivity. Overall, this is achieved by observing the (strain) energy levels within the individual element and the method used to summarise the total strain energy. Equation 15 in (Huang and Xie, 2008) denotes the variation of the objective function as a function of the individual element x:

f  x   lim   Eie  Eie1    Ene n

n 

(2.5)

i 1

In (2.5) E denotes the strain energy, e denotes that the definition is carried out on a single element, i denotes the specific interval on the prescribe design displacement, and n denotes the total number of intervals on the design displacement. Ene is thus the total (numeric) strain energy of the removed element. There are two important aspect to denote in relationship to the above; firstly the i index can be considered to be the equivalent of the timestep t in an FE analysis as the two are coincident. With this in mind the second and more general significance of (2.5) becomes clearer: the algorithm now utilises time-history information; i.e. it utilises the

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value of the previous time-step to assess the impact upon the objective function. Although this information is perhaps of limited use at present it may be worth analysing the potential of expanding this “time-history” further in order to enhance the credibility of the algorithm. (Huang and Xie, 2008a) states: “The Eq. (15) means that the decrease of the total external work due to removing one element is equal to the total strain energy of the element in its final deformed state and irrelevant to the size of displacement intervals” There are two main implications of the above statement. Firstly, the above statement is the basis for the claim in Huang and Xie, 2008) that the algorithm is mesh independent. In the authors opinion this statement is on the face of it quite bold, but in terms of technical significance it is of minor importance as all FEA per definition is mesh dependent. This means that despite the above statement the “input information” is arguably mesh-dependent; questioning the overall importance of the “mesh independence” statement. The second main implication of the above citation lies within the intricate details in the wording, more specifically the term “final deformed state”. This implies that ultimately only the final deformed state is of relevance and the intermediate steps are not. The validity of this statement can be justified in combination with the previous caveat of quasi-static loading, but arguably warrants further investigation in connection with optimisation of e.g. crash structures. A more holistic approach to this topic is of course related to the overall aim of any specific topology optimisation study. For example the design of a front end crash structure (where acceleration levels during the impact must be taken into consideration) is significantly different to a support bracket where only the final deformation is of relevance. In other words the significance of the dynamic nature of any given problem must be carefully considered. Returning to equation (2.5) (Huang and Xie, 2008a) defined the sensitivity of the (existing) element i as:

i  Eni

(2.6)

Equation (2.6) thus confirms the above statement that the sensitivity  i of element i is only dependent upon the last state n, i.e. the final deformed state. (Huang and Xie, 2008a) also states:

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“Note that the sensitivity number is a direct measure of the variation of the objective function due to element removal or addition. To maximize the total external work, the evolutionary process will be conducted by removing elements with the smallest sensitivity numbers and adding the elements with the highest ones. Mathematically such a procedure is known as the “hill-climb” method or the “steepest descent” algorithm.” The above citation thus indicates that the BESO algorithm fundamentally uses the Hill Climbing (HC) optimisation technique which will be subsequently presented in sub-section 2.2.5. This also reinforces the statement relating to the lack of “diversification” of the BESO optimisation algorithm significantly increasing the likelihood of converging towards local optimum solutions as previously discussed in section 2.2.1. The focus of attention is now turned to the “filtering scheme” improvement in (Huang and Xie, 2008), i.e. step 2, Figure 1. As per equation (2.6) the sensitivity number for existing elements are extracted directly from the FEA. This is of course not possible for void elements, but it may be necessary to add elements to the model, e.g. in highly stressed areas. At this point it is convenient to expand the explanation of what constitutes a void element. It is important to remember that BESO optimisation works from a full Design Volume (DV) and subsequently voids (deletes) elements in each iteration as illustrated between Figure 2 A) and Figure 2 B). Finally void elements may be added as illustrated in Figure 2 C).

