Numerical Methods for the Design of Meso-Structures

Proceedings of the ASME 2015 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2015 August 2-5, 2015, Boston, Massachusetts, USA

DETC2015-46289 NUMERICAL METHODS FOR THE DESIGN OF MESO-STRUCTURES: A COMPARATIVE REVIEW Marcus Yoder Research Assistant Mechanical Engineering Clemson University Clemson, SC 29634 [email protected]

Zachary Satterfield Research Assistant Mechanical Engineering Clemson University Clemson, SC 29634 [email protected]

Joshua D. Summers Professor Mechanical Engineering Clemson University Clemson, South Carolina 29634 [email protected]

Mohammad Fazelpour Research Assistant Mechanical Engineering Clemson University Clemson, SC 29634 [email protected]

Georges Fadel Professor Mechanical Engineering Clemson University Clemson, SC 29634 [email protected]

ABSTRACT Over the past decade, there has been an increase in the intentional design of meso-structured materials that are optimized to target desired material properties. This paper reviews and critically compares common numerical methodologies and optimization techniques used to design these meso-structures by analyzing the methods themselves and published applications and results. Most of the reviewed research targets mechanical material properties, including effective stiffness and crushing energy absorption. The numerical methodologies reviewed include topology and size/shape optimization methods such as homogenization, Solid Isotropic Material with Penalization, and level sets. The optimization techniques reviewed include genetic algorithms (GAs), particle swarm optimization (PSO), gradient based, and exhaustive search methods. The research reviewed shows notable patterns. The literature reveals a push to apply topology optimization in an ever-growing number of 3-dimensional applications. Additionally, researchers are beginning to apply topology optimization and size/shape optimization to multiphysics problems. The research also shows notable gaps. Although PSOs are comparable evolutionary algorithms to GAs, the use of GAs dominates over PSOs. These patterns and gaps, along with others, are discussed in terms of possible future research in the design of meso-structured materials.

1

INTRODUCTION

1.1 Motivation Meso-structured materials are a class of meta-materials characterized by a repeating or semi-repeating pattern of solid parent materials and empty space. When viewed on a macroscale, meso-structured materials behave as if they were solid materials with properties, such as density and Young’s moduli, determined by the geometry and arrangement of their unit cells. By tuning the geometry of the unit cells, combinations of effective properties can be achieved that are not found in homogenous materials. The importance of the meso-structures has come from their unique properties; they usually exhibit high strength, high flexibility at low relative density, and/or high conductivity which lead to the effective properties different from the constitutive materials of which they are made [1]. Their applications include light-weight structures with high strength, thermal insulations, packaging, energy and/or sound absorptions [1]–[3]. As an example, the non-pneumatic tire shown in Figure 1 has a meso-structured shear band sandwiched between two membranes [4]–[7]. The shear band is made from aluminum extruded into a honeycomb shape. The honeycomb has approximately the same shear modulus as polyurethane, while keeping the low-to-no damping characteristics of the aluminum material. As the honeycomb deforms and moves as a system, it serves as a replacement for a solid material.

