Hybrid Evolutionary Approach for Level SetTopology Optimization

Mariusz Bujny∗, Nikola Aulig†, Markus Olhofer†, Fabian Duddeck∗∗Technische Universitat Munchen

Arcisstraße 21, 80333 Munich, Germany†Honda Research Institute Europe GmbH

Carl-Legien-Straße 30, 63073 Offenbach/Main, Germany

Abstract—Although Topology Optimization is widely used inmany industrial applications, it is still in the initial phase of de-velopment for highly nonlinear, multimodal and noisy problems,where the analytical sensitivity information is either not availableor difficult to obtain. For these problems, including the highlyrelevant crashworthiness optimization, alternative approaches,relying not solely on the gradient, are necessary. One option areEvolutionary Algorithms, which are well-suited for this type ofproblems, but with the drawback of considerable computationalcosts. In this paper we propose a hybrid evolutionary optimiza-tion method using a geometric Level-Set Method for an implicitrepresentation of mechanical structures. Hybrid optimizationapproach integrates gradient information in stochastic searchto improve convergence behavior and global search properties.Gradient information can be obtained from structural stateas well as approximated via equivalent state or any knownheuristics. In order to evaluate the proposed methods, a minimumcompliance problem for a standard cantilever beam benchmarkcase is considered. These results show that the hybridization isvery beneficial in terms of convergence speed and performanceof the optimized designs.

I. INTRODUCTION

In the last few years, Computer-Aided Design (CAD)methods, together with numerical simulations with the FiniteElement Method (FEM), became a standard for the designand analysis of mechanical structures. Particularly, this ledto a rapid development of methods for structural TopologyOptimization (TO) [5], which aim to develop optimal structuralconcepts within a defined design space and under specifiedboundary conditions and are applied in early phases of aproduct development. TO plays a vital role in many indus-trial branches. In particular, TO methods are widely used inlightweight design of structures in aerospace and automo-tive industry, they have started to gain acceptance in civilengineering, and are applied in materials science to developsmart materials, as well as in biomechanics to simulate boneremodeling when orthopaedic implants are inserted.

In most of the structural TO approaches, the optimizationalgorithms operate on a discrete mesh of cells, which occupiesthe entire design domain. Throughout the optimization pro-cess, the densities of the elements are changed to minimize agiven objective function, subject to a constraint on the totalmass. Typically, boundary conditions such as concentratedloads, body forces, pressures and supports (i.e. constraints onthe nodal displacements of the FEM mesh) are imposed. Fig-

Fig. 1. Compliance minimization of the cantilever beam - typical TO problem.Design domain and boundary conditions (left) and the optimized design [14](right).

ure 1 illustrates an example of a typical structural TO problem,namely the compliance minimization (stiffness maximization)of the cantilever beam.

An alternative class of TO approaches is constituted by so-called Level-Set Methods [8] that use an implicit parameteri-zation of material boundaries. As a result, unlike in the otherTO approaches (e.g. SIMP) [5], the material interface is clearlydefined by the iso-contours of a level-set function, whichis very important in the context of manufacturability of theoptimized designs. For some problems, e.g. crashworthinessTO [5], this property is crucial because intermediate densitiesin the computational model would cause high deviations fromrealistic crash behavior, so in this case the standard density-based approaches should be avoided.

In most of the standard TO approaches, sensitivity analysisis used to obtain analytical gradients and efficient gradient-based methods are applied to optimize the material distribu-tion. However, in case of some objective functions and physi-cal phenomena, e.g. material nonlinearities, contact modelingor material failure, analytical sensitivities are not availableand either stochastic or heuristic methods have to be used.However, the heavy losses of efficiency of the optimizationprocess prohibited currently the application of these algorithmsin industrial relevant optimization problems. In this workwe target to overcome this problem by the integration ofstate information from intermediate solutions in the stochasticsearch process.

We focused mainly on Evolution Strategies that turnedout to be very successful in many industrial applications,because they are gradient-free and work very well evenfor highly nonlinear, noisy or discontinuous problems. Wefollow the assumption that state information of the materialcan be integrated by approximated gradients, which can be

estimated by simplified models or by finite differencing andgeneric models that utilize information obtained from thephysical structural state [2]. Taking advantage of the implicitrepresentation of material boundaries with limited number ofdesign parameters, we combine Level-Set Method with HybridEvolution Strategies, which allow for an efficient search ofglobal optima.

