A Scalable Framework for Large-Scale 3DMultimaterial Topology Optimization with Octree-based

Mesh Adaptation

Ting Wei Chin*, Mark K. Leader*, and Graeme J. Kennedy†

Georgia Institute of Technology,School of Aerospace Engineering, Atlanta, Georgia

Abstract

Advancements in multimaterial additive manufacturing have the potential to enable thecreation of topology optimized structures with both shape and material tailoring. However,multimaterial topology optimization methods that use Discrete Material Optimization (DMO)face three technical challenges for large-scale high-resolution problems: 1) the large-scale de-sign space, since selection variables must be added for each additional candidate material;2) the treatment of numerous sparse partition of unity constraints required in some DMOparametrizations; and 3) the multimaterial design space that has more local minima than anequivalent single material design space. This paper addresses these issues by presenting aparallel, scalable analysis and design optimization framework for multimaterial topology opti-mization that optionally uses local mesh refinement using semi-structured octree meshes. Theadvantages of this framework are demonstrated by showcasing its solution and design scalabil-ity and by efficiently solving large 3D multimaterial compliance-minimization problems withboth isotropic and orthotropic material options on meshes with up to 329 million elements.For the largest case, the adaptive strategy is shown to achieve a compliance objective within1.86% with roughly 1/4 the mesh size.

Keywords: Topology optimization, Multimaterial optimization, Large-scale computation, Adap-tive mesh refinement, Multi-phase topology optimizationDeclarations of interest: None

1 Introduction

Topology optimization, first developed by Bendsøe and Kikuchi [5], is now used in a wide range ofstructural design applications including the design of aerospace structures [50, 35, 15], automotivestructures [47, 26], meta-materials and meta-structures [69, 22, 53] and MEMS devices [67, 12].Recently, advancements in additive manufacturing technologies have further enabled engineering

*Graduate Student†Assistant Professor

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Chin, Leader, Kennedy

applications of topology-optimized designs [79, 10, 31]. Additive manufacturing for both poly-mer and metallic multimaterial structures has also been rapidly maturing, enabling simultaneousshape and material tailoring [34]. As a result, there is increasing interest in multimaterial topologyoptimization methods. Multiphase and multimaterial topology optimization techniques were firstdeveloped by Sigmund and Torquato [69] for thermal coefficient design and later extended by Gib-iansky and Sigmund [33] to the design of extremal composites. Stegmann and Lund [73] developedan extension and generalization of these early multiphase methods, called Discrete Material Opti-mization (DMO), that has proven to be effective in numerous applications including multimaterialdesign and laminate design for composite structures [43, 64, 38, 39, 54, 44, 30, 29]. However,most DMO applications have focused on moderate-scale 2D problems, with fewer large-scale 3Dapplications [42].

Multimaterial topology optimization presents at least three additional difficulties beyond thoseencountered in conventional single-material topology optimization: 1) the design space is largerdue to the addition of material selection variables, 2) a large number of sparse partition of unityconstraints are often required for the most effective DMO parametrizations [38], making com-mon topology optimization algorithms including the Optimality Criteria (OC) method and Methodof Moving Asymptotes (MMA) incompatible with these problems, and (3) it is widely reportedthat the multimaterial design space is more multimodal than the equivalent single material designspace [6, 75]. To address these issues, we describe a scalable analysis and design optimizationframework for multimaterial topology optimization. Within this framework, a hierarchy of analy-sis and design meshes are created from an input CAD geometry by local refinement of an initialcoarse hexahdral mesh. To enable this local refinement, conforming octree meshes are constructedon each element in the coarse hexahedral mesh, thereby facilitating the construction of meshesfor geometric multigrid and local mesh size control around design features of interest. An S`1QPmethod with a trust region globalization is implemented in parallel and is tailored to handle thelarge number of partition of unity constraints that arise from DMO parametrizations. Finally, theadaptive refinement procedures support the capability to study whether finer meshes produce betterresults, revealing potential local optima.

2 Overview

The creation of detailed optimized 3D designs requires finite-element meshes with tens of millionsof elements or more. Several authors have developed methods to tackle large-scale 3D topologyoptimization problems for single material design [1, 2, 3], with the largest design case having 1.1billion elements [3]. Aage et al. observed that topology optimization on large-scale meshes cancapture detailed features which might otherwise be missing in lower resolution results. This scalehas yet to be achieved for multimaterial problems, where the largest cases to date have on the orderof one million elements [42].

Most meshes in topology optimization are generated with an approximately uniform elementsize since, in general, there is no a priori knowledge about the optimal topology. However, uni-form refinement on large-scale meshes produces an inverse cubic dependence between relativeresolution and the number of finite-element nodes. Alternatively, adaptive refinement techniquescan be used to refine the mesh to achieve sufficient resolution thereby breaking this inverse cubicrelationship. Figure 1 shows the number of nodes in the mesh as a function of relative resolution

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Chin, Leader, Kennedy

Aage 2017

Kennedy 2015

Aage 2014

Aage 2014

Maute 2008

Paulino 2007

Borrvall 2001

Additive

Manufacturing

Technologies

Present Work

Figure 1: Number of nodes as a function of relative resolution for topology optimization results.The results shown from this paper are all multimaterial problems.

for uniformly refined meshes (triangles) and for adaptively refined meshes (squares) for a numberof large-scale topology optimization results from the literature. These data points are taken fromreferences [76, 25, 8, 2, 42] as well as five design cases that are presented here, which are a com-bination of uniformly refined and adaptively refined meshes. Here, relative resolution is defined ascube root of the ratio of the smallest element characteristic volume to the characteristic volume ofthe domain. The blue colored band denotes the maximum relative resolution capability of currentmetallic additive manufacturing technologies 1. Even the highest relative resolution mesh usedfor topology optimization has not yet achieved the relative resolution currently available throughadditive manufacturing. Comparing the two data points from the present work that are groupedtogether, both have the same relative resolution but the adaptive mesh refinement design point hasfar fewer nodes compared to the uniformly refined design, thereby requiring less computationalresources.

Within the density-based topology optimization literature there are two common approachesto topology optimization with adaptive mesh refinement: 1) link the design and analysis meshtogether and perform the refinement steps simultaneously on both, or 2) maintain a separate designparametrization mesh and refine only the analysis mesh. Bennett and Botkin [7] pioneered thefirst approach, using adaptive mesh refinement for shape optimization to control the accuracy ofthe finite element solution for planar structures using 2D triangular elements. Around the same

1Relative resolution data obtained from 3D Systems Direct Metal Printers Series using build volume and mini-mum feature size. For example, ProX DMP 200 has minimum feature volume of 2⇥10�13 m3 and build volume of1.96⇥10�3 m3, resulting in a relative resolution of 4.67⇥10�4

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Chin, Leader, Kennedy

time, Kikuchi et al. [46] used r- and h- refinement techniques to generate meshes adaptively witha mixture of 2D triangular and quadrilateral elements. Adaptive mesh refinement can also be usedfor shape optimization with multiple loading conditions as demonstrated by Botkin [9]. In thiscase, the refinement criterion is computed independently for each load case and combined intoa single refinement indicator by finding the most refinement required from all load cases locallyfor each element in the mesh. More recently, Costa Jr and Alves [20] proposed an algorithm thatoptimized the structural layout using an h-adaptive approach with improved definition of materialboundary.

Several authors have used the second approach of keeping the design and analysis mesh sep-arate. Maute and Ramm [55] demonstrated that adaptively refining 2D quadrilateral meshes re-quired less numerical effort but produced similar resolution compared to a uniformly refined meshfor topology optimization with a surface mesh. Maute and Ramm [55] also introduced the con-cept of adaptively changing not only the size of the element but also the orientation of the fi-nite element mesh which reduces the number of optimization variables. These techniques werealso applied on combined shape and topology optimization of shell structures [56]. Other au-thors have characterized the design mesh as a density point grid without connectivity information.Guest and Smith Genut [36] used adaptive mesh refinement on a design variable field consisting ofnodal points that correspond to locations on the finite element mesh. Using a Heaviside projectionmethod, the authors were able to reduce the computational time as well as control the minimumfeature size of the structural layout. Similarly, Wang et al. [77] carried out adaptive mesh refine-ment on the density point grid with a fixed finite element mesh throughout the process. Usinga filter, the cut-off radius was adaptively modified to produce structures with smooth boundaryinterface. In the work of Wang et al. [78], both the analysis and design mesh were refined usingdifferent error control indicators, with the refinement on the analysis mesh used to improve com-putational local accuracy and refinement on the design mesh used to achieve smooth boundaries.Lambe and Czekanski [51] employed a continuous material density field and used adaptive meshrefinement to identify and refine along the boundary of the structural layout without any numericalartifacts such as checkerboarding. Chin and Kennedy [18] utilized feature-based adaptive meshrefinement in the design of thermoelastic structures.

