Research Article Robust Topology Optimization Based on

Research ArticleRobust Topology Optimization Based on Stochastic CollocationMethods under Loading Uncertainties

Qinghai Zhao,1 Xiaokai Chen,1 Zheng-Dong Ma,2 and Yi Lin1

1Collaborative Innovation Center of Electric Vehicles in Beijing, School of Mechanical Engineering,Beijing Institute of Technology, Beijing 100081, China2Department of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48105, USA

Correspondence should be addressed to Xiaokai Chen; [email protected]

Received 14 May 2015; Revised 9 July 2015; Accepted 9 July 2015

Academic Editor: Domenico Mundo

Copyright © 2015 Qinghai Zhao et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A robust topology optimization (RTO) approach with consideration of loading uncertainties is developed in this paper. Thestochastic collocation method combined with full tensor product grid and Smolyak sparse grid transforms the robust formulationinto a weighted multiple loading deterministic problem at the collocation points. The proposed approach is amenable toimplementation in existing commercial topology optimization software package and thus feasible to practical engineeringproblems. Numerical examples of two- and three-dimensional topology optimization problems are provided to demonstrate theproposedRTOapproach and its applications.The optimal topologies obtained fromdeterministic and robust topology optimizationdesigns under tensor product grid and sparse grid with different levels are compared with one another to investigate the prosand cons of optimization algorithm on final topologies, and an extensive Monte Carlo simulation is also performed to verify theproposed approach.

1. Introduction

Structural optimization is an intrinsic element of engineer-ing design for the purpose of improving the structuralperformance and reducing costs. The major ingredients ofstructural optimization are, in general, divided into threelevels, namely, sizing, shape, and topology optimization. Theaim of classic topology optimization is to obtain an optimalmaterial or design parameter distribution at a given structuraldomain under nominal (i.e., deterministic) material proper-ties, geometry, and loading conditions. Until now, a rangeof topology optimization strategies has emerged, includingthe homogenization method, introduced by Bendsøe andKikuchi [1], the density-basedmethod [2], bilinear evolution-ary structural optimization (BESO) [3], and level set method[4]. Details of various representative schemes can be found incomprehensive reviews and text books [5–7].

Traditionally, structural topology optimizations are con-ducted in a deterministic manner, known as determin-istic topology optimization (DTO), where the design is

determined without taking into account various sources ofuncertainties [6]. However, uncertainties are unavoidablyobserved in real-world applications due to insufficient knowl-edge, manufacturing errors, changeable environment, andso forth. This may lead to the vulnerable optimum struc-ture or infeasible topologies due to the fluctuation of thestructure performance. Therefore, there is a strongly increas-ing requirement to take the effect of uncertainty into consid-eration for optimal topologies in structural design.

Recent years have witnessed various statements account-ing for uncertainty in topology optimization process. Inreliability-based topology optimization (RBTO), optimaldesigns with acceptable failure probabilities have beenachieved, where uncertainty is manifested in the probabilityconstraint functions. Many works dealing with applicationsof RBTO in structural design have appeared over the lastdecades. One of pioneeringworks can be traced back toKhar-manda et al. [8], where reliability analysis is decoupled fromoptimization procedures and carried out at the beginning of

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015, Article ID 580980, 14 pageshttp://dx.doi.org/10.1155/2015/580980

2 Mathematical Problems in Engineering

the optimization loops, followed by equivalent DTO. Differ-ent optimization strategies have been increasingly applied forreliable designs in various mechanic andmultiphysical fields.To cite a few, optimization algorithms via the density-basedmethod have been demonstrated formicroelectromechanicalsystems (MEMS) [9], geometrically nonlinear structures [10],and an automotive control arm [11]. Optimization problemsvia the BESO have been reported for a vehicle’s hood rein-forcement [12] and electrothermal-compliant mechanisms[13]. Topology optimization of compliant mechanisms withloads, material properties, and member geometries uncer-tainties is demonstrated for the level set method [14].

