honeycomb wachspress ﬁnite elements for …...struct multidisc optim (2009) 37:569–583 doi...

Struct Multidisc Optim (2009) 37:569–583DOI 10.1007/s00158-008-0261-4

RESEARCH PAPER

Honeycomb Wachspress finite elementsfor structural topology optimization

Cameron Talischi · Glaucio H. Paulino · Chau H. Le

Received: 19 November 2007 / Revised: 7 February 2008 / Accepted: 9 March 2008 / Published online: 22 May 2008© Springer-Verlag 2008

Abstract Traditionally, standard Lagrangian-type fi-nite elements, such as linear quads and triangles, havebeen the elements of choice in the field of topol-ogy optimization. However, finite element meshes withthese conventional elements exhibit the well-known“checkerboard” pathology in the iterative solution oftopology optimization problems. A feasible alternativeto eliminate such long-standing problem consists of us-ing hexagonal (honeycomb) elements with Wachspress-type shape functions. The features of the hexagonalmesh include two-node connections (i.e. two elementsare either not connected or connected by two nodes),and three edge-based symmetry lines per element. Incontrast, quads can display one-node connections,which can lead to checkerboard; and only have twoedge-based symmetry lines. In addition, Wachspressrational shape functions satisfy the partition of unitycondition and lead to conforming finite element ap-proximations. We explore the Wachspress-type hexag-onal elements and present their implementation usingthree approaches for topology optimization: element-based, continuous approximation of material distrib-ution, and minimum length-scale through projectionfunctions. Examples are presented that demonstratethe advantages of the proposed element in achievingcheckerboard-free solutions and avoiding spurious fine-scale patterns from the design optimization process.

C. Talischi · G. H. Paulino (B) · C. H. LeDepartment of Civil and Environmental Engineering,University of Illinois at Urbana-Champaign,Newmark Laboratory, 205 North Mathews Avenue,Urbana, IL 61801, USAe-mail: [email protected]

Keywords Topology optimization · Checkerboard ·Wachspress interpolation functions ·Continuous approximation of material distribution ·Projection functions

1 Nomenclature

c(ρ, u) Compliance of the designci Normalizing shape function coefficientp SIMP penalty exponentq(x) Curve encompassing the points of intersection

of edgesr j Distance from the quadrature points to the

originrmin Radius of projectionw j Weight for the group of quadrature pointsCijkl Components of the elastic stiffness tensorCH

ijkl Components of the homogenized stiffnesstensor

E0 Stiffness of solid phaseNi Shape function corresponding to node iSi Set of elements sharing node iVh Set of all Y-periodic functionsVs Specified volume fractionY Periodic domainα j Angle corresponding to the quadrature groupλi(x) Straight line going through nodes i and i + 1ρe Density of element eρe

i Nodal density corresponding to node i ofelement e

χklp Characteristic displacements

ρmin Density lower bound� Extended design domain�e Regular hexagonal domain

570 C. Talischi et al.

�s Material subset of the extended designdomain

f Global load vectorf(kl) Material load corresponding to test strain klxi Coordinates of node iu Global displacement vectorε0(kl) Test strainB Strain-displacement matrixC0 Constitutive matrix of the solid phaseCe Constitutive matrix for element eK Global stiffness matrixKe Element stiffness matrixK0

e Stiffness matrix for solid reference element

2 Introduction

Topology optimization methods seek to find the opti-mal layout or topology of a fixed amount of materialthat satisfies a required set of design demands. Withsignificant advancements in the recent years and appli-cation to a wide range of practical problems, topologyoptimization has emerged as a powerful and robust toolfor the design of structural, mechanical, and materialsystems. Despite the maturity of the field, however,there remains a class of numerical issues such as thewell-known checkerboard problem that continues to bethe focus of extensive research. This paper introducesa new element for the implementation of topologyoptimization and demonstrates its effectiveness in re-moving the checkerboard pathology.

Traditionally, the topology design is formulated asa material distribution problem, in which every pointof the candidate design domain represents either a ma-terial or a void region. However, the optimal materialdistribution problem as such is ill-posed, lacking solu-tions in the continuum setting (Murat and Tartar 1985;Kohn and Strang 1986; Sigmund and Petersson 1998).The existence of solutions may be achieved through arelaxation of the solid/void formulation. For example,the homogenization method, introduced by Bendsøeand Kikuchi (1988), extends the space of admissibledesigns to include solutions with microstructural fea-tures, whose homogenized properties are used to deter-mine the mechanical behavior of the design. Likewise,the Solid Isotropic Material with Penalization (SIMP)model relaxes the original “0–1” problem by consid-ering a continuous material “density” as the designvariable and a power-law relation for interpolating thematerial properties of the intermediate densities(Bendsøe 1989; Zhou and Rozvany 1991). This inter-

polation also serves as a penalization that steers theoptimization procedure toward a final design withoutintermediate densities. For a discussion on the rela-tionship between the two methods and the physicalinterpretation of the SIMP model refer to Bendsøeand Sigmund (1999). Unfortunately, the topology opti-mization solutions with both methodologies suffer fromthe checkerboard phenomenon, where the optimizeddesigns may contain patches of alternating material andvoid elements.

The checkerboard solutions appear as a result ofinadequate or poor numerical modeling. Diaz andSigmund (1995) attributed the formation of checker-board as a local instability to the error in the finiteelement approximation. The checkerboard pattern hasan artificially high stiffness when modeled by lowerorder finite elements so it is economical in the opti-mization process. In a related investigation, Jog andHaber (1996) addressed general numerical instabili-ties in topology optimization by formulating the cor-responding mixed variational problem and concludedthat insufficient interpolation of the displacement fieldcan lead to unstable modes. Again, it was confirmedthat the degree of approximation and choice of finiteelements plays a crucial role in the appearance of nu-merical anomalies such as checkerboard. Therefore, itis expected that more accurate modeling of the mechan-ical behavior of the design would alleviate the checker-board problem. In fact, higher-order discretizationsusing quadratic displacement elements have beenshown to be more stable even though the final de-signs may exhibit mild forms of checkerboard, de-pending on the severity of the penalization (Diaz andSigmund 1995). Non-conforming elements can also givecheckerboard-free solutions since they correctly cap-ture the vanishing stiffness of checkerboard (Jang et al.2003, 2005). The drawback of using non-conformingshape functions is that they do not preserve the continu-ity of the field across elements; they allow negative fieldapproximation and may suffer from other numericalissues (e.g. lack of convergence).