Figure 2, A) full design volume B) elements void (deleted) C) elements added Two things should be noted when observing Figure 2; firstly that the elements added in C) were originally part of the full DV in A), hence the nodal coordinates are pre-defined. Secondly it should be noted that B) and C) in Figure 2 may occur within the same iteration step, i.e. elements may be both voided and added. The above may seem identical to the VDM / SIMP

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approach, it is however important to remember that the VDM approach does not delete elements from the FE model between iterations, it merely reduces Young’s Modulus. In BESO the elements are completely removed from the stiffness matrix of the FE model, however the nodal coordinates and element definitions remain in the memory of the optimisation software enabling the transition between Figure 2 B) and C); in other words the element is marked as void. It should also be noted that it is possible to initially define void elements. As an example the structure included in the FE analysis at iteration 0 is Figure 2 C), but the full DV of the optimisation is as illustrated in Figure 2 A). The mathematical description for adding elements may be obtained by redefining the sensitivity numbers and simultaneously introducing the filtering scheme. (Huang and Xie, 2008b) therefore defines the sensitivity number for adding elements as:

   r  N

i 

j 1 N

ij

j

(2.7)

 r  j 1

ij

 

In equation (2.7) N is the total number of elements in the FE mesh,  rij is a weighting function defined as equation (2.8), (Huang and Xie, 2008b).

     r 2   ij    exp       r 3     2        3     rij   r 2   3

i  N r

ij

 r

j  1, 2,..., N

(2.8)

In equation (2.8) rij is the direct distance between the centres of elements i and j; r is a user selected filtering radius. Equations (2.7) and (2.8) thus underpin the discussions of section 2.2.1 which highlighted some of the similarities between CA / HCA and BESO; in this case the influences of the “neighbourhood”. In relation to equations (2.7) and (2.8) (Huang and Xie, 2008b) state:

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“The above sensitivity filter serves two purposes: (1) to extrapolate sensitivity number within the full design domain and (2) to smooth the sensitivity number in the neighbourhood of each element and prevent the problem of checkerboard patterns and mesh-dependencies in topology optimization.” With the above in mind the focus of attention can now turn to the 3rd improvement of the BESO algorithm proposed in (Huang and Xie, 2008); the new procedure for adding or removing element, step 5, Figure 1. The intricate details of the algorithm are of less relevance in the current context; what is important is the overall use of 0 (void) and 1 (full) element values. The revised procedure ranks and treats all elements in the model, i.e. both 0 and 1 elements, together. The main reason why this is significant is that Boundary Conditions (BCs) are not initially considered at this step (5, Figure 1), when determining the elements to void or add. This could for example lead to computational instability or even a significant change in the overall physics of the optimisation problem if e.g. an external load or constraint is removed. In an attempt to avoid this scenario step 6, Figure 1, is introduced. In this context (Huang and Xie, 2008c) states: “Once such a problem occurs, using a finer mesh or replacing the removed elements with “soft” elements to preserve the boundary or load can effectively circumnavigate problem.” (Huang and Xie, 2008) do not further specify the definition of a “soft” element, but it can be deduced that this is an element with lower stiffness or indeed lower volumetric mass density. Parallels can thus be drawn between the BESO algorithm and the VDM / SIMP algorithm discussed in section 2.1. The final sections of (Huang and Xie, 2008) contain topology optimisation examples from both linear and non-linear models using BESO. The conclusions contain two separate statements of particular interest, (Huang and Xie, 2008d): “Due to partial unloading of material, the defined optimization problem for nonlinear material is ill-posed and the solution may not always convergent because oscillation between designs of two different deformation modes.” Convergence is always a key aspect of any topology optimisation algorithm; the above statement reveals a very undesirable potential pitfall of utilising BESO in connection with nonlinear material behaviour. Questions can be raised as to why this problem seemingly has not been further addressed since 2008; regardless of what the answer to these are it is certain that

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if BESO is to be further refined for topology optimisation of non-linear structures the above must be a key area of investigation. The second main finding according from (Huang and Xie, 2008d) is: “The examples illustrate substantial differences in the topologies of the linear and nonlinear optimal designs. Unlike the linear design which is independent of the design displacement, the nonlinear optimal design is highly dependent on the specified design displacement. The results show that the nonlinear design at its design displacement is always better than the linear design in terms of the total energy, which is the objective function” This is a very interesting statement which underpins the previous statement relating to the significance of the “final deformed shape”; but the above also clearly indicates that the final deformed shape is heavily dependent upon the previous stages of the deformation, which is hardly surprising from a mechanical engineering viewpoint. It does however also emphasise that the need for analysis into the direct importance and influence of the “intermediate deformation steps” upon the results of the topology optimisation must be further explored. In other words, should the intermediate energy levels, Ei , in equation (2.5) be further implemented into the optimisation algorithm? This will be further addressed in section 4.2.6. The final remark to be noticed in the above citation is that using the BESO algorithm and test cases in (Huang and Xie, 2008) the nonlinear topology optimisation results always outperform the linear counterparts. If this is indeed the case it is suggested that applying non-linear topology optimisation algorithms to linear topology optimisation problems will lead to more optimal solutions. It is understood that this potential advantage comes at the cost of increased CPU time. The review of the BESO algorithms is thereby complete. Despite a number of significant issues relating to the application and results from the BESO algorithm it also raises a number of potential benefits and general areas of interest which should be considered whichever nonlinear topology optimisation algorithm is chosen going forward. The main advantages and disadvantages of the BESO algorithm are summarised in Table 2. .