Distinct from the abovementioned formulations con-sidering the uncertainties in structural topology optimiza-tion problems, robust topology optimization (RTO) hasrecently emerged to optimize the objective performancewhile simultaneously minimizing its sensitivity with respectto uncertainties.When the uncertainty is assigned unknown-but-bounded convex models, the worst case approach isfollowed in general. De Gournay et al. [15] propose theso-called worst case or robust optimal design problem forminimal compliance based on the level set method. Amiret al. [16] demonstrate a fair comparison of a worst caseformulation and a stochastic formulation in robust topologyoptimization procedure. Guo et al. [17] present structuraltopology optimization considering the uncertainty of bound-ary variations via level set approach, where the objectivefunction is modeled by the compliance and fundamentalfrequency of structure enduring the worst case perturbation.

In the context of uncertainty parameters defining prob-ability distributions, the probabilistic description of theperformance function is usually characterized by its statisticalquantities such as mean value and standard deviation. Inthis scenario, a fundamental issue associated with these RTOprocedures is to calculate the statistical moments accuratelyand efficiently. The Monte Carlo simulations (MCS) are themost widely used sampling-based methods in robust designdue to their accuracy and easy implementation. However, itis time-consuming. Schevenels et al. [18] present a robusttopology optimization approach for the design of macro-,micro-, or nanostructures, accounting for spatially varyingmanufacturing errors, where the statistical moments andtheir sensitivities are estimated by means of the MCS. Jansenet al. [19] propose a robust approach to topology optimiza-tion taking into account geometric imperfections, where asampling method (i.e., 100 Monte Carlo samples) is used toestimate these statistics that are defined as a weighted sumobjective function. A new efficient and accurate approachto robust structural topology optimization under loadinguncertainty is developed by Zhao andWang [20]. TheMonteCarlo method and matrix decomposition are employed forconcentrated loads, and the Karhunen-Loeve expansion andits orthogonal properties of random variables are used fordistributed loads. Alternatively, the stochastic collocationmethod has been gaining more attention in robust design.The idea is to calculate the statistical moments by directnumerical integration. Chen et al. [21] employ the univariatedimension-reduction (UDR) method combined with Gausstype quadrature sampling for calculating statistical moments

with consideration of random field uncertainty in loadingand material properties. Lazarov et al. [22] introduce thestochastic collocation methods in topology optimization formechanical systems with material and geometric uncer-tainties. The robust designs are obtained by utilizing well-developed deterministic solvers.

Efficient and accurate generation of statistic moments isa central problem for robust design, especially for structuraltopology optimization with infinite-dimensional property.In this paper, we present a methodology for robust topol-ogy optimization problems with the stochastic collocationmethod to integrate the statistic moments considering load-ing uncertainty. Then, the robust topology optimizationformulation is transformed into a weighted multiple load-ing deterministic problem at the collocation points. Forthe construction of the stochastic collocation method, thecollocation points with corresponding weights are generatedby full tensor product grid and Smolyak sparse grid, respec-tively. Integrations based on full tensor product suffers fromthe curse of dimensionality, while the sparse grid methodoriginating from the Smolyak algorithm will dramaticallyreduce the number of collocation points without losingmuchaccuracy. In addition, existing commercial software packagesare capable of application in robust topology optimization inthese settings.

The paper is organized as follows: A brief description ofthe deterministic and robust topology optimization formu-lation is presented in Section 2. In Section 3, the stochas-tic collocation methods including tensor product grid andsparse grid are demonstrated. The computational resultsfor deterministic and robust topology optimization togetherwith several numerical examples are illustrated in Section 4.Conclusion and future works are discussed in Section 5.

2. Topology Optimization Formulation

2.1. Deterministic Topology Optimization Formulation. Thegeneralized topology optimization problem as shown inFigure 1 is that the predetermined design domain Ω, withrespect to the given loads and boundary conditions, isrepresented by the material distribution function so as tooptimize given objective function 𝑓 (e.g., compliance) underlimited volume constraint.