In order to evaluate and validate proposed hybrid evolu-tionary optimization methods, they are applied to the can-tilever beam benchmark case. The problem of complianceminimization with volume constraint is considered. Since theanalytical gradients for this case are available, the hybridmethods can be directly compared with the gradient-basedmethods and no gradient approximation is necessary. By com-paring different hybrid optimization methods for the cantileverbeam benchmark case, the best hybridization concepts can beidentified and used in the optimization of the problems wherethe analytical gradients cannot be obtained.

The paper is structured as follows. Section II describesthe formulation of the optimization problem and the geo-metric Level-Set Method as well as the sensitivity analysisfor the compliance minimization problem. In Section III thediscretization of the continuous problem is described. SectionIV presents the hybrid evolutionary optimization methods usedin this research. Experimental setup and obtained results aredescribed in Section V. The paper is concluded in Section VI.

II. PROBLEM FORMULATION AND SENSITIVITY ANALYSIS

Let us define a global level-set function having the followingproperty:

Φ(x) > 0, x ∈ Ω,Φ(x) = 0, x ∈ ∂Ω,Φ(x) < 0, x ∈ D\Ω.

(1)

Similarly, let us define a local level-set function of the ith

structural component:φi(x) > 0, x ∈ Ωi,φi(x) = 0, x ∈ ∂Ωi,φi(x) < 0, x ∈ D\Ωi,

(2)

where Ωi is the region occupied by an elementary structuralcomponent. In particular, it holds:

Ω =

m⋃i=1

Ωi, (3)

where m is the number of structural components. Followingthe idea presented by Guo et al. [9], for D = R2, x = (x, y)

T ,we introduce a level-set function (describing an elementarybeam) of the following form:

φi(x) = −((

cos θi (x− x0i) + sin θi (y − y0i)

li/2

)r+

(− sin θi (x− x0i) + cos θi (y − y0i)

ti/2

)r− 1

),

(4)

Fig. 2. Parametrization and a plot of the level-set function given by (4). Forreadability, values φ < 0 were set to 0.

Fig. 3. Possible layout of the elementary beam components for the cantileverbeam problem (left) and the corresponding level-set field (right).

where (x0, y0) denotes the position of the center of a beamwith length l and thickness t. θ is the rotation angle measuredcounterclockwisely from the horizontal axis. Like in the orig-inal approach [9], we take r = 6. The parametrization and aplot of the level-set function described by (4) is shown in theFigure 2.

In order to obtain a structure corresponding to the implicitlevel-set representation, we use a density-based geometrymapping:

E (x) = ρ (x)E0

0 ≤ ρ (x) ≤ 1,(5)

where E0 is the reference stiffness tensor and ρ (x) is thedensity at the point x ∈ D. The density depends in turn onthe value of the level-set function:

ρ (x) = H (Φ (x)) , (6)

where:

Φ (x) = max (φ1(x), φ2(x), ..., φm(x)) , (7)

and H(x) is the Heaviside function:

H(x) =

0, if x < 0

1, if x ≥ 0.(8)

The compliance minimization (stiffness maximization)problem is considered and neglecting body forces, it is for-mulated as follows:

minu∈U,E

C(u) = minu∈U,E

∫Γt

tu dΓ, (9)

Fig. 4. Mechanical problem formulation.

where t are the boundary tractions acting on the surfaceΓt (see Figure 4), U denotes the space of kinematicallyadmissible displacements and E is the stiffness tensor.

Analytical (or approximate) sensitivities are necessary forhybridization of the evolutionary algorithms. By applying theadjoint method [5] and the chain rule to the problem above,we obtain the partial derivative of compliance with respect tothe level-set function:

∂C

∂Φ= −

∫∫D

δ (Φ (x)) εT (x)E0ε (x) dΩ, (10)

where ε is the elastic strain and δ(x) is the Dirac deltafunction:

δ (x) =

+∞, if x = 0

0, if x 6= 0.(11)

The general form of a derivative of the level-set functionwith respect to a design variable pi (e.g. x0i, θi, ti) is givenby the following formula:

∂Φ

∂pi(x) =

∂φi

∂pi(x) , if Φ (x) = φi (x)

0, otherwise,(12)

where the partial derivative ∂φi

∂pican be easily derived analyti-

cally based on (4). Plots of these derivatives (normalized withthe norm of the gradient of the level-set function) are shownin Figure 51.

