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Optimal Design of Nonlinear MultimaterialStructures for Crashworthiness Using Cluster
Analysis
Kai LiuPh.D. Candidate
School of Mechanical EngineeringPurdue University
West Lafayette, Indiana, [email protected]
Duane DetwilerHonda R&D Americas, Inc.
Chief Engineer, Vehicle Research - CAEManager, Strategic Research Department 1 (SR1)
Raymond, Ohio, [email protected]
Andres Tovar∗Associate Professor, Member of ASMEDepartment of Mechanical Engineering
Indiana University-Purdue University IndianapolisIndianapolis, Indiana, USA
This study presents an efficient multimaterial design op-timization algorithm that is suitable for nonlinear struc-tures. The proposed algorithm consists of three steps: con-ceptual design generation, clustering, and metamodel-basedglobal optimization. The conceptual design is generatedusing a structural optimization algorithm for linear mod-els or a heuristic design algorithm for nonlinear models.Then, the conceptual design is clustered into a predefinednumber of clusters (materials) using a machine learning al-gorithm. Finally, the global optimization problem aims tofind the optimal material parameters of the clustered designusing metamodels. The metamodels are built using sam-pling and cross-validation, and sequentially updated usingan expected improvement function until convergence. Theproposed methodology is demonstrated using examples frommultiple physics and compared with traditional multimate-rial topology optimization method. The proposed approachis applied to nonlinear, multi-objective design problems forcrashworthiness.
∗Address all correspondence to this author.
1 IntroductionMultimaterial structural optimization has the potential
to synthesize structures of higher performance than the onesobtained with traditional binary-phase (solid-void) topologyoptimization methods. With the increasing availability ofmultimaterial additive manufacturing technologies, multima-terial topology optimization also becomes increasingly rele-vant.
Most of the available multimaterial topology optimiza-tion methods fall into one of the following classes: homog-enization design method, density-based method, level set-based method, phase field-based method, and heuristic meth-ods. Homogenization design methods assume that the struc-ture is composed of a periodically perforated microstructure,so its bulk mechanical properties can be determined usinghomogenization theory [1,2]. The main drawback of this ap-proach is that the optimal microstructure, which may changeat every point within the structure, is not always known. Thiscan be alliviated by restricting the method to a subclass ofmicrostructure. This approach, referred to as partial relax-ation, has been used to generate conceptual designs of three-phase material composites with extremal thermal expansion
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This is the author's manuscript of the article published in final edited form as:
Liu, K., Detwiler, D., & Tovar, A. (2017). Optimal Design of Nonlinear Multimaterial Structures for Crashworthiness Using Cluster Analysis. Journal of Mechanical Design, 139(10), 101401. https://doi.org/10.1115/1.4037620
[3], piezoelectricity [4], and bulk modulus [5]. An additionalproblem with the homogenization methods is the manufac-turability of the optimized structure. The microscopic holesin the microstructure may be difficult or impossible to fabri-cate.
Density-based methods are an alternative that avoids theapplication of homogenization theory by using a continuousdensity value without a microstructure. In a discretized de-sign domain, the mechanical properties of the material ele-ment (stiffness tensor coefficients) are determined accordingto a rule of mixture between two phases using a power-lawinterpolation function [6,7]. When only a two-phase materialdesign is considered, the power law may implicitly penalizeintermediate density values driving the structure towards abinary configuration. The use of such a penalization proce-dure is referred to as the Solid Isotropic Material with Pe-nalization (SIMP) method [8]. For a three-phase materialdesign the rule of mixtures can be easily extended. Due toits simplicity, the SIMP method has been extensively usedin multimaterial topology optimization [9–11]. The effectiveuse of the SIMP-based methods requires regularization pro-cedures such as filters, which are purely heuristic and verysensitive to changes.
Alternatives to the density-based methods are the levelset-based methods and the phase field-based methods. Thelevel set-based methods use implicit iso-contours of a level-set function to define the boundaries between materialphases. In this sense, these are related to shape optimiza-tion methods. These methods avoid the use of filters andother regularization procedures used in density-based meth-ods and have become popular in structural topology opti-mization [12–14]. The level-set method has been extendedto multimaterial topology optimization following two mainapproaches: the extended variational multilevel sets ap-proach [15–19] and the extended piecewise-constant varia-tional level set approach [20, 21]. Recently, Kriging meta-models have been also incorporated in level set-based topol-ogy optimization [22–24]. The automatic changes of thetopology through breaking and merging also require thelevel-set function to be re-initialized during the update op-eration in order to achieve appropriate numerical accuracy.Although several attempts have been reported to address thisissue [25–28], the re-initialization operation still relies onheuristics.
Phase field-based methods are free boundary trackingmethods that avoid the need for re-initialization [29–34].These methods are capable of handling the motion caused bydomain states and the motion caused by the domain shape,e.g., the temperature and the mean curvature motion, respec-tively, so they are also related to shape optimization methods.Unfortunately, these methods are not easily extended to non-linear problems involving contact, large displacements, andplasticity.
Outstanding heuristic approaches for multimaterialtopology optimization are the bi-directional evolutionarystructural optimization (BESO) method [35,36], the DiscreteMaterial Optimization (DMO) method [37, 38], and the Hy-brid Cellular Automaton (HCA) method [39, 40]. For in-
stance, HCA can be efficient in problems involving a largenumber of design variables in nonlinear multimaterial struc-tures [41]; however, heuristic methods are not applicable togeneral structural optimization problems.
