ARL-RP-0533 ● AUG 2015

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Multiobjective Topology Optimization of Energy Absorbing Materials by Raymond A Wildman and George A Gazonas A reprint from Struct Multidisc Optim. (2015) 51:125–143.

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ARL-RP-0533 ● AUG 2015

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Multiobjective Topology Optimization of Energy Absorbing Materials by Raymond A Wildman and George A Gazonas Weapons and Materials Research Directorate, ARL A reprint Struct Multidisc Optim. (2015) 51:125–143.

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Approved for public release; distribution unlimited. 13. SUPPLEMENTARY NOTES A reprint Struct Multidisc Optim. (2015) 51:125–143.

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A method for the multiobjective optimization of local-scale material topology is presented. The topology optimization scheme is based on a constructive solid geometry-like representation, in which convex polygons---defined as the convex hull of arbitrary-length lists of points---are combined using an overlapping function. This data structure is tree-shaped and so genetic programming is used as the optimizer. The forward problem is solved with a multiscale finite element method with automatic cohesive zone insertion to model damage. As a multiscale method, loads and boundary conditions are applied and objective functions measured at a global scale, while the local scale material structure is optimized. The global scale geometry is assumed fixed. Pareto optimal designs are generated, representing optimal tradeoffs between conflicting goals: quasi-static displacement and dynamic strain energy. Results demonstrate the efficacy of the proposed algorithm.

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topology optimization; Pareto optimization; genetic programming; energy absorbing material

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Struct Multidisc Optim (2015) 51:125–143DOI 10.1007/s00158-014-1117-8

RESEARCH PAPER

Multiobjective topology optimization of energy absorbingmaterials

Raymond A. Wildman · George A. Gazonas

Received: 4 November 2013 / Revised: 21 April 2014 / Accepted: 13 May 2014 / Published online: 24 June 2014© Springer-Verlag Berlin Heidelberg 2014

Abstract A method for the multiobjective optimizationof local-scale material topology is presented. The topol-ogy optimization scheme is based on a constructive solidgeometry-like representation, in which convex polygons—defined as the convex hull of arbitrary-length lists ofpoints—are combined using an overlapping function. Thisdata structure is tree-shaped and so genetic programmingis used as the optimizer. The forward problem is solvedwith a multiscale finite element method with automaticcohesive zone insertion to model damage. As a multiscalemethod, loads and boundary conditions are applied andobjective functions measured at a global scale, while thelocal scale material structure is optimized. The global scalegeometry is assumed fixed. Pareto optimal designs are gen-erated, representing optimal tradeoffs between conflictinggoals: quasi-static displacement and dynamic strain energy.Results demonstrate the efficacy of the proposed algorithm.

Keywords Topology optimization · Pareto optimization ·Genetic programming · Energy absorbing material

1 Introduction

Materials designed to absorb energy have applications innumerous areas: crashworthiness (Mozumder et al. 2012;Guo et al. 2011; Huang et al. 2007; Anghileri et al. 2005),head protection (Rueda et al. 2009), impact resistance (Qiaoet al. 2008), and blast resistance (Main and Gazonas 2008;

R. A. Wildman (�) · G. A. GazonasU.S. Army Research Laboratory,Attn: RDRL-WMM-B, Aberdeen Proving Ground,MD, 21005, USAe-mail: raymond.a.wildman.civ@mail.mil

Qi et al. 2013). There are several ways to design an energyabsorbing material or system of materials, for example,one could layer several different materials in a one dimen-sional sense, while optimizing for material placement andthickness. Another approach is to design the geometry ortopology of a bi-material system or truss structure in twodimensions or three dimensions. In this work, we take thelatter approach, and focus on designing energy absorbingmaterials at a local scale in a multiscale finite element(FEM) setting.