Therefore, using the Lagrangian function of the form,

L = C (u) + λV, (13)

where the volume (area in case of D = R2) occupied by thematerial is

V =

∫∫D

H (Φ (x)) dΩ. (14)

The analytical sensitivities are given by

∂L

∂pi=−

∫∫D

δ (Φ (x))∂Φ

∂pi(x) εT (x)E0ε (x) dΩ

+ λ

∫∫D

δ (Φ (x))∂Φ

∂pi(x) dΩ.

(15)

1The idea of normalization with the norm of the level-set function gradientand the intuitive interpretation of the normalized derivatives is explained inSection III.

Fig. 5. Level-set function and its derivatives with respect to the designvariables. Just the values in the neighborhood of the interface are shown(the rest has been clamped to zero). The derivatives are normalized with thenorm of level-set function gradient

(∇φ =

[∂φ∂x, ∂φ∂y

]).

III. PROBLEM DISCRETIZATION

After discretization, the compliance minimization problemsatisfying the equilibrium conditions can be formulated asfollows:

minu,ρe

fTu

s.t. :

(N∑e=1

ρeKe

)u = f

0 < ρmin ≤ ρe ≤ 1, e = 1, ..., N,

(16)

where f and u are the load and displacement vectors, re-spectively, Ke is the element’s stiffness matrix and ρe is theelement’s density.

In order to estimate the analytical sensitivities of theLagrangian function (15), the discrete form of the gradienthas to be found, as well. Due to the Dirac delta termthe numerical approximation of the integrals is difficult andseveral approaches to solve this problem were proposed. Themost straightforward approach is to introduce an approximateHeaviside function, whose derivative is continuous and istreated as the approximate Dirac delta function [4]. This ap-proach, however, is usually applied to signed distance level-setfunctions2 and did not work correctly for the parametrizationproposed in this work. This is also the case of other approaches

2A signed distance function φ is an implicit function having the followingproperty: |φ (~x)| = d (~x), where d is a distance function defined as d (~x) =min (|~x− ~xI |) ∀~xI ∈ ∂Ω [12]. In particular, for signed distance functions,it holds: |∇φ| = 1.

that either could not be applied to a non-signed distance level-set functions [15] or were giving inaccurate results. As a result,an alternative, simple and accurate method was proposed andis described below.

Analytical sensitivities are given by integrals (15) of thefollowing form:

I =

∫∫R2

δ (Φ (x, y)) f (x, y) dΩ (17)

The integral I , after the change of variables, is rewritten as[11]:

I =

∞∫0

∫K:Φ=t

δ (t)

|∇Φ (x(l, t), y(l, t))|f (x(l, t), y(l, t)) dl dt

(18)The Dirac delta term is nonzero just for one value of t, i.e.

Φ = t = 0. Therefore, it can be written:

I =

∫K:Φ=0

f (x(l), y(l))

|∇Φ (x(l), y(l))|dl (19)

One possible way of estimating the integral above would beto find all the finite elements that the 0th iso-line of the level-set function passes through. Once those elements are found,the value of the fraction f(x, y)/ |∇Φ (x, y)| can be evaluatedin the center of each element and multiplied by the lengthof the interface inside the finite element. By summing up allthose values, the integral (17) can be efficiently approximatedas:

I ≈∑e∈G

f (xCe, yCe

)

|∇Φ (xCe, yCe

)|le, (20)

where G is the set of indices of the elements that the 0th level-set is passing through, (xCe

, yCe) are the coordinates of the

center of the element and le is the length of the interface insidethe element. The formula can be easily applied to calculate theanalytical sensitivities (15)3.

