interpolation scheme for fictitious domain techniques and topology optimization of finite strain...

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Comput. Methods Appl. Mech. Engrg. 276 (2014) 453–472

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Interpolation scheme for fictitious domain techniquesand topology optimization of finite strain elastic problems

Fengwen Wang ⇑, Boyan Stefanov Lazarov, Ole Sigmund,Jakob Søndergaard Jensen

Department of Mechanical Engineering, Technical University of Denmark, Nils Koppels Alle, Building 404, 2800 Kgs. Lyngby, Denmark

Received 25 December 2013; received in revised form 26 March 2014; accepted 27 March 2014Available online 13 April 2014

Abstract

The focus of this paper is on interpolation schemes for fictitious domain and topology optimization approaches withstructures undergoing large displacements. Numerical instability in the finite element simulations can often be observed,due to excessive distortion in low stiffness regions. A new energy interpolation scheme is proposed in order to stabilizethe numerical simulations. The elastic energy density in the solid and void regions is interpolated using the elastic energydensities for large and small deformation theory, respectively. The performance of the proposed method is demonstratedfor a challenging test geometry as well as for topology optimization of minimum compliance and compliant mechanisms.The effect of combining the proposed interpolation scheme with different hyperelastic material models is investigated aswell. Numerical results show that the proposed approach alleviates the problems in the low stiffness regions and for thesimulated cases, results in stable topology optimization of structures undergoing large displacements.� 2014 Elsevier B.V. All rights reserved.

Keywords: Energy interpolation; Large deformation; Fictitious domain; Topology optimization; Hyperelastic material model; Ersatzmaterial models

1. Introduction

To avoid cumbersome meshing and remeshing, different numerical modeling and optimization methods,like e.g. fictitious domain and topology optimization methods make use of fixed, regular finite element meshes.The structures to be modeled are projected or optimized on the fixed mesh and void, non-material, regions aremodeled as very low stiffness regions. Such methods can be very efficient, however, the void regions may insome cases cause trouble and jeopardize convergence of the fixed-grid methods. In the following, we discuss

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0045-7825/� 2014 Elsevier B.V. All rights reserved.

⇑ Corresponding author. Tel.: +45 4525 4266; fax: +45 4593 1475.E-mail address: [email protected] (F. Wang).


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454 F. Wang et al. / Comput. Methods Appl. Mech. Engrg. 276 (2014) 453–472

issues with excessive deformations of void regions for large-displacement fictitious domain and topology opti-mization techniques.

Fictitious domain methods significantly simplify the mesh generation step in finite element analysis andprovide access to solvers that can take advantage of uniform structured grids. The basic idea is to immersethe domain of interest into a simple, geometrically larger one called the fictitious domain, e.g. [1,2]. A regularmesh is usually utilized to discretize the fictitious domain. The boundary of the immersed geometry is notfitted exactly and is padded with weak material which represents the void regions. In some areas the weakvoid region is called “ersatz material”. For geometrically non-linear examples, [3] reports problems withill-convergence in the void regions, i.e. loss of uniqueness of the deformation map within the void (weak)material regions.

Topology optimization [4] is an iterative process which distributes material in a design domain byminimizing a selected objective and fulfilling a set of constraints. The material distribution is updated in eachiterative step using the sensitivity of the objective and the constraints. Over the last two decades, the methodhas been applied in finding optimized designs for a wide range of mechanical problems consideringgeometrically linear modeling. Different techniques have been developed to ensure length scale and to achievemanufacturable designs, e.g. perimeter control [5], filtering [6–8], projection [9,10], robust formulations[11,12], etc. Partly due to large computational cost, partly due to complexity and also due to the numericalinstabilities discussed in this paper, a relatively small number of research works have considered structuralresponse using geometrically non-linear modeling, including minimum compliance problems and compliantmechanism design [13–17].

In topology optimization, the associated mechanical problem is usually discretized using finite elements,and void regions in the design are represented using low stiffness or ersatz material. Such an approach avoidscumbersome re-discretization and has been successfully applied in different optimization schemes like density-based topology optimization [4], bidirectional evolutionary structural optimization [18], optimization usinglevel-set methods [19,20], etc. Studies on structural topology optimization under large deformation [13–16]have shown that excessively distorted elements can be observed in low stiffness regions during topology opti-mization procedures, which leads to numerical instabilities in the Newton–Raphson iterations in the nonlinearfinite element simulations.

To circumvent the numerical instability induced by the low stiffness regions, several different approacheshave been proposed. Buhl et al. [13] and Pedersen et al. [14] avoided the numerical instability by excludingthe internal nodal forces surrounded by low stiffness elements in the Newton–Raphson convergence criterion.Bruns and Tortorelli [15] avoided to deal with low stiffness elements by removing and reintroducing low den-sity elements in the optimization procedure. Yoon and Kim [16] introduced a connectivity parametrization,where all elements were connected by fictitious springs. Recently an element deformation scaling techniquehas been introduced to delay the onset of excessive deformations in low stiffness elements in [21].

In all works mentioned above, the St. Venant–Kirchhoff model is used to describe the material behavior. Itis known that the St. Venant–Kirchhoff model does not provide physically correct response under large com-pression and recently more realistic hyperelastic material models have been employed in structural topologyoptimization [22,23] in order to deal with the numerical instability issues. However, in our topology optimi-zation examples we were not able to achieve satisfactory performance for very large deformations using any ofthe schemes proposed in the literature.