The abovementioned studies were primarily con-cerned with the effects of finite element modeling onthe stability of topology optimization in the contextof element-based formulation, in which the designvariable is the uniform density assigned to each dis-placement element. It turns out that this discontinu-ous representation of the material field is conduciveto the appearance of checkerboard. Considering therepresentation of the design field, a handful of methodsof checkerboard suppression introduce explicit restric-tions on the local variation of the material density so

Honeycomb Wachspress finite elements for structural topology optimization 571

that the undesirable material layouts like checkerboardare avoided. Poulsen (2002a) used a descriptor func-tion to identify corner contacts throughout the designdomain at each iteration step and added a constraintto prohibit their formation. Expanding on this idea,Pomezanski et al. (2005) explored other possible “cor-ner contact” functions that could also eliminate thegrey checkerboards. To achieve geometric control overthe formation of checkerboard, Wavelet methods havealso been applied to topology optimization (Poulsen2002b; Yoon and Kim 2005). Sigmund and Petersson(1998) discussed the addition of slope constraints tocontrol the density gradient and concluded that this ap-proach can establish a well-behaved topology optimiza-tion procedure by arbitrarily weakening the numericalinstabilities.

Other methods have been proposed that constrainthe gradient of material field implicitly by using nodaldensities as the design variables. In one approach(Matsui and Terada 2004; Rahmatalla and Swan 2004),the continuity of the material field is enforced by us-ing finite element shape functions to interpolate thedensity throughout the design domain from nodal den-sities. As a result of this choice of density field rep-resentation, the discontinuous checkerboard patchesare naturally excluded from the design space. How-ever, other forms of numerical instabilities such as“islanding” and “layering” effects have been observedwith these formulations (Rahmatalla and Swan 2004).Alternatively, Guest et al. (2004) used a projectionfunction with an embedded length scale to extractelement densities from nodal densities. This methodhas the added effect of establishing a minimum mem-ber size and generating mesh-independent solutions.The problem of mesh-dependency, linked to the ill-posedness of the continuum problem, arises when theoptimal designs have finer members as more refinedmeshes are used. Since the checkerboard is a fine-scaled feature, it may be removed if a proper lengthscale is imposed on the optimization. Other meth-ods that address the mesh-dependency problem in-clude the perimeter control (Ambrosio and Buttazzo1993; Haber et al. 1996), density and sensitivity filters(Bruns 2005; Wang and Wang 2005), monotonicity-based method (Poulsen 2003), regularized density con-trol (Borrvall and Petersson 2001b), and more recentlythe morphology-based techniques (Sigmund 2007). Al-though these methods have the desirable effects ofgenerating mesh-independent solutions and in somecases improved convergence, there remains interest inobtaining checkerboard-free solutions without impos-ing any further constraints. The topology designs based

on these approaches may be very sensitive to the choiceof parameters (e.g. filter characteristics) and can poten-tially augment the physical model and the optimizationprocess.

It is evident from the above discussion that theapproximation of the two distinct fields of displace-ment and density greatly influences the stability ofthe topology optimization problem. In this work, weaddress the checkerboard issue by introducing theWachspress hexagonal element which possesses de-sirable characteristics in representing both fields: thehexagonal mesh prohibits one-node connections andsubsequently checkerboard patterns, while the interpo-lation functions of the Wachspress element eliminatethe appearance of spurious fine-scale patterns fromthe design optimization. Thus, checkerboard-free solu-tions are obtained without any further restrictions orfiltering.

Topology optimization with honeycomb meshes hasalso been explored by Saxena and Saxena (2007) andLangelaar (2007). For two-dimensional problems, thediscretization is constructed using lower order finiteelements in both cases: Saxena and Saxena (2007) spliteach hexagon into two quads while Langelaar (2007)uses the union of six triangles to achieve the desireddiscretization. We note that the choice of dividing thehexagonal elements in such an approach is not unique.For instance, the hexagonal cells can be either splitvertically, or along the left or right diagonal and thisintroduces ambiguity in the finite element discretization(see Fig. 3 of Saxena and Saxena 2007). Moreover, thisapproach is limited to constant element density formu-lations since T3/T3 and Q4/Q4 elements suffer fromislanding/layering instabilities. Our proposed approachof using Wachspress shape functions circumvents suchissues and can be readily extended for continuous den-sity representations as it defines an actual finite element(see Section 4). The use of Wachspress shape func-tions was first presented in the Multiscale and Func-tionally Graded Materials (M&FGM 2006) Conference(Talischi et al. 2008).

The remainder of the paper is organized as follows:in the next two sections, we discuss the geometricproperties of the new element (Section 3) and the con-struction of Wachspress shape functions (Section 4). InSection 5, we present the numerical integration schemefor the new element. Next we outline the topology opti-mization formulation for the compliance minimizationproblem with different material field representations.In Section 7, we address the stability of the hexagonalWachspress element by investigating its susceptibilityto fine-scale patterns. We show numerical results for


the new implementation in Section 8 to confirm itsrobustness. Finally, we conclude the paper with someremarks in Section 9.

3 Hexagonal elements and the role of meshing

Before discussing the construction and properties ofthe Wachspress hexagonal element, it is constructive tomake a few remarks regarding the role of meshing intopology optimization. First we note that even thoughit is possible to use a non-uniform mesh, it is customaryto model topology optimization problems with uniformmeshes. When employing non-uniform meshes, greatcare must be taken to avoid favoring any part of the de-sign domain because, a priori, one does not know wherethe final solution will lie. For example, in adaptiveschemes where one attempts to obtain high resolutionsolutions inexpensively, the criteria for adaptive refine-ment or de-refinement of the mesh becomes a criticalissue in correctly capturing the optimal solution (Mauteand Ramm 1995; Costa and Alves 2003; Wang 2007).Moreover, the greatest portion of the computationalcost in some topology optimization problems, such ascompliance minimization, is due to solving the equi-librium equations and a uniform mesh eliminates theneed for repeated computation of various local stiffnessmatrices (Borrvall and Petersson 2001a).