Due to the above mentioned simplifications in the numericalestimation of the integral (18), the accuracy of the gradientapproximation might be lowered and would decrease withcoarsening of the FEM mesh. Also in context of the increasednonlinearity of the Lagrangian function introduced by theused parametrization (e.g. corresponding to connecting (dis-connecting) of the elementary beam components), the localvalidity of the gradient information might pose additional

3It should be noted that the term ∂Φ∂p

(x, y)/ |∇Φ (x, y)|, where p is aparameter of a beam (e.g. x0, θ), has a very easy, intuitive interpretation. Asshown in Figure 5, it acts as a weighting factor that takes into account justthose parts of the beam that contribute to a derivative with respect to the givenparameter. For instance, in case of the compliance derivative with respect tothe x0 parameter, this would involve taking strain energies ( 1

2εTE0ε) at one

end of a beam with weight −1 and at the opposite end with weight 1. As aresult, depending on which side of the beam the strain energies are bigger, thederivative is either positive or negative and the optimization algorithm movesthe beam in the direction of higher energies.

difficulties for the optimization algorithm. Nevertheless, sincein this paper we focus mainly on the hybrid algorithms,which in general case should work properly even for veryrough gradient estimates (e.g. crashworthiness), the accuracyof the gradient approximation is not of principal interest andsimplicity of the proposed approach is favored.

IV. HYBRID EVOLUTIONARY METHODS

In this section the hybrid algorithms used in this research aredescribed. Two first hybrid methods are based on the standardEvolution Strategy [3], chosen for simplicity, and having thefollowing core structure:t := 0;initialize P (0) := ~a1(0), ...,~aµ(0) ∈ Iµ;evaluate P (0) : Φ (~a1(0)) , ...,Φ (~aµ(0));while (ι (P (t)) 6= true) do

recombine: P ′(t) := rΘr(P (t));

mutate: P ′′(t) := mΘm(P ′(t));

evaluate P ′′(t) : Φ (~a′′1(t)) , ...,Φ (~a′′λ(t));select: P (t+ 1) := sΘs

(P ′′(t));t := t+ 1;

end whilewhere Φ : I → R is the fitness function (I denotes the

genotype space) and ~a ∈ I = Rn is an individual. Thesize of the parent population is denoted by µ ≥ 1, whereasλ ≥ 1 stands for the offspring population size. A population atgeneration t consists of all the parent individuals ~ai(t) ∈ I inthe current generation, i.e. P (t) := ~a1(t), ...,~aµ(t). The re-combination operator is defined as a mapping rΘr

: Iµ → Iλ,whereas mutation operator takes the form: mΘm

: Iλ → Iλ.Both Θr and Θm are sets of operator parameters controllingrecombination and mutation, respectively. In order to choosethe next generation of the parent population, the selectionoperator sΘs

: Iλ → Iµ is used. The termination conditionis denoted by ι : Iµ → true, false.

All of the strategy parameters are mutated according to theSchwefel’s self-adaptation idea [13]:

σ′ = σ · exp (τ ′ ·N(0, 1))

~a′ = ~a+ ~N(~0, σ′

)τ ′ =

1√2n

(21)

A. Global Hybrid Evolution Strategy with Gradient-basedLocal Search (GHES)

In the GHES method, a local gradient-based improvementstep was introduced into a standard Evolution Strategy. Theimprovement step performs a local search acting on thesolutions produced in the mutation and recombination step.In order to enable the optimization to converge to a minimumeven if the accuracy of the gradient approximation is low, theimprovement acts only on a part of the offspring population.As a result, just the individuals with the best fitness aftera joint recombination-mutation-improvement step survive. Asimplified algorithmic description is presented below:

t := 0;initialize P (0) := ~a1(0), ...,~aµ(0) ∈ Iµ;evaluate P (0) : Φ (~a1(0)) , ...,Φ (~aµ(0));while (ι (P (t)) 6= true) do

recombine: P ′(t) := rΘr(P (t));

mutate: P ′′(t) := mΘm(P ′(t));

improve: P ′′′(t) := iΘi(P ′′(t));

evaluate P ′′′(t) : Φ (~a′′′1 (t)) , ...,Φ (~a′′′λ (t));select: P (t+ 1) := sΘs (P ′′′(t));t := t+ 1;

end whileWhere iΘi

is the improvement operator with parameters Θi,carrying out a gradient step on a subset of individuals (includ-ing necessary evaluations for performing the line search [1]).

In this method, we use an Evolution Strategy as a globaloptimization method and the Steepest Descent method as alocal search algorithm. An optimal step length is found bya quadratic interpolation method [1]. Although the computa-tional cost of a single iteration rises considerably, as can beseen in the Figure 8, the hybrid method can exhibit much betterperformance than each method separately, which justifies theuse of such an approach.