Our approach to designing materials will be to optimizethe topology of a material at a local scale in two dimen-sions, while applying loads and boundary conditions andcomputing objectives at the global scale. A heterogeneousrepresentative volume element (RVE) (strictly two materi-als in our case) will represent the local scale, and the globalscale will be assumed to be statistically homogenous withmaterial properties derived from a homogenization of therepresentative volume element (RVE). At the local scale,boundary conditions will be spatially homogenous usinglinear displacements, while at the global scale, a materialsample will be fixed at one edge, with different loading con-ditions applied depending on whether we are consideringstatic displacement or dynamic energy absorption. The dif-ferent loading configurations are used to simulate possiblescenarios in which such a material would be used, eitheras a structural element or an energy absorbing material forblast-like loading. As will be discussed in more detail later,these two loading conditions will be used as two separategoals in a multi-objective optimization problem. There areseveral options for optimizing such a configuration, the firstchoice being between a simple geometry or shape optimiza-tion method or a topology optimization method. A shapeoptimization method assumes a specific topology (i.e. eachdesign is homeomorphic in that they must have the same

126 R. A. Wildman, G. A. Gazonas

number of shapes and holes), and parameterizes that shapewith some type of function expansion, the simplest beinga linear interpolation between a given number of vertices.An optimization method, either local, derivative-based orglobal heuristic, can then be used to solve for the definingparameters of the shape. An example of this approach can befound for the design of energy absorbing tubular structuresin Chiandussi and Avalle (2002).

Shape optimization is overly restrictive, however, as onemust fix the topology a priori. A popular method for topol-ogy optimization is known as the solid isotropic materialwith penalization (SIMP) method, introduced by Bendsøe(Duysinx and Bendsoe 1998; Bendsoe and Sigmund 2003).This method uses a square grid of unknowns (pixels), whereeach unknown is continuous value representing a gradedmaterial. A local optimization method is used to determinethe optimal value of graded material at each pixel. Thougha structure consisting of graded material is the result, meth-ods exist to regularize the final result to a single material.Level-set methods have also been used to optimize topol-ogy (Sethian and Wiegmann 2000), which use contoursof a higher dimensional function to represent topology inthe plane. Finally, evolutionary methods have also beendesigned for topology optimization (Xie and Steven 1993,1997).

To optimize topology, we use a constructive solid geom-etry (CSG) representation (Requicha and Voelker 1977),which uses Boolean operations to combine shape primitives.While typical implementations of CSG use canonical shapessuch as rectangles and circles, we use convex polygons rep-resented as the convex hull of an arbitrary length list ofpoints. This encoding can represent any physically realiz-able (orientable) topology as a set of line segments. Thoughline segments are used, their length is not restricted, soarbitrarily small line segments can be used to approximatecurves. Genetic programming (GP), a variant of geneticalgorithms (GA) that uses a tree-based chromosome, willbe used and is a natural fit for this topology representa-tion as CSG can be readily expressed as a tree structure.This approach has been used in several applications includ-ing RF microwave inverse scattering (Wildman and Weile2007, 2010), gravitational anomaly inversion (Wildman andGazonas 2009), and phononic bandgap material design(Wildman and Gazonas 2011).

The approach presented here can be compared with cur-rent methods in two ways: geometry representation andoptimization method. Our approach offers an alternative tothe SIMP and level set methods in that it ultimately usesa set of line segments to approximate a topology ratherthan a grid (SIMP) or smooth curves (level-set). As willbe described below, multiple materials are also easily rep-resented in this approach. Further, as automatic meshing isused, the discretization of the topology as represented in

the optimization method is uncoupled from the discretiza-tion used in the forward solver. SIMP and level-set methodstypically use a local optimization method that requires gra-dient information. The genetic programming method usedhere does not require the computation of gradients andit is well-suited for multi-objective problems as it is apopulation-based method. The topology representation usedhere can also be adapted to a local search methodology, asdescribed in Wildman and Gazonas (2009).

Previously, microstructural (or microscale) topologyoptimization has been performed for different goals: max-imum stiffness of a periodic material (Huang et al. 2013),prescribed macroscale constitutive parameters (Sigmund1995, 1994; Zohdi 2002; Mei and Wang 2004), and ex-tremal microstructural properties (Allaire and Kohn 1993;Sigmund 2000). In this paper, we focus on balancing struc-tural stability with energy absorption in a multiobjectivesetting. Typical engineering problems involve balancingconflicting goals, using the present example, if we wish todesign an energy absorbing material and incorporate it intoan overall structure, that material may not be structurallysound and could be unusable in our final design. We couldplace constraints on the optimization problem, such as amass constraint, though this approach may leave out impor-tant information that could be garnered from the Paretofront, or set of multiobjective designs (Cohon 1978; Cohonand Marks 1975; Steuer 1989). The goal of Pareto optimiza-tion is to deliver a set of designs, rather than one singledesign, that represent the optimal trade-offs between two ormore conflicting goals. This set of designs is Pareto opti-mal in that no other design simultaneously outperforms it inall goals. (The Pareto front can be defined as the boundarybetween the infeasible region of designs, and the dominatedregion.) Each design in the Pareto optimal set (Pareto front)can only outperform another design in the Pareto optimal setby at most one goal less than the total being optimized (orone goal in a two-goal problem, two in a three-goal problem,etc.).