In this work, a new energy interpolation scheme is proposed to remove the numerical instability induced byexcessive deformations in low stiffness regions. The elastic energy density for each element is interpolatedbetween the energy density for a large deformation formulation and the one under small deformation. Theinterpolation is performed using a threshold projection function. The proposed interpolation scheme isexpected to work for all topology optimization and fictitious domain approaches that make use of low stiffnessregions to model void domains. The idea of the scheme is simple: ideally, void regions, if modeled by suffi-ciently low stiffness, do not influence the solid (structural) part of the domain. Hence, we can choose to modelthem in any way we like, as long as this does not influence the convergence in the solid regions. If modeled bylinear theory the geometries of the low density elements are not depending on the non-linear deformations, inturn avoiding convergence problems. To validate the generality and the robustness of the proposed approach,a fictitious domain problem as well as several benchmark optimization problems under large deformation are

F. Wang et al. / Comput. Methods Appl. Mech. Engrg. 276 (2014) 453–472 455

considered, including minimum compliance problems [13] and a compliant inverter mechanism [14]. The appli-cability of the proposed approach with different hyperelastic models and the effect of using them are investi-gated as well.

The paper is organized as follows. In Section 2 we outline the idea for simple fictitious domain problemswith discrete and continuous material distributions. In Section 3, density-based topology optimization is pre-sented together with the governing equations for geometrically nonlinear elasticity and finite element formu-lation using hyperelastic material models. In Section 4, the detailed optimization problems, correspondingsensitivity analysis and numerical implementation are described. Section 5 demonstrates the performance ofthe proposed interpolation scheme by the optimization of several benchmark examples.

2. Fictitious domain approach

In finite element analysis, a structure is usually modeled by a conformal mesh. Depending on structuralcomplexity, building the mesh may be challenging and tedious. Alternatively, one may project the structureon a fixed mesh which may degrade the solution quality but greatly enhance efficiency. Whereas the latter,fictitious domain approach, may work well for linear problems, the following example will showcase its lim-itations for geometrically non-linear problems. We consider the C-shape problem as illustrated in Fig. 1 [16].The structure is supported at the left edge. The dimensions are 10 m � 10 m with a thickness of 1 m, and it isdiscretized using square elements of a mesh size of 1 m as in [16]. The simulations are first performed using alinear finite element analysis with plane stress assumption. The structural deformations using a conformingmesh and a fictitious domain approach for modeling the void region are shown in Fig. 2. Here we use Young’smodulus E1 ¼ 1 for the solid region and E0 ¼ 10�9 for the void region. It is seen that the difference between thesystem response obtained by removing void regions or using low stiffness material in the void region isnegligible.

Using geometrically non-linear modeling, the low stiffness elements may become excessively distorted whichleads to numerical instability in finite element simulations. This phenomenon has been reported in severalstudies [13–15,3]. To illustrate the problem, the C-shape problem is reconsidered using a finite deformationformulation and nine times bigger loads than the case presented in Fig. 2. The structure is simulated using4 equally sized load increments. Fig. 3(a) shows the structural deformation for the conforming mesh. Forthe fictitious domain approach Fig. 3(b) it is seen that the low stiffness elements are distorted significantly,and in addition some of them become inverted. Although the deformation of the solid domain is undisturbedhere, the ill-convergence of the void region may in other cases disturb the overall convergence – especially ifthe distinction between solid and void regions is not clear as e.g. seen in topology optimization or in fictitiousdomain approaches that model boundaries by gray-scale interpolations.

2.1. Discrete material distributions

The distorted and ill-converged void region mesh in Fig. 3(b) can be alleviated by a simple idea which pro-vides the foundation of this paper. If we base the analysis in the solid region on the non-linear stored energyand we base the analysis in the void regions on linear stored energy we get a deformation pattern as seen in

1m10m

f1 = 0.002

f2 = 0.003

10m

Fig. 1. Schematic illustration of the C-shaped problem presented in [16].

E1 = 1, ν = 0.3

E1 = 1, ν = 0.3

E0 = 10−9, ν = 0.3

(a)

(b)

Fig. 2. (a) Modeling the solid region (in black) and corresponding deformation for linear analysis. (b) Fictitious domain approach –modeling the void region (in white) using low stiffness material and corresponding deformation for linear analysis.

(a)

(b) (c)

Fig. 3. Deformation of the C-shape problem under large deformation with nine times bigger loads. (a) Modeling the solid region (in black)using the geometrically nonlinear elements. (b) Modeling both the solid region and the void region (in white) using geometrically nonlinearfinite elements. (c) Modeling the solid region using geometrically nonlinear elements and the void region using linear elements.


Fig. 3(c). Here, the void mesh is deformed in a smooth fashion and although elements still seem to be inverted,this is not seen by the solver, since these elements are only evaluated in the un-deformed geometry due to thelinear theory employed for these elements.

The exact formulations for the energy terms discussed above will be given in subsequent sections. For now,however, to conceptualize the idea, we write the energy interpolation form for element e as

/e ueð Þ ¼ / ceueð Þ � /L ceueð Þ þ /L ueð Þ½ �Ee ð1Þ

where Ee is Young’s modulus of element e (i.e. E1 for solid material and E0 ¼ E1 � 10�9 for the void region), ue

is the element nodal displacement vector, / �ð Þ refers to the stored elastic energy density function in the basematerial for unit Young’s modulus and /L �ð Þ is the stored elastic energy density under small deformation (withunit Young’s modulus). The interpolation factor ce is 1 for solid elements and 0 for void elements. Using thisscheme, it is clear that for solid elements (ce ¼ 1), the stored elastic energy corresponds simply to the


non-linear one (i.e. /e ¼ /ðueÞE1) and for void elements (ce ¼ 0), the stored elastic energy corresponds simplyto the linear one (i.e. /e ¼ /LðueÞE0).