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Table 2, advantages and disadvantages of BESO algorithm. Advantages Strong, direct and relatively simple implementation into FEA. Local and global constraint control.

Disadvantages Quasi-static loading assumption. Non-convergent and potential oscillation between “equal” design solutions.

Direct link between objective function and

No direct consideration of “intermediate”

FEA.

deformation modes / levels.

Neighbourhood influence.

Relatively high computational cost.

Inclusion of gradient information (time-

Significant issues with high levels of non-linear

history)

material behaviour. Low dispersion of search within potential solution space, high probability of “converging” towards local optimum points.

All discussions throughout Contributions 1-4 of this PhD portfolio, section 2.1, sub-sections 2.2.1 and 2.2.2 of this monologue have predominately been focused on algorithms which are currently being used for structural optimisation on a large scale, e.g. in commercial FE software. All previous discussions, models and results have clearly indicated that none of these approaches clearly meets all the requirements of a topology optimisation algorithm for structures exposed to large deformations. The overarching requirements can in general be listed as: 1. Proof of convergence towards an optimum solution. 2. Reasonable overall computing requirements (CPU time, memory etc.). 3. Appropriate level of ability to cater for the natural complexity of the load scenario such as: a. Time dependent loading. b. Non-linear system response. c. Variations in material characteristics as a function of deformation. d.

Overall system response versus localised effects.

e. Relevance and feasibility of results to “real world” engineering.

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The above list of “requirements” is by no means exhaustive and is in general open to interpretation and discussion. It is however clear that all individual algorithms discussed so far have severe limitations with one or multiple aspects of the above list. Furthermore it is unlikely that any one algorithm will be able to “fully” meet all requirements; ironically this problem can in itself be considered to be a multi-objective optimisation problem. With this in mind this review will now focus on optimisation techniques and algorithms in general. This means that the majority of algorithms presented will not previously have been applied to topology optimisation of structures exposed to large deformations using a rigorous and well documented basis. The potential application of the algorithms and the feasibility to meet the above list of requirements will be discussed throughout the following sections. 2.2.3

Combinatory Optimisation Problems

A Combinatorial Optimisation Problem (COP) can in general be perceived as an optimisation problem containing only discrete variables. Finite Element based topology optimisation falls into this category as each individual element should either be included or excluded from the structure, this could e.g. be done using the Boolean operators 1 (included) or 0 (excluded). As presented in Contribution 1, the introduction of the Variable Density Method (VDM) turns the problem into a continuous one. A further general requirement of COPs is that the solution space should contain a finite number of entries, i.e. discrete or alternatively a countable infinite set (Schrijver, 2003). Due to the previously discussed curse of dimensionality it is clear that the reference to a “finite solution space” above rapidly becomes violated with respect to topology optimisation, but this may not necessarily be the case for the “countable infinite set”. It is clear that most “real world” topology optimisation problems will have an infinite number of solutions, and so the “reduction” to a “countable number of infinite solutions” will most likely influence the results of such an algorithm. With this in mind, the study into COPs and the applicability to non-linear topology optimisation may continue. COPs may further be divided into two main categories; Deterministic Combinatorial Optimisation Problems (DCOP) or Stochastic Combinatorial Optimisation Problems (SCOP). The difference between the two types of problems is easily understood through the following example. Consider the optimisation problem of creating packed lunches for a set upper sale price with maximum nutrition. The packed lunch may consist of eggs, meat and bread. Knowing the cost as well as the nutritional value of each item the DCOP can be solved relatively easily by creating an objective function of nutrition and a single constraint relating to