While an individual design on the Pareto front may bedetermined using constraints on one or more goals, theshape of the Pareto front can provide information as well.For example, the front’s shape may indicate areas of dimin-ishing returns, whereby minuscule improvement in one goalonly comes at the severe detriment of another. Here, wewill optimize for two goals: a quasi-static loading problemrepresenting a material’s structural qualities, and a dynamicloading problem to measure energy absorption. These goalsare conflicting because a structural material will be massiveand stiff, while an energy absorbing material tends to be softand compliant.

Three objectives, though only ever two simultaneously,will be considered in a multi-objective setting: Static dis-placement under uniaxial load, dynamic energy absorption

Multiobjective topology optimization of energy absorbing materials 131

space. Each chromosome i with goal vector xi is assigned asharing value by first finding all other chromosomes withina radius r in goal space. The sharing value for chromosomei is then given by

si = Ni −Ni∑

j=1

∥∥xi − xj∥∥

r, (2)

where xj is the goal vector of chromosome j within sharingradius r of chromosome i and Ni is the number of chromo-somes within the sharing radius. This approach is describedin more detail in Weile et al. (1996).

Finally, a single objective function value that will bemaximized can be assigned to each chromosome using itsPareto rank and sharing value. First, all rank one chromo-somes are given an objective function value of f 1i = 1/si .Next, the minimum objective function value of the rank onechromosomes is used as the starting value for the rank twochromosomes, i.e. now, for rank two, f 2i = φ1/si , whereφ1 = mini f 1i . This process is repeated for rank three andso on. After all chromosomes are assigned an objectivefunction value, standard roulette wheel selection is used.

2.2.3 Crossover

After selection, a new population is generated with an aver-age fitness better than the previous generation. Crossover isthen used to combine traits of those surviving chromosomesto hopefully generate even better performing chromosomes.In a standard GA, crossover is typically performed by firstrandomly choosing two chromosomes from the new popula-tion. A gene, say g3 in Table 1, and hybridized in some way.The genes following (g4 and g5) are then swapped betweenthe two chromosomes.

In GP, crossover is performed in a similar way: Twonodes in a tree are chosen at random and hybridized ifapplicable, and their subtrees are swapped. Given that ourfunction nodes are identical, there is no hybridization tobe done, however, if two terminal nodes are chosen, theirpoint lists and material properties can be hybridized. Asthe point lists are of arbitrary length, they can grow with-out bound in some situations. This is related to the issueof tree bloat (Banzhaf et al. 1999), in which tree sizesgrow without bound as protection against harmful muta-tions and crossover. Larger trees (or point lists) can containmore redundant information, so that crossovers and muta-tions with the potential to harm performance are minimized.This can lead to population stagnation, so it is undesirable.For point lists, we then use a two point crossover, ensuringthat the number of points exchanged between two termi-nal nodes is equal so that each point list remains the samelength. An example of this process is shown in Table 3: Thedouble-horizontal lines and text in bold in the point lists

Table 3 Example of crossover between two terminal nodes

C D

Mat ID Mat ID

k l

x y x y

x0 y0 u0 v0

u1 v1 x1 y1u2 v2 x2 y2u3 v3 x3 y3

x4 y4 u4 v4

u5 v5

show the two crossover points, and the coordinates xi , yiand ui , vi will be swapped between nodes C and D. Thecrossover points are chosen at random. Additionally, thepoints at the crossover locations are hybridized, i.e. mixedtogether by choosing a number t from a uniform randomdistribution between 0 and 1 as

p∗ = tp + (1 − t) qq∗ = tq + (1 − t) p, (3)where p and q are the points at the crossover locations.Finally, because we are using a database of material prop-erties, we do not hybridize materials, although this is nota restriction, as randomly generated constitutive parametersmay benefit from hybridization.