Keywords: Meso-Structures, Numeric Design, Size, Shape, Topology Optimization

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smaller. For any particular problem, macro and meso-scale are not defined relative to an absolute size, but in relation to each other and to the scale of the problem. Typically meso-scale refers to the scale of a single unit cell. A meso-structure is a pattern of repeating or semi-repeating meso-scale unit cells made of a parent material. Macro-size and macro-shape refer to the size and shape of the space that the meso-structure exists in. At least one dimension of the macro-size is at least an order of magnitude larger than the dimensions of the meso-structure unit cells. The topology of a structure deals with the shapeconnectivity properties of the structure that are not affected by continuous deformations. Two unit cells have the same topology if one of them could be bent and stretched to become the other without cutting the edges, letting the edges cross or touch, or creating any new holes. Meso-size describes the size of a meso-structure unit cell. Meso-shape describes the different shapes that can exist inside the unit cell without changing topology or shape. Figure 1 shows an example of three configurations of a nonpneumatic tire with a shear bands made of different mesostructured materials. All three examples have the same macrosize and macro-shape. The regular hexagonal and auxetic hexagonal structures have different meso-sizes and shapes, but they have the same topology. The chiral structure has different meso-size, shape, and topology. Periodic Cellular Materials (PCM) are a class of mesostructured materials that are defined by a strictly repeated pattern in two or three dimensions. Honeycombs are a well-known example of a PCM. Meso-scale lattice structures (MSLS) are a meso-structured material with hundreds of meso-scaled trusses in a semirepeating pattern. Each truss can have a different size and orientation angle. This type of truss structure will behave similarly to a functionally graded solid material that fills the same space. Traditional truss structures have trusses on a macroscale, not a meso-scale, and do not fit this definition. Truss cored sandwich panels are PCMs instead of MSLSs because they have a strictly repeating pattern.

Figure 1: Example Use of Meso-structures in a NonPneumatic Tire In most cases, the effective properties and geometric parameters are fully coupled, that is, changing any geometric parameter will affect many effective properties. A common solution to this problem is to run an optimization routine to search for the best possible solution. Although optimizing mesostructures presents challenges for the designer, they are challenges that have been successfully overcome in the literature many times. This paper reviews this literature and presents a comprehensive comparison of optimization methods and applications in the design of meso-structures. The literature on meso-structure optimization covers a wide variety of physical goals, unit cells and optimization methods. Despite this diversity, there are common patterns linking the physical problem to the best method to use to optimize the mesostructure. Understanding these patterns can guide a designer to a better method to find an optimal design.

2

PROBLEM TYPE

2.1 Topology The topology of a 2D or 3D structure can be defined as the arrangement of material therein. A thorough review of mathematical formulations in the field of Topology Optimization (TO) are derived and applied to general and periodic topology problems in [8]–[10]. Two specific methods used to design topologies are discussed in detail in this paper: Homogenization Method (including several variants) and Level Set Method. The aim of this section is to briefly introduce these methods without detailing their mathematical formulations. However, sources are provided that derive the formulations.

1.2 Meso-Structure Terms and Definitions In the meso-structure design literature, terms are not always used consistently. This paper will use words as they are defined in this section. Macro properties are those exhibited by the overall structure. Meso refers to a scale much larger than the micro-scale but much smaller than the macro-scale. Typically it is on the order of 1-10 mm, but depending on the problem it could be larger or

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2.1.1 Homogenization Method Homogenization theory mathematically relates macro-level properties of a medium to its micro-level properties [11]. Homogenization theory has been adapted for use in TO by determining the effective, or bulk, material properties of heterogeneous media by enabling it to be solved using the finite element method [12]. The Homogenization Method (HM) uses homogenization theory linked with the finite element method and an optimization algorithm to determine the optimal topology of a structure. Homogenization theory was first applied to TO by [13]. A thorough mathematical formulation of homogenization theory and its application to TO is found in [14]–[16]. The HM can generally be described as optimizing the material distribution by adding or removing material in microscopic voids across the design domain. Optimization problems using HM begin by defining a design domain with a particular macro geometry, boundary conditions, homogenous material properties, and optimality target. The material is discretized into unit cells with microscopic voids in them, defined by the voids’ height, width, and rotation as shown in Figure 2. The dimensions of each void are independent variables and each unit cell can be completely solid, completely void, or anywhere in between. An optimization algorithm iteratively adds and removes material by changing the dimensions of the voids, and runs FEA to see how that affects the target. Eventually the optimization algorithm settles on an optimized solution [17].