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the cost. Imagine now that the “packed lunch business” will be selling lunches over the coming school year. No single supplier is however able to commit to a set price or even a consistent product. This means that both the price as well as the nutritional value of each item will vary. The optimisation problem is now more complex, as a certain level of “randomness” or “uncertainty” has been introduced; the nature of the problem has become stochastic, and the overall optimisation problem is now a Stochastic Combinatory Optimisation Problem (SCOP). Leaving the world of “packed lunches” and returning to the world of structural optimisation it is obvious that there are direct parallels between the above example and that of linear vs. nonlinear topology optimisation. Although there are several practical differences the essence of the DCOP problem is (in its basic form) similar to that of linear optimisation; it is possible with great accuracy to “predict” the outcome of the optimisation and it can be justified that an optimum solution has been obtained, e.g. via the use of gradient based optimisation. The essence of the SCOP problem also holds parallels to the non-linear topology optimisation problem; it is very difficult if not impossible to predict the variations in the response of the system. Consider for example the task of consistently and repeatedly predicting the changes of structural response of a FE crash model when adding or deleting elements, this is the equivalent of adding uncertainty into the system. The stochastic nature of this system is most likely not completely random which indicates that it may be possible to introduce probabilities into the optimisation algorithm. With this in mind it is now useful to review algorithms which in general focus on SCOP; for simplicity it is however more convenient to introduce these in terms of DCOP and subsequently address the additional complications with respect to SCOP. (Bianchi et. al., 2008) completed a significant survey of meta-heuristic algorithms. This survey forms the basis of the explanations presented below. 2.2.4

Ant Colony Optimisation

Ant Colony Optimisation (ACO) is as the name suggests based on the natural behaviour of ant colonies and their individual worker ants. When ants forage they naturally seem to find a ”logical” and “effective” route between their nest and the food source; in other words they seem to determine an optimum route; this observed behaviour is the basis of ACO, (Bianchi et. al., 2008). Imagine two ants walking from the nest to a food source using two different routes. As the ants walk they release pheromones which naturally decay over time. The ant which (randomly) selected the short route will commence the return leg of the journey quicker than the other ant, thus reinforcing the pheromone trace on the shorter route. Other ants will instinctively follow the stronger pheromone path, reinforcing and even adding to this.

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According to (Bianchi et. al., 2008) the ACO algorithm contains three major steps which constitute the central optimisation loop of the algorithm: 1. Construct Ants Solutions. This is the procedure for which “ants” incrementally and stochastically construct paths, i.e. solutions in a wider optimisation context. 2. Evaporate Pheromone. This is the process in which pheromone for certain solutions are decreased using “local” information, therefore this step is also often referred to as local update. This step is pivotal to ensure that the ACO algorithm does not prematurely converge to a single solution. 3. Deamon Actions. This step refers to decisions made based on global information relating to the optimisation problem. Note the difference between local in step 2 and global in step 3. Analogue to step 2, step 3 is also often referred to as global update. The three steps highlighted above are repeated until the optimisation problem has converged or is otherwise terminated via a pre-specified termination condition. Even though the above description is general and perhaps somewhat abstract in relation to non-linear topology optimisation and the much higher level of details in sub-section 2.2.2 the basic ideology of the ACO algorithm can be understood based on the above. Based on the above it is clear that significant development and analysis work is required before ACO could be applied to the non-linear topology optimisation problems of concern. It is also clear that it is unlikely for the ACO algorithm to be a “stand-alone solution” to the non-linear topology optimisation problem. It is perhaps more appropriate to see the ACO algorithm as something which could be merged into other structural optimisation algorithms, for example as an “evolution” of BESO, i.e.an algorithm which could be combined with that presented in section 2.2.2. Based on this relatively short review of ACO it is not feasible to detail a specific list of advantages and disadvantages of ACO for non-linear topology optimisation, quite simply because additional analysis and development work is required. It should however be noted that ACO has the potential to be combined with e.g. BESO in order to overcome the previously stated concerns with respect to the diversity of the search area for BESO algorithms. 2.2.5

Stochastic Hill Climbing

Stochastic Hill Climbing (SHC) is a variation of the more general optimisation search technique denoted Hill Climbing (HC). As presented in sub-section 2.2.2 HC is the fundamental search