B. Concurrent Hybrid Evolution Strategy with Gradient Indi-vidual (CHES)

In the CHES method, a concept of a gradient individual wasutilized. A similar approach was proposed by Woo et al. [16].An outline of the algorithm is presented below:

t := 0;initialize P (0) := ~a1(0), ...,~aµ−1(0),~ag(0) ∈ Iµ;evaluate P (0) : Φ (~a1(0)) , ...,Φ (~aµ−1(0)) ,Φ (~ag(0));while (ι (P (t)) 6= true) do

improve gradient individual: ~a′g(t) := iΘi(~ag(t));

recombine: P ′(t) =~a′1(t), ...,~a′λ−1(t)

:=

rΘr

(~a1(t), ...,~aµ−1(t),~a′g(t)

);

mutate: P ′′(t) =~a′′1(t), ...,~a′′λ−1(t)

:=

mΘm

(~a′1(t), ...,~a′λ−1(t)

);

evaluate P ′′(t) ∪ ~a′g(t) :Φ (~a′′1(t)) , ...,Φ

(~a′′λ−1(t)

),Φ(~a′g(t)

);

select: P (t+ 1) := sΘs

(P ′′(t) ∪ ~a′g(t)

);

select gradient individual:~ag(t+ 1) := argmin~a∈P (t+1)Φ (~a);t := t+ 1;

end while

The optimization method works as follows: (µ − 1) in-dividuals follow a standard optimization procedure with theEvolution Strategy. Additionally, a gradient individual (~ag),undergoing a single Steepest Descent step (as in Section IV-A)in each optimization iteration, is introduced. As the individualto improve, always the best offspring from the whole popu-lation is chosen. However, in this approach, neither mutationnor recombination is applied to the improved individual, whichfollows a pure gradient-based optimization path. In this waya strategy that, in each iteration, chooses between evolu-tionary and gradient-based optimization method is obtained.

This seems to be reasonable since the performance of bothmethods at different optimization stages might be different. Itis commonly believed that the Evolutionary Algorithms arebetter for identifying good areas of the search space, whilethey are worse at the end of optimization, in the vicinity of anoptimum, due to the stochastic character of the optimization.In this approach merits of both methods can be combined.

C. Hybrid CMA-ES Method (HCMA-ES)

The Covariance Matrix Adaptation Evolution Strategy(CMA-ES) [10] turned out to be very successful in manypractical applications. An interesting idea for hybridization ofthe CMA-ES was proposed by Chen et al. [7].

In this method, the gradient information is incorporatedby adjusting the weighted mean 〈~x〉(g)w along the gradientdirection, i.e.:

〈~q〉(g)w = 〈~x〉(g)w − α∇f(〈~x〉(g)w

)(22)

In this work, a dynamic adaptation of the step length αaccording to the line search algorithm as in the previoussections was additionally introduced.

Besides replacing 〈~x〉(g)w with 〈~q〉(g)w , the method remainsunchanged with respect to the standard CMA-ES.

V. EXPERIMENTS

A. Setup

In this paper the cantilever beam benchmark test wasconsidered. The beam is fixed at one end and a static, point,unit load is applied in the middle of the opposite end. Theobjective function minimized in this case is the compliance ofthe structure. The design domain and boundary conditions areshown in the Figure 6. A simplified representation of the initialdesign and the corresponding mapping to the FEM mesh areshown in the Figure 7.

Fig. 6. Design space and loads of the cantilever beam test case.

Exact configuration of the case is shown in Table I.

Property Symbol Value UnitYoung’s modulus E 2.1 · 105 MPa

Poisson’s ratio ν 0.3 -Load F 1 N

Mesh resolution - 100 x 50 -

TABLE ICONFIGURATION OF THE CANTILEVER BEAM TEST CASE.

Fig. 7. Initial layout of the beams and the corresponding mapping to theFEM mesh.

Method DescriptionSD Steepest Descent optimization method (as a reference).

ES Standard (20,100) Evolution Strategywith a single standard deviation.

CMA-ES Covariance Matrix Adaptation (8,17) Evolution Strategy.

GHESGlobal Hybrid (20,100) Evolution Strategy

with a single standard deviationand 10% of improved individuals.