The structural topology optimization considering thegeneral elasticity problem is reformulated in the uniformcon-text of the principle of virtual displacement.The optimizationproblem for deterministic design can be written in a weakform as

min 𝑙 (u) = ∫Ω

𝜀𝑇(u)D (𝜌) 𝜀 (k) 𝑑Ω

s.t. 𝑎 (u, k) = 𝑙 (k) , ∀k ∈ K

∫Ω

𝜌 𝑑Ω ≤ 𝑉∗

𝜌min ≤ 𝜌 ≤ 𝜌max,

(1)

where K is the space of kinematically admissible displace-ment fields. u is the equilibrium displacement field, and v is

Mathematical Problems in Engineering 3

Design domain

Void material domain

Solid material domain

t

y

xo

F

Ω

ΓtΓu

Figure 1: Generalized topology optimization problem.

the arbitrary virtual displacement in the space K. 𝑉∗ is theprescribed volume constraint. The energy bilinear form andthe load linear form are defined as follows:

𝑎 (u, k) = ∫Ω

𝜀𝑇(u)D (𝜌) 𝜀 (k) 𝑑Ω,

𝑙 (k) = ∫Ω

F ⋅ k 𝑑Ω + ∫Γ𝑡

t ⋅ k 𝑑Γ,(2)

where 𝜀 is the strain tensor. F and t are the body forcesand the boundary tractions on the boundary Γ

𝑡, respectively.

Considering linear isotropic materials, Young’s modulus isa function of design variables 𝜌 given by the density-basedmethod [6] as follows:

𝐸 = 𝐸 (𝜌) = 𝜌𝑝𝐸0, (3)

where 𝐸0is Young’s modulus of solid material and 𝑝 is the

penalization power to ensure solid-void solutions.By using FEM discretization for the linear elasticity, the

optimization problem can be rewritten as follows:

min 𝑓 (𝜌) = FTU = UTKU =

𝑁𝑒

∑

𝑒=1

𝐸𝑒(𝑥𝑒) u𝑇𝑒k0u𝑒

s.t. 𝑉 = ∑

𝑒∈𝑁

𝜌𝑒V𝑒≤ 𝑉∗

K (𝜌)U = F


(4)

where 𝑓(𝜌) is the compliance of the continuum structure, Uis the displacement vector, and 𝑁 is a set of finite elements.u𝑒and k

0are the element displacement and the element

stiffness matrix with unit Young’s modulus, respectively, V𝑒

is the element volume, and the density of each elementis represented by one density variable 𝜌

𝑒. Therefore, the

standard FEM global stiffness matrix K is expressed as

K (𝜌) = ∑

𝑒∈𝑁

K𝑒(𝜌𝑒) = ∑

𝑒∈𝑁

∫Ω𝑒

B𝑇D (𝜌𝑒)B 𝑑Ω, (5)

f(𝜉)

𝜉

B

A

2Δ𝜉2Δ𝜉

Figure 2: The concept of robust design model.

where K𝑒(𝜌𝑒) is the element stiffness matrix, B is the strain-

displacement matrix of shape function derivatives, Ω𝑒is

element domain, andD(𝜌𝑒) is the element constitutivematrix,

defined in 2D as

D (𝜌𝑒) =

𝐸 (𝜌𝑒)

1 − ]2[[[[

[

1 ] 0

] 1 0

0 01 − ]2

]]]]

]

, (6)

where ] denotes Poisson’s ratio. In general, the optimalsolution obtained by direct implementation of the methodis prone to problems with checkerboards and mesh depen-dency. A large number of restriction schemes have been pro-posed to ensure manufacturability and mesh independency,such as perimeter control method, mesh-independent fil-tering methods, and Heaviside projection. A comprehensivecomparison of the existing restriction schemes is provided inthe paper of Sigmund [23].