To combat tree bloat, we use a crossover probabilitybased on geometric similarity. Typically, the crossover prob-ability is a constant around 80 %, and subtrees are chosenat random between two random chromosomes. Our imple-mentation sets a crossover probability by first selecting achromosome for crossover. A mate is chosen out of a pool ofrandom chromosomes, the size of which is an input parame-ter, typically chosen as ten. Each chromosome in the matingpool is assigned a probability by randomly choosing a givennumber of subtrees (again, usually chosen to be at most ten)in each and comparing their decoded geometries as

pc =[

�(C1

⋂C2

)

�(C1

⋃C2

)]s

, (4)

where �(C1) indicates the area of the topology generatedfrom subtree C1 and s is a biasing exponent typically cho-sen as 2.5. Consequently, if C1 does not overlap C2, then pcis zero, and if C1 = C2 then pc is one. Now, pc is computedfor a given number of randomly chosen subtrees of a mat-ing pair, and the maximum is saved. After the maximum pcis found for each mate in the pool, a mate is chosen at ran-dom using a weighted roulette wheel with the weights givenas the crossover probabilities. In other words, we choose

132 R. A. Wildman, G. A. Gazonas

a random mate from the pool, favoring mates with highercrossover probabilities. Crossover is then performed usingthe chosen mating pair and subtrees with the maximum pc.

While this scheme is somewhat convoluted, it is designedto match similar pairs of subtrees, but with some random-ness. Matching similar subtrees helps ensure populationconvergence and randomness aids diversity. Striking a bal-ance between the two is important in the design of a GA, sothat it can cover a large search space and resolve fine detailssimultaneously. These issues are not present with a fixedlength chromosome representation as we know a priori thefunction of each gene in a chromosome, thus ensuring thatcrossover is meaningful.

2.2.4 Mutation

The final genetic operator, mutation, is used to inject newgenetic information into a population, preventing prema-ture convergence or stagnation. In our implementation, theflexibility in the chromosome allows for a large numberof options for mutation. We can, on one hand, manipulatesolely the encoded chromosome (tree structure with pointlists at the terminal nodes) and on the other hand, adjust thedecoded topology. The mutation rates for the examples inSection 4 are constant throughout, so they are listed here.

First, single points from terminal node point lists aredeleted with a given probability. This rate can be set some-what high (for mutation rates) as terminal nodes tend togrow in length for reasons discussed earlier. Here we usea point deletion rate of 2 % per point. There is a mini-mum length for terminal nodes of three, as three points arerequired for a 2D simplex. We also duplicate points in a ter-minal node with a 0.5 % rate per point. New points are firstduplicated then shifted by a random amount according toa Gaussian distribution with standard deviation of 10 % ofthe largest dimension of the design bounds (20 mm in allexamples from Section 4).

Points are also added by splitting existing convex hullsegments. First, a terminal node is decoded by taking itsconvex hull. Next, for each line segment in the hull, we ran-domly insert a new point at a random location between 10 %and 90 % of its length. This mutation is performed with arate of 0.5 % per segment.

In a mutation most analogous to a standard GA, pointsare shifted by a random amount. The shift amount is chosenfrom a Gaussian distribution with zero mean and standarddeviation of 10 % of the region size (as above for pointduplication), and the mutation rate is 1 % per point.

One issue with the above point mutation method is thata non-coding point (a point not on a convex hull) canbe moved to another non-coding position, meaning thatultimately the mutation had no effect on the decoded topol-ogy. A convex hull aware point mutation was designed to

remedy this situation. Given a point p inside a convexhull, the closest segment on the hull is found. Two vec-tors are formed pointing from the point inside the hull, tothe two points q1 and q2 defining the closest segment. Thepoint inside the hull is then moved inside the parallelogramformed by the addition of these vectors as

p∗ = r1 (q1 − p) + r2 (q2 − p) + p, (5)where r1 and r2 are two random values from a uniform dis-tribution between 0 and 1. This mutation then has a 50 %probability of altering the convex hull and is used with anoverall rate of 1 % per point.