Figure 3. Example TO Solution using HM [18] The HM in its original form will result in solutions with gradient materials. An example solution of the first formulation of the HM is shown in Figure 3 where black indicates solid material, white indicates void of material, and shades of grey indicate a gradient of material and voids. Because manufacturing constraints make material gradients difficult or impossible, variants on HM have been developed to eliminate these grey areas. Periodic boundary conditions allow for the optimization to converge on meso-structures. These boundary conditions have successfully been implemented in the reviewed papers discussed in Section 4 with an example of TO converging on an auxetic honeycomb [19] in Figure 4.

Figure 4. Periodic Structure from TO using HM [19] 2.1.2 Variants of the Homogenization Method The HM is a powerful TO tool that has laid the foundation for several variants. These variants are typically motivated to remedy some undesirable characteristic of the original method or its solution. Reasons for varying the original HM include limits on enforcing constraints, limits in the size of the repeated unit cell compared to the design domain, the computational cost, and manufacturability of the solution. Two such variants will be introduced in this section. The Solid Isotropic Material with Penalization model (SIMP) is a widely-used variant of the HM. This variant introduces a continuous density variable in the mathematical formulation that is exponentially raised to a user-defined term “𝑝.” If chosen correctly, this term will cause any element density between 0 and 1 to be effectively penalized, resulting in a topology with no gradient material (see [20] and sources therein for details). The primary purpose of the SIMP method is to eliminate topologies that contain gradients with microscopic voids in an attempt to make them manufacturable. Another variant of the HM is the Volume Averaging method. The original HM is mathematically formulated on the

Figure 2. Unit Cell Variables for Square Void using HM in TO [17] There are three generic problem formulations that the HM can be applied to. The first formulation, stated previously, optimizes the material distribution between a single material and voids. The second formulation optimizes the material distribution of two or more distinct materials. The third formulation combines the first two as optimizing two or more distinct materials and voids. All three formulations follow the same underlying procedures, stated previously, to achieve a distribution of the distinct materials and/or voids.

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assumption that the unit cell is microscopic in comparison to the overall domain. This presents an upper limit on the size of periodic unit cells. However, Volume Averaging uses a representative volume element to replace a periodic unit cell. This volume element is defined by requiring the strain energy of the system to be equivalent regardless of an applied traction or displacement boundary condition. The use of representative volume elements enables the required number of periodic unit cells to decrease, or become larger, in comparison to the overall domain (see [19] and sources therein for details).

All of the research reviewed by this paper optimized size and shape at the same time, therefore this review paper will only discuss size and shape together. An example of size optimization of meso-structures is to find an optimized honeycomb by varying the angle and thickness to maximize specific energy absorption under in-plane crushing [2]. The optimized honeycomb is shown in Figure 5

2.1.3 Level-Set Method Level Set Method (LSM) is a TO method that is derived from a mathematical tool, of the same name, for numerically tracking fronts and free boundaries. A zero level set of a function is the set of all points that will give a value of zero when they are input into the function. For TO, a level set will form a number of closed loops for a 2-dimensional problem or closed surfaces for a 3-dimensional problem [21]. LSM as applied to TO creates a function on the problem space. The zero level set of this function is the boundary between solid material and empty space. With each iteration, an FEA analysis finds the sensitivity to removing or adding any element of material, by using a shape derivative, and uses this to change the function. As the function moves the zero level set moves with it [21], [22]. An early paper on LSM claimed it was about as computationally expensive as the SIMP method when the SIMP method is available and that it produced similar results [23]. More recent work has shown that LSM can have significant problems with convergence [24]. The Level Set Method was original explored by [25] and applied to TO by [26]. An easy to follow, excellent, simple working example of LSM is presented in [22].