If we restrict ourselves to uniform meshes, there areonly three possible regular tessellations in two dimen-sions, namely those generated by equilateral triangles,squares, and hexagons (see, for example, Chavey 1989).Since the first two have been widely utilized in topologyoptimization, it is reasonable to explore the other pos-sibility as a feasible alternative. In fact, we recognizethat the hexagonal tessellation is distinguished from theother two in that it does not allow for corner contacts.That is, two connected hexagonal tiles must share anedge and have two common vertices (see Fig. 1). Con-sequently, unlike triangular and quadrilateral grids, the

Fig. 1 Quadrilateral meshes can display one-node connections,while in a hexagonal mesh two connected elements always sharetwo nodes through an edge connection

Fig. 2 The Q4 element has two edge-based symmetry lines, whilethe hexagonal element has three edge-based symmetry lines

hexagonal tessellation, by the virtue of its geometry,constrains the material layout and naturally excludesthe unwanted formation of checkerboard and one-nodehinges. Note that higher-order triangular and quadri-lateral elements may suffer from one-node hinges eventhough the more accurate approximation of the dis-placement field may mitigate the checkerboard prob-lem. Employing hexagonal meshes, on the other hand,simply eliminates the possibility of checkerboard andone-node hinge formations without the need for impos-ing any further restrictions. Another appealing featureof the hexagonal element is that it has more lines ofsymmetry per element compared to the triangular andsquare elements and, consequently, suffers from lessdirectional constraint and allows for a more flexiblearrangement of the final layout in the optimizationprocess (Fig. 2).

We must point out that in order to model a domainwith straight boundaries using a hexagonal mesh, it isnecessary to insert one layer of triangular and quadri-lateral elements along the boundary (see Fig. 3). Lin-ear triangular (T3) and bilinear quadrilateral elements(Q4) are used in this study. The difference in elementsize must be considered when enforcing the volumeconstraint, and the related parameters must be adjustedaccordingly. Since only one layer of these elements isneeded to straighten the boundary, their effect on theoptimization procedure is expected to be negligible. Infact our numerical experiments show that the optimalsolutions remain qualitatively unchanged when theseelements are placed around the boundary.

4 Wachspress shape functions

In this work, we adopt Wachspress rational interpo-lation functions for the proposed hexagonal element.Wachspress introduced general interpolants for convexpolygons, and his pioneering work provided a basis forfurther development of polygonal finite element for-mulations (Wachspress 1975). Wachspress interpolants


Fig. 3 Domain discretizationwith Wachspress hexagonalmesh: a domain without“boundary” elements bdomain with “boundary”elements (standard quads andtriangles)

a b

were developed using concepts of projective geometryand are the lowest order functions that satisfy theconditions of boundedness, linear precision, and globalcontinuity (Warren 2003; Sukumar and Malsch 2005).Although the Wachspress method can be extended toobtain higher order shape functions, the element exam-ined in this work is a first-order element.

For an n-sided polygon, the Wachspress shape func-tions Ni for i = 1,...,n are given by the ratio of twopolynomials, with degree (n – 2) for the numerator, and(n – 3) for the denominator (e.g. Dasgupta 2003):

Ni (x1, x2) = Pn−2 (x1, x2)

Pn−3 (x1, x2)(1)

In the following paragraphs, we discuss a geometricconstruction of these shape functions based on the alge-braic equations of the edges of the polygonal domain.Alternatively, it is possible to compute the coefficientsof the numerator and the denominator of Wachspressfunctions symbolically (Dasgupta 2003) or numerically(Dalton 1985).

Let �e denote the regular hexagonal domain (seeFig. 4). The shape function Ni, corresponding to node i,is given by:

Ni (x) = ciλi+2 (x) λi+3 (x) λi+4 (x) λi+5 (x)

q (x)(2)

where λi+1(x) = 0 represents the straight line goingthrough nodes “i” and “i + 1” while q(x) = 0 is theequation of the circle encompassing the points of inter-section of the extensions of the edges. It is understoodthat λ7 = λ1, λ8 = λ2 and so on. In other words, thenumerator is the product of the equations of the edgesnot intersecting the given node. The ci coefficient is anormalizing factor, which is given by:

ci = q (xi)

λi+2 (xi) λi+3 (xi) λi+4 (xi) λi+5 (xi)(3)

Here xi represents the nodal coordinates. A typicalconforming shape function is shown in Fig. 5.

Wachspress rational shape functions satisfy the nec-essary conditions for conforming Galerkin approxi-mations (Sukumar and Tabarraei 2004; Sukumar andMalsch 2005). First, these shape functions are bounded,non-negative and form a partition of unity:

6∑

i=1

Ni (x) = 1, 0 ≤ Ni (x) ≤ 1 (4)

Since the Wachspress shape functions are non-negative, they can be used to interpolate the densityfield (see discussion on CAMD in Section 6). This isnot possible with higher order (e.g. Q8 and Q9 quads)or non-conforming elements. Furthermore, they exhibitthe Kronecker-delta property which simplifies applyingthe necessary boundary conditions:

Ni(x j

) = δij ={

0, i �= j1, i = j

(5)

Also, these shape functions can reproduce a linearfunction (exhibit linear precision), and thus satisfy the

Fig. 4 Hexagonal element domain illustrated for the construc-tion of Wachspress shape functions


Fig. 5 Typical Wachspress shape function (the value of the shapefunction is raised on the edges for better visualization)

sufficient condition of convergence for second-orderpartial differential equations:

6∑

i=1

Ni (x) xi = x (6)

Finally, the Wachspress shape functions provide C0

continuous field approximations over the domain, andthus lead to a conforming representation. The elementperformance in solving second-order boundary-valueelliptical problems has been studied by Gout (1985),who compared this element with polynomial finite el-ements of the same degree.