Points are also deleted by pruning non-coding points thatare inside a point set’s convex hull. As an extreme example,a point list may contain hundreds of points, but only threeactually on the convex hull that defines its decoded shape.These redundant points can be harmful to a population’sprogress, and so they can be pruned. Points to be pruned arechosen by first computing the convex hull of the point listand scaling it by a random amount between 0.4 and 0.8. Allpoints within this scaled convex polygon are removed fromthe point list. This mutation is performed at a rate of 5 %per terminal node.

We can also apply affine transformations to the terminalnode point lists, effectively altering the overall geometry.Here, we separately apply scaling, rotation, or translationmutations with a rate of 1 % per terminal node. If a node ischosen for affine transformation, scaling, rotation, or trans-lation is chosen at random with equal probability. Eachtransformation has its defining parameter chosen from aGaussian distribution: Scaling uses a mean value of 1 anda standard deviation of 0.1, rotation uses a mean of 0 andstandard deviation of π /10, and translation uses a mean ofzero and standard deviation of 10 % of the region size forboth coordinates.

Material properties can also be mutated. Here we use adatabase of materials and so, with a given probability of1 % per terminal node, we simply choose a new material atrandom from the database.

Finally, we can alter the tree structure of a chromosome.The first type of tree mutation is standard in GP, subtreedeletion and regrowth. In this mutation, a subtree is deletedand replaced with a randomly generated subtree with a givenprobability. This type of mutation can be destructive so itis applied with a low probability of 0.5 % to each node ina tree. Another way of altering the tree structure is to splitterminal nodes into a function node with two new terminalnodes. There are a few ways of accomplishing this, first,we can simply split a terminal node’s point list at an arbi-trary point and separate it into two new nodes. Next, we canseparate points internal to a list’s convex hull, much likethe pruning operation described above. Here, we again scaledown the convex hull of a point list, and separate points

Multiobjective topology optimization of energy absorbing materials 135

Fig. 10 Algorithm flow chart

Initialization

Selection

Crossover

Mutation

Evaluation

Decode (CGAL Operations)

Mesh (CGAL)

Multiscale FEM (Multimech)

For all chromosomes

Compute Pareto ranks

Compute sharing values

Roulette wheel selection

Choose set of random mates

Compute crossover probabilities (CGAL)

Roulette wheel selection to choose possible mate

Hybridize subtreeson successful coin flip

For all chromosomes

Point deletion

Point duplication

Line segment splitting

Point translation

For all chromosomes

Convex hull mutation

Affine transformation

Material mutation

Delete node

Split list

Split hole

Split line

Aggregate lists

and y directions along the y = 0 edge. The size of the RVEis 20 mm-by-20 mm and the problem is plane stress. Weapply a load in the −ŷ direction with a value of 2 Pa alongthe top edge of the structure. The goal of the optimizationproblem is to minimize the displacement along the load-ing area, so the objective function is simply the sum of thedisplacement along the top edge:

D =∫

�

u (x, T ) dx, (6)

200 mm

Fig. 11 Static loading problem

where u is the displacement, T is the time at the final timestep, and � is the boundary where loading is applied at theglobal scale. Without the inclusion of damage, the prob-lem is linear so the magnitude of the load is not importantbecause it will be normalized in the objective function. Withdamage, we use total time of 0.1 s with 200 steps in a quasi-static solver. Another goal used in the static loading casewill be total mass of a local scale RVE:

m =∫

�l

ρ (x) dA, (7)

where �l represents the local scale region, and ρ is thedensity.

3.2.2 Dynamic loading

The setup of the dynamic problem (shown in Fig. 12) issimilar to that of the static problem, though we apply a hor-izontal load along the x = 0 and x = 200mm edges as welland with a pressure-time history shown in Fig. 13. Here,we use a time step size of 1 μs over a total of 300 μs. The

136 R. A. Wildman, G. A. Gazonas

Fig. 12 Dynamic loading problem

objective function is the strain energy at the global scale,summed over the entire run time, as

U = 12

∫ T0

∫

�g

σij (x, t) ij (x, t) dAdt, (8)

where σ is the stress tensor, is the strain tensor, �g isthe global scale region, and Einstein summation is assumed.This objective will be maximized as we seek a designthat absorbs energy. It represents a conflicting goal withthe static loading problem, and so we must solve in aPareto-optimal sense.