Figure 5: Hexagonal honeycomb before (left) and after (right) optimization for maximum energy absorption [2] 3

OPTIMIZATION ALGORITHMS

3.1 Gradient Based Methods (GBM) Gradient Based Methods (GBM) form a class of optimization algorithms which finds the gradient of the objective function at the current point and then moves along the gradient direction or along some direction estimated using gradient information. A function increases at its fastest pace along the gradient direction [29]. The gradient based methods include sequential linear/quadratic programming [30], quasi-Newton [31], and feasible directions [32]. 3.2 Exhaustive Search (ExS) An Exhaustive Search (ExS) is the most straightforward search method available. It works by filling the entire solution space with solutions, evaluating all of them and choosing the best one. It should be noted that ExS is not a numerical method to intelligently optimize parameters of interest but instead a method to more fully search all possible combinations within the design domain. In fact, this method is so simple that the authors reviewed by this paper who used ExS ([33]–[36]) did not give the method a name. Thus, ExS is presented in this paper as a rudimentary method for comparison to the algorithms that converge on an optimal solution in some intelligent manner. If the solution space is large, the objective function is too sensitive to the input variables, or if a single evaluation of the function is computationally expensive, the ExS method can quickly become prohibitively expensive. The authors who used this method all solved problems where analytical formulas were available, or where the FEA was simple enough to be run in seconds.

2.1.4 Discrete Topology Optimization (DTO) Discrete Topology Optimization (DTO) searches for an optimal set of trusses in a pre-defined ground base network [27]. Each individual truss is assigned a different thickness. With each iterative step, trusses are removed in areas with low stresses and the thickness of the remaining trusses are determined through optimization algorithms. Reviews on DTO can be found in [27], [28]. DTO problems usually have been solved through mathematical programming approach such as method of modified feasible directions and sequential quadratic programming [27]. 2.2 Size and Shape Size and shape optimization (SSO) searches for the optimal dimensions, orientation, and/or curvature of the unit cell without changing the unit cell’s topology. The topology and variables to be optimized are selected by the designer. SSO iterates an optimization algorithm coupled with an analytical tool, such as finite element analysis, to calculate the objective function until it finds an optimal solution.

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3.3 Genetic Algorithm (GA) Genetic Algorithms (GA) are a solution method that was inspired by natural selection in biology. An initial population of candidate solutions is created randomly in the search space. The solutions are encoded with genes that describe the characteristics of the solution. Each solution is evaluated according to the fitness function that is to be optimized. A fraction of the solutions that have the highest fitness are kept and the rest are discarded. Two main methods are used to create each generation of solutions. The first, typically more common, method is to create one or more “child” solution(s) by randomly selecting the genes of two or more “parent” solutions. The second, typically less common, method is to randomly mutate the genes of a single parent solution to create a child solution. The process of generation, evaluation and selection is repeated until an end criterion has been reached [29], [37].

the objective function or constraint functions. It works by generating random points throughout the design domain, evaluating them with the function, and fitting a hyper-surface to the data points. Using more data points generally increases the accuracy of the response surface. Once the response surface has been generated, the optimum point can be found using a wide variety of search algorithms [40], [41]. Response surfaces combine well with many different types of search algorithms. It is common practice for GAs and PSOs to be combined with response surfaces since both require a large number of objective function evaluations. Response surfaces can always estimate those function evaluations in a fraction of the time that would be required to run a full analysis. However, RSM can only be used when the number of design variables are manageable. Thus, using RSM with methods containing a large number of design variables such as TO (thousands to millions of design variables) is infeasible.

3.4 Particle Swarm Optimization (PSO) Particle Swarm Optimization is a solution method that was inspired by the natural phenomena such as the movement of a school of fish or flock of birds. An initial population of candidate solutions is created randomly in the search space. Each solution changes as it moves through the design space with a random velocity. At regular time intervals, each solution is evaluated, and the results are stored. Each solution remembers the location of the point that it has individually been at and the location of the best point that the entire population has been at. The solution accelerates towards both points until an end criterion has been reached [29], [37].