CHES Concurrent Hybrid (20,100) Evolution Strategywith a single standard deviation.

HCMA-ES Hybrid CMA-ES(8,17) Method.

TABLE IITESTED OPTIMIZATION ALGORITHMS.

Parameter Symbol ValueLagrange multiplier for volume λ 0.02

Reference step length in the line search α 0.2Initial standard deviation in the mutation step σinit 0.1

TABLE IIIPARAMETRIZATION OF THE OPTIMIZATION ALGORITHMS.

The FEM mesh consists of 5000 four-node shell finite ele-ments that, depending on the value of the level-set function, areassigned Young’s modulus either equal to the value specified inTable I (material) or equal to 1% of this value (non-material).Material properties as for steel are introduced.

The optimization framework was implemented in Python.A python implementation of the CMA-ES algorithm [10] wasused and modified for implementing the HCMA-ES, as well.The free, open-source CalculiX solver was used to carry outthe finite element analyses.

In order to compare different optimization methods, anextensive statistical testing was carried out. Comparison ofthe means was based on the two-sample t-test for normallydistributed random variables with unequal variances. Equalityof variances was checked with use of the two-sample F-test.Normality of the compared random variables was checked withthe Jarque-Bera test. For all tests, a 5% significance level wasassumed.

B. Results

In this section a comparison of different optimization al-gorithms for the geometric Level-Set Method is presented.The evaluation of the methods is based on the results of 30optimization runs for each of the algorithms presented in TableII.

In Table III the parametrization of the optimization algo-rithms is presented.

Convergence of the averaged Lagrangian function for dif-ferent optimization methods is shown in the Figure 8. Corre-

Fig. 8. Convergence of the Lagrangian function.

sponding statistical evaluation of the methods is presented inthe Figure 9.

The results show that the GHES method has superiorconvergence properties compared to both methods composingit, i.e. ES and SD. This is true both in terms of optimizationiterations and number of evaluations of the objective func-tion (each involving one static FEM analysis). It should benoted that the increase of numerical costs associated with thehybridization is relatively low and requires just 20% moreevaluations than in case of the ES. The GHES much morefrequently leads to the best topologies (Table IV) and involvesmuch lower variance of the Lagrangian function (see Figure9). The method converges fast at the beginning, which is acrucial property in practical applications, in which the numberof available evaluations is often heavily limited, and muchmore often tends to the global optimum. The main advantageof the GHES method is that it does not only switch betweendifferent methods composing it, but it can indeed benefit froma joint mutation-improvement phase by finding better starting

Fig. 9. Statistical evaluation of the evolutionary optimization methods.

Topology type ES CMA-ES GHES CHES HCMA-ES

7% 0% 80% 37% 23%

27% 63% 0% 43% 50%

10% 10% 17% 7% 0%

3% 0% 3% 3% 3%

TABLE IVFOUR MAIN TOPOLOGY TYPES AND THE FREQUENCY WITH WHICH THEY

WERE DEVELOPED. TOPOLOGY TYPES ORDERED ACCORDING TO THEINCREASING VALUE OF THE MINIMIZED LAGRANGIAN FUNCTION.

points for the Steepest Descent improvements. In this way,additional speedup is achieved.

In case of the CHES method, the hybridization also leads

Fig. 10. Topologies optimized with use of the SIMP approach [14] (left) anda standard Level-Set Method [6] (right).

to significant improvements with respect to ES and SD. Thistime, however, the algorithm just chooses between ES and SD,what no longer leads to the additional increase of convergencespeed associated with hybridization. As a result, the algorithmis as good as the best of the methods composing it. However,this is still a desirable behavior since a method performingbetter in a given search space region can be chosen. Moreover,the CHES method needs just one additional evaluation ofthe objective function per iteration than the ES. Therefore,the computational cost associated with the hybridization isnegligible compared to the speedup that is obtained. As inthe GHES method, CHES much more frequently leads to thebest topologies.

The last hybrid approach, HCMA-ES, at least in the initialphase of the optimization process, exhibits the best perfor-mance among all of the hybrid approaches as long as thenumber of FEM evaluations is considered4. Nevertheless, themethod exhibits much higher variance of the Lagrangianfunction throughout the whole optimization process and isconsiderably more noisy than any other method consideredin this section. The method seems to be very sensitive to anyinaccuracies of the gradient approximation, which might be asevere drawback when applying it to the problems where theanalytical sensitivities are not available.