Recently, clustering or related techniques have been ex-plored in topology optimization to reduce the computationalcost of the optimization algorithm or to reduce the dimen-sion of the design space. Using a modified P-norm distance,stress functions have been grouped to reduce the number ofconstraints in stress-constrained topology optimization [42].A clustering method in a genetic algorithm is also being re-ported for the design of rotor topologies [43, 44]. Cluster-ing in topology optimization has been utilized by researchersat the Honda Research Institute Europe GmbH to reducethe dimension in a (multi-dimensional) local state featuresspace [45]. Finally, in our previous work [46], K-meansclustering is tailored to reduce the dimension of the designspace and allow the application of a heuristic multiobjective,metamodel-based optimization algorithm. This approach hasbeen developed over the years and the current state is sum-marized in this work.
Here, we propose a method for multimaterial structureoptimization that incorporates clustering and is suitable toefficiently solve large-scale optimization problems involv-ing nonlinear structures. The resulting structures are well-defined multimaterial designs without artificial (intermedi-ate) materials. The proposed design optimization algorithmconsists of three steps: conceptual design generation, de-sign clustering, and metamodel-based global optimization.During the first step, the conceptual design is generated us-ing a structural optimization algorithm for linear models ora heuristic design algorithm, such as HCA, for nonlinearmodels. During the second step, the dimensionality of theproblem is reduced by clustering the continuous field vari-able. To this end, unsupervised machine learning, i.e., K-means clustering, is implemented to optimally group thecontinuous field variable into a reduced number of clusters.The number of clusters determines the number of materi-als within the structure. During the third step, metamodelsare built using appropriate curve fitting or interpolation func-tions, e.g., Kriging metamodels. Then, the metamodels aresequentially updated using an expected improvement func-tion. The global optimum, which corresponds to the presentbest function value, is obtained once the expected improve-ment reaches a sufficiently small value.
2 Conceptual DesignThe first step of the proposed design strategy is to gen-
erate a conceptual multimaterial design of the structure. Ifthe structure’s finite element model is linear and numericallytractable, a structural optimization algorithm can be used togenerate the conceptual design. Otherwise, a heuristic designalgorithm can be used instead. The heuristic design algo-rithm used in this work is HCA [39]. HCA is commerciallyavailable in LS-TaSC for LS-DYNA (LSTC, California). Ineither case, the material property is characterized by the de-sign variable xe ∈R, where 06 xe 6 1, for e= 1, . . . ,n, where
n is the number of finite elements in the structure’s finite el-ement model. The conceptual design problem is to find thedistribution of all possible materials x∈Rn that minimize theobjective function f (x) :Rn→R subjected to a set of (equal-ity and inequality) constraints. The optimization problem isexpressed as:
find x ∈ Rn
minimize f (x,U(x, t))subject to h(x,U(x), t) = 0
g(x,U(x), t)6 00 6 xe 6 1, e = 1, . . . ,n,
(1)
satisfying the finite element equilibrium equations. Ifthe problem involves microstructured multi-phased isotropicmaterials, then it must also consider limits such as theHashin-Shtrikman bounds [47]. It becomes apparent thatthe mapping between the design variable xe and the mate-rial property must be correctly selected in order to avoid anon-physical distribution of artificial materials.
Representing the material property by a single scalar isgenerally possible with the use of homogenization methods.In that way, two or multiple phases are blended to form a new(composite) material. To generate the conceptual design, thiswork makes use of a linear interpolation:
Pe = Pmin +(P0−Pmin)xe, (2)
where Pe is the element material property tensor. The op-timal material distribution (or an approximation) x∗ ∈ Rn
from Eq. (1) is used as the conceptual design. Generally,the desired number of materials K in the final design is sig-nificantly smaller than n—the value of n is in the order of 103
to 106. Therefore, the dimension of the problem is reducedby mapping the n optimal values into K clusters as describedin Sec. 3.
3 ClusteringThe dimension reduction problem consists on drawing
inferences from datasets consisting of input data without la-beled responses. The solution to this problem, usually ad-dressed by cluster analysis in unsupervised machine learn-ing, leads to groups of observations in such a way that theobservations in the same group are more similar to eachother than to those in other groups. Cluster analysis has beenused in global optimization to identify promising design re-gions [48] and eliminate near-duplicate designs [49]. In con-trast to this prevailing use, this work applies cluster analysisto downsize the dimension of the design space reducing thenumber of the materials from the conceptual design.
Due to its effectiveness and simplicity, K-means is oneof the most popular and widely used cluster analysis algo-rithms [50]. K-means cluster analysis is employed in thiswork. The input to this cluster analysis algorithm is a set of
n observations and the desired number of clusters K, where1 6 K 6 n; usually, K � n. In this application, an obser-vation consists of a single attribute, namely, the normalizedmaterial parameter xe, where 0 6 xe 6 1 and e = 1, . . . ,n. Adiscussion on the optimal value of K can be found in [51].
To start the clustering process, the algorithm distributesK cluster seeds within the set of all observations xe, wheree = 1, . . . ,n. According to the distance between the observa-tions and each seed, the set is partitioned into Voronoi cells(clusters) Sk, where k = 1, . . . ,K. From this point, the algo-rithm finds the cluster centroid values µk that minimize thewithin-cluster sum of squares. Mathematically, this problemis expressed as follows:
find µµµ ∈ RK
minimize J(µµµ) =K
∑k=1
∑xe∈Sk
(xe−µk)2.
(3)
In order to solve the clustering problem in Eq. (3), aniterative refinement algorithm is utilized [52] (Algorithm 1).Due to the nonlinearity of the optimization problem, the finalclustering depends heavily on the initial cluster seed distri-bution. While global optimization methods may be suitable,this work addresses this limitation by using 1000 randomly-generated initial seeds. After the clustering, the value ofxe ∈ Sk is replaced by µk. The result is a K-dimensional clus-tered design suitable for building metamodels and perform-ing global optimization. The computational cost of solvingEq. (3) is usually a small fraction of the cost of a finite ele-ment analysis.