4 Results

In this section, results of the optimization are presented.Several design cases are presented, each with the samematerial system, (a dense, stiff linear elastic material andvoid), though with varying damage models. To summarize,first, in Subsection 4.1, a baseline linear elastic model atthe local scale with no damage modeling is given. Next, inSubsection 4.2, we consider the same linear elastic material,

0 50 100 150 200 250 3000

100

200

300

400

500

600

Time (µs)

Pre

ssu

re (

MP

a)

Fig. 13 Pressure load for the dynamic problem

but allow damage in the form of cohesive zone inser-tion at the local scale only. Finally, in Subsection 4.3, thesame local scale model (linear elastic with cohesive zoneinsertion) is used, but global scale damage is consideredusing XFEM and a multiscale localization method describedbelow. In each case, we will use two objectives, forming ageneric goal vector as:

f = [f1, f2] , (9)with f1 and f2 being one of the objectives defined above,U , D, or m.

In addition, each example was run twice (with differ-ent seeds for the random number generator) and comparedto demonstrate stability of the stochastic method. As thisis a multiobjective optimization that does not iterativelystep through one design goal as a constraint, any compar-ison between optimization results must consider the curveof the approximate Pareto fronts. To that end, we definethe difference in results as the integral of the differencein approximate Pareto fronts, using a linear interpolationbetween rank-one chromosomes, given by the piece-wiseparameterization

F (t) = (n − t + 1) fn + (t − n) fn+1, n ≤ t ≤ n + 1 (10)where n = 1, . . . , N1 and N1 is the number of rank-onedesigns. While the results from two runs may have differentvalues of N1 and different spacings between all f1, (10) cannonetheless be converted from a parametric form as, by def-inition of Pareto optimality, it can be defined as a function.Given a suitable mapping to a function g (x), a relative errormeasure can then be defined as

Errora,b :=∫ f1,max

1 |ga (x) − gb (x)| dx√∫ f1,max1 |ga (x)| dx

∫ f1,max1 |gb (x)| dx

, (11)

where f1,max = min(max fa,1, max fb,1

)is the minimum

of the maximum values of goal f1 of both runs, fa,1 is thefirst goal of one run and corresponds to function ga, and fb,1and gb represent a second run. In practice, the two functionsare simply discretized on an even, fine grid and the integralis computed numerically.

4.1 Linear elastic

The first results use a linear elastic material in the RVE,with Young’s modulus 65 GPa, Poisson’s ratio 0.2, and den-sity 2235 kg/m3. The remaining material was assumed tobe void. The global scale geometry is given in Figs. 11and 12, with loads and boundary conditions as discussed inSubsections 3.2.2 and 3.2.1. Here, the goal vector is given as

f = [∣∣D · ŷ∣∣ , U] , (12)and we wish to minimize f1 and maximize f2.

142 R. A. Wildman, G. A. Gazonas

that it may be possible to forgo the use of XFEM for thisdesign study.

5 Conclusions

A method for the multiobjective design of energy absorb-ing materials was presented that uses a GP-based topologyoptimization method and a multiscale FEM code. Topologywas represented using a CSG-like approach, where convexpolygons defined by the convex hulls of lists of points wereused as primitives and combined with an overlapping func-tion. This approach is flexible in that convex polygons withany number of sides can be combined together generatingcomplicated topologies. Material properties are embeddedin the convex polygons as well. A commercially availablemultiscale FEM code was used that models damage usingautomatic insertion of cohesive zone elements at the localscale and XFEM at the global scale. Pareto optimal designswere generated for several different cases, including quasi-static and dynamic loadings, with and without local scalecohesive zone insertion, and with and without global scaleXFEM insertion.

Future work includes extending the algorithm to 3Ddesigns. The method is easily implemented in 3D as allconcepts–convex hulls and CSG–are equally valid in 3D.We also plan to incorporate a hierarchical design scheme,where the global structure is optimized along with thelocal structure. This can be done again with a homogenousassumption of global scale material properties, or we couldenable multiple local scale designs that can be used at theglobal scale.

Acknowledgments This research was supported by grants of com-puter time and resources on HPC systems under the Departmentof Defense High Performance Computing Modernization Program(HPCMP) Challenge Project ARLAP02331C5V.