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method used in the BESO optimisation algorithm. HC is generally considered to belong to the “local search” techniques, the basic principles of HC is as follows. A starting solution is required, i.e. an initial guess for a viable solution is required; in the case of BESO this constitutes an initial FE analysis. Based on this initial guess, variations to the current solution are introduced based on some pre-defined rules and the newly “updated” solution is then analysed. The performance, i.e. objective function, of the “updated” solution is then compared to the “original” solution, if the former out-performs the latter it will be accepted as the “current” solution, and the iterative process will repeat. If the “original” solution out-performs the “updated” solution the former will remain the “current” solution and further variations of the “original” solution will be made. It is of course necessary to introduce termination criteria, e.g. maximum number of iterations. As discussed in discussed in sub-section 2.2.2, the specific implementation of the “pre-defined rules” constitutes the filtering and averaging of the element sensitivity numbers, i.e. steps 2 and 3, Figure 1. As highlighted in Table 2, questions about the “low dispersion of search within the solution space” for this type of algorithm. This statement is supported by (Bianchi et. al., 2008a): “Such a simple and very local general local search behaves quite poorly in practice, particularly because when a local optimum is found, the algorithm stops improving, and combinatorial problems often have a local optima whose objective values are much worse than that of the global optimum.” SCH attempts to reduce this problem by not only “obeying” the pre-defined rules for variation, but also introduce a certain level of “randomness”, hence stochastic HC, by including larger (global) variations not governed by the local variation laws. The ideology behind this is an attempt to avoid converging towards local optimum points. It should however be noted that this “randomness” will still be governed by some rules in order to enable the implementation into numerical software. A very crude example of SHC adaptation of the BESO algorithm would be to “at random” select a pre-defined number of elements or perhaps percentage of the design volume to remove per iteration. Although the SHC adaptation introduces a potential method for avoiding local optimum solutions it also has the unfortunate likelihood of introducing numerical instability into the FE model, let alone the optimisation problem in general. A perhaps more refined and systematic variation of HC is denoted Tabu Search (TS).

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2.2.6

Tabu Search

As mentioned above TS is in essence a variation of HC. The ideology of TS is very much the same as SHC; to avoid premature convergence towards a local optimum solution. There are 3 general TS concepts, (Bianchi et. al., 2008): 1. Best Improvement. Consider a cluster of solutions, which are variations of each other as was the case with the HC above. Imagine a simple minimisation problem, where the current solution AA provides the objective value 5. Variations of S1, i.e. the “neighbours” AB and AC provide the values 6 and 7 respectively. AB is then chosen as the solution to replace AA because it is the “best neighbour”, despite the fact that AA is overall a better solution. The reason behind this is simply to avoid local optimums by expanding the search area inside the solution space. 2. Tabu Lists. Following on from point 1 AB is now the current solution, the search is repeated. The neighbours of AB are AA and AC. Under the rules of “best improvement” AA will once again be chosen as the new solution quite simply because it is the best neighbour to AB. This is of course detrimental to the efficiency of the algorithm, as the solution will now simply enter into a cycle of swapping between AA and AB. This general phenomenon was also observed in Huang and Xie, 2008) where the convergence of the BESO algorithm was hindered by the oscillation between two separate solutions, as discussed in sub-section 2.2.2. TS attempts to avoid this by storing information about each solution, i.e. AA, AB and AC in a so-called Tabu list. Due to the scale of information involved and limits of memory available it is unfeasible to store each solution in its entirety, and therefore the “changes” from one solution to another is typically stored. By utilising a Tabu list moves to previous solutions can be prohibited. 3. Aspiration Criteria. The introduction of Tabu lists may therefore reduce or even remove the likelihood of oscillation between two solutions; it may however also prevent solutions which have not yet been explored. In extreme events this may lead to better local or even the global optimum solution being discarded from the search space. This may occur as it may not be possible to distinguish a previous solution from an “unexplored” one due to the fact that not all details of the individual solutions are stored in the Tabu lists. Therefore an “aspiration criteria” is introduced, a typical example of which is to stipulate that any new solution found must be better than the best solution found from the start of the algorithm, (Bianchi et. al., 2008).