The topologies optimized with the hybrid methods areconsistent with the designs obtained with use of the state-of-the-art methods for structural TO (see Figure 10). How-ever, those approaches cannot be used to optimize arbitraryobjective functions, which was the main motivation for usingevolutionary optimization methods in this work.

C. Discussion

The results show that with hybridization significant im-provements of convergence properties and global search ca-pabilities can be achieved. The GHES has proven that thehybridization can lead to methods not only as good as theircomponents, but also having superior convergence speed withrespect to every method composing them. This was the maindifference between GHES and CHES approach, where thealgorithm can just switch between the methods and is as goodas its best component, but involves considerably lower compu-tational costs. Finally, the HCMA-ES was the most promising

4Ideally, when comparing different optimization methods, just the algo-rithms in their optimal configurations should be considered. This can resultin completely different population sizes for the ES and the CMA-ES. Inthis paper, however, the main focus is to compare the standard optimizationalgorithms with their hybrid versions, justifying the choice of differentpopulation sizes for the ES and the CMA-ES.

method in terms of the number of the FEM evaluations, butinvolved an increased noisiness of the Lagrangian functionduring the optimization.

It seems that the GHES represents the best hybridizationconcept also for the optimization problems where the sensi-tivities cannot be derived analytically. This was demonstratedalso in the considered test case, where due to the inaccuracyof the gradient estimation caused either by an insufficientmesh resolution or by an imprecise minimum estimation in theline search, both SD and HCMA-ES were strongly disturbed,while GHES remained unaffected. In GHES, the evolutionaryprocess can decide when to use the external knowledge sourcesand therefore, more local improvement techniques can beincorporated without having negative influence on the opti-mization process. Finally, similar hybridization concept couldbe used together with the CMA-ES, which turned out to bevery successful in solving the optimization problem consideredin this paper.

VI. CONCLUSION

The main focus of this paper was a development of efficient,hybrid, level-set-based evolutionary methods for structural TO.For evaluation, a standard cantilever beam benchmark case wasused. However, the principal goal was to find a hybridizationconcept that could be further used for optimization of prob-lems, where analytical sensitivities are not available.

In particular, the application of the proposed methods tocrashworthiness TO seems to be very interesting and iscurrently under investigation. The crashworthiness problemsare characterized by very strong nonlinearities, discontinu-ities and noise, arising from modeling of highly complexphysical phenomena ranging from contact to material failure.As a result, in general case, analytical sensitivities for crashproblems cannot be easily found. However, simplified modelsor generic models utilizing local state features [2] can beused to obtain approximations of the gradients. The hybridevolutionary algorithms proposed in this paper offer ways touse these rough approximations and enhance the performanceof the evolutionary search.

Within the research, performance of the following hybridoptimization algorithms was evaluated:• Global Hybrid Evolution Strategy with Gradient-based

Local Search (GHES), where after the mutation step, apart of the offspring population is improved by a singlestep in the negative gradient direction.

• Concurrent Hybrid Evolution Strategy with Gradient In-dividual (CHES), where after each selection step, the bestindividual becomes the gradient individual, optimizedwith the Steepest Descent method.

• Hybrid CMA-ES Method (HCMA-ES), where the meanof the search distribution in the CMA-ES is moved alongthe direction of the negative gradient.

The hybrid methods were compared with both standard Evolu-tion Strategy and the Covariance Matrix Adaptation EvolutionStrategy.

The hybrid approaches presented in this paper showed thatthe evolutionary methods can benefit from incorporation of theexternal knowledge sources. Due to the increased capabilityof finding global optima, they might be considered as anattractive alternative even for the gradient-based methods incase if the analytical sensitivities are available. Nevertheless,even if the information that should support the convergenceprocess (e.g. gradient estimate) is incorrect, evolution canuse the selection mechanism to eliminate the negative effects(e.g. GHES, CHES). Those properties broaden significantlythe scope of possible extensions and make the further researchvery promising.

Finally, the hybridization concept used in the GHES methodseems to be the most promising in terms of convergence speed,ability to find global optima and applicability to the problems,where only a rough approximation of the gradient informationcan be obtained.

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