Different indicators can be used to drive the adaptive mesh refinement process. These indica-tors often involve estimating the solution error contribution from each element. One of the morepopular error estimators was introduced by Zienkiewicz and Zhu [80], which uses the norm of thedifference between stresses interpolated from quadrature points and stresses computed from thefinite-element interpolation. Alternatively, feature-based techniques drive mesh refinement basedon the design. Stainko [72] used a filter to identify and refine elements along the solid-void bound-ary. Bruggi and Verani [11] combined two different refinement heuristics for topology optimiza-tion: one based on compliance and a second based on element density. For a compliance objective,they showed that using a combination of these two indicators produced a structural layout with afine interface between the solid and void region and a better design objective compared to an adap-tive strategy that used only a density-based indicator. de Sturler et al. [21] identified the need for adynamic adaptive mesh refinement strategy that allows for both refinement and coarsening of themesh. They found that both refinement and coarsening are required for the optimizer to generatestructural layouts that are equivalent to the optimal design on a uniformly fine mesh. Wang et al.[78] used a strain energy error indicator to refine its displacement field while a functional graytransitional region measure (GTR) to refine the density field.

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(a) Analysis mesh (b) Design mesh (c) Interpolated design variables

Figure 2: Illustration of the construction of the design parametrization. The analysis mesh iscoarsened to create the design mesh. Design variables are defined at the nodes. Density values areinterpolated back to the elements on the analysis mesh.

DMO is an effective parametrization technique for multimaterial design. Sørensen et al. [71],Sørensen and Lund [70] recently extended DMO to include both thickness and material selectionsvariables in the design optimization of laminated composites. Alternative multimaterial optimiza-tion methods have also been proposed. Bendsøe and Sigmund [6] extended the SIMP approachfor single material design to a multiple material formulation that remains a commonly used tech-nique for multimaterial design. Bruyneel et al. [14] and Bruyneel [13] demonstrated the use ofshape functions with penalization (SFP) to ply selection in laminated composite design problems.The formulation for SFP can be extended to different number of materials by selecting appropriateshape functions. James [40] recently applied the SFP parametrization to a multimaterial problemwith a constraint on the cardinality set of candidate materials.

In this work, we describe and demonstrate the capability of the proposed scalable multimaterialtopology optimization framework with adaptive mesh refinement based on domains parametrizedby CAD geometry. Multimaterial designs with tens of millions of elements are created throughfeature-based adaptive mesh refinement using the DMO parametrization and optimized using anS`1QP trust region method. In the following sections, we describe the key components of thisoptimization and mesh generation framework.

3 Multimaterial Topology Optimization

In this section, we describe the multimaterial topology optimization parametrization techniqueused for this work. To achieve an efficient and consistent design representation between mesheswith different levels of local refinement, we use a node-based parametrization [37], where thedesign field is defined on a coarsened copy of the analysis mesh. This node-based parametrizationis well-suited for the geometric multigrid solution strategies that are described in Section 4.1.Figure 2 illustrates the process whereby the initial analysis mesh is coarsened, and the designdensity associated with each element in the finer analysis mesh is interpolated from the coarsedesign mesh [68] through a linear interpolation that acts as a filter. This linear interpolation process

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is written asx = Fx (1)

where x are the design variables on the coarser design mesh, x are the interpolated design variableson the finer analysis mesh and F is the matrix that interpolates the design variables across thecoarser design mesh to the finer analysis mesh. Note that a trilinear interpolation is employed, sothe rows of F satisfy a partition of unity property. This trilinear interpolation technique eliminatescheckerboarding [68] and allows for the creation of mesh-independent designs if the design meshis fixed independent of mesh adaptation. While here we form the filter, F, using a straightforwardinterpolation between mesh levels, F could also be formed implicitly through a PDE-based filter,such as a Helmholtz filter [52].

Extending this parametrization to multimaterial design, each node in the design mesh has M+1 design variables, consisting of one topology optimization variable, x1, associated with a voidmaterial, and M candidate material selection variables, xi, for i = 2, . . . ,M + 1. For each node,the void or candidate material is selected if x j = 1, while xi = 0 for i 6= j. After the interpolationstep (1), each element has M + 1 interpolated values, corresponding to the interpolated topologyvariable and candidate material selection variables. For the stiffness properties, we use a DMO-type parametrization [73, 44] that takes the following form for each element in the analysis mesh

D(x) =M

Âi=1

✓xi+1

1+q(1�xi+1)+k0

◆Di. (2)

where k0 = 10�6/M is a small constant term added to ensure that the stiffness matrix is non-singular. The density of the element is also determined from the interpolated variables as

r(x) =M

Âi=1

xi+1ri.

Further details of this method can be found in [44]. To ensure a reasonable interpolation of in-termediate designs, DMO methods often impose a partition of unity constraint on the interpolatedelement variables, x , used in the stiffness interpolation (2). However, this constraint can instead beimposed on the nodal variables, x, due to the properties of the partition of unity interpolation (1),such that

M+1

Âi=1

xi = 1, (3)

for every node in the design mesh. The partition of unity constraint (3) results in many sparse,linear constraints: one constraint for each node in the design mesh. These sparse constraints aredenoted

Awx = e, (4)

where the matrix Aw is sparse and consists of unit entries, and e is a vector of all ones. Note thatthe constraint (3), automatically imposes that each design variable will be bounded from above byunity, so the upper design variable bound can be omitted.

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3.1 S`1QP Trust Region Method

Efficient, parallel optimization algorithms are required to solve large-scale topology and multima-terial optimization problems. Parallel implementations are required, since the design vector sizescales with the size of the analysis problem and optimization operations would otherwise consti-tute a computational bottleneck. The Method of Moving Asymptotes (MMA) is commonly usedto solve topology optimization problems [74], and has been parallelized for large-scale applica-tions [2, 3]. However, topology optimization benchmark studies have demonstrated that MMAis not always the best choice, and sequential quadratic optimization or interior-point algorithmsmay be superior [62]. Furthermore, the presence of the partition of unity constraints (3), meansthat MMA cannot be applied without modification to these problems. To address these issues, wehave developed an open source optimizer, called ParOpt2, that is designed to take advantage of thespecial sparsity structure of the partition of unity constraints (3). In this section, we describe thekey algorithmic components of this optimizer and the details of its parallel implementation.

A general formulation for mulitmaterial topology optimization problems takes the form

minx

f (x)

such that Awx = e

c(x)� 0x� 0

(5)

where x is the distributed design vector. The constraints, consist of a small number of dense con-straints, c(x), whose values are duplicated on all processors and a large number of sparse partitionof unity constraints (3), that are distributed across all processors. Note that common multimaterialoptimization problems such as mass-constrained compliance minimization and stress-constrainedmass minimization problems fit within this formulation.

In this work, the topology optimization problem (5) is solved using an S`1QP method witha trust-region globalization [28]. In this technique, at each iteration k, a candidate step, pk, isobtained by solving a non-smooth optimization problem that consists of a quadratic approximationof the objective about the point xk, combined with an `1 penalty function of the linearized constraintviolation, leading to the following problem

minp

hk(p) = f (xk)+gTk p+

12

pT

Bkp+ gk [c(xk)+Akp]1

such that Awp = e�Awxk = 0kpk• Dk

xk +p� 0

(6)

where [x]1 = Âi max{0,�xi}, denotes the `1 norm of the constraint violation. Here, Dk is thetrust region radius, gk is the penalty parameter for the `1 penalty function and hk(p) is the trustregion model function. The linearized constraint term contains the constraint Jacobian, denotedAk = —xc(xk). The model function, hk, is formed from the objective gradient, gk = —x f (xk), anda quasi-Newton Hessian approximation, Bk. Note that we always start the problem from a point

2https://github.com/gjkennedy/paropt

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Chin, Leader, Kennedy

that is feasible with respect to the partition of unity constraints (4) so that the constraint, Awxk = e,is satisfied at every iteration. In this work, a quasi-Newton Hessian approximation, Bk is formedbased on compact limited-memory BFGS update [17]

Bk = b0I�WkMkWTk , (7)

where b0 is a scalar, Mk 2 R2m⇥2m is a small dense matrix and Wk is a matrix with 2m columnsthat is stored as a series of vectors. The approximate Hessian is updated such that it approximatesthe Hessian of the Lagrangian, Bk ⇡H(xk), —2

x�

f (xk)� zT

c(xk)�.