3.7 Neural Network (NN) A Neural Network (NN) is a tool to estimate an objective or constraint function. A NN consists of a parallel interconnected network of processors (neurons) [29]. The neurons are organized into an input layer, an output layer, and one or more “hidden” layers which lie in between the input and output. The neurons in the input layer take the input variables, perform calculations on them according to a preprogrammed activation function and passes the results to the first hidden layer. Those nodes do further calculations and pass the results to the next layer. This repeats until the signal reaches the output layer. Each connection between neurons has a pre-determined weight that scales the signal as it passes [29]. For a NN to give accurate results, it must be “trained,” using an optimization algorithm such as steepest descent method. The optimization algorithm searches for the minimum mean square error between the calculated and desired outputs by changing the weights within the network [29].

3.5 Physical Heuristic Method (PHM) Physical Heuristic Methods (PHM) are heuristics for size and shape optimization with a rule derived from a physical understanding of the problem rather than a mathematical understanding. Heuristic methods for topology optimization are not included in this definition. Genetic algorithms, particle swarm optimization and other search methods are based on mathematical heuristics and therefore not included in this definition. Every PHM is different. The authors reviewed by this paper who used heuristic methods ([38], [39]) each used a different heuristic that they had created with a particular problem in mind. Although each method was generalizable to a certain set of related problems, it is not likely that for any given meso-structure problem there will be an applicable heuristic method described in the literature. The PHM papers showed that their methods, found solutions close to the global optimum, were far more efficient for their cases than the traditional alternative searches, and that the advantage in efficiency increased as the problem grew more complex.

4

APPLICATIONS

4.1 Topology Optimization Table 1 shows 27 papers that used TO to design mesostructures. They are organized by problem type and optimization algorithm. The HM, or some variant, is used in 77% of these papers. The majority of applications use a variant of the HM since they expand the limits and usefulness of the original HM, as discussed in Section 2.1.2. Level Set and other methods account for the method used of the remaining 23% of reviewed papers in TO. The most commonly used type of algorithm in TO was GBM, accounting for 27% of applications. GA’s and MMA both accounted for 15% each with Simplex accounting for 4%. 38% of applications did not state the algorithm used. GBM, more specifically SLP, was probably used most since it was applied by early researchers in TO that developed the HM, such as in [42]– [44]. Their methods were adapted to new problems still

3.6 Response Surface Method (RSM) The Response Surface Method (RSM) is a mathematical tool to estimate the relationship between the design variables and

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Table 1. Applications of TO: Method and Algorithm Breakdown Homogenization Original

SIMP

Other

Volume Averaging

GA GA+NN

[47], [48]

Other Penalized

Level Set

[53]

[23]

Other Method

[47], [48]

PSO SLP

[43], [44], [51], [57]

[42]

[45], [46], [51], [54]

Simplex

[55]

RSM MMA

[52]

[50], [56]

[49], [50]

[58]

[59]

[60]–[62]

BF Unknown

[63]

Not Required

[21], [64]–[66]

stiffness with thermal heat transfer coefficient (1 demonstration study).

Property Optimized

implementing SLP by [45], [46]. Note that PSOs, RSMs, and ExSs were not used in any TO application in the reviewed research. It is far too computationally expensive to use ExS for a TO application due to the sheer size of the design domain. The RSM could be used in parallel with another algorithm since it is an approximation method as NNs were coupled with GAs (see [47], [48]). PSOs are comparable evolutionary heuristic algorithms to GAs in both flexibility and cost [37], so it is unclear why none of the reviewed TO applications used one. A majority (52%) of the reviewed TO papers were not motivated by an application but rather an extension in TO theory. Topologies for hydrophones [43], [46] and permeable materials used in fluid transport [49], [50] are both optimized in two papers each with the remaining papers having unique applications. These unique applications include materials with extreme thermal expansion [45], cathodes in batteries [51], piezoelectric ultrasonic transducers [43], robust properties for manufactured materials [52], fiber reinforced polymeric bridge decks [53], shear layers of a non-pneumatic wheel [47], multiphase composites with extreme shear moduli [54], microelectromechanical systems [55], and materials with negative acoustic permeability [56]. The properties that were optimized vary widely across 6 general fields of study: 65% were mechanical in nature, 15% piezoelectric or electrical, 12% were thermal, and the remaining 6% were fluid, and 3% were acoustic in nature. This is depicted in Figure 6 where this information is further divided into singleand multi-physics optimizations which account for 71% and 29% of case studies, respectively. The most common multiphysics optimization combined mechanical and fluid properties with 2 demonstration studies examining the optimal properties of elastic stiffness and permeability. Other multi-physics optimizations included electrical and thermal conductivity (1 case study), and mechanical elastic stiffness with electric conductivity (1 demonstration study), and mechanical elastic