In the energy interpolation scheme in Eq. (1) we assume that the Young’s modulus is separable from theenergy functional. For some material models (like e.g. the Mooney–Rivlin model for rubber) this is not thecase. Nevertheless, it should be straight forward to modify the interpolation scheme to handle such casesas well.

The proposed scheme is almost trivial to implement for fictitious domain and topology optimizationapproaches where the structure is described by discrete material distributions, i.e. where mesh elements onlycan be solid or void. However, in many fictitious domain approaches and in most topology optimizationapproaches, (boundary) elements may take any stiffness values between solid and void and hence the interpo-lation factor ce, discussed above must become a function of the local element stiffness. This aspect is discussedin the following.

2.2. Continuous material distribution

In most topology optimization approaches as well as in some fictitious domain approaches, structuraltopologies are described by continuously varying stiffness distributions, controlled by a spatially dependentdesign variable. Examples are density-based topology optimization [4], level set methods [19,20] and fictitiousdomain approaches [1,3].

In general, the topology of a structure can be represented with the help of a scalar function w yð Þ, defined as

a function of location, y, over the design domain and bounded in the interval w; wh i

. The material distribu-

tion may be defined as

x ¼ HðwðyÞ � gÞ ð2Þ

where Hð�Þ is the Heaviside function, g is a prescribed threshold value and hence x takes value one if the pointis occupied with material (w yð Þ > g) and zero if the point is void (w yð Þ < g). For discrete material distributionschemes, the Heaviside function is sharp whereas for most gradient-based optimization approaches, the Heav-iside is smooth, allowing a smaller or larger transition region between solid and void domains with interme-diate values of x. Eq. (2) holds for standard level set approaches with g ¼ 0 and unbounded level-set functionw as well as for projection-type, density-based topology optimization approaches with g varying between 0and 1 and w bounded to the interval ½0; 1�, c.f. [12]. Level-set-based approaches typically operate on the inter-face between solid and void regions. On the other hand, density-based topology optimization approaches pro-vide optimized designs by controlling the scalar function w yð Þ over the whole design domain, c.f. [24].

The following derivations hold for any topology optimization approach or fictitious domain approach withvarying material coefficients, however, to simplify the description we concentrate the developments to theSIMP approach (Simplified Isotropic Material with Penalization) used in density-based topology optimization[25,26].

The Young’s modulus Ee for element e is interpolated as

Ee ¼ xe yð Þð ÞpðE1 � E0Þ þ E0 ð3Þ

where E1 is the Young’s modulus of the base material (in black), E0 P 0 is the Young’s modulus of the voidregion (typically E0 ¼ 10�9 � E1 and p P 1 is the penalization exponent used to ensure black-and-white designsin density based topology optimization (typically p ¼ 3).

As for the discrete case discussed in the previous subsection we want low density elements to be modeled bylinear analysis and high density elements by non-linear analysis. In order to differ between the two regions forthe continuous case, and at the same time ensuring a smooth and differentiable transition we suggest to modelthe threshold parameter ce in Eq. (1) by a smoothed Heaviside projection function, stated as

ce ¼tanh b1q0ð Þ þ tanh b1 xp

e � q0

� �� tanh b1q0ð Þ þ tanh b1 1� q0ð Þð Þ ð4Þ


here q0 is a threshold used to determine the element behavior. When xpe < q0 and b1 is sufficiently large, the

stored energy density in the low-density element is identical to the stored energy density under smalldeformation with ce � 0. Hence the element is modeled as a linear element. When xp

e > q0, the stored energydensity is identical to the one under large deformation and the element is modeled as a nonlinear element withce � 1. It can be seen from Eqs. (1) and (4) that elements with 0 < ce < 1 are modeled with a reducedgeometrically nonlinear contribution. Remark that the density value xe is raised to the power p, indicating thatit is element stiffness and not just density that must be thresholded for the scheme to work properly. Based onthe energy density in Eq. (1), the finite element formulation can be derived correspondingly and details will bepresented in the next section. It should be noted that a related approach, where the element residual isinterpolated using the element residual under large deformations and the one under small deformationswas proposed in [27]. A short description and a detailed comparison between the two approaches is givenin Appendix A.

To validate the proposed model, the C-shape problem is analyzed using the proposed model with q0 ¼ 0:01and b1 ¼ 500. The Newton–Raphson method converges within few iterations for each load increment.Fig. 3(c) displays the structural deformation. Comparison of the structural deformations using the proposedmodel and the standard fully-nonlinear model (i.e. without the energy interpolation scheme as seen inFig. 3(b)) demonstrates that the proposed model can circumvent excessively distorted elements in low stiffnessregions and leads to smooth displacement fields. Hence it can be used to remove the numerical instabilities infinite element simulations for finite strain elasticity during topology optimization.

The proposed scheme consisting of the energy interpolation Eq. (1) and the projection Eq. (4) is general andcan be used for all imaginable fictitious domain and topology optimization schemes based on fixed meshes.The only challenge is to find appropriate values of the threshold variable q0 and the sharpness parameterb1. For simple fictitious domain approaches the choice of parameters is trivial (one can simply use an if-state-ment, i.e. a sharp projection), however, for optimization approaches the choice of parameters is open and maydepend on the optimization strategy and problem considered. In the remainder of the paper we apply thescheme to density-based topology optimization. Here we find that the parameter values can be chosen onceand for all (i.e. q0 ¼ 0:01 and b1 ¼ 500). These values should hold for other approaches as well although alter-native values or even continuation strategies may result in improved convergence for some cases.