2.2. Robust Topology Optimization Formulation. The topol-ogy optimization of linear elastic structures subject to loadinguncertainties is considered in this paper. Due to the variationof uncertain parameters 𝜉, the performance function 𝑓(𝜌, 𝜉)has probability distribution description.Thus, the robustnessof the design objective can be achieved by optimizing themean performance and minimizing the standard deviationof the performance simultaneously, as shown in Figure 2.In Figure 2, point 𝐴 denotes the deterministic optimum notconsidering the uncertainty, while point 𝐵 denotes the robustoptimum. The performance of deterministic optimum isbetter than that of the robust optimum.However, its variationis wider than the robust optimum.

Therefore, a main challenge in performing RTO for real-istic problems is the implementation of an algorithm, whichinherently represents a multiobjective optimization problem.One efficient way is to combine all objective functions into


a single weighted function. In mathematical terms, a generalrobust topology optimization task can be stated as

find: 𝜌

min: 𝜇𝑓(𝜌, 𝜉) + 𝜅𝜎

𝑓(𝜌,𝜉)

s.t.:𝑉 (𝜌)

𝑉∗− 1 ≤ 0


(7)

where 𝜅 is the weight to be determined by the designerthat satisfies 𝜅 ⩾ 0. Note that the weights have to be set apriori, which make a trade-off decision between an optimalmean performance and minimization of standard deviationof the performance to obtain a compromise solution. 𝜇

𝑓

and 𝜎𝑓are the mean value and the standard deviation of

the performance function, respectively. 𝑉∗ is the materialresource constraint. The design variables 𝜌 are bounded bytheir lower and upper bounds 𝜌min and 𝜌max, respectively. Ina probabilistic setting, the mean value 𝜇

𝑓and the standard

deviation 𝜎𝑓can be expressed as

𝜇𝑓(𝜌, 𝜉) = E [𝑓 (𝜌, 𝜉)] = ∫

Ω

𝑓 (𝜉) 𝑝 (𝜉) 𝑑𝜉,

𝜎𝑓(𝜌, 𝜉) = √E {[𝑓 (𝜌, 𝜉) − 𝜇

𝑓(𝜌, 𝜉)]

2

}

= {∫Ω

[𝑓 (𝜉) − 𝜇𝑓]2

𝑝 (𝜉) 𝑑𝜉}

1/2

,

(8)

where E is the expectation operator and 𝑝(𝜉) is the jointprobability density function (PDF) of the random variables.In practice, it is very difficult and even impossible to calculatethemean value and the variation of the performance functionthrough the multidimensional integration in (8). Such diffi-culties have motivated the development of various numericalmethods such as simulation-basedmethods, local expansion-based methods, and the stochastic collocation methods. Thereader is referred to Lee and Chen [24] for a more detailedreview.

3. Stochastic Collocation Method

The main idea of stochastic collocation method is to samplethe quantity of interest at specific collocation points in therandom space, where the sample points are deterministic andare associated with quadrature formula for the evaluationof integral statistics such as mean and standard deviation.Examples of such quadrature rules include the Newton-Cotes (midpoint, rectangle, and trapezoidal) rule, the Gaussquadrature (Legendre, Chebyshev, Laguerre, Hermite, Jacobi,Kronrod, and Patterson) rule, and the Clenshaw-Curtis rule[25–27]. In the case of a one-dimensional random space,these points are quadrature abscissas selected according tothe probability measure of the random variable. Tensorproduct grid and sparse grid constructions are then utilizedto generate the multidimensional abscissas.

3.1. Tensor Product Grid. Without loss of generality, considerthe class of function 𝑓(𝜉) over the 𝑑-dimensional unit hyper-cubeΩ := [−1, 1]

𝑑 with numerical integration representation

𝐼𝑑(𝑓) = ∫

Ω

𝑓 (𝜉) 𝑑𝜉. (9)

In the one-dimensional case, a sequence of univariate quadra-ture operators by 𝑚1

𝑙-point quadrature formulas with level

𝑙 ∈ 𝑁 is

𝑄1

𝑙(𝑓) =

𝑚1

𝑙

∑

𝑖=1

𝑓 (𝜉𝑖

𝑙)𝑤𝑖

𝑙, (10)

where 𝜉𝑖

𝑙and 𝑤

𝑖

𝑙are the collocation points and weights,

respectively. To obtain a quadrature formula for the multi-variate case 𝑑 > 1, the full tensor product formula is definedas