5 Numerical integration for hexagonal elements

For uniform meshes, the constant element density im-plementation using the SIMP model requires the com-putation of the stiffness matrix only once, while theCAMD approach (see the following section for moredetails) requires that the element stiffness matrices beevaluated at each iteration step. Therefore the effi-ciency of the integration method must be consideredin choosing the appropriate scheme. One possibilityis to partition the element into triangular regions andapply the commonly used quadrature rules on eachtriangle (Sukumar and Tabarraei 2004). An alternativeis to use the quadrature rules developed for polygonaldomains, specifically the fully symmetric quadraturefor regions with regular hexagonal symmetry given byLyness and Monegato (1977). We have adopted the

Fig. 6 Schematic illustrationof the quadrature rule for theregular hexagonal element:integration points are shownwith “x” marks

second approach since it is more practical and uses theleast number of quadrature points for a given degree ofaccuracy.

The method is illustrated in Fig. 6. The quadraturerule is invariant under 60˚ rotation due to the hexag-onal symmetry of the integration region, and may beexpressed in the following form:

∫

�e

f d� ≈ w0 f (0, 0) +N∑

j=1

6∑

i=1

w j f(

rj, α j + π i3

)(7)

Here w j represents the weight corresponding to thequadrature point at distance r j from the origin andangle α j + π i

3 from the horizontal axis for each 1 ≤i ≤ 6. The list of quadrature points for various valuesof j can be found in the original reference (Lyness andMonegato 1977), and for completeness, are providedin Table 1 of the Appendix. In our implementation,we used N = 1 (i.e. 7 quadrature points includingthe origin) which corresponds to degree 5 polynomialaccuracy.

6 Topology optimization formulation

The performance of the proposed hexagonal elementis assessed through the implementation of benchmarkcompliance minimization problems. In this class ofproblems, the objective is to find the least compli-ant (i.e. stiffest) layout of a fixed volume of mater-ial within a pre-defined design domain, subjected togiven traction and displacement boundary conditions.As mentioned before, the topology or layout of thestructure is commonly described by material “density”design variables. By convention, density value of oneat any point in the design domain signifies a materialregion while the voids are represented by zero density.Subsequently, the designed structure �s is defined asthe material subset of the design domain �:

ρ (x) ={

0, x /∈ �s

1, x ∈ �s(8)


As mentioned before, the problem is relaxed to allowfor continuous variation of density in [ρmin, 1]. Placingthe positive lower bound ρmin helps prevent the sin-gularities of the global stiffness matrix. In this work,we use SIMP as the material model which gives thefollowing power-law relation to define the stiffness ofintermediate densities:

E (x) = ρ (x)p E0, p > 1ρmin ≤ ρ (x) ≤ 1

(9)

where E0 denotes the stiffness of the solid phase. Withvalue of p greater than 1, the stiffness of the interme-diate densities becomes small compared to their con-tribution to total volume of the structure, making themunfavorable in the optimization process. Therefore, thispenalization steers the optimization process to a 0–1design.

Using this density parameter as the design vari-able, the minimum compliance problem in the discreteform is formulated as (see, for example, Bendsøe andSigmund 2003):

minρ,u

c (ρ, u) = fTu

s.t. K (ρ) u = f∫�

ρdV = Vs

(10)

Here c(ρ, u) is the objective function (i.e. the com-pliance of the structure) and f and u are the globalforce and displacement vectors. Moreover, K repre-sents the global stiffness matrix, which is dependenton the density distribution. The parameter Vs is thespecified maximum volume of structural material.

In order to solve this optimization problem, we mustchoose a proper discretization of the design field. Weconsider the following three different approaches forimplementation of the Wachspress hexagonal element:

1. Element-based2. Continuous Approximation of Material Distribu-

tion (CAMD)3. Projection Method to achieve minimum length

scale

Although these approaches do not exhaust the pos-sible discretizations of the density field, we have lim-ited our investigation of the Wachspress element tothese cases in order to assess its performance againstthe corresponding numerical problems, namely thecheckerboard and islanding/layering instabilities, andmesh-dependency.

6.1 Element-based approach

In the element based approach, a uniform densityparameter ρe is assigned to each displacement finite

element. The element densities become the design vari-ables, and their sensitivities are calculated using theadjoint method:

∂c∂ρe

= −uTe

∂Ke

∂ρeue = −pρ p−1

e uTe K0

eue (11)

As discussed previously, the element-based im-plementation using linear triangular and bilinearquadrilateral displacement elements suffer from thecheckerboard.

6.2 Continuous Approximation of MaterialDistribution (CAMD)

Alternatively, we can define the design parameters tobe the nodal densities, from which the density throughthe domain is interpolated. An appealing feature ofthis density parameterization is that, irrespective of theinterpolation scheme, the local variation of density isrestricted. Since adjacent elements share nodal densi-ties, the change from solid to void must occur acrossat least one element, thus making the checkerboardformation impossible. We consider two possible in-terpolation schemes. Based on the concept of gradedelements (Kim and Paulino 2002; Silva et al. 2007), weuse shape functions to obtain the density within eachelement and throughout the design domain:

ρ(x) =n∑

e=1

6∑

i=1

Nei (x) ρe

i (12)

Here ρei denotes the nodal density of element e, which is

taken to be coincident with the corresponding displace-ment node (Fig. 7). Incidentally, this condition is notnecessary and one may explore the cases where the dis-placement and density meshes are not coincident. Thisapproach for topology optimization is referred to as theContinuous Approximation of Material Distribution

a bFig. 7 A schematic illustration of the displacement and densityapproximation: a H6/U element b H6/H6 element (CAMD).Larger circle represents displacement nodes, while the smallercircle represents the density design variable


(CAMD; Matsui and Terada 2004). In an investigationby Rahmatalla and Swan (2004) it was discovered thateven though the solutions with Q4/Q4 do not exhibitthe checkerboard patterns, they may suffer from othernumerical instabilities such as “islanding” and “layer-ing.” This observation is also confirmed by Jog andHaber’s (1996) study, in which they determined theQ4/Q4 implementation to be unstable.