3. Topology optimization formulation

In density-based topology optimization the topology is typically described by element-wise constant densityvariables xe interpolating between solid (xe ¼ 1) and void (xe ¼ 0) regions. The optimization problem is for-mulated as:

min06x61

c xð Þ

s:t: r x; uð Þ ¼ 0

v xð Þ � v�ð5Þ

where x is the design variable vector, c xð Þ the objective function to be minimized, r the residual force vector ofthe structural equilibrium, u the equilibrium displacement vector and v xð Þ and v� the actual and the prescribedmaterial volume fractions, respectively.

Using a projection filtering approach [12], the density field corresponding to the scalar function w yð Þ intro-duced in the previous sections is obtained by convoluting the design field x yð Þ with a filter function w yð Þ and inthe discrete case the operation is given as

exe ¼P

j2New yj

� �vjxjP

j2New yj

� �vj

ð6Þ

where xj is the design variable associated with element j; yj is the center of element j; vj is corresponding vol-ume, N e is the neighborhood of element e within a certain filter radius r specified by N e ¼ fjjkyj � yek 6 rg,and w yj

� �is the weighting factor defined as w yj

� �¼ r � kyj � yek.


The smoothed Heaviside projection in Eq. (2) is approximated here with the help of a tanh function [12] as

xe ¼tanh bgð Þ þ tanh b exe � gð Þð Þtanh bgð Þ þ tanh b 1� gð Þð Þ ð7Þ

where b controls the sharpness of the projection. As b!1 the expression given by Eq. (7) approaches thediscrete Heaviside projection. Note that the projection Eq. (7) is associated with the density projectionapproach for general topology optimization and has nothing to do with the similar looking projection func-tion Eq. (2) which controls the energy interpolation scheme as proposed in this paper.

3.1. Finite element formulation

The displacements of the structures considered here are large and a linear formulation is not capable ofrepresenting the real physical behavior. Therefore, the total Lagrangian finite element formulation is utilizedto model the deformation in solid regions. The nonlinear strains are given in tensor form as

E ¼ 1

2FT F � I� �

ð8Þ

where I is the unit tensor, F denotes the deformation gradient tensor, defined as F ¼ I þ @u=@y and the super-script T indicates the transpose. The conjugate stress tensor (the second Piola–Kirchhoff stress tensor) isdefined as

S ¼ @/ uð Þ@E

ð9Þ

here / uð Þ is a stored elastic energy density function (see later).Using a finite element discretization, the structural equilibrium in Eq. (5) is written as,

r ¼ f ext � f int ¼ 0 ð10Þ

where f ext is the external nodal load vector and f int ¼P

efinte is the internal nodal load vector. In this paper, we

assume that the external forces do not depend on the displacement u or on the design variables x. Based on theenergy interpolation scheme in Eq. (1), the element internal force vector is expressed by

f inte ¼

@R

Xe/e ueð Þ dv

@ueð11Þ

The equilibrium, Eq. (10), is solved using the Newton–Raphson method with the incremental equation givenas

K tDu ¼ r ð12Þ

where K t is the tangent stiffness matrix defined as

K t ¼ �@r=@u: ð13Þ

The nodal displacement vector is updated by u ¼ uþ Du. The detailed calculation of K t and f int can be derivedfollowing e.g. [28] and will not be stated here.

3.2. Hyperelastic material models

Several elastic material models described in details below are utilized in this paper. These are: the St.Venant–Kirchhoff, the modified St. Venant–Kirchhoff and a modified neo-Hookean material model. Thestored elastic energy density in the St. Venant–Kirchhoff model is expressed as:

/ ¼ 1

2kE2

kk þ lEijEij ¼1

8k IC � 3ð Þ2 þ l

4I2

C � 2IIC � 2IC þ 3� �

ð14Þ


where Eij are the components of the non-linear strain tensor from Eq. (8), k and l are the Lame parameters,

IC ¼ tr Cð Þ is the first invariant of C , with C ¼ FT F as the right Cauchy strain, and IIC ¼ tr Cð Þ2 � tr C2� ��

=2

is the second invariant of C . The relations between Young’s modulus E, Poisson’s ratio m and the Lameparameters are k ¼ mE=ð1� m2Þ and l ¼ E=ð2ð1þ mÞÞ.

The stored elastic energy density under small deformation is similarly given by

Fig. 4.clampe

/L ueð Þ ¼1

2ke2

kk þ leijeij ð15Þ

where e is the Cauchy strain tensor.The St. Venant–Kirchhoff model may fail to represent the physics under large compression since the slope

of the stress-stretch curve may go to zero or even change sign in certain compressive strain situations. Hencedifferent modifications have been proposed to circumvent the drawback of this model. In one modified St.Venant–Kirchhoff model [22], the stored elastic energy density is given as

/ ¼ 1

2k J � 1ð Þ2 þ l

4I2

C � 2IIC � 2IC þ 3� �

ð16Þ

with J ¼ det Fð Þ.Another type of material model widely applied in the literature and utilized in the examples presented later

in the paper is the modified neo-Hookean model [22] with elastic energy density given by

/ ¼ 1

2k J � 1ð Þ2 þ l

2IC � 3ð Þ � l ln Jð Þ ð17Þ

4. Test problems

In order to validate the proposed energy interpolation, a few benchmark topology optimization problemsunder large deformations are studied in this paper, i.e. the minimum compliance and minimum complemen-tary elastic work of a cantilever and a clamped beam from [13] as well as the compliant inverter mechanismfrom [14], as sketched in Fig. 4.