(𝑄1

𝑙1

⊗ ⋅ ⋅ ⋅ ⊗ 𝑄1

𝑙𝑑

) (𝑓)

=

𝑚1

𝑙1

∑

𝑙1=1

⋅ ⋅ ⋅

𝑚1

𝑙𝑑

∑

𝑙𝑑=1

𝑓 (𝜉𝑖1

𝑙1

, . . . , 𝜉𝑖𝑑

𝑙𝑑

) ⋅ (𝑤𝑖1

𝑙1

⊗ ⋅ ⋅ ⋅ ⊗ 𝑤𝑖𝑑

𝑙𝑑

) .

(11)

Then, the mean and standard deviation of the objectiveperformance are derived from the following equations:

𝜇𝑓= E [𝑓] = ∫

Ω

𝑓 (𝜉) 𝑝 (𝜉) 𝑑𝜉

=

𝑚1

𝑙1

∑

𝑙1=1

𝑤𝑖1

𝑙1

⋅ ⋅ ⋅

𝑚1

𝑙𝑑

∑

𝑙𝑑=1

𝑤𝑖𝑑

𝑙𝑑

𝑓 (𝜉𝑖1

𝑙1

, . . . , 𝜉𝑖𝑑

𝑙𝑑

) ,

𝜎𝑓= √E {[𝑓 − E (𝑓)]

2}

= {∫Ω

[𝑓 (𝜉) − 𝜇𝑓]2

𝑝 (𝜉) 𝑑𝜉}

1/2

=

{{

{{

{

𝑚1

𝑙1

∑

𝑙1=1

𝑤𝑖1

𝑙1

⋅ ⋅ ⋅

𝑚1

𝑙𝑑

∑

𝑙𝑑=1

𝑤𝑖𝑑

𝑙𝑑

[𝑓 (𝜉𝑖1

𝑙1

, . . . , 𝜉𝑖𝑑

𝑙𝑑

) − 𝜇𝑓]2}}

}}

}

1/2

.

(12)

Clearly, the tensor product grid requires 𝑀 = (𝑚1

𝑙1

, . . . , 𝑚1

𝑙𝑑

)

collocation points, which are sampled on the full grid.Assuming the same level accuracy in each dimension—thatis, 𝑚1𝑙1

= ⋅ ⋅ ⋅ = 𝑚1

𝑙𝑑

= 𝑚—the total number of points is𝑀 =

𝑚𝑑. Thus, the computational burden increases significantly

with the dimension 𝑑. This is often referred to as the curse ofdimensionality [28].

3.2. Sparse Grid. The sparse grid method is a special dis-cretization technique, which can be traced back to theSmolyak algorithm [29]. It is based on hierarchical basis, arepresentation of a discrete function space which is equiv-alent to the conventional nodal basis, and a sparse tensorproduct construction [30, 31]. The readers are referred to


the review paper by Xiong et al. [32] and the referencestherein for more information. The one-dimensional differ-ence quadrature formula is defined as

Δ1

𝑘(𝑓) = (𝑄

1

𝑘− 𝑄1

𝑘−1) (𝑓) , with 𝑄

1

0(𝑓) = 0. (13)

Then the Smolyak quadrature formula for 𝑑-dimensionalfunctions 𝑓 with level 𝑙 ∈ 𝑁 is given by

𝑄𝑑

𝑙(𝑓) = ∑

|k|≤𝑙+𝑑(Δ1

𝑘1

⊗ ⋅ ⋅ ⋅ ⊗ Δ1

𝑘𝑑

) (𝑓) . (14)

Note that |k| denotes the summation of the multi-indices(|k| = 𝑘

1+ ⋅ ⋅ ⋅ + 𝑘

𝑑). Alternatively, the above formula can

be written as

𝑄𝑑

𝑙(𝑓) = ∑

𝑙+1≤|k|≤𝑙+𝑑(−1)𝑙+𝑑−|k|

⋅ (

𝑑 − 1

𝑙 + 𝑑 − |k|)

⋅ (𝑄1

𝑘1

⊗ ⋅ ⋅ ⋅ ⊗ 𝑄1

𝑘𝑑

) (𝑓) .