Algorithm 1: Iterative K-means clustering algorithm:
1 Randomly initialize K cluster centroids, µ1, . . . ,µK ;2 while stopping criterion has not been met do3 for k : 1 to K do4 Sk←{} ;5 end6 for e : 1 to n do7 j← argmin j′ (xe−µ j′ )
2 ;8 S j← S j ∪{xe} ;9 end
10 for k : 1 to K do11 µk← 1
|Sk| ∑xe∈Skxe ;
12 end13 end
A rigorous method to find K has not yet found and,therefore, a parametric study is utilized as shown in the nu-merical examples (Secs. 6.3 and 6.4). The metamodel-basedglobal optimization strategy presented in this work consistsof two steps: generating initial metamodels via sampling andcross-validation (Sec. 4), and finding the global optimum de-sign via expected improvement maximization (Sec. 5).
4 Generation of the Initial MetamodelsDynamic models involving geometric, material, and
contact nonlinearities are commonly found in crash simu-lations [53]. For such models, the computational cost of afunction evaluation is considerably high and it is impracti-cal to use traditional gradient-based optimization methodsdue to the lack of reliable sensitivity coefficients. As an al-ternative, metamodels can be derived by sampling the dy-namic, nonlinear finite element model. The resulting meta-models are numerically inexpensive and allow to find near-optimal solutions through the use of global multi-objectivealgorithms [51]. The key aspect to using metamodels forglobal optimization lies in balancing between global explo-ration and local exploitation. It is desirable to generate anaccurate metamodel that explores a large portion of the de-sign space with a few sampling points.
Several metamodels have been evaluated for designproblems in crashworthiness including: polynomial responsesurface, radial basis functions, and Kriging [51]. Based oncross-validation errors, Kriging is the preferred metamodeland it is used in this work. The general form of the Krigingmetamodel f is of a function f is
f (S) = E[ f (S)]+P
∑p=1
ωp
{f (S(p))−E[ f (S(p))]
}(4)
where E[·] is the expected value (mean), S(p) are the pthsampled designs, and ωp are the Kriging weights, which arederived from a covariance function. This metamodel can befound implemented in MATLAB [54]. One metamodel isbuilt for each function in the optimization problem.
4.1 SamplingComputer design of experiments is the selection proce-
dure for finding the points in the design space that must besimulated. Many strategies can be used to sample the designpoints including factorial designs, D-optimal designs, andLatin hypercube sampling (LHS) [55]. LHS is used in thiswork to generate the initial metamodel. This provides de-signs that are independent of the mathematical model of theapproximation and allow the estimation of the main effectsof all factors in the design in an unbiased manner. Anotheradvantage of LHS is the number of points to be evaluated canbe directly defined. For K clusters (design variables) and Pdesign points, the LHS provides a P×K matrix S that ran-domly samples the entire design space broken down into Pequal-probability regions. The LHS matrix components aredefined as
Spk =ηpk−0.5
P, (5)
where η1k, . . . ,ηPk are uniform random permutations of theintegers 1 through P. However, the LHS design points gen-erated using Eq. 5 might be highly correlated, which is un-desirable (Fig. 1a). Approaches to avoid high correlation in-
cluding correlation minimization methods [56] and the max-imization of the minimum inter-site distances [57]. The laterone is used in this work. In this approach, the inter-site dis-tance between two samples S(r) and S(t) is the defined as
d(
S(r),S(t))=
[K
∑j=1
(Srk−Stk)2
]1/2
, (6)
for r, t ∈ {1,2, . . . ,P}, where S(r) is the rth row of the sam-pling matrix S, and Srk indicates the kth column of the rowvector S(r). Figure 1b shows an optimal LHS generated fromthe use of the maximize the minimum inter-site distance cri-terion.
0 1 2 3 4 50
1
2
3
4
5
(a)
0 1 2 3 4 50
1
2
3
4
5
(b)
Fig. 1: Latin hypercube samplings with P = 5 and (a) highcorrelation and (b) low correlation achieved using the maxi-mization of the minimum inter-site distances.
4.2 Cross-validationDespite the number and optimal distribution of the sam-
pled designs, the resulting metamodel may not be sufficientlyaccurate to provide meaningful predictions. To estimate theaccuracy of the metamodel, this work uses the leave-one-out cross-validation. In this cross-validation approach, onedesign is left out from the P sampled designs; then, themetamodel is re-generated using the remaining P−1 designs(Fig. 2). If S(p) is the pth sampled design that has been leftout, f (S(p)) the function value, and f−p(S(p)) is the cross-validated prediction of f (S(p)), then one can plot f−p(S(p))
against f (S(p)). If the metamodel fits perfectly, these pointsshould lie on the 45◦ line [58]. If this diagnostic plot lookssatisfactory, e.g., high coefficient of determination R2, thenthe metamodel is also considered satisfactory; otherwise, themetamodel is refit with a log or inverse transformation to thedependent variable. If one of these transformations gives sat-isfactory diagnostic plot, then this transformation function isused in the rest of the analysis.
In addition, root-mean-square error (RMSE) from thepredicted residual error sum of squares (PRESS) vector canbe used to estimate the cross-validation error. Using theleave-one-out cross-validation, the PRESS vector e is formedwith the errors ep = | f (S(p))− f−p(S(p))| obtained when the
pth sampled design is left out. Then, the RMSE from thePRESS vector is given by [59]:
PRESSRMS =
√1P
eTe. (7)
0 0.2 0.4 0.6 0.8 1
S
-5
0
5
10
15
20
!"
Fig. 2: Leave-one-out cross-validation at the second sampledpoint exemplified by fitting a function with a Kriging meta-model.