References

Ahmed F, Bhattacharya B, Deb K (2013) Constructive solid geom-etry based topology optimization using evolutionary algorithm.In: Bansal JC, et al. (eds) Proceedings of seventh internationalconference on bio-inspired computing: theories and applications(BIC-TA 2012), Volume 201 of advances in intelligent systemsand computing, Berlin, Germany, pp 227–238. Springer-Verlag,Berlin

Allaire G, Kohn R (1993) Optimal-design for minimum weight andcompliance in plane-stress using extremal microstructures. Eur JMech A-Solids 12(6):839–878

Allen D, Searcy C (2001) A micromechanical model for a viscoelasticcohesive zone. Int J Fract 107(2):159–176

Anghileri M, Chirwa E, Lanzi L, Mentuccia F (2005) An inverseapproach to identify the constitutive model parameters for

crashworthiness modelling of composite structures. ComposStruct 68(1):65–74

Banzhaf W, Nordin P, Keller RE, Francone FD (1999) Geneticprogramming: an introduction. Morgan Kauffman, Inc., San Fran-cisco

Bendsoe M, Sigmund O (2003) Topology optimization: theory, meth-ods and applications. Engineering Online Library. Springer

Bieri H (1995) Nef polyhedra: A brief introduction. In: Hagen H,Farin G, Noltemeier H (eds) Geometric modelling, Volume 10 ofComputing Supplement, pp 43–60. Springer, Vienna

CGAL (2007) Computational Geometry Algorithms Library. http://www.cgal.org

Qi C, Yang S, Yang L-J, Wei Z-Y, Lu Z-H (2013) Blast resistance andmulti-objective optimization of aluminum foam-cored sandwichpanels. Compos Struct 105:45–57

Chessa J, Smolinski P, Belytschko T (2002) The extended finite ele-ment method (XFEM) for solidification problems. Int J NumerMethods Eng 53(8):1959–1977

Chiandussi G, Avalle M (2002) Maximisation of the crushing perfor-mance of a tubular device by shape optimisation. Comput Struct80(27–30):2425–2432

Cohon J (1978) Multiobjective programming and planning. Academic,New York

Cohon JL, Marks DH (1975) A review and evaluation of multiobjectiveprogramming techniques. Water Resour Res 11(2):208–219

Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and eli-tist multiobjective genetic algorithm: NSGA-II. IEEE Trans EvolComput 6(2):182–197

Duysinx P, Bendsoe M (1998) Topology optimization of continuumstructures with local stress constraints. Int J Numer Methods Eng43(8):1453–1478

Goldberg DE (1987) Genetic algorithms in search, optimization, andmachine learning. Addison-Wesley, Reading

Hamza K, Saitou K (2004) Optimization of constructive solid geome-try via a tree-based multi-objective genetic algorithm. In: Deb K,et al. (eds) Genetic and Evolutionary Computation Gecco 2004, Pt2, Proceedings, Volume 3103 of lecture notes in computer science,Berlin, Germany, pp 981–992. Springer-Verlag, Berlin

Huang X, Xie YM, Lu G (2007) Topology optimization ofenergy-absorbing structures. Int J Crashworthiness 12(6):663–675

Huang X, Zhou SW, Xie YM, Li Q (2013) Topology optimiza-tion of microstructures of cellular materials and composites formacrostructures. Comput Mater Sci 67:397–407

Koza JR (1999) Genetic programming III: Darwinian invention andproblem solving. Morgan Kauffman, Inc, San Francisco

Guo L, Tovar A, Penninger C, Renaud J (2011) Strain-based topologyoptimisation for crashworthiness using hybrid cellular automata.Int J Crashworthiness 16:239–252

Main JA, Gazonas GA (2008) Uniaxial crushing of sandwich platesunder air blast: influence of mass distribution. Int J Solids Struct45(7–8):2297–2321

Mei Y, Wang X (2004) A level set method for microstructure designof composite materials. Acta Mechanica Solida Sinica 17(3):239–250

Mozumder C, Renaud JE, Tovar A (2012) Topometry optimisation forcrashworthiness design using hybrid cellular automata. Int J VehDes 60(1–2):100–120