Algorithm 1 S`1QP method with trust region globalization.1: Input: Initial design point x1 and initial trust region radius D12: Set k = 13: while Optimality criteria not satisfied do

4: Compute the candidate step pk by solving (6)5: Evaluate the ratio rk using Equation (9)6: if rk � h then

7: Accept step pk and set xk+1 = xk +pk8: end if

9: if rk < rl then

10: Set Dk = max(Dk/4,Dmin) . Shrink the trust region11: else if rk > ru then

12: Set Dk = min(2Dk,Dmax) . Expand the trust region13: else

14: Set Dk+1 = Dk . Keep the same trust region15: end if

16: Update quasi-Newton Hessian approximation17: k k+118: end while

Once a candidate step, pk, is computed as a solution of the quadratic optimization problem (6),its acceptance is based on the merit function

f1(x;gk) = f (x)+ gk [c(x)]1 , (8)

and conventional trust region acceptance criteria [19]. The trust region update criteria are based onthe ratio of the actual improvement in the merit function (8), to the improvement predicted by themodel. This ratio, denoted rk, is defined as follows

rk =f1(xk,gk)�f1(xk +pk,gk)

f1(xk,gk)�hk(pk). (9)

Based on the value of rk, a candidate step is either accepted or rejected and the model function hkis improved through a quasi-Newton update procedure. The overall S`1QP trust region algorithmis shown in Algorithm 1. The acceptance of the step is governed by the criteria that it must makesufficient improvement such that rk � h where the parameter takes a value of h = 1/4. The trustregion radius is also updated based on the ratio rk, such that values below rl = 1/4 result in adecrease of the trust region radius, while values which are indicative of good progress exceedingru = 3/4 result in an increase in the trust region radius.

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3.2 Parallel interior-point method for convex subproblems

The step, pk, required in the outer S`1QP method is computed using a parallel interior-point methoddescribed in the following section. Since the quasi-Newton Hessian approximation used in the sub-problem is positive definite, and the constraints are linearized, the subproblems are convex. Withineach subproblem, the non-smooth `1 penalty term is reformulated using an elastic programmingtechnique, generating an equality-constrained quadratic optimization problem with bound con-straints [58, Ch 18, pg 549]. This problem always has a feasible solution, since the constraintviolation is treated through a penalty term. Throughout the remainder of this section, we drop theiteration subscript k to simplify notation.

The efficient implementation of the parallel interior-point method leverages the special struc-ture of the partition of unity constraints (3). The structure of the sparse constraint Jacobian, Aw, issuch that the matrix AwDA

Tw is diagonal whenever the matrix D is a diagonal matrix. In ParOpt,

the rows of the dense constraint Jacobian, A, are stored as a series of distributed vectors, while Awis accessed in a matrix-free manner through matrix-vector operations.

The interior point algorithm approximately solves a sequence of barrier problems that are de-signed to approach the true minimizer of the subproblem (6) in the limit. The barrier problem isformed by adding a interior log-penalty function to the objective to account for the bound con-straints. This barrier function is designed to keep the iterates strictly in the interior of the feasibleregion. The barrier problem corresponding to (6) at design point xk is

minp,s,t

j(p,s, t; µ) = gT

p+12

pT

Bp+ geT

t�µi [logs+ log t+ log(p� l)+ log(u�p)]

such that Awp = 0c+Ap = s� t

(10)where s and t are slack variables associated with the dense constraints. Here the components of thelower-bound vector l are given by li = max{0,xi�Dk}, and the components of the upper boundvector u are given by ui = xi +Dk, to conform with the variable bounds from the subproblem (6).The function log(·) is the component-wise sum of the logarithm of each vector component such thatlogs = Âi lnsi. As the barrier parameter, µi, decreases, the minimizer of the barrier problem (10)approaches the KKT solution. For this work, the barrier parameter are computed by a constantfactor decrease, known as the Fiacco–McCormick approach, as follows

µi+1 = siµi (11)

where si = 0.25.The barrier problem (10) is related to the perturbed KKT conditions for the optimization prob-

lem (5). Introducing Lagrange multipliers for the inequality cosntraints, z, partition of unity con-straints, zw, and the upper and lower bounds, zl , and zu, respectively, the perturbed KKT conditions

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Chin, Leader, Kennedy

can be written as follows

r ,

2

66666666664

rprtrwrzrsrzt

rzlrzu

3

77777777775

=

2

66666666664

g+Bp�AT

z�Awzw� zlge� z� zt

Awp

c+Ap� s+ t

Sz�µie

Tzt�µie

(P�L)zl�µie

(U�P)zu�µie

3

77777777775

= 0 (12)

where P = diag{p}, T = diag{t}, S = diag{s}, L = diag{l}, and U = diag{u}. At each iterationof the interior-point method, an update to the design variables, slacks, and Lagrange multipliers iscomputed based on the solution to a linearization of the perturbed KKT system (12) as follows

JBD =

2

66666666664

B 0 0 �ATw �A

T 0 �I I

0 0 0 0 �I �I 0 0Aw 0 0 0 0 0 0 0A I �I 0 0 0 0 00 0 Z 0 S 0 0 00 Zt 0 0 T 0 0 0Zl 0 0 0 0 (P�L) 0Zu 0 0 0 0 0 (U�P)

3

77777777775

2

66666666664

Dp

Dt

Ds

DzwDz

DztDzlDzu

3

77777777775

=�r. (13)

The solution of the system of equations (13) is the single most computationally expensive oper-ation in the optimization algorithm itself. A solution for this update step, D, can be obtained byexpressing the update equation as

JBD =⇥J0 +WMW

T ⇤D =�r, (14)

where J0 is a matrix obtained by replacing B in (13) with b0I. The matrix W is

W =⇥W

T 0 0 0 0 0 0 0⇤T

,

where W is from the compact BFGS formula (7). Linear systems of the form, J0y = b, can besolved efficiently in parallel using a bordering method described in Appendix A.

The solution of the update equation (14) can be obtained using using the Sherman–Morrison–Woodbury formula [58, Ch 19, pg 597]. such that

D = J�10 WC

�1W

TJ�10 r�J

�10 r

where the matrix C 2 R2m⇥2m is given as follows:

C , WT

J�10 W�M.

As a result, a solution of the linear system (14) can be obtained from the solution of 2m+1 linearsystems of the form J0y = b. Since the matrix W is stored as a series of column vectors, the opera-tions needed to compute the solution consist primarily of dot-products of distributed vectors with a

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Chin, Leader, Kennedy

(a) Initial (b) Unbalanced (c) Re-balanced

Figure 3: Sequence of meshes demonstrating differences between balanced and non-balancedmeshes.

small number of matrix operations on small dense matrices. Since the vector operations parallelizeefficiently, and the matrix operations constitute a small contribution to the overall computationaltime, the optimization subproblem exhibits good parallel performance.

The termination criterion for each quadratic subproblem is based on the `1 norm of the per-turbed KKT conditions (12) as follows:

max���rp

��1 , krtk1 , krwk1 , krzk1

etol. (15)

For this work we use a tight stopping tolerance of etol = 10�7 for the subproblems. Based on ourexperience, a tight tolerance is desirable since the bound multipliers may be very small, necessi-tating a small value of µi to achieve full convergence.

4 Mesh Generation and Adaptivity

In this section, we describe the mesh generation and adaptive refinement methods that are usedwithin the topology optimization framework. These methods are implemented within an opensource tool, called TMR3. Within this tool, a coarse, geometry-conforming hexahedral mesh isgenerated from the geometry as an initial step. Next, a connected forest of octrees is createdand refined on each element within the coarse hexahedral mesh, in a similar manner to the workof Burstedde et al. [16]. The octrees are stored using an encoding that tracks the element size andlocation without storing the entire tree data structure [27, 57]. All octree operations are distributedacross all processors and are performed in parallel, enabling the construction of large hierarchicalmeshes that are compatible with geometric multigrid. The resulting meshes must be balanced,meaning that there is no more than one level of difference in refinement across edges or faces.Figure 3 illustrates the process of balancing a mesh, where an initially balanced mesh, shown inFigure 3a, is refined, creating an non-balanced mesh shown in Figure 3b. Restoring the balancedproperty requires adding elements in the vicinity of the refinement, shown in Figure 3c.

TMR uses OpenCASCADE [59] as the underlying geometry kernel, enabling the use of ge-ometries generated from CAD. Hexahedral meshes can be either imported or generated directlyusing a sweeping method between source and target surfaces with the same quadrilateral meshtopology [60]. Within TMR, the source surface meshes are generated using an implementation

3https://github.com/gjkennedy/tmr

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Chin, Leader, Kennedy

(a) Mesh distribution across 4processors

(b) Refinement of mesh is bal-anced across processors

(c) Refined mesh repartitionedacross processors

Figure 4: Illustration of balancing and repartitioning across processors.

0.70 0.75 0.80 0.85 0.90 0.95 1.00Hexahedron Shape Metric

0.0%

2.5%

5.0%

7.5%

10.0%

12.5%

Figure 5: Mesh quality for the orthogonal bracket geometry.

of the Blossom-Quad algorithm [61, 32], which first generates a triangular mesh with an evennumber of elements, and recombines them into quadrilaterals using the Blossom perfect-matchingalgorithm [24, 23, 49].

To achieve good parallel scalability, the elements within the mesh must also be distributedacross all processors. This is achieved by first ordering the coarse hexahderal mesh to promotea good overall hexahdral mesh partition using METIS [41]. Next, fine-grained partitioning isachieved by a repartitioning operation on the octrees such that the number of elements assigned toeach processor is approximately equal. Figure 4a shows a 2D L-bracket mesh that is divided across4 processors, each denoted by a different color. When the mesh is refined, as shown in Figure 4b,it is automatically balanced, which changes the number of elements assigned to each processor.Figure 4c shows the refined mesh repartitioned across the 4 processors to distribute the workload.