For the compliance minimization and mechanism design problems, the objective function can be written as

c ¼ lT u ð18Þ

1 m

0.25 m

fthickness = 0.1 mE1 = 3 GPa ν = 0.4

3 m

1 m fthickness = 0.1 m

E1 = 3 GPaν = 0.4

300 µm

150µm

koutkin fin uout

thickness = 7µm

E1 = 180 GPaν = 0.3

(a)

(b)

(c)

(a) Design domain and boundary conditions for a cantilever beam. (b) Design domain and boundary conditions for a doubled beam. (c) Design domain and boundary conditions for compliant inverter design.


where l ¼ f ext for the minimum compliance problems, and for the compliant inverter design, l is a zero vectorexcept the element corresponding to the output degree of freedom which is one. As will be seen later, thecomplementary work objective can be written as a sum over incremental values of u and hence in complexitycorresponds to the mechanism design problem. In all the design problems, there is only one design constraint,i.e. prescribed material volume fraction.

The adjoint method is utilized for computing the sensitivity of the objective, c, with respect to a design var-iable xe

@c@xe¼ @c u; xð Þ

@xeþ kT @r

@xeð19Þ

with k being the adjoint variable vector, which is obtained as

K tð ÞT k ¼ @c u; xð Þ@u

� �T

ð20Þ

where K t is the tangent stiffness at the converged solution. See additional details in [14]. For the consideredproblem, the tangent stiffness matrix is symmetric and hence the factorized stiffness matrix from the forwardproblem (Eq. (13)) can be reused for solving the adjoint problem in Eq. (20).

In the proposed model, the derivative of the nodal residual force, r, with respect to design variable, xe, iscalculated following the chain rule

@r

@xe¼ �

Xj2Ne

@f intj

@xjþ@f int

j

@cj

@cj

@xj

!@xj

@~xj

@~xj

@xeð21Þ

Based on the sensitivity analysis, the structure is iteratively updated using the Method of Moving Asymptotes(MMA) [29]. The numerical implementation of the optimization problems follows the standard procedure out-lined in many references, e.g. [12,14,17] and is not presented here.

As far as possible we use the same parameter settings as in the benchmark problem references. The voidmaterial stiffness is set to E0 ¼ 10�9E1 for all examples. We use continuation strategies for all the standardtopology optimization settings. The description and implementation of these may seem rather cumbersome,however, to ensure convergence to good solutions and make the detailed comparisons between other problemsettings easier. We remark that the two projection parameters b1 and q0 are not subjected to continuation andhence their “optimal values” should be problem independent.

In the minimum compliance design problems, numerical solutions are obtained using equally sized loadincrements with the maximum load increment being 10 kN under plane strain assumption. The base materialis Nylon with E1 ¼ 3:00 GPa and m ¼ 0:4. A continuation scheme is applied on the penalty factor, p, as in [13].It is updated with p ¼ 1þ Dp with a maximum value of p ¼ 3. The parameter p is updated every two iterationswhen p < 2:0 and updated every five iterations when p P 2:0. In order to ensure discrete designs, a continu-ation scheme is employed to increase b in Eq. (7) after the penalty factor p reaches the maximum value. b isupdated every 10 iterations using b ¼ 2� b, with a maximal value of bmax ¼ 64 and an initial value of b ¼ 4.

In the compliant inverter design, the input and output springs are set to kin ¼ 4000 Nm�1 andkout ¼ 40 Nm�1, respectively. The actuating force is fin ¼ 20 mN. Numerical solutions in this case are obtainedusing 2 equally sized load increments under plane stress assumption. The base material is Silicon withE1 ¼ 180:00 GPa and m ¼ 0:3. The penalty factor, p, is fixed to be p ¼ 3. In this case b is updated following[17]. It is updated every 20 iterations using b ¼ bþ 1 and is doubled when b P 16, with a maximal valueof bmax ¼ 32 and an initial value of b ¼ 1.

The threshold parameters in Eq. (4), are fixed to be q0 ¼ 0:01 and b1 ¼ 500 for all examples, unless other-wise stated. Whereas the best choice of q0 is discussed in the example section, we, based on extensive numericaltests, suggest to fix b1 to 500. Selecting a value lower than this seems to result in unwanted slight increases instiffness of low density regions and selecting a value larger than this makes the transition too sharp, in turnmaking the optimization problem non-smooth and jeopardizing convergence. However, the exact choice ofb1 value is not critical and we have successfully solved problems with fixed b1 values between 200 and 1000.


5. Results

In this section, the proposed model is validated by solving the aforementioned benchmark optimizationproblems. In the first part, the robustness is investigated by optimizing the cantilever, the clamped beamand the compliant inverter mechanism, where the stored energy density of the base materials is describedby the St. Venant–Kirchhoff model given by Eq. (14). In the second part, the generality of the proposed modelis demonstrated by optimizing the cantilever beam using different hyperelastic models, i.e. the modified St.Venant–Kirchhoff model in Eq. (16) and the modified neo-Hookean model stated in Eq. (17).

5.1. Cantilever beam

The first example considers optimization of the cantilever beam sketched in Fig. 4(a). The beam is sup-ported at the left edge and loaded downwards in the midpoint at the right edge. The dimensions of the beamare 1.00 m � 0.25 m with a thickness of 0.10 m. It is discretized using a mesh of 120 � 30 4-node finite ele-ments. The prescribed material volume fraction is 50%. In this problem, the updating step in the penalty factoris Dp ¼ 0:05. The optimized structures are generated using g ¼ 0:5 in Eq. (7) with a filter radius of r ¼ a=8with a being the height of the structure.