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Based on the above, the introduction of TS into the BESO algorithm holds the potential to reduce some of the disadvantages of the algorithm, namely the dispersion of the search as well as the oscillation between competing solutions. 2.2.7

Simulated Annealing

Simulated Annealing (SA) is an optimisation method which has received a lot of recent attention and general development within a wide range of different optimisation applications, e.g. (Bureerat and Limtragoool, 2008), (Garcia-Lopez et. al., 2011) and (Barati, 2014). The principle of SA based optimisation was first presented by (Kirkpatrick et. al., 1983) in order to physically model the temperature controlled annealing process. In other words, the SA optimisation principle has its roots in statistical mechanics a commonality it shares with entropy, which shall be presented in section 2.3.3. The SA algorithm may be classed as a “random search method” (Garcia-Lopez et. al., 2011), indicating that the fundamental algorithm has similarities with e.g. ACO, SHC and TS as previously presented. Analogue to e.g. SHC SA commences with a “random solution”, i.e. an initial “guess” of a solution. SA then attempts to gradually obtain a more optimum solution by searching for alternative solutions within the neighbourhood of the current solution. This approach is equivalent to the controlled cooling required for the physical annealing process. The first step is also analogue to many other optimisation algorithms including SHC and TS. The method in which alternative solutions may be accepted as the “current” solution is however what sets SA apart from e.g. SHC and TS. There are two paths in which this can happen, (Garcia-Lopez et. al., 2011): 1. Improved fitness value. In this rule a new solution “B” is accepted to replace the current solution “A” if the objective function value of B is less than A (for a minimisation problem). 2. Reduced fitness value. In this rule a new solution “B” is accepted to replace the current solution “A” if the objective function value of B is larger than A (for a minimisation problem), given a certain probability. The rule of point 1 above is clear and intuitive, but if used in isolation may lead to premature convergence towards a local optimum as extensively discussed throughout this monologue. The rule in point 2 above is perhaps counterintuitive. Firstly it is important to recognise that this rule is introduced to avoid premature convergence as per point 1 above. The probability

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mentioned in point 2 above is where the direct link to the annealing process lies, as the probability P  t  is calculated using equation (2), (Garcia-Lopez et. al., 2011a ):

P  t   exp  T 1Ft 

(2.9)

In equation (2.9) Ft is the difference in objective function value between the two steps (A and B using the above notation).T is the current “temperature”, i.e. scaling parameter which is calculated as equation (3), (Garcia-Lopez et. al., 2011a):

T  t    jT0

0 (22 September 2014) Chuang, C.H., Yang, R.J. (2012a) “Benchmark of Topology Optimization Methods for Crashworthiness Design” (online) available from: page 3 (22 September 2014) Garcia-Lopez, M. et. al. (2011) “A hybrid topology optimization methodology combining simulated annealing and SIMP” (online) available from: (10 November 2014) Garcia-Lopez, M. et. al. (2011a) “A hybrid topology optimization methodology combining simulated annealing and SIMP” (online) available from: page 1514 (10 November 2014) Garcia-Lopez, M. et. al. (2013) “An improved robust topology optimization approach using multiobjective evolutionary algorithms” (online) available from: < http://www.sciencedirect.com/science/article/pii/S0045794913001569> page 1 (01 November 2014)

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Garcia-Lopez, M. et. al. (2013a) “An improved robust topology optimization approach using multiobjective evolutionary algorithms” (online) available from: < http://www.sciencedirect.com/science/article/pii/S0045794913001569> (01 November 2014) Hosny, M. I. (2010) page 16 “Investigating Heuristic and Meta-Heuristic Algorithms for Solving Pickup and Delivery Problems” (online) available from: (03 September 2014) Huang, X. and Xie Y.M. (2008) “Topology optimization of nonlinear structures under displacement loading” (online) available from: (02 May 2014) Huang, X. and Xie Y.M. (2008a) “Topology optimization of nonlinear structures under displacement loading” (online) available from: page 2059 (02 May 2014) Huang, X. and Xie Y.M. (2008b) “Topology optimization of nonlinear structures under displacement loading” (online) available from: page 2060 (02 May 2014) Huang, X. and Xie Y.M. (2008c) “Topology optimization of nonlinear structures under displacement loading” (online) available from: page 2061 (02 May 2014) Huang, X. and Xie Y.M. (2008d) “Topology optimization of nonlinear structures under displacement loading” (online) available from: page 2065 (02 May 2014) Huang, X. and Xie, Y. M. (2010) "A further review of ESO type methods for topology optimization", Structural and Multidisciplinary Optimization, volume 41, issue 5, pp. 671683 Jaynes, E.T. (1957) “Information Theory and Statistical Mechanics” (online) available from: (18 July 2014)