5 Metamodel-based Global OptimizationOnce the initial metamodels are built, the global opti-
mization problem can be solved. The global optimizationproblem is to find the material parameters that minimize theobjective function vector f(µµµ) : RK → Rn f , where n f is thenumber of objective functions. The input to the metamodelsf is the vector of clustered design variables µµµ. The outputis the predicted values of the finite element models f. Theoptimal design can be found using the Efficient Global Op-timization (EGO) algorithm [58, 60]. During the search forthe global optimum, the EGO algorithm balances betweenglobal exploration and local exploitation.
5.1 Expected ImprovementThe expected improvement function calculates the
amount of improvement one can expect at a given point S(p).The expected improvement function is defined as
E[I(S(p))] = E[max( fpbs−F,0)
], (8)
where fpbs = min{ f (S(1)), . . . , f (S(p)), . . . , f (S(P))} is thepresent best function value (pbs) and F is a normally dis-tributed random variable with mean and standard deviationdefined by the Kriging metamodel. With F ∼N ( f ,σ2), onecan express the expected improvement as follows [58]:
E[I(S(p))] =(
fpbs− f)
Φ(u)+σ(S(p))φ(u), (9)
where u =(
fpbs− f)/σ, f = f (S(p)) is the predicted value
at point S(p), σ2(S(p)) is the variance of the Kriging predic-tor, and Φ(·) and φ(·) are the cumulative density function(CDF) and probability density function (PDF) of a normaldistribution, respectively.
For constrained problems, one must ensure the feasibil-ity of the newly selected points S(p). If the finite elementmodel used to evaluate the constraint is computationally ex-pensive, the constraint can be approximated using a meta-model. In this case, the probability of feasibility needs tobe considered along with the expected improvement on theobjective function [60]. If the new point is not feasible, itscorresponding expected improvement must be zero.
The expected improvement function is maximized us-ing an evolutionary algorithm or a branch-and-bound algo-rithm. If the maximum expected improvement is less than0.1% of the present best function value in two consecutiveiterations, then convergence is achieved and the metamodelneeds no further improvement; otherwise, the point wherethe expected improvement is maximized is added to the sam-pled set and the metamodel is updated. If no convergence isachieved in 100 iterations, the algorithm is terminated.
5.2 Multi-objective Expected ImprovementFor a multi-objective optimization problem, a multi-
objective expected improvement function needs to be de-fined. Consider an optimization problem that minimizes twoobjectives f1(x) and f2(x), with the set of m Pareto points
f∗1,2 ={(
f ∗(1)1 , f ∗(1)2
), . . . ,
(f ∗(m)1 , f ∗(m)
2
)}, (10)
where f ∗(i)j = f j
(S∗(i)
)and S∗(i) is a Pareto design. The ex-
pected improvement for this multi-objective problem is de-fined as [60]:
E[I(S∗(p))] = P[I(S∗(p))] min{d1, . . . ,dm}, (11)
where P[I(S∗(p))] is the probability of improving both func-tions f1 and f2 at the Pareto design S∗(p). The probability ofimprovement is defined as:
P[I(S∗(p))] = Φ(u11)+
m−1
∑i=1
[Φ(ui+1
1 )−Φ(ui1)]
Φ(ui+12 )+
[1−Φ(um1 )]Φ(um
2 ), (12)
where uij = ui
j(S∗(p)) =(
f ∗(i)j − f j(S∗(p)))/σ j(S∗(p)).
In Eq. (11), di for i= 1, . . . ,m is the distance between thevectors (F1, F2) and
(f ∗(i)1 , f ∗(i)2
), where (F1, F2) is the cen-
troid of the probability integral used to calculate E[I(S∗(p))]:
F1(S∗(p)) =1
P[I(S∗(p))]
[z1
1 +m−1
∑i=1
(zi+1
1 − zi1)
Φ(ui+12 )+
zm1 Φ(um
2 )] (13)
where zij = zi
j(S∗(p)) = f j(S∗(p))Φ(uij) − σ j(S∗(p))φ(ui
j).F2(S∗(p)) is defined similarly. Details on the derivation of themulti-objective expected improvement formula can be foundin [60]. The use of the multi-objective expected improve-ment is illustrated in Secs. 6.3 and 6.4.
6 Numerical ExamplesFour examples with different physics are presented in
this section. The first two examples involve linear finite ele-ment models under a static load and the evaluation of thermaland mechanical compliance. The last two examples involvenonlinear finite element models under a dynamic load (armorplate and vehicle S-rail structure).
In the first two examples, the results are comparedwith an alternative optimization method, i.e., the alternat-ing active-phase multimaterial topology optimization algo-rithm [61], which is available in MATLAB. In the last twoexamples, a parametric study on the number of clusters isincluded.
6.1 Thermal ComplianceThe minimum thermal compliance example is adopted
from [61]. The design problem is illustrated in Fig. 3. Dueto the axis symmetry of the model, only a quarter of the de-sign domain is considered. The reduced design domain isdiscretized into 50×50 Q4 elements. The remainder of thissection follows the proposed three-step design optimizationapproach (3SDO).
!!"#
!"#
!"#
!"#
Fig. 3: Thermal compliance problem—Design domain andDirichlet boundary condition: constant surface temperatureu = 0.
Step 1: Conceptual design Without a constraint in thenumber of materials, the optimization problem is stated asfollows:
find x ∈ Rn
minimize f (x,U(x)) = FTU(x)
subject to h(x) =1n
n
∑e=1
xe−m f = 0
0 6 xe 6 1, e = 1, . . . ,n,
(14)
where U(x) = K(x)−1F under plane stress. f is the thermalcompliance function, U(x) denotes the finite element globalnodal temperature vector, F denotes the global thermal loadvector, and K(x) denotes the global thermal conductivity ma-trix. A relative thermal conductivity of 1.0 is defined forthe most dense material and 10−3 for the least dense mate-rial. The element conductivity matrix is interpolated usingEq.(2). The mass constraint limit (or mass fraction) is pre-scribed to be m f = 0.4. The conceptual design is generatedusing topology optimization and achieved in 20 iterations.The convergence criterion is ‖xk−xk−1‖ ≤ 10−3, where k isthe iteration number. The corresponding objective value isf = 1.94×106. The conceptual design is shown in Fig. 4.