(15)

To compute 𝑄𝑑𝑙(𝑓), a specific tensor product rule of sparse

grid is needed, defined as

𝑈𝑑

𝑙= ⋃

𝑙+1≤|k|≤𝑙+𝑑(𝑈1

𝑘1

⊗ ⋅ ⋅ ⋅ ⊗ 𝑈1

𝑘𝑑

) , (16)

where 𝑈1

𝑘1

denotes the set of collocation points for one-dimensional 𝑙-level accuracy. With the Smolyak algorithm,the weight 𝑤

𝑖corresponding to the 𝑖th collocation point 𝜉

𝑖

is defined as

𝑤𝑖= (−1)

𝑙+𝑑−|k|(

𝑑 − 1

𝑙 + 𝑑 − |k|) (𝑤𝑖1

𝑘1

⊗ ⋅ ⋅ ⋅ ⊗ 𝑤𝑖𝑑

𝑘𝑑

) . (17)

Then, the mean and standard deviation of the objective per-formance by sparse grid method can be computed by

𝜇𝑓= E [𝑓] =

𝑛𝑑

𝑙

∑

𝑖=1

𝑤𝑘𝑓 (𝜉𝑘) ,

𝜎𝑓= √E {[𝑓 − E (𝑓)]

2}

={

{

{

𝑛𝑑

𝑙

∑

𝑖=1

𝑤𝑘[𝑓 (𝜉𝑘) − 𝜇𝑓]2}

}

}

1/2

,

(18)

where 𝑚𝑑𝑙is the total number of points in sparse grid that is

determined as

𝑚𝑑

𝑙= ⋃

𝑙+1≤|k|≤𝑙+𝑑(𝑚1

𝑘1

⊗ ⋅ ⋅ ⋅ ⊗ 𝑚1

𝑘𝑑

) . (19)

Suppose that the same quadrature rule is chosen underthe dimension 𝑑 with same level in each dimension, and𝑚 represents the collocation points in each dimension andthe tensor-product grid scales as 𝑚𝑑, whereas the sparsegrid scales as 𝑚

log 𝑑, significantly mitigating the curse ofdimensionality. Accurate quadrature can be created by usinghigher values of the level 𝑙; however, this also leads to moresampling points.

The sparse grid method discretizes the stochastic spacein hierarchical structure based on nested collocation nodes,such as the Chebyshev nodes or Gauss-Patterson nodes,leading to Clenshaw-Curtis rule or Gauss-Patterson rule,respectively. Here, the Clenshaw-Curtis type sparse griddesign 𝐻CC, with nonequidistant nodes, is constructed [33–35]. The collocation points 𝜉𝑖

𝑘are defined as

𝜉𝑖

𝑘

={

{

{

− cos(𝜋 𝑖 − 1

𝑚𝑘− 1

) , for 𝑖 = 1, . . . , 𝑚𝑘, if 𝑚

𝑘> 1,

0, if 𝑚𝑘= 1.

(20)

The sequence𝑚𝑘is chosen as

𝑚𝑘={

{

{

1, if 𝑘 = 1,

2𝑘−1

+ 1, if 𝑘 > 1

(21)

and the corresponding weights are given by

𝑤𝑖

𝑘=

{{{{{{{{

{{{{{{{{

{

1

𝑚𝑘(𝑚𝑘− 2)

, for 𝑖 = 1,𝑚𝑘, if 𝑚

𝑘> 1,

2

(𝑚𝑘− 1)

(1 −cos𝜋 (𝑖 − 1)𝑚𝑘(𝑚𝑘− 2)

− 2

(𝑚𝑘−3)/2

∑

𝑗=1

1

4𝑗2 − 1⋅ cos

2𝜋𝑗 (𝑖 − 1)

𝑚𝑘− 1

) , for 𝑖 = 2, . . . , 𝑚𝑘− 1, if 𝑚

𝑘> 1,

2, if 𝑚𝑘= 1.