The sensitivities of the objective function with re-spect to the nodal densities in the CAMD implemen-tation can be computed as follows:

∂c∂ρe

i= −

∑

e∈Si

uTe

∂Ke

∂ρie

ue (13)

Here Si is the set of all elements sharing node i. Theelement stiffness matrix is given by:

Ke =∫

�e

⎛

⎝6∑

j=1

Nejρ

ej

⎞

⎠p

BTC0Bd� (14)

where B denotes the strain-displacement matrix and C0

is the constitutive matrix of the solid phase. Using thisrelation, we can compute the sensitivity of the stiffnessmatrix with respect to the nodal densities:

∂Ke

∂ρei

=∫

�e

pNei

⎛

⎝6∑

j=1

Nejρ

ej

⎞

⎠p−1

BTC0Bd� (15)

6.3 Projection method

The other scheme explored in this work is the use ofprojection functions with a fixed length scale. Proposedby Guest et al. (2004) for Q4 discretization, the methodalso uses nodal densities as design variables, and assignsto each element a uniform density based on a projec-tion of nodal densities surrounding that element. Bychoosing a fixed physical radius rmin independent of themesh, one can obtain mesh-independent designs withprescribed minimum member size. The element densityis given by a weighted average of nodal densities thatare within radius rmin from the centroid of that element:

ρe =∑

i∈Sewiρi∑

i∈Sewi

(16)

Here we have implemented linear weight functions(Fig. 8), which are given by:

wi = rmin − ri

rmin, ri ≤ rmin (17)

where ri is the distance of the node i from the centroidof element e. However, other weight functions can alsobe explored.

a

b

Fig. 8 Projection function: illustration of a the domain ofinfluence and b the linear weight function

We must point out that it is possible to couple theprojection scheme with the CAMD approach by apply-ing the projection function on the nodal densities in-stead of the element densities. The shape functions canbe used to interpolate the density within each element,guaranteeing a prescribed level of smoothness of thedensity field throughout the domain, while observingthe required length-scale. This is especially useful inimposing minimum length scale for topology optimiza-tion design of functionally graded structures, where itis necessary to have C0 continuity of density field tocapture the gradation of material properties. Topologyoptimization of graded structures can be accomplishedby means of the FGM-SIMP (Functionally GradedMaterial–Solid Isotropic Material with Penalization)formulation by Paulino and Silva (2005).

7 Discussion on stability of Wachspress elements

As discussed before, two elements in a hexagonal mesheither share one edge or are not connected at all. There-fore, the geometric nature of the hexagonal Wachspress


Unit Cell

Pattern 1

Pattern 2

Pattern 3

Pattern 4

Checkerboard

Pattern

Fig. 9 Small scale patterns of hexagonal elements and their corresponding unit cells


element can eliminate the severe error in the densityrepresentation that is observed with the Q4 elementsin the form of checkerboard. However, undesired smallscale patterns may still be possible with the hexagonalmesh. In this section, we demonstrate that with theproposed formulation, such patterns do not appear aseasily as the checkerboard of Q4 elements, establish-ing the Wachspress element as a more stable elementfor topology optimization. We accomplish this task bystudying the overall stiffness of candidate patterns ofhexagonal elements using numerical homogenization.As discussed by Diaz and Sigmund (1995), the ap-pearance of checkerboard in topology stiffness designcan be linked to the poor finite element modelingthat overestimates its stiffness. As a result of this arti-ficially high stiffness, the checkerboard is stiffer thanother arrangements of material and is favored in theoptimization process. Following this reasoning, we in-vestigate the susceptibility of topology optimizationformulation with Wachspress elements to similar anom-alies by considering the behavior of the possible finescale patterns in the hexagonal mesh (see Fig. 9). Inthis section, we discuss the numerical procedure for de-termining the homogenized stiffness of these patternsand compare the results with those obtained for the Q4element.

The overall stiffness of the patterns may be rep-resented by the homogenized stiffness of repetitiveunit cells. According to the theory of numerical ho-mogenization (see, for example, Bendsøe and Sigmund2003), the homogenized stiffness is calculated as:

CHijkl = 1

|Y|∫

Y

(Cijkl − Cijpq

∂χklp

∂yq

)dY (18)

where Cijkl are elastic stiffness coefficients at a givenpoint in the unit cell; CH

ijkl are homogenized stiffnesscoefficients; Y is the periodic domain, which is the areaof the unit cell shown in Fig. 9; and χkl

p are characteristicdisplacements obtained by solving the following equa-tion:∫

Y

Cijpq∂χkl

p

∂yq

∂vi

∂y jdY =

∫

Y

Cijkl∂vi

∂y jdY, ∀v∈Vh (19)

Here Vh is the set of all Y-periodic functions.Using finite element discretization, one obtains the

characteristic displacement by solving the followingequation:

Kχ(kl) = f(kl) (20)

where K is the standard stiffness matrix; and f(kl) is thematerial load corresponding to test strain kl, calculatedfrom:

f(kl) =∑

e

∫

�e

BTe Ce (y) ε0(kl)dy (21)

In this expression, Be is the standard kinematic matrix;Ce is the constitutive matrix, and ε0(kl) is the test strain.The homogenized stiffness tensor is given by:

CHijkl = 1

|Y|∑

e

(χ0(ij) − χ(ij)) Ke

(χ0(kl) − χ(kl)) (22)

It is worth noting that for the imposition of periodicboundary conditions, we have used regular node num-bering along with multipoint constraints (e.g., Cooket al. 2002). This is different from the traditional treat-ment of periodicity that involves the use of repeatednode numbers. Such an approach in the context of

Fig. 10 Comparison betweenthe stiffness of hexagonalelement patterns,checkerboard pattern, andhomogeneous distribution ofmaterial (SIMP with p = 3 isused unless otherwise noted)


Fig. 11 Schematicrepresentation of the designdomain, loading andboundary conditions forMBB (Messerschmitt-Bölkow-Blohm) beamproblem. Notice that thenon-hexagonal elementson the boundary are eithertriangles or quads

Ω

homogenization with polygonal cells is discussed byDiaz and Benard (2003).