To demonstrate the capability of the mesh generation techniques in TMR, we plot the hexahe-dron shape metric, first proposed by Knupp [48], to evaluate the mesh quality. The shape metricis a scale-invariant value and ranges from 0 to 1, with the value 1 denoting a cube element and0 denoting a degenerate element. Based on CUBIT [65], an automated mesh generation toolkitused in Sandia National Laboratories, elements with the value between 0.3 and 1 are considered

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Chin, Leader, Kennedy

(a) Fine analysis mesh (b) Medium mesh (c) Coarsest mesh

Figure 6: Illustration of the construction of the meshes for geometric multigrid. The analysis meshis the first geometric multigrid level and the subsequent geometric multigrid levels are obtained bycoarsening the previous mesh through TMR.

to be good quality. Figure 5 shows the mesh quality metric for the orthogonal bracket presentedSection 5.2.2. The shape metric varies between 0.68 and 1, with all elements having good quality.In the results presented here, with up to 329 million elements, all octree meshes are generated inunder 20 seconds.

4.1 Geometric Multigrid

The finite-element analysis and derivative evaluation is performed using the open source finite-element tool TACS4 [45]. At each optimization iteration, the large linear system of govern-ing equations arising from the finite-element discretization is solved using a parallel multigrid-preconditioned Krylov subspace method [42]. Other authors [4, 2] have also applied multigrid tolarge-scale topology optimization problems. The advantage of multigrid methods is that they offerthe potential of both algorithmic scalability and parallel efficiency.

In this work, the hierarchy of meshes used within the multigrid preconditioner are generatedusing TMR. An illustrative example of the mesh generation procedure is shown in Figure 6, wherethe coarse mesh levels are created automatically by successive coarsening of the fine, adaptivelyrefined analysis mesh. Each mesh level has an associated stiffness matrix, Ki(xi), and a solutionvector ui where the subscript i denotes the mesh level, ranging from i = 1, ...,N, where i = 1 is thefinest mesh and i = N is the coarsest mesh. In addition, xi is the design vector restricted from thenext finest design mesh. On all but the coarsest mesh, a block symmetric Gauss–Seidel smoothingoperation is performed to damp out the high frequency solution error. This operation is written as

ui Gi(ri), (16)

where ri is the residual on the i-th mesh level. The restriction of the solution between mesh levelsis written as

ui+1 = Riui, (17)

where Ri is the restriction operator from the finer mesh level i to the coarser mesh level i+ 1. Inthis work, the interpolation operator is formed as the transpose of the restriction operator. On the

4https://github.com/gjkennedy/tacs

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Chin, Leader, Kennedy

coarsest mesh level, i = N, the system of equations uN = KN(xN)�1rN is solved using a parallel

direct Schur-complement method [45]. Note that Aage et al. [2] used an algebraic multigrid methodon the coarsest geometric multigrid problem.

Algorithm 2 Geometric multigrid algorithm with a symmetric block Gauss–Seidel smoother.1: function MULTIGRID(ri)2: if i = N then

3: return KN(xN)�1rN . Direct solve

4: end if

5: ui Gi(ri) . Pre-smooth6: ri ri�Ki(xi)ui . Compute the residual7: ri+1 Riri . Restrict the residual8: ui+1 MULTIGRID(ri+1) . Apply multigrid at the next level9: ui ui +R

Ti ui+1 . Interpolate the solution

10: ui Gi(ri) . Post-smooth11: return ui12: end function

The complete V-cycle multigrid preconditioning algorithm is shown in Algorithm 2. We usemultigrid-preconditioned GMRES(100) [63], with a maximum of two restarts to solve the gov-erning equations, unless stated otherwise. We have found that this combination of multigrid pre-conditioning and GMRES is sufficiently robust to handle the large variability in the local elementstiffness inherent to topology optimization applications.

The restriction of the design vector between mesh levels utilizes the same operations that arerequired for multigrid. The design vector on the next coarsest design mesh is obtained as xi+1 =Rixi, where Ri denotes the restriction operator for the nodal design mesh. These design restrictionoperations utilize the same underlying routines as the multigrid method, enabling code reuse.

4.2 Adaptive Mesh Refinement Heuristic

As discussed in Section 2, there are different adaptive mesh refinement indicators that have beenemployed for topology optimization such feature-based methods [72], solution-based methods [80],and functional output-based methods [78]. For this work, feature-based refinement is demon-strated. In the feature-based adaptive refinement approach, elements are refined or coarsened basedon the optimized material distribution. Here we use a simple heuristic where the mesh is refinedwhere the optimizer has placed material, and coarsened where the optimizer has allocated void.The parameter xrl defines an upper limit on the void material, above which elements with voidmaterial selection variables are coarsened whenever x1 � xrl . For this work, we use a value ofxrl = 0.95, so that the void material selection variable is close to unity. The parameter xru definesa lower limit on the void material, below which the element is refined. For this work, we use avalue of xru = 0.75, which refines the mesh in areas where there is either intermediate or convergedmaterial. After the mesh adaptation step is invoked, the design on the new mesh is interpolatedfrom the previously optimized design using TMR. The overall feature-based adaptive refinementheuristics used in this work shown in Algorithm 3.

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Chin, Leader, Kennedy

Algorithm 3 Topology optimization with adaptive mesh refinement1: Input: CAD geometry; initial mesh spacing2: Generate initial analysis and design meshes in TMR3: Optimize to obtain initial topology4: while Termination criterion not satisfied do

5: Identify elements to coarsen with filtered density x1 � xrl6: Identify elements to refine with filtered density x1 xru7: Regenerate the mesh in TMR based on updated refinement8: Interpolate design variables onto the new mesh9: Restart the topology optimization

10: end while

5 Results

In this section, we apply the proposed framework to mass-constrained compliance-minimizationdesign problems formulated as follows:

minx

f (x) = uT

f

such that x� 0Awx = e

c(x) = mfixed�m(x)� 0governed by K(x)u = f

(18)

where x are the design variables, f (x) is the compliance of the structure, u are the displacementstate variables, f is the force vector, and Awx = e is the partition of unity constraint (4).

The results obtained in subsequent sections were run on a dedicated cluster within the Partner-ship for an Advanced Computing Environment (PACE) at the Georgia Institute of Technology. Thecluster has 19, 2.50GHz Intel Xeon CPU E5-2680-v3 compute nodes. Each node has 24 processorcores with a total of 128 GB of RAM per node.

5.1 Scalability Demonstration

Before examining topology optimized designs from the proposed framework, we first demonstratethe scalability of the algorithms on a topology optimization problem with 18.9 million elements,19.2 million nodes, and 7.32 million design variables. For this study, the number of optimizationiterations is fixed at 50 and we vary the number of processors to investigate the computationalefficiency of individual components of the framework. The elements are distributed equally acrossall processors on all geometric multigrid levels, except for the coarsest level, where they are dividedamong 24 processors. Figure 7 shows the fraction of the ideal cost of each optimization componentas a function of the number of processors. Figure 7a and 7c show the computational efficiencyof GMRES and the geometric multigrid preconditioner, respectively. These components of theframework scale well from 48 to 144 processors where the multigrid preconditioner scales up to144 processors with 90.8% efficiency, and GMRES overall scales with 90.5% efficiency. FromFigure 7a and 7b, we note that the small inefficiency arises from the increase in the fraction of

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Chin, Leader, Kennedy

48 72 96 120 144Number of processors

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Frac

tion

ofid

ealt

ime

PreconditioningOrthogonalization

(a) Solution Efficiency

48 72 96 120 144Number of processors

50

100

250

500

1000

1500

Cum

ulat

ive

Wal

ltim

e(s

)

TotalPreconditioningOrthogonalizationIdeal

(b) Cumulative Solution Cost

48 72 96 120 144Number of processors

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Frac

tion

ofid

ealt

ime

FinestLevel 1+2

Coarsest

(c) Geometric Multigrid Effi-ciency

48 72 96 120 144Number of processors

10

100

250

500

10001500

Cum

ulat

ive

Wal

ltim

e(s

) TotalFinestLevel 1+2

CoarsestIdeal

(d) Cumulative GeometricMultigrid Cost

48 72 96 120 144Number of processors

50

100

200

300

Cum

ulat

ive

time

(s)

Quasi-Newton Subspace Size = 25Quasi-Newton Subspace Size = 10Ideal

(e) Cumulative ParOpt Cost

Figure 7: Scalability of the individual framework components as a function of the number ofprocessors.

time spent in the orthogonalization step of the solution. Figure 7c shows that the overall geometricmultigrid cost scales close to ideally as the number of processors increase. Figure 7d shows thatthe fraction of time taken on the coarsest level increases for an increasing number of processors,due to the use of the direct solver. The slight increase in computational time for the direct solveris due to additional communication costs, since the computation at this level is restricted to 24processors.