MultiMech Research & Development (2012) MultiMech. http://www.multimechrd.com/

Nguyen VP, Stroeven M, Sluys LJ (2011) Multiscale continuous anddiscontinuous modeling of heterogeneous materials: a review onrecent developments. J Multiscale Model 3(4):1–42

Qiao P, Yang M, Bobaru F (2008) Impact mechanics and high-energyabsorbing materials: review. J Aerosp Eng 21(4):235–248

Multiobjective topology optimization of energy absorbing materials 143

Requicha A, Voelker HB (1977) Constructive solid geometry, TM-25.Technical memorandum. Production Automation Project, Univer-sity of Rochester

Rueda MAF, Cui L, Gilchrist MD (2009) Optimisation of energyabsorbing liner for equestrian helmets. Part I: layered foam liner.Mater Des 30(9):3405–3413

Sethian J, Wiegmann A (2000) Structural boundary design via levelset and immersed interface methods. J Comput Phys 163(2):489–528

Sigmund O (1994) Materials with prescribed constitutive parame-ters - an inverse homogenization problem. Int J Solids Struct31(17):2313–2329

Sigmund O (1995) Tailoring materials with prescribed elastic proper-ties. Mech Mater 20(4):351–368

Sigmund O (2000) A new class of extremal composites. J Mech PhysSolids 48(2):397–428

Srinivas N, Deb K (1994) Multiobjective optimization using non-dominated sorting in genetic algorithms. Evol Comput 2(3):221–248

Souza FV, Allen DH (2010) Modeling failure of heterogeneous vis-coelastic solids under dynamic/impact loading due to multipleevolving cracks using a two-way coupled multiscale model. MechTime-Dependent Mater 14(2):125–151

Souza FV, Allen DH (2011) Modeling the transition of microcracksinto macrocracks in heterogeneous viscoelastic media using atwo-way coupled multiscale model. Int J Solids Struct 48(22–23):3160–3175

Souza FV, Allen DH (2013) Verification of a two-way coupled multi-scale finite element code for dynamic/impact problems. Mech AdvMater Struct 20(1):74–90

Souza FV, Allen DH, Kim YR (2008) Multiscale model for predictingdamage evolution in composites due to impact loading. ComposSci Technol 68(13, SI):2624–2634

Steuer R (1989) Multiple criteria optimization: theory, computation,application. Krieger, Malabar

Weile D, Michielssen E, Goldberg D (1996) Genetic algorithm designof Pareto optimal broadband microwave absorbers. IEEE TransElectromagn Compat 38(3):518–525

Wildman RA, Gazonas GA (2009) Gravitational and magneticanomaly inversion using a tree-based geometry representation.Geophysics 74(3):I23–I35

Wildman RA, Gazonas GA (2011) Genetic Programming-basedPhononic Bandgap Structure Design. Technical Report ARL-TR-5733, US Army Research Laboratory

Wildman RA, Weile DS (2007) Geometry reconstruction of conduct-ing cylinders using genetic programming. IEEE Trans AntennasPropag 55(3):629–636

Wildman RA, Weile DS (2008) Greedy search and a hybrid localoptimization/genetic algorithm for tree-based inverse scattering.Microw Opt Technol Lett 50(3):822–825

Wildman RA, Weile DS (2010) Inverse scattering of dielectric cylin-drical targets using genetic programming. Electromagnetics 30(1–2, Sp. Iss. SI):222–236

Xie Y, Steven G (1993) A simple evolutionary procedure for structuraloptimization. Comput Struct 49(5):885–896

Xie Y, Steven G (1997) Evolutionary structural optimization. SpringerXu X, Needleman A (1994) Numerical simulations of fast crack-

growth in brittle solids. J Mech Phys Solids 42(9):1397–&Xu X, Needleman A (1996) Numerical simulations of dynamic crack

growth along an interface. Int J Fract 74(4):289–324Yamagiwa M, Uehara M, Murakami M, Yoneyama M (2010) A

reconstruction method for ultrasonic deterioration image by thecombination of constructive solid geometry and strongly typedgenetic programming. Comput Syst Sci Eng 25(2, SI):161–170

Zohdi T (2002) On the tailoring of microstructures for prescribedeffective properties. Int J Fract 118(4):L89–L94

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