Finite Element Model Identification for Structure ...

Technical Report

Substructure Analysis Framework for Accelerated Finite Element Modeling

By Kai Zhou, Ph.D, Research Intern Majid Cashany, M.Sc., Research Intern Zheng Yi Wu, Ph.D, Bentley Fellow Applied Research, Bentley Systems, Incorporated Watertown, CT 06795, USA

Note: This report summarizes the research project undertaken by kai Zhu and Majid Cashany under

the supervision of Zheng Yi Wu at Bentley, Systems, Incorporated, Watertown, CT office from 2014 to 2015. The work can be cited as follows. Zhou, K., Cashany, M. and Wu, Z. Y. (2015) “Substructure Analysis Framework for Accelerated Finite Element Modeling”, Technical Report, Bentley Systems, Incorporated, Applied Research Group, Watertown, CT, USA.

Substructure Analysis Framework for Accelerated Finite Element Modeling

Executive Summary Finite Element (FE) model has been widely employed by structural engineers to analyze a structural system for achieving a good design and also to simulate an in-service or as-built structure for performance assessment. Both design analysis and as-built simulation require for iterative executions of a FE model. For instance, design optimization and model calibration involve tens or hundreds of thousands of FE model runs. Therefore, it is essential that an efficient FE analysis method is developed and adopted for those applications that iteratively call on FE solvers. Substructure analysis, especially component modal synthesis (CMS), has been well established method for accelerated FE modeling. Some FE analysis software packages, such as NASA STRucture ANalysis or NASTRAN based FE products, have provided the substructure analysis function to accelerate FE model run. This report sums up our research endeavor in developing a CMS-based FE analysis framework. CMS approach is to solve each component or substructure of the whole structure, perform the order (degree of freedom – DOF) reduction, and finally assemble and solve the orderreduced structure system. The method has been implemented as a class library with OpenSTAAD and can be easily adapted for other structure analysis packages. Three examples, including a hypothetical truss structure model, the 3D FE model for an approach span of Verrazano-Narrows (VZN) Bridge in New York City and the 3D FE model for Devon Railroad Bridge in Milford Connecticut, have been tested with the developed CMS framework. The analysis results by substructure analysis are compared with those obtained with the original whole structure analysis. The comparison shows that the CMS method is able to achieve the consistently accurate results for both frequencies and modal shapes, and significantly accelerate the FE analysis. It has been demonstrated that the developed substructure analysis is 3 times, 22 times and 26 times faster respectively for three examples than the original whole structure analysis. This indicates that the greater speedup can be achieved with the CMS analysis for the greater size (more elements) of structural model. It is believed that the applications with iterative FE model runs will be dramatically accelerated by embedding the developed substructure analysis framework.

Acknowledgements This research project has been solely sponsored by Bentley Systems, Incorporated while the research interns were employed by Bentley. The research team is greatly indebted to the Bentley’s sponsorship, and the technical supports by Bentley colleagues including Sudip Chakraborty, Sumanta Chakraborty, Bulent Alemdar and others from structural group for providing OpenSTAAD libraries and their insightful suggestions. 2

Substructure Analysis Framework for Accelerated Finite Element Modeling

Table of Contents Executive Summary ................................................................................................................................................................ 2 Acknowledgements ................................................................................................................................................................ 2 List of Figures ............................................................................................................................................................................ 4 List of Tables.............................................................................................................................................................................. 5 1 Introduction ........................................................................................................................................................................... 6 1.1.

Brief Review ........................................................................................................................................................... 7

2 Substructure Analysis Formulation ............................................................................................................................. 8 2.1

Overview of CMS methods................................................................................................................................ 8

2.2

Fixed Interface Method ...................................................................................................................................... 8

2.3

Free Interface Method ...................................................................................................................................... 10

3 Implementation .................................................................................................................................................................. 14 3.1

Overall Architecture .......................................................................................................................................... 14

3.2

Input and Output ................................................................................................................................................ 16

3.3

Fixed Interface ..................................................................................................................................................... 17

3.4

Free Interface ....................................................................................................................................................... 18

3.5

Exported APIs ...................................................................................................................................................... 19