We study four arbitrary small scale patterns ofhexagonal mesh, which may appear in topology op-timization results. These patterns are those with po-tentially high stiffness and are shown in Fig. 9. Theresults of this investigation are illustrated in Fig. 10where the stiffness of hexagonal patterns along with thecheckerboard of Q4 elements and homogenous mate-rial distribution are plotted. Also included in the plotis the stiffness of the homogenous material distributionsubjected to no penalization. In all other cases, thepenalization parameter is taken as 3.

For each pattern, the density of “black” elements isincreased from 0 to 1. The horizontal axis shows thetotal volume fraction of the pattern. Note that each plotends at the point corresponding to the configurationshown in Fig. 9 with black elements having density ofone. Therefore, the checkerboard plot terminates atvolume fraction 0.5 while the plot for pattern 1 ends atvolume fraction of 2/3 (this pattern has 4 black elementsand 2 white elements in its unit cell). Only the resultof CH

1111is calculated. Other stiffness coefficients can becomputed in a similar manner.

As shown in Fig. 10, the black–white checkerboardpattern avoids penalization: the black–white checker-board (volume fraction 0.5), modeled by Q4 elements,has stiffness equal to the homogenous distribution withno penalization (p = 1). This is in agreement with theresults presented by Diaz and Sigmund (1995). Thepatterns of hexagonal elements, however, are not asoverly stiff as the checkerboard pattern because theirstiffness curves lie below the line of no penalization.Moreover, the homogenized stiffness of the hexagonalpatterns is closer to the stiffness of penalized homoge-neous distribution (obtained using SIMP with p = 3).These results demonstrate that the formation of un-desired small scale patterns in the results of topol-ogy optimization is alleviated when using Wachspresshexagonal elements. As we shall see in the next section,the topology optimization results obtained with theWachspress elements confirm this conclusion.

8 Numerical results

The benchmark MBB-beam problem (see, for exam-ple, Olhoff et al. 1991) is solved using the Wachspresshexagonal element and results are compared with the

Fig. 12 MBB beam designwith element-basedformulation: a–c results withQ4 elements and d–f resultswith H6/U (hexagonal)elements. Boundary elementswere added to achievedomain closure. The meshdiscretization is 60 × 20,90 × 30, 120 × 40 from top tobottom, respectively, for bothimplementations

a d

b e

c f


Fig. 13 MBB beam designwith CAMD approach:a–c results with Q4/Q4elements and d–f results withH6/H6 (hexagonal) elements.Boundary elements wereadded to achieve domainclosure. The meshdiscretization is 60 × 20,90 × 30, 120 × 40 from top tobottom, respectively for bothimplementations

a d

b e

c f

corresponding Q4 implementation. Due to the sym-metry of the problem, only half of the MBB-beam isconsidered (Fig. 11). The beam has an aspect ratio of6:1, and three levels of mesh discretization are used.The Poisson’s ratio of the material is taken to be 0.3,while Vs is 50% of volume of the design domain. Wesolved the minimum compliance problem using theMethod of Moving Asymptotes (MMA) developed bySvanberg (1987). In addition, we used a continuationmethod on the value of p to avoid converging to localminima. The value of p was gradually increased (usingincrements of 0.5) from 1 to 4 after sufficient conver-gence for each value of p. The optimization results forthe element-based and projection schemes are plottedusing the element densities ρe. For the CAMD results,the continuous density ρ(x) is shown in an averagesense: the average of ρe

i is obtained and plotted for eachhexagonal element.

a

b

cFig. 14 MBB beam design with projection approach (rmin = 0.15hwhere h is the height of the beam): a–c results with hexagonalelements. The mesh discretization is 60 × 20, 90 × 30, 120 × 40from top to bottom, respectively

In Fig. 12, the results of the element-based formu-lation for the Q4 element and the hexagonal elementfor various levels of mesh refinement are shown. Thesolutions with Q4 implementation contain patches ofcheckerboard while no such fine scale patterns are ob-served with the Wachspress implementation. Note thatno filtering technique or density gradient was imposedand thus the checkerboard-free property of the hexag-onal element is attributed essentially to its geometricfeatures and interpolation characteristics.

Figure 13 shows the results of CAMD approach forthe MBB-beam design for both Q4 and Wachspressimplementation. Note that in this case, the density anddisplacement fields are interpolated using the sameshape functions for each element discretization. We canobserve that the Q4/Q4 results suffer from spuriousislanding and layering patterns, which is in agreementwith the findings of Rahmatalla and Swan (2004). Thedesigns using Wachspress elements, however, show nosigns of such instability. Therefore, the Wachspresselement performance in this case is attributed to its in-terpolation characteristics. We note that our numericalresults for the MBB-beam agree closely with the exactanalytical solutions derived by Lewinski et al. (1994).

Finally, the results using projection scheme are pre-sented in Fig. 14. The radius of the projection rmin

is taken to be 0.15 of the height of the beam andindependent of the mesh size. We can see that de-spite the change in the level of mesh refinement, thesame design is obtained in all cases. The length scaleimposed on the optimization through rmin guaranteesmesh-independent solutions that satisfy the requiredminimum member size.

9 Concluding remarks

In this work, the checkerboard pathology in topologyoptimization is addressed and circumvented by means


of a new finite element. The proposed hexagonal ele-ment with Wachspress-type shape functions is shown topossess advantages over conventional finite elements.Geometric properties of the hexagonal element, suchas two-node connections and symmetry in three edge-based directions, are among its distinguishing features.As discussed and demonstrated by examples, the useof hexagonal elements eliminates the formation ofcheckerboard and other numerical anomalies, and pro-vides a robust and stable means for solving topologyoptimization problems.