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Kennedy, J. and Eberhart, K. (1995) “Particle Swarm Optimization” (online) available from: (30 July 2014) Kennedy, J. and Eberhart, K. (1995a) “Particle Swarm Optimization” (online) available from: page 1943(30 July 2014) Kirkpatrick, S., Gelatt Jr., C. D., Vecchi, M. P. (1983) “Optimization by Simulated Annealing” (online) available from: (30 July 2014) Lin, C.Y., Lin, S.H. (2005) “Artificial neural network based hole image interpretation techniques For integrated topology and shape optimization” (online) available from: < http://www.sciencedirect.com/science/article/pii/S0045782504004657#> (23 October2014) LSTC (2011) “Topology and Shape Computations for LS-Dyna” (online) available from: < http://www.lsoptsupport.com/documents/manuals/ls-tasc/ls-tasc-2.0manual.pdf/view > (June 2014) LSTC, (2011a) “Topology and Shape Computations for LS-Dyna” (online) available from: < http://www.lsoptsupport.com/documents/manuals/ls-tasc/ls-tasc-2.0manual.pdf/view > page 11 (June 2014) Luh, G.C., Lin, C.I., Lin, Y. S. (2011) “A binary particle swarm optimization for continuum structural topology optimization” (online) available from: page 2 (12 June 2014) Schrijver, A. (2003) “Combinatorial Optimization” Springer: London. Singh, S. (2014) Re-design of car seat rails and its height elevation system. Unpublished MEng Thesis. Coventry: Coventry University.

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Sorensen, R. and Lund, E. (2012) “Ply-based Optimization of Laminated Composite Shell Structures under Manufacturing Constraints” [online] available from: [23 August 2014] Tovar, A. (2004) “Bone Remodelling as a Hybrid Cellular Automaton optimization Process ” (online) available from: (2 August 2014) Vazirani, V. (2001) “Approximation Algorithms” (online) available from: (23 August 2014)

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Appendix A MatLab MEP code: clear; clc; %Maximum entropy example - 1 constraint only. %Reset error warnings ERROR = 0; %Initial problem definitions: %The sum of probabilities must equal 1: Psumresidual = 1E-5; %Residual for final check %Design parameters: e.g. "individual" element density D(1) = 1; D(2) = 1; D(3) = 1; D(4) = 1; %Constraint 1: e.g. overall strain energy Con1 = 2.5; Con1residual = 2;%E-5; %Residual to determine ERROR termination or not. %Known quantities (e.g. from FEA) Q is a known quantity; e.g. the combined srain energy must equal 0.3 --> Q = 0.3; qA(i) is the quantity associated with the individual parameter / option; i.e.: Q = Con1; qD(1) = 1; qD(2) = 2; qD(3) = 3; qD(4) = 8; %Solution methodology: %Step 1: Setup and solve f_Beta = SUM((qAi-Q)*2^(-Beta(qAi-Q)))i = 0 --> maximum entropy S (by finding the Lagrangian multipliers Alpha and Beta): %Define initial Beta guess Beta=1;%0.25; %Newton-Raphson %Define maximum number of allowed iterations maxiter = 10000; %Define termination tolerance termtol = 0.000005; %NR outer loop: for j = 1:maxiter %Define/reset f_Beta and df_dBeta functions

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f_Beta = 0; df_dBeta = 0; for i = 1:4 f_Beta = f_Beta + (qD(i)-Q)*2^(-Beta*(qD(i)-Q)); end %Check for convergence if ((f_Beta)^2)^(1/2) = Psumresidual) str = ('ERROR: The sum of probabilities does not equal 1; the actual value is: ',num2str(Psum)) ERROR=1; end CCon1 = ((Con1sum-Con1)^2)^(1/2); if (CCon1 >= Con1residual) str = ('ERROR: The residual tolerance set for constraint 1 of ',num2str(Con1residual),' has been violated; the current value is: ',num2str(CCon1)) ERROR=1; end if (ERROR ==0) str=('Program successfully completed, the resulting values are:') str=('Alpha = ',num2str(Alpha)) str=('Beta = ',num2str(Beta)) str=('Entropy S = ',num2str(S)) end if (ERROR==1) str=('ERROR TERMINATION, please see ERROR message(s) above') end Pd % Write outcome to screen

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CONTRIBUTION

6

BEETS Optimisation software Completed December 2014 [The software can be accessed on the enclosed on the CD]

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