Fig. 4: Thermal compliance problem—Conceptual designwith f = 1.94×106.
Step 2: Clustered design In this example, let us considerfour clusters, this is K = 4. The K-means clustering opti-mization in Eq. (3) leads to the optimal material parametersµk and corresponding volume fractions m f k summarized inTable 1, for clusters k = 1, . . . ,4. The corresponding valueof the thermal compliance is f = 1.97× 106. The clustereddesign is shown in Fig. 5.
Step 3: Metamodel-based global optimization Given theinitial material parameters µk, for k = 1, . . . ,4, the final opti-
Table 1: Thermal compliance problem—Optimal materialparameters of the clustered design.
k Color µk m f k
1 Red ( ) 0.98 0.19
2 Green ( ) 0.31 0.30
3 Blue ( ) 0.57 0.13
4 Black ( ) 0.13 0.38
Fig. 5: Thermal compliance problem—Clustered designwith f = 1.97×106.
mization problem is the following:
find µµµ ∈ RK(K = 4)
minimize f (x(µµµ)) = FTU(x(µµµ))
subject to h(x(µµµ)) =1n
K
∑k=1
∑xe∈Sk
µk−m f = 0
0.10 6 µk 6 1.00, k = 1, . . . ,4,
(15)
where xe = µk for all xe ∈ Sk. The mass function h is lin-ear and no metamodel is required. A Kriging metamodel isbuilt only for the thermal compliance f using the method de-scribed in Sec. 4. The Kriging metamodel is trained with 40samples generated by an optimal Latin hypercube sampling.In this initial metamodel, R2 = 0.98 and PRESSRMS = 0.078.The constrained expected improvement function is used tosearch for the global optimum using the EGO algorithm.Converges is achieved in four iterations. In the final meta-model, R2 = 0.99 and PRESSRMS = 0.054. The optimalmaterial parameters µ∗1 = 1.00, µ∗2 = 0.33, µ∗3 = 0.43, andµ∗4 = 0.16 as summarized in Table 2. The objective value isimproved to f = 1.96×106.
For comparison, this problem is also solved using thealternating active-phase multimaterial topology optimization(MTOP) algorithm [61]. The MTOP algorithm consists ofouter and inner iterations. Each outer iteration involves thesolution of K(K− 1)/2 alternating active-phase inner itera-tions, i.e., binary topology optimization subproblems. It also
requires the definition of the number of materials and theircorresponding material properties.
Table 2 summarizes the results obtained by the MTOPalgorithm and the proposed three-step design optimizationmethod (3SDO). Both results have the same material param-eters µµµ∗k and mass fractions m∗µk. As stated in [61], the filteris a key success of the optimization procedure and shouldbe gradually reduced during the multimaterial topology op-timization; hence, the filter is applied and gradually reducedin MTOP. A penalization power is also utilized. As it canbe seen in Table 2, although the solutions from two differentapproaches have similar objective value, the structures arequite different. Due to the use of alternative phases, designspace dimension R is considerable higher than the proposed3SDO method.
Table 2: Thermal compliance problem—Comparison of thefinal objective value.
f ∗ µ∗k m∗µk R
3SDO
1.96×106
1.000.330.430.16
0.190.300.130.38
4
MTOP
1.95×106 2672
Using the proposed 3SDO approach, the objective func-tion is evaluated only at the conceptual design generationand the metamode-based global optimizationl. The concep-tual design is generated in 20 topology optimization iter-ations. The metamodel is built with 40 training samplesinitially, and converged in four iterations. On the otherhand, the MTOP converged in 156 outer iterations, whichrequires 936 alternating active-phase (inner) iterations. Eachinner iteration evaluate the objective function twice. There-fore, the total number of function evaluations for MTOP is156×6×2 = 1872. Table. 3 summarizes the number of iter-ations (# iter) and the computational cost in number of func-tional evaluations (# feval) in every step of the 3SDO methodand the MTOP algorithm.
Scalability The scalability of the optimization method canbe assessed from the number of function evaluations. For theproposed 3SDO, the number of function evaluations is givenby
N3SDO = Nc +NsK +Ne, (16)
where Nc is the number of function evaluations in concep-tual design step. NsK is the initial function evaluations to
Table 3: Thermal compliance problem—Comparison of thecomputational cost.
Step # iter # feval
3SDO
1. Conceptual 20 20
2. Clustering 7 0
3a. Sampling 40 40
MTOP
3b. Optimization 4 4
Outer 156 0
Inner 936 1872
generate the metamodel and Ne is the number of functionevaluations in the EGO.
On the other hand, the number of function evaluationsfor the MTOP is given by
NMTOP = No×K× (K−1), (17)
where No is the outer iteration number. From Eq. (16) andEq. (17), the proposed 3SDO method is linear with numberof clusters K while the MTOP is quadratic with K. A plotof number of function evaluations with up to 10 materials isgiven in Fig. 6 with Nc = 100,Ns = 10,Ne = 100 and No =500.
2 3 4 5 6 7 8 9 10
# materials
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
pre
dic
ted
# f
ev
al
104
3SDO
MTOP
Fig. 6: Thermal compliance problem—Comparison of thepredicted number of function evaluations.
6.2 Mechanical Compliance under a Static LoadLet us consider the optimal distribution of three different
materials (K = 3) in a 3D beam in cantilever (Fig. 7). The ob-jective is to minimize the mechanical compliance subjected
to a mass constraint. The design domain is discretized into60× 20× 8 H8 identical finite elements. The three-step de-sign optimization approach is applied as follows.
x
y
z
Fig. 7: Mechanical compliance problem—Design domainand boundary conditions for the 3D beam in cantilever.