Figure 7e shows the computational cost of the ParOpt optimizer for different quasi-Newtonsubspace sizes. The computational time increases with subspace size due to the additional op-erations required for each additional set of quasi-Newton vectors. The optimization steps scaleat better than the ideal rate between 48 and 72 processors, which we attribute to better cache ef-ficiency. The optimization steps scale at a near ideal rate between 72 to 144 processors. Notethat the computational cost of the optimizer is no more than 20% of the computational cost of thesolution method.

5.2 3D problems

Three different 3D design problems are investigated to demonstrate the capability of the proposedframework for large-scale topology optimization: a cantilever, an orthogonal bracket, and an or-thotropic box. The first two problems investigated are isotropic material selection problems, wherethere are two candidate materials, while the third problem involves the selection of the principaldirection of an orthotropic material with 13 candidate orientations. For the isotropic cases, the twocandidate material properties are listed in Table 1, while the material properties for the orthotropic

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Chin, Leader, Kennedy

Material Properties M1 M2

Young’s Modulus E (GPa) 70 35Density r (kg/m3) 2600 1300Poisson’s ratio n 0.3 0.3

Table 1: Candidate isotropic material properties.

Material Properties M3

Young’s Modulus Exx (GPa) 70Young’s Modulus Eyy, Ezz (GPa) 23.3Shear Modulus Gyz (GPa) 10Density r (kg/m3) 2600Poisson’s ratio nxy, nxz, nyz 0.3

Table 2: Orthotropic material properties.

material are listed in Table 2.

5.2.1 Cantilever Beam

The first isotropic example is a cantilever beam of overall length L shown in Figure 8. A hole witha diameter of 0.1L is cut out at a distance of 0.85L from the root of the beam. A traction is appliedin the vertical direction throughout the hole. In this work, the value of L is 1m and the magnitudeof the applied traction is 5.8MPa. The mass is fixed at 0.2raverageV , where V is the volume of thedomain and raverage is the average density of the candidate materials.

For this problem, an external planar mesh, generated in NX [66], is imported into TMR whichis used to generate the hexahedral mesh via volume mesh sweeping. Three different meshes areused for this study: a medium adaptive mesh, a fine adaptive mesh, and an ultra fine fixed mesh.The initial mesh for the medium adaptive case has 5.14 million hexahedral elements, 5.27 mil-lion nodes, and 2.02 million design variables. After the adaptive refinement step, the final meshcontains 11.3 million elements, 11.0 million nodes, and 4.59 million design variables. The initialmesh for the second adaptive case has 41.1 million elements, 41.6 million nodes, and 15.8 milliondesign variables. After the adaptive refinement step, the final mesh contains 81.8 million elements,

L

0.25L

0.25L

0.85L

0.1L

Figure 8: Dimensions for the cantilever beam problem.

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Chin, Leader, Kennedy

(a) View from the root (b) View from the root (c) View from the root

(d) View from the front (e) View from the front (f) View from the front

(g) View from the side (h) View from the side (i) View from the side

(j) View from the top (k) View from the top (l) View from the top

Figure 9: Final design of the beam from the three cases with M1 denoted in orange and M2denoted in gray. From left, center to right are the medium resolution case, the fine resolution, andthe ultra-fine resolution.

80.3 million nodes and 31.8 million design variables. The fixed mesh without adaptive refinementhas 329 million elements, 331 million nodes, and 125 million design variables. For the ultra fineproblem, we used multigrid-preconditioned GMRES(50) with a maximum of five restarts to solvethe governing equations [63] to save memory. The final design for each case is shown in Figure 9.

Figure 9 shows the resulting designs for the three cases with M1 denoted in orange and M2denoted in gray. For all three cases, the optimizer produced a multi-section beam design with alarge central web and two thinner outer webs arranged symmetrically about the center line of thebeam. The stiffer M1 material extends from the cantilever root along the top and bottom of thestructure, increasing its bending stiffness. The softer and lighter M2 material forms the shear websof the beam, connecting the stiffer M1 material at the top and bottom of the structure and also formsthe structure surrounding the load application point. There is a slight difference in the distributionof M1 at the cantilever root for the medium case compared to the fine and ultra-fine case. As theresolution of the initial mesh becomes finer, the design at the front of the structure towards the loadapplication point becomes more intricate and slender. The meshes for the medium and fine design

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Chin, Leader, Kennedy

Optimization Outcome Medium Fine Ultra-fine

Normalized Compliance 0.295 0.268 0.263CPU Cores 96 456 456Walltime (h) 30.7 49.2 77.0Number of elements in initial mesh (million) 5.14 41.1 329Number of elements in adaptively refined mesh (million) 11.3 81.8 -Relative resolution of final design (10�3) 3.06 1.53 1.53CPU Days 122.7 934.4 1463.2

Table 3: Summary of optimization outcomes for each design case.

cases do not have the resolution to form the structure as shown in the ultra-fine case.Figure 10 shows the convergence of the three cases as a function of iteration history. The com-

pliance value shown is normalized by the initial compliance of the medium resolution design. Foran accurate comparison, the final adaptive mesh refinement design is projected onto the uniformlyfine mesh for compliance evaluation. We found that the effect of the mesh size on complianceat this resolution is on the order of 0.1%. For the medium resolution case, the problem is solvedusing 96 processors, requiring 1000 optimization iterations over 30.7 hours. For the fine resolutioncase, the problem is solved using 456 processors, requiring 1000 optimization iterations over 49.2hours. For the ultra-fine case, the problem is solved using 456 processors, requiring 500 optimiza-tion iterations over 77.0 hours. To compare the computational effort for each case, we computethe CPU days, which is the product of the number of processors with the optimization wall time.The convergence history of the compliance of these three cases are plotted as a function of its CPUdays as shown in Figure 10c. A summary of the optimization outcome for the three cases are inTable 3.

From Figure 10a, we observe that there are some slight differences in the normalized com-pliance of the three different designs. These differences are illustrated in a close-up as shown inFigure 10b and summarized in Table 3. The compliance of the medium resolution case is 10.8%larger when compared to the ultra-fine resolution case, while the compliance of the fine resolu-tion design is 1.86% larger when compared to the ultra-fine resolution case. This is not surprisingwhen accounting for the larger difference in the distribution of the material between the mediumdesign and the ultra-fine design, compared to the difference between the fine and ultra-fine de-signs. Figure 10c illustrates the computational time advantage of the adaptive mesh refinementapproach over the fixed mesh. The ultra-fine design took 1463.2 CPU days whereas the adaptivefine design took only 934.4 CPU days, with both meshes having the same design feature size.This design required 36.1% less computational resources with only a 1.86% different in terms ofcompliance. Finally, the medium resolution design took only 122.7 CPU days with a final relativeresolution twice that of the fine and ultra-fine resolution designs, a difference of 91.6% in termsof computational resources used. For all three designs, the mass constraint is satisfied to machineprecision.

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Chin, Leader, Kennedy

(a) Compliance convergence history for thebeam cases.

(b) Zoom view for the compliance convergencehistory for the beam cases.

(c) Compliance convergence for the beam cases as a function of CPU days

Figure 10: Convergence history of the beam design for the three resolution cases.

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Chin, Leader, Kennedy

A

BC

L

0.1L

0.75L

0.75L

0.75L

0.25L 0.25L

0.1L

0.1L0.15L

Figure 11: Orthogonal bracket problem domain

5.2.2 Orthogonal Bracket

The geometry of the orthogonal bracket is shown in Figure 11. It consists of 3 identical orthogonalbeam members of length L, each with a cylindrical hole cut at 0.15L from the free ends of eachmember. Hole A at the top of the vertical member is completely clamped, and traction loads ofmagnitude 3.2MPa are applied to holes B and C. The loads on B and C create a bending moment ineach of the horizontal members, and a combined bending and torsional load in the vertical member.Here, the length L shown in the diagram below is 1.0m. The mass constraint of the compliance-minimization problem is fixed at 0.2raverageV , where V is the volume of the domain and raverage isthe average density of the candidate materials used.

The mesh used in this study has 77.0 million elements, 77.9 million nodes, and 29.5 milliondesign variables. No adaptive mesh refinement is used for this problem. The optimization required58 hours on 192 processors with 500 optimization iterations. For this problem, we used 5 multigridlevels. Figure 12 shows the resulting design with M1 denoted in orange and M2 denoted in gray.Note that the stiffer material M1 is placed primarily along the vertical member and the re-entrantcorner to counteract the combined bending and torsional loads. Along the horizontal memberswhere the primary loading is due to bending, M2, which is lighter and less stiff, is used to minimizethe deflection as well as satisfy the mass constraint. Figure 13 shows the convergence of theoptimization and the overall design history with the compliance value normalized by its initialvalue. The normalized compliance rapidly decreases over the first 50 iterations, while the massconstraint violation initially exceeds 50%. From iteration 100 to 330, significant changes in thedesign occurred mainly along the horizontal members, with more truss-like members forming asthe optimizer seeks to reduce the mass of the design to satisfy the constraint.

Figure 14 shows detailed designs at design iterations: 150, 300, and 400, respectively. Fig-ures 14a and 14b, show where more truss members started to form. From iteration 330 to 500,with the constraint satisfied, the optimizer created designs with intricate trusses, as shown in Fig-ure 14c. We note that without this mesh resolution, the optimizer would not have been able to formsuch slender members.