The present approach may be extended to three-dimensional topology optimization by noting the fea-tures employed to eliminate the checkerboard andother instabilities. Such extension requires: (i) a meshthat excludes point and edge contact, i.e., a mesh inwhich two connected elements share a face; and (ii)selection of an appropriate finite element interpolationor enrichment scheme.

A few remarks regarding the use of honeycomb finiteelements are in order. Reliance on a particular finiteelement formulation may naturally impose limitationsfrom practical perspective. Such objection can also bemade to the use of nonconforming or higher orderelements. Indeed there is a trade-off: conventional el-ements like quads and triangles are widely used butsuffer from serious instabilities unless additional con-straints are imposed. The implementation of these con-straints is not always straightforward or desirable. Onthe other hand, particular formulations, such as the onepresented in this paper, may be better suited for topol-ogy optimization, even though they are less commonlyused. Moreover, there have been several recent papersin the literature (see review paper by Sukumar and

Malsch 2005) that address polygonal finite elements. Asthese elements become more popular and widespread,solutions such as the one contributed by our work havethe potential to become more popular and practical.

The honeycomb element together with the topol-ogy optimization formulation has promising exten-sions, such as design of microelectromechanical systemsand piezoelectric actuators (Carbonari et al. 2007a, b;Sigmund 2001). Future investigations include the useof higher-order Wachspress elements (see, for exam-ple, Gout 1985). In addition, the CAMD approachmay be investigated for meshes in which the displace-ment and density locations are not coincident in theelement, which would allow for a more flexible den-sity field discretization. Furthermore, nonlinear weightfunctions can be studied in conjunction with the projec-tion method (see Section 6.3). Such considerations havethe potential to lead to enhanced Wachspress elementsfor high-fidelity topology optimization.

Acknowledgements We acknowledge the support by the De-partment of Energy Computational Science Graduate Fellow-ship Program of the Office of Science and National NuclearSecurity Administration in the Department of Energy undercontract DE-FG02–97ER25308. We are grateful to Prof. KristerSvanberg for providing his MMA (Method of Moving Asymp-totes) code, which was used to generate the examples in thispaper. We acknowledge the anonymous reviewer for pointingout the references by Saxena and Saxena (2007) and Langelaar(2007). Finally, we thank Dr. Ivan F. M. Menezes for insightfuldiscussions and suggestions.

Appendix

Quadrature rule for Wachspress hexagonal element(from Lyness and Monegato 1977): Table 1

Table 1 Quadratureparameters as definedin Section 5

N j α j r j w j

1 0 0.000000000000000 0.000000000000000 0.2559523809523811 0.000000000000000 0.748331477354788 0.124007936507936

0 0.000000000000000 0.000000000000000 0.1745886843250772 1 0.000000000000000 0.657671808727194 0.115855303626943

2 0.523681372148045 0.943650632725263 0.021713248985544

0 0.000000000000000 0.000000000000000 0.1108265472286611 0.000000000000000 0.792824967172091 0.037749166510143

3 2 0.523598775598299 0.537790663359878 0.0824197053505903 0.523598775598299 0.883544457934942 0.028026703601157

0 0.000000000000000 0.000000000000000 0.0870055490948081 0.000000000000000 0.487786213872069 0.071957468118574

4 2 0.000000000000000 0.820741657108524 0.0275001856508663 0.523598775598299 0.771806696813652 0.0452489321316634 0.523598775598299 0.957912268790000 0.007459892497607


References

Ambrosio L, Buttazzo G (1993) An optimal design problem usingperimeter penalization. Calc Var Partial Differ Equ 1:55–69

Bendsøe MP (1989) Optimal shape design as a material distribu-tion problem. Struct Optim 1:193–202

Bendsøe MP, Kikuchi N (1988) Generating optimal topology instructural design using a homogenization method. ComputMethods Appl Mech Eng 71:197–224

Bendsøe MP, Sigmund O (1999) Material interpolation schemesin topology optimization. Arch Appl Mech 69:635–654

Bendsøe MP, Sigmund O (2003) Topology optimization-theory,methods and applications. Springer Verlag, New York

Borrvall T, Petersson J (2001a) Large-scale topology optimiza-tion in 3D using parallel computing. Comput Methods ApplMech Eng 190(46–47):6201–6229

Borrvall T, Petersson J (2001b) Topology optimization usingregularized intermediate density control. Comput MethodsAppl Mech Eng 190:4911–4928

Bruns TE (2005) A re-evaluation of the SIMP method with fil-tering and an alternative formulation for solid-void topologyoptimization. Struct Multidisc Optim 30:428–436

Carbonari RC, Silva ECN, Paulino GH (2007a) Multi-actuatedfunctionally graded piezoelectric micro-tools design: a multi-physics topology optimization approach. Int J Numer Meth-ods Eng (in press)

Carbonari RC, Silva ECN, Paulino GH (2007b) Topology opti-mization design of functioanlly graded bimorph-type piezo-electric actuators. Smart Mater Struct 16(6):2605–2620

Chavey D (1989) Tilings by regular polygons 2. A catalog ofTilings. Comput Math Appl 17(1–3):147–165

Cook RD, Malkus DS, Plesha ME (2002) Concepts and applica-tions of finite element analysis, 4th. edn. Wiley, New York

Costa JCA, Alves MK (2003) Layout optimization with h-adaptivity of structures. Int J Numer Methods Eng 58(1):83–102

Dalton GR (1985) Automatic indexed calculation ofWachspress’ rational finite element functions. ComputMath Appl 11(6):621–623

Dasgupta G (2003) Interpolants within convex polygons:Wachspress’ shape functions. J Aerosp Eng 16(1):1–8

Diaz A, Benard A (2003) Designing materials with prescribedelastic properties using polygonal cells. Int J Numer MethodsEng 57(3):301–314

Diaz A, Sigmund O (1995) Checkerboard patterns in layout op-timization. Struct Optim 10:40–45

Gout JL (1985) Rational Wachspress-type finite elements onregular hexagons. IMA J Numer Anal 5(1):59–77

Guest J, Prevost J, Belytschko T (2004) Achieving minimumlength scale in topology optimization using nodal designvariables and projection functions. Int J Numer MethodsEng 61(2):238–254

Haber RB, Jog CS, Bendsøe MP (1996) A new approach tovariable-topology shape design using a constraint on perime-ter. Struct Optim 11:1–12

Jang G, Jeong H, Kim Y, Sheen D, Park C, Kim M (2003)Checkerboard-free topology optimization using non-conforming finite elements. Int J Numer Methods Eng 57:1717–1735

Jang G, Lee S, Kim Y, Sheen D (2005) Topology optimizationusing non-conforming finite elements: three-dimensionalcase. Int J Numer Methods Eng 63:859–875.