Step 1: Conceptual design The mechanical compliance isa scalar counterpart of thermal compliance. The optimiza-tion problem is stated as in Eq. (14) except that U(x) is thenodal displacement vector, F is the external load vector, andK(x) is the stiffness matrix. The mass fraction constraint isprescribed to be m f = 0.3. The relative elastic modulus ofthe most dense material is 1.0 and the one of the least densematerial is 10−3. The Poisson’s ratio is 0.3.
The optimal density distribution x∗ (Fig. 8) is achievedin 64 iterations using Top3d [62] with a control-based strat-egy [63]. The convergence criterion is ‖xk− xk−1‖ ≤ 10−3,where k is the iteration number. The optimal design variablesare distributed from 1.0 (stiff material in black color) to 0.0(compliant material in white color) with 8320 distinct values(using 16 digits of precision). The corresponding objectivefunction value is f = 2217.
Fig. 8: Mechanical compliance problem—Conceptual de-sign with 8320 distinct density values and f = 2217.
Step 2: Clustered design The conceptual design is thenclustered into three clusters (Fig. 9). As in the previous ex-ample, contour indicates the cluster distribution. The clus-tered design has compliance value f = 2432, which is, asexpected, higher than the one of the conceptual design.
Fig. 9: Mechanical compliance problem—Clustered designwith three clusters and f = 2432.
Step 3: Metamodel-based global optimization The op-timization problem is to minimize mechanical compliancesubject to a mass constraint with three design variables. Theinitial Kriging metamodel of the mechanical compliance isbuilt with 30 LHS samples. In this metamodel, R2 = 0.99and PRESSRMS = 0.037. The final design is obtained af-ter three EGO iterations using the constrained expected im-provement function. In the final metamodel, R2 = 0.99 andPRESSRMS = 0.034. The comparison between 3SDO andMTOP on the final objective value and computational cost issummarized in Tables 4 and 5, respectively. Results demon-strate the effectiveness and efficiency of the proposed algo-rithm: the 3SDO requires only 3% of functional evaluationand it is able to achieve result 11% better than the MTOP.
Table 4: Mechanical compliance problem—Comparison ofthe final objective value.
f ∗ µ∗k m∗µk R
3SDO
23501.000.100.48
0.160.150.69
3
MTOP2645 6025
Table 5: Mechanical compliance problem—Comparison ofthe computational cost.
Step # iter. # feval
3SDO
1. Conceptual 64 64
2. Clustering 3 0
3a. Sampling 30 30
MTOP
3b. Optimization 3 3
Outer 480 0
Inner 1440 2880
6.3 Minimum Penetration and Mass under a DynamicLoad on an Armor Plate
Let us consider the thickness (topometry) optimizationof an armor plate impacted by a rigid ball (Fig. 10). The plateundergoes large displacement and plasticity. Nonlinear finiteelement analysis is utilized. The goal is to minimize both theimpact penetration and the mass of the plate. The dimensionof the plate is 300 mm × 300 mm. The displacement of theplate is constrained along its four edges. In the numericalanalysis, the plate is discretized into 30× 30 identical finiteelements. The rigid ball impacts the plate in a perpendiculardirection at a speed of 10 m/s. The base material propertiesare listed in Table 6. The initial design has uniformly elementthickness distribution of 5 mm as shown in Fig. 10. Theinitial design has maximum penetration f1 = 12.05 mm andmass f2 = 0.50.
10 m/s
constrained edges
Fig. 10: Armor plate problem—Finite element model.
Step 1: Conceptual design The simulation is performedusing explicit nonlinear finite element analysis in LS-DYNA.The conceptual design is obtained with one iteration of theHCA algorithm using element internal energy as the fieldvariable [39]. The result is shown in Fig. 12. The contour in-dicates the value of the element thickness from 1 mm (white)to 10 mm (black). The average thickness is kept at 5 mm.The corresponding penetration is f1 = 9.33 mm and the massfraction is f2 = 0.50.
Step 2: Clustered design The conceptual design is clus-tered using the K-means algorithm. A parametric study is
Table 6: Armor plate problem—Base material properties.
Property Value
Density 7830 kg/m3
Elastic Modulus 207 GPa
Poisson’s Ratio 0.3
Yield stress 200 MPa
Tangent modulus 2.0 GPa
Fig. 11: Armor plate problem—Initial design (left) andimpact simulation (right) with maximum penetration f1 =12.05 mm and mass fraction f2 = 0.50.
Fig. 12: Armor plate problem—Conceptual design (left) andimpact simulation(right) with maximum penetration f1 =9.33 mm and mass fraction f2 = 0.50.
performed to determine the influence of the K value. Fig-ure 13 shows the clustered designs corresponding to K =1, . . . ,18. Figure 14 shows the values of the the maximumpenetration f1 and mass fraction f2 for each clustered de-sign. As the number of clusters increases, the value of f1tends to decrease. The value of f2 remains relatively con-stant. For the clustered design, it is desirable to keep thelowest K value and the lowest values for f1 and f2. Fromthese results, K = 4 can be identified as a potential optimaldesign.
Step 3: Metamodel-based global optimization The ob-jective is to minimize both the maximum penetration f1 andthe mass fraction f2 of the armor plate. The multi-objective
optimization problem is:
find µµµ ∈ RK(K = 3)minimize f 1(x(µµµ)) : maximum penetrationminimize f2(x(µµµ)) : mass fractionsubject to 1 6 µk 6 10, k = 1,2,3.