Single Physics Optimization Acoustic

Multi Physics Optimization

Fluids Thermal Piezo/Electrical Mechanical 0

5

10

15

Case Studies

20

25

Figure 6. Optimized Property Classification Breakdown One limitation to performing TO on a multi-physics problem is the complexity in setting up the governing equations. The reviewed papers that accomplished this went into great detail into their model setup, see [49]–[51], [60], [67] for several approaches. The reviewed case studies were also examined in terms of the number of properties that were simultaneously optimized. Single-objective case studies optimizing 1 property account for 59%, and multi-objective case studies optimizing 2 and 3 properties account for 31% and 10%, respectively. Singleobjective optimizations are the simplest to set up while multiobjective optimizations can range from simple to very complex depending on the number of objectives and the number of physical phenomena being captured. Though less computationally demanding, TO in 2dimensions accounted for 53% of all TO case studies examined. Thus, TO is useful and being applied in applications in 2- and 3dimensions. In fact, several reviewed papers tested algorithms in 2-dimensions first before applying in the desired 3-dimensions (see [23], [44], [49], [54], [59], [60], [63], [64], [66]).

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surface was fit to this data and genetic algorithms were used to search it. Heat transfer combined with stiff mechanical properties was an optimization target of two papers [34], [80]. Both papers were designing heat exchangers that would also serve as structural components. The paper that ran the longest running simulation, 30 hours, also had the most detailed description of their optimization method. This analysis studied the crushing energy absorption of honeycomb structures in 3-dimensions. The authors only had time to run 9 full simulations. They went to considerable effort to construct and verify a multi-fidelity response surface model, which combined the results of 9 fully detailed simulations with those from a response surface made with less detailed simulations. This allowed them to find a better optimum result with less computer time, than if they had simply made a response surface from 10 full simulations [74]. The strongest pattern seen in the SSO literature reviewed is that the longer a single analysis takes to run, the fewer total analyses are run. When analyses take a long time, the researchers spend considerable effort to use a method that does not require many analyses. When analyses can be run in only a few seconds, the authors select methods that are either easy to implement, such as ExS, or robust in finding solutions, such as GAs. The authors do not typically present much justification for their choice of optimization method. It is possible that many authors chose methods that were less efficient than others they could have used. The physical phenomenon being studied is a strong predictor of how long an analysis will take to run. The dynamic simulations required to study crushing energy absorption are time consuming, especially if they have to be run in fullydetailed 3-dimensions. By contrast, the simulations required to predict static mechanical properties are relatively simple. Analytical formulas are sometimes available. Even if they are not, the FEA analyses can often be run quickly. Other considerations that go into the simulation time, such as required level of detail, ability to employ symmetry and/or reduce the problem to two dimensions, appear to be of lesser importance.