3.6

Use APIs.................................................................................................................................................................. 20

4 Validation .............................................................................................................................................................................. 23 4.1

Evaluation Criteria ............................................................................................................................................. 23

4.2

2D Truss ................................................................................................................................................................. 24

4.3

Verrazano-Narrows (VZN) Bridge Approach Span .............................................................................. 28

4.4

Devon Railway Bridge ...................................................................................................................................... 31

5 Conclusions and Recommendations .......................................................................................................................... 35 6 References ............................................................................................................................................................................ 35 Appendix I: Simplification of Devon Railway Bridge Finite Element Model ................................................. 37 Appendix II: Eigen Analysis in Free Interface Method ........................................................................................... 40 Appendix III: Inversion of Badly-conditioned Matrices in Free Interface Method .................................... 41

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Substructure Analysis Framework for Accelerated Finite Element Modeling

List of Figures Figure 1 Overview of general architecture of CMS implementation ................................................................ 15 Figure 2 Flow chart of the fixed interface method .................................................................................................. 18 Figure 3 Flow chart of free interface method ............................................................................................................ 19 Figure 4 Sample output files of CMS run...................................................................................................................... 20 Figure 5 GUI developed to validate and test the API of CMS DLL ...................................................................... 23 Figure 6 CMS evaluation criteria ..................................................................................................................................... 23 Figure 7 2D Truss with node labels ............................................................................................................................... 24 Figure 8 Graphical demonstration of sub-structure divisions of 2D Truss ................................................... 25 Figure 9 Substructure and interface data for 2D truss structure example.................................................... 26 Figure 10 Efficiency comparison of 2D truss: CPU clock time and speedup ................................................. 28 Figure 11 VZN Bridge approach span: connection to pier EB-14 of VZN Bridge ........................................ 28 Figure 12 Six substructures of VZN Bridge approach span ................................................................................. 29 Figure 13 VZN Bridge approach span model with interfaces at boundaries of diagonal bracings ..... 29 Figure 14 Efficiency comparison of VZN Bridge approach span: CPU clock time and speedup ........... 31 Figure 15 Devon Railway Bridge: view from inside (a), view from downside (b) ..................................... 32 Figure 16 Finite element model of Devon Railway Bridge and its substructure divisions..................... 32 Figure 17 Efficiency comparison of Devon Railway Bridge: CPU clock time and speedup .................... 34 Figure 18 Original finite element model of Devon Railway Bridge (a) and simplified model (b) ....... 37 Figure 19 Modes with largest participation factor have the same frequency .............................................. 38 Figure 20 Major steps of simplification of Devon Railway Bridge finite element model ........................ 39

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Substructure Analysis Framework for Accelerated Finite Element Modeling

List of Tables Table 1 Substructure and interface definition .......................................................................................................... 16 Table 2 Substructure divisions of 2D Truss ............................................................................................................... 24 Table 3 2D Truss: accuracy of frequencies ................................................................................................................. 27 Table 4 2D Truss: accuracy of modal shapes ............................................................................................................. 27 Table 5 VZN Bridge approach span: accuracy of frequencies ............................................................................. 30 Table 6 VZN Bridge approach span: accuracy of modal shapes ......................................................................... 30 Table 7 Devon Railway Bridge: accuracy of frequencies ...................................................................................... 33 Table 8 Devon Railway Bridge: accuracy of modal shapes .................................................................................. 33

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Substructure Analysis Framework for Accelerated Finite Element Modeling