(18)
A Kriging metamodel is only required for f1. The num-ber of LHS samples to build the f1 initial metamodel is tentimes the number of clusters, this is: P = 10K. The massfraction function f2 is linear and no metamodel is required;therefore, σ2 = 0 and the multi-objective expected improve-ment Eq. (11) can be simplified to E[I] as defined by Eq. (9).
The Pareto front of all clustered designs is shown inFig. 15. As observed, Pareto designs for K = 4 dominatesmost of the other Pareto designs as well as the initial design,the conceptual design, and clustered design (K = 4). In thisobjective space, the conceptual design and the clustered de-sign (K = 4) are close to each other. The computational costof this structural optimization problem is summarized in Ta-ble 7.
Table 7: Armor plate problem—Computational cost of the3SDO method (K = 4).
Step # iter. # feval
1. Conceptual 0 1
2. Clustering 12 0
3a. Sampling 40 40
3b. Optimization 100 100
6.4 Thin-walled S-rail crashworthiness designThin-walled S-rails are essential components of the pro-
gressive crushing zone in a vehicle. During a frontal or rearcollision, S-rails are designed to absorb a high amount of theimpact kinetic energy through plastic deformation. In thisexample, the geometry of the thin-walled S-rail of squarecross section is defined as shown in Fig. 16, where L= 1.0 m.The S-rail is crushed in the axial direction by a rigid walltraveling at a constant speed of 5 m/s. The crushing distanceis prescribed to be 0.5L, which occurs 100 ms after the im-pact.
The objective is to maximize specific energy absorption(energy absorption per unit mass) and minimize peak crush-ing force. The design variable in this example is the shellelement thickness of the S-rail. The material properties aresummarized in Table 8.
K = 1 K = 2 K = 3 K = 4 K = 5 K = 6
K = 7 K = 8 K = 9 K = 10 K = 11 K = 12
K = 13 K = 14 K = 15 K = 16 K = 17 K = 18
1 18
Fig. 13: Armor plate problem—Clustered designs with K = 1, . . . ,18.
2 4 6 8 10 12 14 16 18
number of clusters K
9
9.5
10
10.5
11
11.5
12
12.5
maxim
um
penetr
ation (
mm
) f
1
0.4
0.45
0.5
0.55
0.6
mass fra
ction f
2
Fig. 14: Armor plate problem—Maximum penetration andmass fraction as functions of the number of clusters.
The crashworthiness indicators of the S-rail are the spe-cific energy absorption (SEA) the peak crushing force (PCF).These crashworthiness indicators are defined as follows:
SEA(x) =∫
δP(x,δ)dδ
m(x), (19)
PCF(x) = maxδ
P(x,δ), (20)
where P(x,δ) is the reaction force of the thin-walled com-ponent at a crushed distance δ. The simulation is per-formed with the non-linear finite element analysis softwareLS-DYNA.
0 0.2 0.4 0.6 0.8 1
mass fraction f2
5
10
15
20
25
30
max p
enetr
ation f
1 (
mm
)
Initial Design
Conceptual Design
Clustered Design (K = 4)
K = 1
K = 2
K = 3
K = 4
K = 5...18
Fig. 15: Armor plate problem—Pareto front of the designoptimization problem and the optimized clustered design(K = 4) with maximum penetration f1 = 9.46 mm and massfraction f2 = 0.50.
Step 1: Conceptual design The conceptual design of theS-rail aims to trigger its progressive collapse. To this end,principles of compliant mechanism design are utilized: giventhe displacement of input ports, the objective is to find thethickness distribution that maximizes the displacement ofoutput ports [3]. In this case, the input ports are prescribed atthe contact nodes with a rigid wall. The output ports are de-fined by the wavelength λ of the progressive buckling corre-sponding to an ideal axial crushing condition [64] (Fig. 17).The optimization is to maximize the mutual potential energy
!"#"$%"&"#"'($!
'()*&&'(*&
'($)*!
!
Fig. 16: S-rail problem—Geometry of the thin-walled S-rail(side view). The cross section is squared of dimensions H ×H and thickness xe.
Table 8: S-rail problem—Material properties of the finite el-ement model.
Property Value
Density 7800 kg/m3
Elastic Modulus (E) 207 GPa
Poisson’s Ratio 0.29
Yield stress (σY ) 253 MPa
Tangent modulus
(σY/E < ε≤ 0.048) 2437 MPa
(0.048 < ε≤ 0.108) 883 MPa
(0.108 < ε≤ 0.148) 550 MPa
(0.148 < ε≤ 0.208) 433 MPa
(0.208 < ε≤ 0.407) 281 MPa
(0.407 < ε≤ 0.607) 185 MPa
(0.607 < ε≤ 0.987) 124 MPa
of the structure subjected to a mass constraint [65]. This is:
find x ∈ Rn
minimize f (x,U(x)) =−LTU(x)
subject to h(x) =1n
n
∑e=1
xe−m f = 0
xLe 6 xe 6 xU
e , e = 1, . . . ,n
(21)
where x is the thickness distribution, U(x) is the nodal dis-placement vector, and L is the nodal (dummy) force vectorwith zeros at all degrees of freedom except for the ones cor-responding to the output ports where the value is one. Thelower thickness bound is xL
e = 0.6 mm and the upper thick-ness bound is xU
e = 6.0 mm. The mass fraction is set tobe m f = 0.50. The initial design has a constant thicknessof 3.0 mm for all the finite elements. The corresponding
crash simulation shows Euler-type buckling with two plastichinges (Fig. 18). The conceptual design is obtained with oneiteration of the HCA algorithm using element mutual poten-tial energy as the field variable [39]. Progressive folding isobserved in this conceptual design. The corresponding thick-ness distribution and crash simulation are shown in Fig. 19.