The optimized cantilever beams for different final loads are summarized in Fig. 5. Compared to the originalwork on geometrical non-linear topology optimization [13] we are, with the proposed energy interpolationscheme, able to solve problems with much higher external loads (c.f. 500 kN compared to 240 kN in the ori-ginal work), without encountering convergence problems. However, as also discussed in the original work, theoptimized structures for larger loads seem somewhat odd since the rightmost bar clearly will undergo large(rotary) deformations for small loads. This can be explained by the chosen objective function: end-compliance.If structures are wanted, that also are efficient for smaller loads, one may minimize the integrated work ofexternal forces instead [13], i.e. the complementary elastic work. The corresponding objective is expressed as

c ¼ Df T 1

2u f 1ð Þ þ

Xn�1

j¼2

u f j

� �þ � � � þ 1

2u f nð Þ

!ð22Þ

where n is the number of load increments and Df the size of the increments determined by Df ¼ f =n. The sen-sitivity of the objective can be calculated using the adjoint method presented in Eq. (19).

Fig. 5. Optimized cantilever beams for different loads.


Fig. 6 shows the optimized cantilever beam for f ¼ 300 kN by minimizing the complementary elastic workwith 10 load increments. It is seen that the optimized cantilever topology is similar to the one optimized forf ¼ 144 kN using the end-compliance objective, i.e. without the hanging rightmost member. However, forhigher load levels, even the complementary elastic work objective results in isolated rightmost members. Inthe following, we keep the focus on minimizing the end-compliance since this results in the most numerically

Fig. 6. Structural deformation of the optimized cantilever beam obtained by minimizing the complementary elastic work.

f = 144 (kN) c = 21.2785 (kJ)

f = 300 (kN) c = 84.9594 (kJ)

(a)

(b)

Fig. 7. Structural deformations of the optimized cantilever beams simulated using the fully non-linear model. (a) Optimized cantileverbeam for f ¼ 144 kN. (b) Optimized cantilever beam for f ¼ 300 kN.

c = 84.9684 (kJ)

c = 84.9684 (kJ)

(a)

(b)

Fig. 8. Structural deformations of the optimized structure for f ¼ 300 kN without void regions. (a) Structural deformation using theproposed model. (b) Structural deformation using the standard fully non-linear model.


challenging problems concerning mesh distortion in low density regions. We believe that if we can prove goodbehavior for the end-compliance and mechanism test cases then the energy interpolation scheme may work forother objective functions as well.

In order to evaluate the proposed model in further details, the optimized cantilever beams for f ¼ 144 kNand f ¼ 300 kN are also simulated using the standard, fully non-linear model, see Fig. 7. The objectives of the

(a)

(b)

Fig. 9. (a) Optimized double clamped beam under small deformation. (b) Structural deformation of the optimized clamped beam whenconsidering large deformation modeling.

Fig. 10. Optimized clamped beams using different q0.


optimized structures are slightly lower when using the proposed model than using the fully non-linear model,which indicates that the proposed model results in optimized structures which are slightly stiffer. However, thedifference is less than 0.04% which is negligible. Moreover, no visible difference can be observed in thedeformed structures for f ¼ 144 kN. For the larger final load, a significant difference can be observed inthe deformation of the void regions (in white), despite similar deformation of solid regions (in black). A closeinspection reveals that excessively distorted elements appear in the void regions using the fully non-linearmodel Fig. 7 while the void regions show smooth deformations using the proposed model Fig. 5.

To evaluate the influence of the void region on the performance, the optimized structure for f ¼ 300 kN isalso simulated with the elements of xe < 0:5 removed in the finite element simulation. Fig. 8 presents thestructural deformations of the optimized cantilever beam obtained using the proposed model and the fullynon-linear model. It can be seen that both the objectives and the structural deformations are identical using

Fig. 12. (a) Optimized compliant inverter mechanism. (b) Deformation of the optimized compliant inverter mechanism using the proposedmodel, c ¼ �18:0188 lm. (c) Deformation of the optimized design using the standard fully non-linear model, c ¼ �18:0488 lm.

Fig. 11. Deformed optimized clamped beam for q0 ¼ 0:01.


both models. As expected, the optimized structure is slightly softer without void regions when simulated usingthe proposed model as well as the fully non-linear model. The objective difference with and without voidregions in both cases is less than 0.04% and can be considered negligible.

The deformations of the optimized beams in Fig. 5 illustrates that the proposed model results in smoothdeformations in low stiffness regions. Furthermore, it can be seen that excessive deformations occur mainlyin the void regions and that the deformation in the solid regions for all load cases are relatively small. Hencethe St. Venant–Kirchhoff model is still valid for describing the material behavior in the solid regions (see alsoSection 5.4 for further discussions and demonstrations of this aspect).

5.2. Double clamped beam

The study in [13] has shown that the optimized topologies of the double clamped beam in Fig. 4(b) obtainedunder small deformation and large deformation are completely different due to buckling effects. The doubleclamped beam example is used to demonstrate the influence of q0 on the optimized topologies.

The beam is supported at the left and right edges and loaded downwards in the midpoint at the upper edgewith a load of f ¼ 230 kN. The dimensions of the beam are 3.00 m � 1.00 m with a thickness of 0.10 m. It isdiscretized using a mesh of 120 � 40 4-node finite elements. The prescribed material volume fraction is 10%. Abigger updating step is used in the penalty factor of Dp ¼ 0:1 to speed up the optimization procedure in thiscase. The optimized structures are generated using g ¼ 0:5 with a filter radius of r ¼ a=20.

Fig. 9 shows the optimized structure under small deformation using linear finite element analysis and thecorresponding deformation when considering the large deformation formulation. It can be seen that the opti-mized structure snaps through due to the buckling of the two compressed beams in Fig. 9(b), resulting in ahuge objective value of the optimized structure under large deformation of c ¼ 379:91 kJ.