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Chin, Leader, Kennedy

(a) Isometric view of design (b) Side view of design

(c) Front view of design (d) Top view of design

Figure 12: Final design of the orthogonal bracket with M1 denoted in orange and M2 denoted ingray.

5.2.3 Orthotropic Box

For the last example, we examine the placement and orientation of an orthotropic material withina rectangular domain. The multimaterial design in this case requires the optimizer to select thealignment of the material along the different possible directions. Figure 15 shows the 13 possibledirections for each element. The orthotropic properties of the material are listed in Table 2.

The rectangular design domain is shown in Figure 16. The bottom 4 corners of the domain arepinned and a traction load is applied on the bottom of the box over a small square section, withan area of 0.125L2, centered on the bottom surface. The dimension L is 0.1m and the tractionload applied is 60kPa. Since the problem is symmetric, we impose symmetry and design only aquarter of the domain. The initial mesh has 1.57 million hexahedral elements, 1.61 million nodes,and 2.90 million design variables. The mass constraint is fixed at 15% mass fraction. One cycleof adaptive mesh refinement is used and in this case and no mesh coarsening is applied. Theproblem is solved using 72 processors, requiring 1000 optimization iterations over 9.5 hours. Forthis problem, we used 4 multigrid levels. Figure 17 shows the final design, with the gray regionsdenoting the face directions, blue regions denoting the edge directions and orange regions denoting

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Chin, Leader, Kennedy

Figure 13: Orthogonal bracket convergence and design history

the corner directions. The lines on the structure denote the principal direction of the material.After one cycle of adaptive mesh refinement, the final mesh had 3.19 million elements, 3.15

million nodes, and 5.91 million design variables. The final design utilized 12 out of the 13 or-thotropic directions, leaving out only the edge direction in the positive x-y plane. Figure 18 showsthe convergence history of the design as a function of the wall time, where every 50th iteration isindicated by a symbol. Figure 19 shows the design history as a function of optimization iteration.The compliance value shown is the compliance normalized by the initial compliance.

The total wall time for this optimization is 9.5 hours, with the adaptive mesh refinement steptaking place at 4.1 hours into the optimization. Similar to the other design problems, the nor-malized compliance and the infeasibility of the design quickly converges. The design becomesfeasible within the first 40 iterations and remains feasible for the rest of the optimization process.However, at this point the structure has not yet formed, as shown by Figure 19a. From iteration 60onwards, the change in the normalized compliance becomes small and the structure starts to takeshape, as shown in Figure 19b and 19c. After 500 design iterations, the mesh is adaptively refinedand the optimization is continued on this new mesh. The normalized compliance increases on thefiner mesh and remains at a larger value, due to the larger discretization error on the coarse meshfor the orthotropic problem.

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Chin, Leader, Kennedy

(a) Design at 150-th iteration (b) Design at 300-th iteration (c) Design at 400-th iteration

Figure 14: Designs of the orthogonal bracket at various optimization iterations

(a) Face Directions

(b) Edge Directions

(c) Corner Directions

Figure 15: Schematic of the 13 orthotropic directions

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Chin, Leader, Kennedy

L

2p2L

2p2L

Figure 16: Schematic of orthotropic domain problem

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Chin, Leader, Kennedy

(a) Isometric view of design (b) Top view of design

(c) View from the x-z plane (d) View from the y-z plane

Figure 17: Final design of the orthotropic box domain.

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Chin, Leader, Kennedy

Figure 18: Orthotropic problem convergence history

(a) Design at 40-th iteration (b) Design at 60-th iteration (c) Design at 250-th iteration

Figure 19: Designs of the rectangular domain at various optimization iterations

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Chin, Leader, Kennedy

6 Conclusion

In this paper, we developed and demonstrated methods for efficiently solving large-scale multima-terial topology optimization problems. High-resolution topology optimization designs are desir-able, but difficult to create using uniform meshes due to the high computational cost of large-scalefinite-element computations. Therefore, a reasonable approach is to use adaptive mesh refinementduring the design process to achieve the desired resolution, while coarsening void regions to reducecomputational cost. The proposed adaptive octree-based framework has the capability to gener-ate large-scale, hierachical meshes directly from CAD geometry that are well-suited for multigridsolution methods, and can perform multimaterial topology optimization with mesh adaption. Theproposed parallel S`1QP trust region method used for design optimization was tailored to effi-ciently handle the sparse partition of unity constraints associated with the DMO parametrization.The mesh generation and adaptivity algorithms that allowed for large-scale mesh generation anddesign were demonstrated to achieve greater than 90% parallel efficiency. The framework was thenused to solve some of the largest multimaterial designs performed to date, including a case with329 million elements and 125 million design variables.

Acknowledgement

The authors gratefully acknowledge the funding provided by NASA through the TransformativeTools and Technologies program with grant number NNX15AU22A with Technical Monitor SteveMassey.

A Appendix

The KKT system J0pk = b can be solved efficiently in parallel using a series of variable elimina-tions. This series of computation can be reduced to parallel vector operations and a small numberof operations on small dense matrices. As a result, the factorization and the application of the fac-torization scales efficiently. The solution procedure begins by obtaining the solution for the slackvariables and lower and upper Lagrange multiplier bound variables

pzl = P�1(bzl �Zlpp),

ps = Z�1(bs�Spz),

pt = Z�1t (bzt +T(pz +bt))

(19)

Next, using the first two equations gives

b0pp�AT

pz�pzl = bp,

App�ps +pt = bz,(20)

Substituting the expressions for the slack and Lagrange multiplier updates (19) into the expressionfor the first two linearized KKT conditions (20) yields the following

Dpp�AT

pz = bp +P�1

bzl ,

App +(Z�1S+Z

�1t T)pz = bz +Z

�1bs�Z

�1t (bzt +Tbt)

(21)

28

Chin, Leader, Kennedy

where the diagonal matrix D is defined as follows:

D =⇥b0I+P

�1Zl⇤.

Finally, pp can be eliminated for pz as follows

(Z�1S+Z

�1t T+AD

�1A

T )pz = dz, (22)

The right hand side dz is

dz , bz +Z�1

bs�Z�1t (bzt +Tbt)�AD

�1 �bp +P

�1bzl

�.

This solution procedure is invoked each time a solution of the form J0pk = b is required.

References

[1] N. Aage and B. S. Lazarov. Parallel framework for topology optimization using the methodof moving asymptotes. Structural and Multidisciplinary Optimization, 47(4):493–505, Apr2013. ISSN 1615-1488. doi:10.1007/s00158-012-0869-2.

[2] N. Aage, E. Andreassen, and B. Lazarov. Topology optimization using PETSc: An easy-to-use, fully parallel, open source topology optimization framework. Structural and Multidis-ciplinary Optimization, 51(3):565–572, 2015. ISSN 1615-147X. doi:10.1007/s00158-014-1157-0.

[3] N. Aage, E. Andreassen, B. S. Lazarov, and O. Sigmund. Giga-voxel computational mor-phogenesis for structural design. Nature., 550(7674):84–86, 2017. ISSN 0028-0836.doi:10.1038/nature23911.

[4] O. Amir, N. Aage, and B. Lazarov. On multigrid-CG for efficient topology optimiza-tion. Structural and Multidisciplinary Optimization, 49(5):25–57, 2014. ISSN 1615-147X.doi:10.1007/s00158-013-1015-5.

[5] M. P. Bendsøe and N. Kikuchi. Generating optimal topologies in structural design using ahomogenization method. Computer Methods in Applied Mechanics and Engineering, 71(2):197 – 224, 1988. ISSN 0045-7825. doi:10.1016/0045-7825(88)90086-2.

[6] M. P. Bendsøe and O. Sigmund. Material interpolation schemes in topology optimiza-tion. Archive of Applied Mechanics, 69(9):635–654, Nov 1999. ISSN 1432-0681.doi:10.1007/s004190050248.

[7] J. Bennett and M. Botkin. Structural shape optimization with geometric description andadaptive mesh refinement. AIAA Journal, 23(3):458–464, 1985. doi:10.2514/3.8935.

[8] T. Borrvall and J. Petersson. Large-scale topology optimization in 3D using parallel comput-ing. Computer Methods in Applied Mechanics and Engineering, 190(46–47):6201 – 6229,2001. ISSN 0045-7825. doi:10.1016/S0045-7825(01)00216-X.

29

Chin, Leader, Kennedy

[9] M. Botkin. Shape optimization with multiple loading conditions and mesh refinement. AIAAJournal, 28(5):922–927, 1990. doi:10.2514/3.25140.

[10] D. Brackett, I. Ashcroft, and R. Hague. Topology optimization for additive manufac-turing. In Solid Freeform Fabrication Symposium, volume 1, pages 348–362, Austin,TX, August 2011. URL http://sffsymposium.engr.utexas.edu/Manuscripts/2011/2011-27-Brackett.pdf.