Inp
ut
po
rts0.5λλ
Output ports
Fig. 17: S-rail problem—Locations of input and output portsfor a thin-walled S-Tube following the wavelength λ cor-responding to the progressive buckling after an ideal axialcrushing condition.
Fig. 18: S-rail problem—Initial design represented by a uni-form thickness distribution in the “unfolded” thin-walledstructure (left). The initial design depicts Euler-type buck-ling (right). The corresponding crashworthiness indicatorsare SEA= 3.39 kJ/kg and PCF= 267 kN.
Fig. 19: S-rail problem—Conceptual design represented thethickness distribution in the “unfolded” thin-walled struc-ture (left). The conceptual design depicts progressive fold-ing (right). The corresponding crashworthiness indicatorsare SEA= 5.05 kJ/kg and PCF= 359 kN.
Step 2: Clustered design A parametric study is performedon the number of clusters. Figure 20 shows the “unfolded”clustered designs for K = 1, . . . ,12. The corresponding spe-cific energy absorption (SEA) and peak crushing force (PCF)values are summarized in Figure 21. While a rigorous
K = 1 K = 2 K = 3 K = 4
K = 5 K = 6 K = 7 K = 8
K = 9 K = 10 K = 11 K = 12
1 12
Fig. 20: S-rail problem—Clustered designs with K = 1, . . . ,12.
method to find the optimal number of K has not yet beenfound, in this example K = 2 depicts a desirable high SEAand low PCF values. This design can be idenfied as the po-tential optimal design.
0 2 4 6 8 10 12
number of clusters K
3.6
3.8
4
4.2
4.4
4.6
4.8
5
SE
A (
kJ/k
g)
260
280
300
320
340
360
380
400
420
440
PC
F (
kN
)
Fig. 21: S-rail problem—SEA and PCF values as a functionof the number of clusters K.
Step 3: Metamodel-based global optimization The de-sign objectives are to maximize the specific energy absorp-tion (SEA) and minimize the peak crushing force (PCF). Themulti-objective optimization problem can be stated as fol-
lows:
find µµµ ∈ RK
maximize SEA(x(µµµ))minimize PCF(x(µµµ))subject to 0.6 mm 6 µk 6 6.0 mm, k = 1, . . . ,K,
(22)where µk is the thickness of the kth cluster. Kriging meta-
models are built for both objective functions using 40 initialLHS samples. The EGO algorithm with multi-objective ex-pected improvement is utilized. Figure 22 shows the result-ing Pareto front for K = 1, . . . ,4. All Pareto fronts dominatethe initial design, the conceptual design, and clustered de-signs. The best clustered design is for K = 2; however, thebest Pareto fronts correspond to K = 3 and K = 4. Table 9summarizes the computational cost.
Table 9: S-rail problem—3SDO computational cost
step # iter. # feval
Conceptual 1 2
Clustering 8 0
Sampling 40 40
Optimization 100 100
7 Summary and DiscussionThis work presents a design strategy to solve multi-
material structural optimization problems that consists of
-7 -6 -5 -4 -3 -2 -1 0
-SEA (kJ/kg)
0
100
200
300
400
500
600
700
PC
F (
kN
)
Initial Design
Conceptual Design
Clustered design (K = 1)
Clustered design (K = 2)
Clustered design (K = 3)
Clustered design (K = 4)
K = 1
K = 2
K = 3
K = 4
Fig. 22: S-rail problem—Pareto fronts for K = 1, . . . ,4. Ini-tial, conceptual, and clustered design are dominated. Clus-tered designs have the following (−SEA, PCF) coordinates:for K = 1: (−3.82,274), for K = 2: (−4.81,349), for K = 3:(−3.98,423), and for K = 4: (−3.90,381).
three steps: conceptual design generation, clustering, andmetamodel-based global optimization. The conceptual de-sign is a continuous design variable distribution generated bystructural optimization in linear models (Secs. 6.1 and 6.2) ora heuristic design method such as HCA for nonlinear models(Secs. 6.3 and 6.4). Unsupervised machine learning tech-niques such as the K-means clustering algorithm are utilizedto reduced the dimension of the optimization problem fromthousands of design variables to a lower number of clusters.With the reduced number of design variables, metamodel-based global optimization can be performed. To this end,this work uses EGO. The proposed method is demonstratedthrough four examples: thermal compliance minimization,mechanical compliance minimization, mass and impact pen-etration minimization, as well as specific energy absorptionmaximization with peak crushing force minimization. Thefirst two examples use linear finite element models under astatic load. The last two examples use nonlinear finite ele-ment models under a dynamic load.
Aspects of the proposed structural optimization ap-proach can be modified to solve specific problems. For ex-ample, instead of using topology optimization as illustratedin the paper, topograghy optimization can be applied. Forclustering, K-means can be replaced by Principle Compo-nent Analysis or another unsupervised machine learning al-gorithm. In any case, the designer can supervise the designclustering and select the final number of clusters (materials)depending on the type of metamodel used in the final op-timization step and/or the manufacturing process of the op-timized design. Our ongoing research explores the use ofsupervised clustering methods.
In this work, the results of the linear models are com-pared to a multimaterial topology optimization algorithm,demonstrating lower computational cost in the proposed al-gorithm. For the nonlinear model, the results for different
number of clusters are compared. While more research isneeded, the parametric studies in this work show that theanalysis (crash simulation) of the cluster designs is a suit-able indicator to approximate the optimal number of clusters.Other approaches such as the elbow method, information cri-terion approach, and the Silhouette method [51] are compu-tationally less expensive but not as effective as the paramet-ric study shown in this work. Ongoing investigation aimsto control the structural complexity of the clustered designgenerated by the proposed approach.
AcknowledgementsHonda R&D Americas supported this research effort.
Any opinions, findings, conclusions, and recommendationsexpressed in this investigation are those of the writers and donot necessarily reflect the views of the sponsors.
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