1 Introduction Infrastructure asset is the foundation of human society, and essential for sustaining the prosperity and civilization around the world. Infrastructure assets, e.g. buildings, bridges, dams, water systems, etc., need to be cost-effectively designed, well-constructed, proactively maintained and adequately upgraded, as needed, for meeting the requirement of increased capacity, preventing from or minimizing the damage of infrastructural deterioration. To design a large structural system requires engineers to undertake comprehensive analysis via computer model such as finite element model. A good design solution usually arrives only after conducting many runs of the model to evaluate the design alternatives. The more design alternatives can be analyzed and evaluated, the better solution can be achieved. It is important that the finite element model can run as efficiently as possible with adequately accurate results so that design engineers can attain a cost-effective design. To maintain a large civil structural system requires engineers to assess the infrastructure performance that is quantified by a set of indicators predefined for a particular type of infrastructure assets. Achieving a sound evaluation of a given infrastructure asset, a computer model is usually constructed to simulate an as-built or in-service infrastructure system. Such a simulation model is often different from a design model, which is constructed and adopted for design analysis to check if a design meets the code requirements for safety, functionality and serviceability. A design model is built with assumptions of perfect material attributes (as new), and exact geometric dimensions of the elements. A simulation model needs to adequately emulate the existing conditions of an in-service infrastructure system, e.g. bridge, building, drinking water system etc., which may very well be in imperfect conditions, including likely material deterioration, capacity loss, corrosions, damages etc. Therefore, a simulation model must be calibrated so that it can sufficiently represent the conditions of the in-service infrastructure. Model calibration is accomplished by manually or automatically undertaking numerous model runs, each of which is an updated model with the adjusted model parameters. The faster a model can run, the more efficiently a model can be calibrated. As elaborated above, designing and maintaining large civil structural systems involve iterative execution of a finite element (FE) model. The computational efficiency is the key for engineers to obtain good solutions. For instance, if one finite element model run takes 10 seconds, 100,000 runs of such a model would take about 35 days. Although cloud computing seems to offer the affordable and unlimited computation resources, it is imperative to accelerate every single analysis. Therefore, it is desirable and important that a new analysis 6

Substructure Analysis Framework for Accelerated Finite Element Modeling

framework, such as substructure finite element analysis, can be developed to speed up each and every FE model run. 1.1.

Brief Review

Finite element model enables engineers to analyze the dynamic behavior of structures in a numerical manner instead of bulky experimental instruments (Ern and Guermond, 2004). However, for large-scale structures which oftentimes need to be characterized by high mesh density models, the computational cost, e.g., eigen-analysis, is quite prohibitive (Masson et al, 2006) for efficiently embedding a FE model into an iterative process, such as design optimization and model calibration, to achieve satisfactory solution in a rapid manner. In practice, the structures are inevitably subject to uncertainty or the properties of structures always require to be iteratively calibrated. It makes abovementioned issue more challenging (Mares et al, 2006; Zhou et al, 2013). There have been continuing efforts on improving the computational efficiency in general finite element analysis with large number of degree of freedoms (DOFs). Among the approaches developed, component mode synthesis (CMS) based order-reduction approach is a well-known technique (Castanier et al, 1997; D’Souza and Saito 2012). The pronounced advantage of this technique is its capability in efficient response evaluation based on orderreduced model. The underlying idea and analysis procedures of CMS will be detailed in next subsection. CMS essentially is one category of dynamic condensation/order-reduction approaches. Many mechanical or civil structures can be deemed as an assembly of smaller substructures or components. The underlying idea of CMS is to analyze a number of substructures collectively rather than analyzing the structure as a whole. From eigenvalue problem perspective, this scheme will lead to a substantial reduction in computational cost. For example, we have a structure that can be decomposed into N substructures, in which i-th substructure has ni DOFs. Solving the eigenvalue problem of the whole structure yields the N

computational complexity of O(( ni )3 ) . Under CMS framework, smaller-size eigenvalue i 1

problems are computed first for all substructures, with a significantly reduced N

computational complexity of O( ni 3 ) (Xia and Tang, 2010). The typical procedures of CMS i 1

includes four basic steps: Step 1. Step 2. Step 3. Step 4.