[11] M. Bruggi and M. Verani. A fully adaptive topology optimization algorithm with goal-oriented error control. Computers & Structures, 89(15–16):1481 – 1493, 2011. ISSN 0045-7949. doi:10.1016/j.compstruc.2011.05.003.

[12] T. Bruns and D. Tortorelli. Topology optimization of non-linear elastic structures and com-pliant mechanisms. Computer Methods in Applied Mechanics and Engineering, 190(2627):3443–3459, 2001. ISSN 0045-7825. doi:10.1016/S0045-7825(00)00278-4.

[13] M. Bruyneel. SFP – a new parameterization based on shape functions for optimal mate-rial selection: application to conventional composite plies. Structural and MultidisciplinaryOptimization, 43(1):17–27, 2011. ISSN 1615-147X. doi:10.1007/s00158-010-0548-0.

[14] M. Bruyneel, P. Duysinx, C. Fleury, and T. Gao. Extensions of the shape functions withpenalization parameterization for composite-ply optimization. AIAA Journal, 49(10):2325–2329, 2011. doi:10.2514/1.J051225.

[15] S. Buchanan. Development of a wingbox rib for a passenger jet aircraft using design op-timization and constrained to traditional design and manufacture requirements. In 10thUK Altair Technology Conference, Warwickshire, UK, September 2007. URL https://altairatc.com/EventPage.aspx?event_id=97&name=Conference+Papers+-+2007.

[16] C. Burstedde, L. C. Wilcox, and O. Ghattas. p4est: Scalable algorithms for parallel adaptivemesh refinement on forests of octrees. SIAM Journal on Scientific Computing, 33(3):1103–1133, 2011. doi:10.1137/100791634.

[17] R. H. Byrd, J. Nocedal, and R. B. Schnabel. Representations of quasi-Newton matrices andtheir use in limited memory methods. Mathematical Programming, 63(1-3):129–156, 1994.ISSN 0025-5610. doi:10.1007/BF01582063.

[18] T. W. Chin and G. J. Kennedy. Efficient large-scale thermoelastic topology optimization ofCAD geometry with automated adaptive mesh generation. In 59th AIAA/ASCE/AHS/ASCStructures, Structural Dynamics, and Materials Conference, AIAA SciTech, Kissimmee, FL,January 2018. doi:10.2514/6.2018-1381. AIAA 2018-1381.

[19] A. R. Conn, N. I. Gould, and P. L. Toint. Trust region methods, volume 1. SIAM, 2000.

[20] J. C. A. Costa Jr and M. K. Alves. Layout optimization with h-adaptivity of structures.International Journal for Numerical Methods in Engineering, 58(1):83–102, 2003. ISSN1097-0207. doi:10.1002/nme.759.

30

Chin, Leader, Kennedy

[21] E. de Sturler, G. H. Paulino, and S. Wang. Topology optimization with adaptive meshrefinement. In 6th International Conference on Computation of Shell and Spatial Struc-tures, Ithaca, New York, May 2008. URL https://www.math.vt.edu/people/sturler/publications/Proc-IASS-IACM_TopOptAMR_SturlerPaulinoWang_2008.pdf.

[22] A. R. Diaz and O. Sigmund. A topology optimization method for design of negative perme-ability metamaterials. Structural and Multidisciplinary Optimization, 41(2):163–177, Mar2010. ISSN 1615-1488. doi:10.1007/s00158-009-0416-y.

[23] J. Edmonds. Maximum Matching and a Polyhedron with 0,1 Vertices. Journal of Researchof the National Bureau of Standards, 69:125–130, 1965. URL https://nvlpubs.nist.gov/nistpubs/jres/69B/jresv69Bn1-2p125_A1b.pdf.

[24] J. Edmonds, E. Johnson, and S. Lockhart. Blossom I: a computer code for the matchingproblem. IBM TJ Watson Research Center, Yorktown Heights, New York, 1969.

[25] A. Evgrafov, C. J. Rupp, K. Maute, and M. L. Dunn. Large-scale parallel topology optimiza-tion using a dual-primal substructuring solver. Structural and Multidisciplinary Optimization,36(4):329–345, Oct 2008. ISSN 1615-1488. doi:10.1007/s00158-007-0190-7.

[26] J. f. Shi and J. h. Sun. Overview on innovation of topology optimization in vehicle CAE.In 2009 International Conference on Electronic Computer Technology, pages 457–460, Feb2009. doi:10.1109/ICECT.2009.60.

[27] R. A. Finkel and J. L. Bentley. Quad trees: a data structure for retrieval on composite keys.Acta Informatica, 4(1):1–9, 1974. ISSN 1432-0525. doi:10.1007/BF00288933.

[28] R. Fletcher. A model algorithm for composite nondifferentiable optimization problems.In D. C. Sorensen and R. J.-B. Wets, editors, Nondifferential and Variational Tech-niques in Optimization, pages 67–76. Springer Berlin Heidelberg, Berlin, Heidelberg, 1982.doi:10.1007/BFb0120959.

[29] T. Gao and W. Zhang. A mass constraint formulation for structural topology optimizationwith multiphase materials. International Journal for Numerical Methods in Engineering, 88(8):774–796, 2011. ISSN 1097-0207. doi:10.1002/nme.3197.

[30] T. Gao, P. Xu, and W. Zhang. Topology optimization of thermo-elastic structures with mul-tiple materials under mass constraint. Computers & Structures, 173:150 – 160, 2016. ISSN0045-7949. doi:10.1016/j.compstruc.2016.06.002.

[31] N. Gardan and A. Schneider. Topological optimization of internal patterns and support inadditive manufacturing. Journal of Manufacturing Systems, 37(Part 1):417 – 425, 2015.ISSN 0278-6125. doi:10.1016/j.jmsy.2014.07.003.

[32] C. Geuzaine and J.-F. Remacle. Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities. International Journal for Numerical Methods in Engineering,79(11):1309–1331, 2009. ISSN 1097-0207. doi:10.1002/nme.2579.

31

Chin, Leader, Kennedy

[33] L. V. Gibiansky and O. Sigmund. Multiphase composites with extremal bulk modulus.Journal of the Mechanics and Physics of Solids, 48(3):461 – 498, 2000. ISSN 0022-5096.doi:10.1016/S0022-5096(99)00043-5.

[34] I. Gibson, D. W. Rosen, and B. Stucker. The Use of Multiple Materials in Additive Man-ufacturing, pages 436–449. Springer US, Boston, MA, 2010. ISBN 978-1-4419-1120-9.doi:10.1007/978-1-4419-1120-9 17.

[35] S. Grihon, L. Krog, and K. Hertel. A380 weight savings using numerical structural opti-mization. In 20th AAAF Colloqium ”Material for Aerospace Application”, Paris, France,November 2004.

[36] J. K. Guest and L. C. Smith Genut. Reducing dimensionality in topology optimization usingadaptive design variable fields. International Journal for Numerical Methods in Engineering,81(8):1019–1045, 2010. ISSN 1097-0207. doi:10.1002/nme.2724.

[37] J. K. Guest, J. H. Prevost, and T. Belytschko. Achieving minimum length scale in topologyoptimization using nodal design variables and projection functions. International Journal forNumerical Methods in Engineering, 61(2):238–254, 2004. doi:10.1002/nme.1064.

[38] C. Hvejsel and E. Lund. Material interpolation schemes for unified topology and multi-material optimization. Structural and Multidisciplinary Optimization, 43(6):811–825, 2011.ISSN 1615-147X. doi:10.1007/s00158-011-0625-z.

[39] C. Hvejsel, E. Lund, and M. Stolpe. Optimization strategies for discrete multi-materialstiffness optimization. Structural and Multidisciplinary Optimization, 44(2):149–163, 2011.ISSN 1615-147X. doi:10.1007/s00158-011-0648-5.

[40] K. A. James. Multiphase topology design with optimal material selection using an inversep-norm function. International Journal for Numerical Methods in Engineering, 114(9):999–1017, 2018. doi:10.1002/nme.5774.

[41] G. Karypis and V. Kumar. A fast and high quality multilevel scheme for partitioning irreg-ular graphs. SIAM journal on scientific computing, 20(1):359–392, 1998. ISSN 1064-8275.doi:10.1137/S1064827595287997.

[42] G. J. Kennedy. Large-scale multimaterial topology optimization for additive manufacturing.In 56th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference,Kissimmee, FL, January 2015. doi:10.2514/6.2015-1799. AIAA 2015-1799.

[43] G. J. Kennedy and T. W. Chin. A sequential convex optimization method for multimaterialcompliance design problems. Computers & Structures, 212:110 – 124, 2019. ISSN 0045-7949. doi:10.1016/j.compstruc.2018.10.007.

[44] G. J. Kennedy and J. R. R. A. Martins. A laminate parametrization technique for discreteply-angle problems with manufacturing constraints. Structural and Multidisciplinary Opti-mization, pages 1–15, 2013. ISSN 1615-147X. doi:10.1007/s00158-013-0906-9.