Fig. 13. Optimized cantilever beams for different loads using modified St. Venant–Kirchhoff model.


Fig. 10 summarizes the optimized clamped beams under large deformation for different Heaviside thresh-olds q0. It can be seen that the optimized topologies are similar for q0 6 0:05 and that the objective differencesare negligible for the optimized structures obtained for q0 6 0:01. For q0 > 0:05 the influence of the linearmodeling in the void regions becomes noteworthy and actually destroys convergence as seen for q0 ¼ 0:1.The deformed state of the optimized clamped beam obtained for q0 ¼ 0:01 is shown in Fig. 11.

From this numerical study, it can be concluded that too big q0 restricts the influence of the large deforma-tion formulation on the optimized topologies. Oppositely, too small q0 cannot circumvent the numerical insta-bility due to large deformations in low stiffness regions. This is not noted for the double-clamped beamexample, however for other examples considered in the paper we found that q0 6 0:001 jeopardizesconvergence.

5.3. Compliant mechanism design

In order to demonstrate the general applicability of the proposed method, the design of a compliant inver-ter mechanism sketched in Fig. 4(c) is considered. The dimensions of the design domain are 300 lm � 150 lmwith a thickness of 7 lm. The design domain is discretized using a mesh of 160� 80 4-node finite elements.The given material volume fraction is set to be 20%. To illustrate the optimized design clearly, only the defor-mations of the solid regions are presented.

Fig. 12 shows the optimized compliant inverter generated using g ¼ 0:5 with a filter radius of r ¼ a=10. Theoptimized design shows hinge connections, i.e. low stiffness element connections and length scale cannot beobserved in the optimized design. This can be alleviated using the robust formulation in [12]. Howeverthis is not important for demonstrating the proposed approach and is left out for simplicity. Fig. 12(b) and

Fig. 14. Optimized cantilever beams for different loads using modified neo-Hookean model.

Fig. 15. Influence of hyperelastic material models on the performance of optimized cantilever beams. Deformations of the optimizeddesigns simulated using three different models are presented in each row, where c0 indicates the original objective of the optimized designsin each row.


(c) present the deformation of the optimized compliant inverter mechanism simulated using the proposedmodel and the standard fully non-linear model, respectively. Similar to the cantilever beam problem, thedeformations of the optimized design in Fig. 12(b) and (c) demonstrate that the deformed states of the solidregions using the proposed model and the standard fully non-linear are close. The objective using the proposedmodel is c ¼ �18:0188 lm, and the one using the standard fully non-linear model is �18:0488 lm. The objec-tive difference is 0.2%.

5.4. Cantilever beam optimized using hyperelastic material models

In this subsection, the applicability of the proposed model to hyperelastic material models is validated usingthe minimum compliance problem of the cantilever beam presented in Fig. 4(a).

Fig. 13 summarizes the deformed optimized structures for different loads using the modified St. Venant–Kirchhoff model in Eq. (16). Fig. 14 presents the deformed optimized structures using the modified neo-Hook-ean model in Eq. (17). It can be observed that different hyperelastic models may result in different optimizedtopologies. However, the optimized structures perform almost equally well. Only small differences are seen interms of objectives. This indicates that even though different elasticity models may result in similar finaldesigns, subtle differences during the optimization process may cause convergence to different (but similarlyperforming) local minima.

In order to investigate the influence of various hyperelastic material models on the performance of the opti-mized cantilever beams, the cantilever beams optimized using three different material models for f ¼ 500 kNare considered. The three optimized structures simulated using the three different material models are shown inFig. 15. Hence each row shows one optimized structures modeled by all three models. Objective values are


given as compliance (c) values, with c0 indicating the optimized objective. As expected, all the optimizeddesigns perform slightly different when simulated using different material models. It can be seen that the opti-mized structure based on the modified St. Venant–Kirchhoff model in the second row performs better than theothers when simulated using the same hyperelastic model for all the three material models. This observationimplies that the optimized topologies in rows 1 and 3 are local minima.

Numerical instabilities are observed using the St. Venant–Kirchhoff model when optimizing structures forloads for f > 500 kN. We observe that the numerical instability is associated with intermediate density ele-ments undergoing large compressive strains as could be expected with this material model. In contrast, thecantilever beam can be optimized for higher loads using the two other hyperelastic models. Fig. 16 showsthe optimized cantilever beams for f ¼ 1000 kN using the modified St. Venant–Kirchhoff model and the mod-ified neo-Hookean model.

Apart from the reduced stability of the St. Venant–Kirchhoff model, our results indicate that the resultingoptimized topologies are quite independent on the hyperelastic model used. In order to understand this aspectin more details we perform an additional study on one of the optimized topologies. First, Fig. 17 (top) shows

Fig. 17. Histogram of the principle stretches in the optimized design for f ¼ 500 kN. Top: Second Piola–Kirchhoff stress versus principlestretch under homogeneous uniaxial deformation. Bottom: Histogram of the principle stretches in the optimized design from Fig. 15(upper right).