(22)


Table 1: Comparison of robust and deterministic topology optimization design results.

Tests Number of points Mean 𝜇𝑓

Standard deviation 𝜎𝑓

𝑉/𝑉∗ Iterations Time

Deterministic 1 22.36 12.03 0.25 54 35 sMonte Carlo simulation 1000 21.37 11.13 0.25 55 302 sRobust

Tensor product grid (𝑙 = 1) 9 22.26 11.85 0.25 51 59 sTensor product grid (𝑙 = 2) 25 21.87 11.66 0.25 54 90 sTensor product grid (𝑙 = 3) 81 21.77 11.34 0.25 54 147 sSparse grid (𝑙 = 1) 5 22.23 11.84 0.25 53 59 sSparse grid (𝑙 = 2) 13 21.92 11.70 0.25 54 68 sSparse grid (𝑙 = 3) 29 21.52 11.27 0.25 54 71 s

Examples of two-dimensional collocation points of fulltensor product grid and Smolyak sparse grid based onClenshaw-Curtis rule with level 𝑙 = 3 are shown in Figure 3.It is clearly observed that, compared with full tensor productgrid, the sparse grid dramatically reduces the number ofcollocation points, while preserving a high level of accuracy.

4. RTO Algorithm and Numerical Examples

4.1. Optimization Algorithm. The implementation of opti-mization algorithm for RTO is based on the commercialsoftware package Optistruct 11.0 of Altair Hyperworks, wherethe solid isotropic material with penalization is employedas material interpolation model and convex programmingmethod as optimization solver. In addition, the coordinatesand corresponding weights of the collocation points forrandom variables are evaluated in an external environmentof Optistruct for auxiliary conditions. The optimizationalgorithm can be summarized as follows:

(1) State problem discretization and initialization,including initial design domainΩ and boundary con-ditions D.

(2) Identify the random loading uncertainty 𝜉 as well asthe related loading angle and magnitude probabilitydistributions.

(3) Compute collocation points 𝜉𝑖𝑘and the corresponding

weights 𝑤𝑖𝑘by tensor product gird and sparse grid,

according to the dimension 𝑑 and level 𝑙 information.(4) Define multiple loading cases through mapping the

loading uncertainty with the collocation points.(5) Establish the mean 𝜇

𝑓and standard deviation 𝜎

𝑓

of the performance function through specifying theuser-defined equations, according to the prescribedformat.

(6) Implement the weighted function by combining themean and standard deviation functions with pre-defined weight 𝜅, and choose the function as theobjective function to be minimized.

(7) Define the manufacturing restriction schemes, opti-mization strategy, and volume fraction constraint.

(8) Run the optimization loop until the convergencecriterion is satisfied.

4.2. Numerical Examples. In this section, we will illustratethe proposed RTO approach using two numerical exampleproblems—a 2D simple supported beam and a 3D L-shapedstructure—to demonstrate the effectiveness and efficiencyfor tensor product grid and sparse grid. A simple Monte-Carlo simulation is also performed to verify the accuracy ofthe statistical moment calculated by the proposed approach.The procedure of topology optimization is conducted on aWindows 7 workstation (Intel Xeon (R) E5440, 2.83GHz,4.00GBRAM, 8 cores).

4.2.1. A 2D Simply Supported Beam. As the first example,the design domain, the boundary, and loading conditions ofthe simply supported beam are illustrated in Figure 4. Thedimensions of the beam are 𝐿 = 90mm and𝐻 = 30mm andthe thickness is 𝑇 = 1mm.The isotropic material is assumedto have Young’s modulus 𝐸

0of 1MPa and Poisson’s ratio ] of

0.30. A single random load case is considered with which theangle and magnitude are characterized by two independentrandom variables. The angle is assumed to follow a uniformdistribution with the interval of [−3𝜋/4, −𝜋/4]. The mag-nitude follows a Gaussian distribution with mean of 1 andstandard deviation of 0.3. In robust design, the optimizationproblem is to minimize the weighted summation of themean 𝜇


𝑓of the compliance of the

structure under the volume constraint, where the weight 𝜅 isset to be 1.The design domain is discretized by 2700 (90 × 30)four-node quadrilateral elements.