32

Chin, Leader, Kennedy

[45] G. J. Kennedy and J. R. R. A. Martins. A parallel finite-element framework for large-scalegradient-based design optimization of high-performance structures. Finite Elements in Anal-ysis and Design, 87(0):56 – 73, 2014. ISSN 0168-874X. doi:10.1016/j.finel.2014.04.011.

[46] N. Kikuchi, K. Y. Chung, T. Toshikazu, and J. E. Taylor. Adaptive finite element methods forshape optimization of linearly elastic structures. Computer Methods in Applied Mechanicsand Engineering, 57(1):67 – 89, 1986. ISSN 0045-7825. doi:10.1016/0045-7825(86)90071-X.

[47] S. I. Kim, S. W. Kang, Y.-S. Yi, J. Park, and Y. Y. Kim. Topology optimization of vehiclerear suspension mechanisms. International Journal for Numerical Methods in Engineering,113(8):1412–1433, 2018. ISSN 1097-0207. doi:10.1002/nme.5573.

[48] P. M. Knupp. Algebraic mesh quality metrics for unstructured initial meshes. Finite Elementsin Analysis and Design, 39(3):217 – 241, 2003. ISSN 0168-874X. doi:10.1016/S0168-874X(02)00070-7.

[49] V. Kolmogorov. Blossom V: a new implementation of a minimum cost perfect matchingalgorithm. Mathematical Programming Computation, 1(1):43–67, 2009. ISSN 1867-2957.doi:10.1007/s12532-009-0002-8.

[50] L. Krog, A. Tucker, M. Kemp, and R. Boyd. Topology optimisation of aircraft wing box ribs.In 10th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference, Albany, NY,August 2004. doi:10.2514/6.2004-4481. AIAA 2004-4481.

[51] A. B. Lambe and A. Czekanski. Topology optimization using a continuous density fieldand adaptive mesh refinement. International Journal for Numerical Methods in Engineering,2017. ISSN 1097-0207. doi:10.1002/nme.5617.

[52] B. S. Lazarov and O. Sigmund. Filters in topology optimization based on Helmholtz–typedifferential equations. International Journal for Numerical Methods in Engineering, 86(6):765–781, 2011. doi:10.1002/nme.3072.

[53] L. Lu, T. Yamamoto, M. Otomori, T. Yamada, K. Izui, and S. Nishiwaki. Topologyoptimization of an acoustic metamaterial with negative bulk modulus using local reso-nance. Finite Elements in Analysis and Design, 72:1 – 12, 2013. ISSN 0168-874X.doi:10.1016/j.finel.2013.04.005.

[54] E. Lund. Discrete Material and Thickness Optimization of laminated composite structuresincluding failure criteria. Structural and Multidisciplinary Optimization, Dec 2017. ISSN1615-1488. doi:10.1007/s00158-017-1866-2.

[55] K. Maute and E. Ramm. Adaptive topology optimization. Structural optimization, 10(2):100–112, 1995. ISSN 1615-1488. doi:10.1007/BF01743537.

[56] K. Maute and E. Ramm. Adaptive topology optimization of shell structures. AIAA Journal,35(11):1767–1773, 1997. doi:10.2514/2.25.

33

Chin, Leader, Kennedy

[57] D. Meagher. Geometric modeling using octree encoding. Computer Graphics and ImageProcessing, 19(2):129 – 147, 1982. ISSN 0146-664X. doi:10.1016/0146-664X(82)90104-6.

[58] J. Nocedal and S. J. Wright. Numerical Optimization. Springer Series in Operations Researchand Financial Engineering. Springer, 2nd edition, 2006.

[59] Open CASCADE S. A. S. Open CASCADE Technology, 3D modeling & numerical simula-tion, 2010. URL http://www.opencascade.org/.

[60] S. J. Owen. A survey of unstructured mesh generation technology. In IMR, pages 239–267,1998. URL http://ima.udg.edu/~sellares/ComGeo/OwenSurv.pdf.

[61] J.-F. Remacle, J. Lambrechts, B. Seny, E. Marchandise, A. Johnen, and C. Geuzainet.Blossom-Quad: A non-uniform quadrilateral mesh generator using a minimum-cost perfect-matching algorithm. International Journal for Numerical Methods in Engineering, 89(9):1102–1119, 2012. ISSN 1097-0207. doi:10.1002/nme.3279.

[62] S. Rojas-Labanda and M. Stolpe. Benchmarking optimization solvers for structural topologyoptimization. Structural and Multidisciplinary Optimization, 52(3):527–547, 2015. ISSN1615-1488. doi:10.1007/s00158-015-1250-z.

[63] Y. Saad and M. H. Schultz. GMRES: A generalized minimal residual algorithm for solvingnonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 7(3):856–869, 1986. doi:10.1137/0907058.

[64] E. D. Sanders, M. A. Aguilo, and G. H. Paulino. Multi-material continuum topology opti-mization with arbitrary volume and mass constraints. Computer Methods in Applied Mechan-ics and Engineering, 340:798 – 823, 2018. ISSN 0045-7825. doi:10.1016/j.cma.2018.01.032.

[65] Sandia National Laboratories. CUBIT, the Sandia National Laboratory automated mesh gen-eration toolkit, 2018. URL https://cubit.sandia.gov/public/15.3/help_manual/WebHelp/cubithelp.htm.

[66] Siemens PLM Software. Siemens NX 12.0, 2017. URL https://www.plm.automation.siemens.com/global/en/products/nx/.

[67] O. Sigmund. On the Design of Compliant Mechanisms Using Topology Op-timization. Mechanics of Structures and Machines, 25(4):493–524, 1997.doi:10.1080/08905459708945415.

[68] O. Sigmund and J. Petersson. Numerical instabilities in topology optimization: A surveyon procedures dealing with checkerboards, mesh-dependencies and local minima. Structuraloptimization, 16(1):68–75, 1998. ISSN 0934-4373. doi:10.1007/BF01214002.

[69] O. Sigmund and S. Torquato. Design of materials with extreme thermal expansion using athree-phase topology optimization method. Journal of the Mechanics and Physics of Solids,45(6):1037 – 1067, 1997. ISSN 0022-5096. doi:10.1016/S0022-5096(96)00114-7.

34

Chin, Leader, Kennedy

[70] S. N. Sørensen and E. Lund. Topology and thickness optimization of laminated compositesincluding manufacturing constraints. Structural and Multidisciplinary Optimization, 48(2):249–265, 2013. ISSN 1615-147X. doi:10.1007/s00158-013-0904-y.

[71] S. N. Sørensen, R. Sørensen, and E. Lund. DMTO – a method for discrete material andthickness optimization of laminated composite structures. Structural and MultidisciplinaryOptimization, 50(1):25–47, 2014. ISSN 1615-147X. doi:10.1007/s00158-014-1047-5.

[72] R. Stainko. An adaptive multilevel approach to the minimal compliance problem in topologyoptimization. Communications in Numerical Methods in Engineering, 22(2):109–118, 2006.ISSN 1099-0887. doi:10.1002/cnm.800.

[73] J. Stegmann and E. Lund. Discrete material optimization of general composite shell struc-tures. International Journal for Numerical Methods in Engineering, 62:2009–2027, 2005.ISSN 1097-0207. doi:10.1002/nme.1259.

[74] K. Svanberg. The method of moving asymptotes: a new method for structural optimization.International Journal for Numerical Methods in Engineering, 24(2):359–373, 1987. ISSN1097-0207. doi:10.1002/nme.1620240207.

[75] M. Y. Wang and X. Wang. “Color” level sets: a multi-phase method for structural topologyoptimization with multiple materials. Computer Methods in Applied Mechanics and Engi-neering, 193(6):469 – 496, 2004. ISSN 0045-7825. doi:10.1016/j.cma.2003.10.008.

[76] S. Wang, E. de Sturler, and G. H. Paulino. Large-scale topology optimization using pre-conditioned Krylov subspace methods with recycling. International Journal for NumericalMethods in Engineering, 69(12):2441–2468, 2007. doi:10.1002/nme.1798.

[77] Y. Wang, Z. Kang, and Q. He. An adaptive refinement approach for topology optimizationbased on separated density field description. Computers & Structures, 117:10 – 22, 2013.ISSN 0045-7949. doi:10.1016/j.compstruc.2012.11.004.

[78] Y. Wang, Z. Kang, and Q. He. Adaptive topology optimization with independent error controlfor separated displacement and density fields. Computers & Structures, 135:50 – 61, 2014.ISSN 0045-7949. doi:10.1016/j.compstruc.2014.01.008.

[79] T. Zegard and G. H. Paulino. Bridging topology optimization and additive manufacturing.Structural and Multidisciplinary Optimization, 53(1):175–192, Jan 2016. ISSN 1615-1488.doi:10.1007/s00158-015-1274-4.

[80] O. C. Zienkiewicz and J. Z. Zhu. A simple error estimator and adaptive procedure for practicalengineerng analysis. International Journal for Numerical Methods in Engineering, 24(2):337–357, 1987. ISSN 1097-0207. doi:10.1002/nme.1620240206.

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