Decomposition of a whole structure into substructures, Modal analysis of individual structures, Selection of kept component modes for constructing transformation basis Assembly of substructures into an order-reduced model (Xia and Tang, 2011). 7

Substructure Analysis Framework for Accelerated Finite Element Modeling

There are a number of variants in the general category of CMS methods, primarily depending on the choice of substructure-to-substructure boundary conditions utilized when the substructure-level modal analysis is performed. They basically include fixed interface (Rixen, 2004; Craig and Kurdila, 2006), free interface (Craig and Chang, 1976; Zhou and Tang, 2012), hybrid interface (MacNeal, 1971; Soucy and Humar, 2014) CMS methods. The merit of free interface CMS over the fixed interface lies in its easier implementation of experiments and fewer DOFs that participate in the synthesis of order-reduced model. In last several decades, CMS methods have become a standard practice in structural dynamic analysis for a variety of research objectives. Tran (2001) developed an algorithm for reduction of CMS interface coordinates, which can be further extensively applied into free or hybrid interface methods. Its effectiveness has been validated through a case demonstration of cyclic structures. Tournour et al (2001) investigated the convergence behavior of free interface CMS method with respect to the number of retained substructure modes, and verified its performance using several design optimization examples. More recently, Papadimitriou and Papadioti (2013), Liu et al (2013) integrated CMS methods into model updating or calibration framework for the purpose of structural health monitoring (SHM) of real bridge systems.

2 Substructure Analysis Formulation 2.1 Overview of CMS methods As mentioned in Section 1, there have been several classifications for current CMS methods in term of interface assumption among adjacent substructures. As hybrid interface method generally is an optimal combination of fixed and free interface methods regarding how to deal with interface DOFs, without loss of generality here we only introduce the detailed formulation of fixed and free interface methods. 2.2 Fixed Interface Method The fixed interface method is firstly developed by Hurty (1965) and refined by Craig and Bampton (1968, 1976) that requires all of the original interface displacement coordinates be retained in the final coupled structure. The mathematical derivation will be detailed below. The equation of motion for undamped free vibration of a typical substructure is written as Mx  Kx  f

(1)

where M and K are the mass and stiffness matrices of certain substructure that can be obtained from finite element assembly, f is the forced vector caused by adjacent 8

Substructure Analysis Framework for Accelerated Finite Element Modeling

substructures. The generalized coordinates of such substructure generally consists of interface and interior coordinates, converting the Eq. (1) into the form of  Mii M  ji

Mij    xi    K ii    M jj   x j   K ji

K ij    xi    0      K jj   x f  j    j 

(2)

where subscript i and j indicate the interior and interface coordinates. Since free vibration analysis is only considered, the force at interior coordinates is zero. The normal modes Φii that can be obtained by imposing x j  0 in Eq. (2) satisfy the ortho-normalization conditions below, ΦTii Mii Φii  I

(3a, b)

ΦTii K ii Φii  Λii

where I is an identity matrix, and Λii is diagonal eigenvalue matrix. We further neglect the inertia effect in approximating the dynamic displacement (Eq. (4a)), and apply the Guyan reduction (Panayirci et al, 2011), yielding Eq. (4b)  K ii K  ji

K ij   xi   0     K jj  x j  f j 

(4a, b)

xi  (K ii1K ij )x j

It is worth nothing here K ii1 exists as K ii is a definite symmetric matrix due to the constraints on interface DOFs. Here we consider Φij  (K ii1K ij ) and then construct transformation basis in combination with previously obtained Φii , shown as Φ Φ   ii  0

Φij  I 

(5)

Let x  Φp . Substituting it into Eq. (2) and pre-multiplying ΦT at the both sides of Eq. (2), resulting in Eq. (6) in modal space,  Mii  M ji

Mij    pi    K ii    M jj   p j   K ji

K ij    pi    0       K jj   p j    f j  

(6)

where Mii  I , Mij  MTji  ΦTii (Mii Φij  Mij ) , M jj  Mij  ΦTij (Mii Φij  Mij )  M ji Φij K ii  Λii , K ij  KTji  0 , K jj  K jj  K jiΦij , f j  Φij f j

It is worth mentioning that in this method the interface is assumed to be completely fixed, such that the interface force cannot influence the dynamic response at interior coordinates. For the sake of simplicity, we can consider no interface force for the free vibration scenario. 9

Substructure Analysis Framework for Accelerated Finite Element Modeling

Based on Eq. (6), the order-reduction of this substructure can be subsequently implemented. The detailed procedures are summarized as follows: Sort the eigenvalues in matrix Λii (assume its dimension is s) in ascend order; keep the lowest r nominal modes (r