Resolution, filtering, and length scale control are very important aspects of TO, especially in the convergence to mesostructures. If the resolution is not high enough or if filtering is not done properly for a given application, important physical behaviors may go unexplored and sub-optimal solutions will be obtained. A complete discussion on these topics is out of the scope of this paper. 4.2 Size and Shape Optimization (SSO) Table 2 organizes the 25 papers that used SSO in the reviewed literature. If a single paper used two methods, the primary method used is the one referenced in the table. Objectives related to mechanical properties of a static structure accounted for 16 papers (64%) of SSO literature. FEA analysis of static structures is relatively computationally inexpensive. This allows the analysis to be repetitively run in search of the best possible solution. The optimization methods that require the most simulations, PSOs/GAs (without RSM or NN) and ExSs, were only used for static mechanical properties. Of the papers optimizing for static properties, high stiffness (or low deflection) was the most common target with 7 papers or 44% [34]–[36], [38], [39], [53]. The second most common target was high strength with 4 papers or 25% [33], [68]–[70]. Other papers optimized for targeted stiffness against specific loads [39], [71], [72], combined stiffness and heat transfer [34], [52], combined high shear strength and high yield shear strain [33], and low rolling resistance of a non-pneumatic tire [3]. Optimization of crushing energy absorption accounted for 4 papers or 16% of SSO literature reviewed. Crushing simulations are typically very computationally expensive. This makes running a large number of simulations completely infeasible. All 4 of these papers used response surfaces to reduce the number of required simulations. Three papers or 12% of this literature, studied acoustic insulation. These FEAs take a moderate amount of time to run. One researcher [73] reported that he ran the simulations in 10 minutes on his computer. This researcher ran 274 simulations, a moderate number compared to other papers. Then, a response

Table 2: Uses of Size and Shape Optimization Static Mechanical Properties PCM GAs

[53], [71]

GAs + RSM/NN

[3]

Other

Crushing Energy Absorption

MSLS

PCM

[72]

[2]

MSLS

PCM

MSLS

[73]

PSO PSO + RSM/NN

[74], [75]

RSM

[76]

GBM

[30], [70], [77]

[31], [78]

ExS

[33]–[35], [68], [69]

[34], [79]

PHM

[32]

[38], [39]

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REFERENCES

For these reasons, there is a strong connection between the type of problem being optimized and the best method to optimize that problem. 5

CONCLUSIONS This paper was a literature review of the computational design of meso-structures. Key terms and definitions were presented along with brief explanations on TO methods, SSO methods, and optimization algorithms used in this field. Relevant literature for TO and SSO were critically compared. The review of papers in TO revealed several interesting findings regarding optimization algorithms. PSOs are currently not being used as of the writing of this paper and implementation would be a useful comparison for future TO applications. Closely related, RSMs are not used as approximation tools in TO algorithms. Though the implementation of RSMs may be rigorous in the mathematical formulation for TO, it also has the potential to have a positive impact on computational expense. The review also showed promise in expanding the usefulness of TO as 54% of papers focused on expanding the capabilities of the fundamental theory. Researchers are attempting to capture more complex topologies with 47% of TO case studies using 3-dimensional models. Researchers are also making TO more applicable to a growing number of problems as 31% of TO case studies involved multi-physics applications. The future of topology optimized meso-structures promises to continue on this trend of increasing usefulness as more case studies push the limits of its mathematical theory and robustness. By comparison, the review of papers in SSO did not expand the mathematical theory, as it is already well established. Instead, they focused more on the selection of optimization algorithms. Analysis showed that the physical nature of the problem is indirectly linked to the method that will be used. The physical nature of the problem strongly predicts how long it will take to run a single simulation and whether analytical formulas can be used. The time required to run a single simulation or analysis is a strong predictor of the optimization algorithm that will be used. Other factors that influence the simulation time were not significantly addressed by many of the reviewed papers. When a single simulation takes a long time to run, a method will be chosen that requires few simulations. The reverse is true to a lesser extent. With 80% of SSO papers dealing with either static or dynamic mechanical properties, there is an opportunity to explore meso-structured optimization in other fields, such as acoustic, thermal, or electrical applications. ACKNOWLEDGEMENTS The authors would like to recognize the contributions in review and discussion on the paper by Professor Lonny Thompson of Clemson University. This research was supported, in part, by SC Space Grant and the Automotive Research Center. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the sponsors.

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