(a) (b)

Fig. 16. Optimized cantilever beams for f ¼ 1000 kN. (a) Optimized design using modified St. Venant–Kirchhoff model. Top: Optimizeddesign; Bottom: Structural deformation. (b) Optimized design using modified neo-Hookean model.


the stress-stretch curve for the different hyperelastic models when the base material undergoes homogeneousuniaxial deformation (stretch is here defined as k ¼ ðl� l0Þ=l0). Here it is clearly seen that the St. Venant–Kirchhoff curve flattens for decreasing stretch values. The histogram in Fig. 17 (bottom) displays the principlestretches in all the elements with xe 0:5 of the deformed, optimized design from the upper right corner ofFig. 15 (i.e. the cantilever optimized using the St. Venant–Kirchhoff model and modeled using the neo-Hookean model). It can be seen that all the principle stretches of the solid structure are close to one and withinthe range of 0:8; 1:2½ �. In this interval, the stress-stretch curves for the different models are almost coinciding.Several conclusions can be drawn based on this study: (a) Even though the optimized structures undergo verylarge displacements, the material stretches in the solid parts are moderate, indicating that material non-lineareffects play a small role; (b) The histogram shows a “double Gaussian” shape which is to be expected since noelements should have unity stretch (i.e. are unloaded) in an optimized structure; (c) Apart from stability issueswith the St. Venant–Kirchhoff model for very large loadings, the particular hyperelastic model is unimportantfor the problems considered here; (d) If the low density regions were modeled by the non-linear hyperelasticmodel and not by the linear model as proposed here, the choice of model would play a big role.

6. Conclusion

A new energy interpolation scheme is proposed to alleviate the numerical instability induced by excessivedeformation in low stiffness regions during topology optimization and fictitious domain analysis of structuresundergoing large displacements. The stored elastic energy density is interpolated between the stored elasticenergy density under large deformation and the one under small deformation using a threshold projectionfunction. In low stiffness regions, the stored elastic energy density is modeled using small deformation theory,and in the rest of the elements, the stored elastic energy density is computed using a large deformation elas-ticity formulation.

The proposed energy interpolation scheme is validated by several benchmark optimization problems forlarge deformation. Cantilever, clamped beam and the compliant inverter mechanism designs are demonstratedusing the proposed model. Numerical results illustrate that the proposed model leads to smooth deformationsin low stiffness elements and results in stable finite element simulations. Optimized topologies for cantileverbeams are obtained for loads much higher than any results found in the literature. Moreover, numerical exam-ples using hyperelastic material models demonstrate that the proposed energy interpolation scheme can effec-tively be combined with different hyperelastic material models, which can be used for optimizations for verylarge displacements. We remark that even though the structures undergo large displacements, the strains insolid regions are still relatively small. Hence different choices of hyperelastic material laws do not have a note-worthy influence on optimized device performance. Even though the proposed energy interpolation scheme isvalidated using only 2D examples, it should be applicable to the 3D case without further modification.

The proposed scheme is based on two parameters that control the sharpness and transition value of theinterpolation, namely b1 and q0, respectively. Although these values may be tuned for specific cases to getthe best optimization performance, we find that fixed values of b1 ¼ 500 and q0 ¼ 0:01 result in good and sta-ble convergence for all the examples that we considered. This makes the scheme simple to implement and indi-cates that the scheme should be simple to apply to other optimization techniques and fictitious domainapproaches.

Acknowledgments

This work was financially supported by an ERC Starting Grant (INNODYN), and by Villum Fondenthrough the NextTop project.

Appendix A. Comparison between two interpolation approaches

In [27], a somewhat similar approach was proposed based on direct interpolation of the element residual. Inthis approach, the element residual is interpolated using a homotype between the geometrically nonlinear andlinear elastic residuals, written as

Fig. 18(A.2).


re ¼ aerNLe ueð Þ þ 1� aeð ÞrL

e ueð Þ; 0 6 ae xeð Þ 6 1 ðA:1Þ
where rNL
e is the standard residual of the nonlinear finite element problem, rLe is the residual of the correspond-

ing linear finite element, a is the homotype parameter. Eq. (A.1) can be rewritten in terms of element elasticenergy density as

/e ¼ Ee ae/ ueð Þ þ 1� aeð Þ/L ueð Þð Þ; 0 6 ae xeð Þ 6 1 ðA:2Þ
In this formulation, the element stiffness is correspondingly interpolated between the nonlinear element tan-gent stiffness matrix and the linear element stiffness matrix following the same interpolation scheme as in Eq.(A.1). The conclusion in [27] stated that the choice of a is nontrivial and can vary from problem to problem.
In order to compare the proposed approach in Eq. (1) with the one in Eq. (A.2), we here set the homotypeparameter, ae, in Eq. (A.2) identical to ce in Eq. (1). Close inspection of Eqs. (1) and (A.2) shows that these twoapproaches are identical for the elements with ce ¼ 1 or ce ¼ 0, but that there are significant differencesbetween these two approaches for elements with 0 < ce < 1. When the elements with 0 < ce < 1 undergo largedeformation, the approach in Eq. (A.2) can still lead to a negative determinant of the deformation gradient, J,when elements are inverted. In contrast, our approach in Eq. (1) can, to certain extent, circumvent such a sit-uation by rescaling the element displacement using ce, hence outperforming Eq. (A.2) for the chosen param-eters. Due to the nonlinearity of Eq. (1), we believe it is impossible to build an analytical relation between ae

and ce such that Eq. (A.2) performs identically to Eq. (1).To numerically demonstrate the difference between these two approaches, the cantilever beam is optimized

using Eq. (A.2) for k ¼ 500 kN. Fig. 18 presents the structural deformations obtained using the two interpo-lation approaches for a selected intermediate design. It can be seen that these two approaches lead to differentdeformations in the low stiffness regions. The Newton–Raphson iterations do not converge when using Eq.(A.2), due to excessively distorted elements with 0 < ce < 1, as shown in the subplot in Fig. 18(a). In contrast,the numerical simulation using Eq. (1) leads to a converged numerical solution as shown in Fig. 18(b).

(a)

(b)

. Comparison between two different interpolation approaches. (a) Structural deformation using the interpolation scheme in Eq.(b) Structural deformation using the interpolation scheme in Eq. (1).


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