Jog CS, Haber RB (1996) Stability of finite element modelsfor distributed-parameter optimization and topology design.Comput Methods Appl Mech Eng 130:203–226

Kim J, Paulino GH (2002) Isoparametric graded finite elementsfor nonhomogeneous isotropic and orthotropic materials.J Appl Mech Trans ASME 69(4):502–514

Kohn RV, Strang G (1986) Optimal design and relaxation ofvariational problems. Commun Pure Appl Math 39, Part I:1–25, Part II: 139–182, Part III: 353–377

Lewinski T, Zhou M, Rozvany GIN (1994) Extended exact least-weight truss layouts—part II: unsymmetric cantilevers. Int JMech Sci 36:399–419

Lyness JN, Monegato G (1977) Quadrature rules for regionshaving regular hexagonal symmetry. SIAM J Numer Anal14(2):283–295

Matsui K, Terada K (2004) Continuous approximation of ma-terial distribution for topology optimization. Int J NumerMethods Eng 59:1925–1944

Maute K, Ramm E (1995) Adaptive topology optimization.Struct Optim 10:100–112

Murat F, Tartar L (1985) Optimality conditions and homogeniza-tion. Proceedings of Nonlinear variational problems (Isolad’Elba, 1983), Res. Notes in Math., vol 127. Pitman, Boston,pp 1–8

Olhoff N, Bendsøe MP, Rasmussen J (1991) On CAD-integratedstructural topology and design optimization. ComputMethods Appl Mech Eng 89:259–279

Paulino GH, Silva ECN (2005) Design of functionally gradedstructures using topology optimization. Materials ScienceForum, 492–493, 435–440. Trans Tech Publications,Switzerland

Petersson J, Sigmund O (1998) Slope constrained topology opti-mization. Int J Numer Methods Eng 41(8):1417–1434

Pomezanski V, Querin OM, Rozvany GIN (2005) CO-SIMP: ex-tended SIMP algorithm with direct Corner Contact Control.Struct Multidisc Optim 30:164–168

Poulsen TA (2002a) A simple scheme to prevent checkerboardpatterns and one-node connected hinges in topology opti-mization. Struct Multidisc Optim 24:396–399

Poulsen TA (2002b) Topology optimization in wavelet space. IntJ Numer Methods Eng 53:567–582

Poulsen TA (2003) A new scheme for imposing minimum lengthscale in topology optimization. Int J Numer Methods Eng57(6):741–760

Langelaar M (2007) The use of convex uniform honeycomb tes-sellations in structural topology optimization. Proceedings ofthe 7th world congress on structural and multidisciplinaryoptimization, Seoul, South Korea, 21–25 May 2007

Rahmatalla SF, Swan CC (2004) A Q4/Q4 continuum structuraltopology optimization. Struct Multidisc Optim 27:130–135

Saxena R, Saxena A (2007) On honeycomb representation andSIGMOID material assignment in optimal topology syn-thesis of compliant mechanisms. Finite Elem Anal Des 43:1082–1098

Sigmund O (2001) Design of multiphysics actuators using topol-ogy optimization—part II: two-material structures. ComputMethods Applied Mech Eng 190(49–50):6605–6627

Sigmund O (2007) Morphology-based black and white filters fortopology optimization. Struct Multidisc Optim 33:401–424

Sigmund O, Petersson J (1998) Numerical instabilities in topol-ogy optimization: a survey on procedures dealing withcheckerboard, mesh-dependence and local minima. StructOptim 16:68–75


Silva E, Carbonari R, Paulino GH (2007) On graded elements formultiphysics applications. Smart Mater Struct 16:2408–2428

Sukumar N, Malsch EA (2005) Recent advances in the construc-tion of polygonal finite element interpolants. Arch ComputMethods Eng 11:1–38

Sukumar N, Tabarraei A (2004) Conforming polygonal finiteelements. Int J Numer Methods Eng 61:2045–2066

Svanberg K (1987) The method of moving asymptotes—a newmethod for structural optimization. Int J Numer MethodsEng 24:359–373

Talischi C, Paulino GH, Le CH (2008) Topology optimizationusing Wachspress-type interpolation with hexagonal ele-ments. In: Paulino GH et al (ed) Multiscale and function-ally graded materials 2006 (M&FGM 2006), AIP conferenceproceedings, vol 973. American Institute of Physics, Melville,New York, pp 309–316

Wachspress EL (1975) A rational finite element basis. AcademicPress, New York

Wang S (2007) Krylov subspace methods for topology optimiza-tion on adaptive meshes. PhD dissertation, University ofIllinois at Urbana-Champaign, Urbana, Illinois, USA.

Wang M, Wang S (2005) Bilateral filtering for structural topologyoptimization. Int J Numer Methods Eng 63:1911–1938

Warren J (2003) On the uniqueness of barycentric coordinates.Contemporary mathematics: proceedings of AGGM’02,pp 93–99

Yoon GH, Kim YY (2005) Triangular checkerboard controlusing a wavelet-based method in topology optimization. IntJ Numer Methods Eng 63:103–121

Zhou M, Rozvany GIN (1991) The COC algorithm, part II:topological, geometry and generalized shape optimization.Comput Methods Appl Mech Eng 89:197–224

honeycomb wachspress ﬁnite elements for …...struct multidisc optim (2009) 37:569–583 doi...

Documents