The optimal topologies obtained by deterministic andthe proposed robust topology optimization approaches arepresented in Figures 5 and 6, and the corresponding resultsare compared in Table 1, including the required number ofpoints, statistical information of the performance function,volume fraction constraint, iterations, and computing time.Here, full tensor product grid and sparse grid under level𝑙 = 1, 2, and 3 are investigated in this section. Extensive1000 Monte Carlo simulations are also performed to verifythe accuracy of the proposed robust approach.

For comparison purposes, the deterministic designis performed using the mean values of the load and


−1

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ghts

(i) Level: 1, points: 9

(ii) Level: 2, points: 25

(iii) Level: 3, points: 81

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Figure 3: Continued.


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(i) Level: 1, points: 5

(ii) Level: 2, points: 13

(iii) Level: 3, points: 29

(b) Sparse grid

Figure 3: Two-dimensional collocation points and weights by Clenshaw-Curtis rule in full tensor product grid (a) and Smolyak sparse grid(b) with level 𝑙 = 1, 2, and 3.


Table 2: Comparison of robust and deterministic topology optimization design results.

Tests Number of points Mean 𝜇𝑓

Standard deviation 𝜎𝑓

𝑉/𝑉∗ Iterations Time

Deterministic 1 241.08 131.42 0.06 44 7min 14 sMonte Carlo simulation 1000 182.20 111.34 0.06 51 28min 27 sRobust

Tensor product formula (𝑙 = 1) 9 183.95 121.71 0.06 46 8min 19 sTensor product formula (𝑙 = 2) 25 182.62 119.30 0.06 49 9min 44 sTensor product formula (𝑙 = 3) 81 182.91 113.72 0.06 49 17min 26 sSparse grid method (𝑙 = 1) 5 183.20 122.66 0.06 47 8min 18 sSparse grid method (𝑙 = 2) 13 182.71 119.24 0.06 50 8min 47 sSparse grid method (𝑙 = 3) 29 182.93 114.02 0.06 50 10min 25 s

F

H

L

Figure 4: Design domain, boundary conditions, and loading conditions of a 2D simply supported beam.

Contour plotElement densities (density)

1.0e + 008.9e − 017.8e − 016.7e − 015.6e − 014.5e − 013.4e − 012.3e − 011.2e − 011.0e − 02

Max = 1.000E + 002D 5572

Min = 1.000E − 022D 5576

No result

(a) Deterministic design


1.0e + 008.9e − 017.8e − 016.7e − 015.6e − 014.5e − 013.4e − 012.3e − 011.2e − 011.0e − 02

Max = 1.000E + 002D 2371

Min = 1.000E − 022D 2375

No result

(b) Monte Carlo simulation

Figure 5: Optimal designs of the simply supported beam.

the robust design is conducted with multiple load cases dueto randomness of load angle and amplitude. The DTO andRTO resulted in significantly different topologies, as shown inFigure 5. The remarkable difference lies in the robust designexhibiting an asymmetric layout compared with the deter-ministic design characterized by symmetric configuration. Itcan be attributed to the fact that the deterministic designsuffers from symmetric vertical constraints at the lowerleft and the lower right corner. However, in robust design,the structure possesses an asymmetrical degree of freedomconstraint in horizontal direction under the condition ofhorizontal component of load.

As the results listed in Table 1 show, the robust designspresent more robust topologies than the deterministic

approach, according to the results summary of the statisticalinformation. This emphasizes the importance of consideringthe uncertainties in load for structure design. It is also notedthat the corresponding computation cost for robust designsis larger than for the deterministic design since multiple loadcases are required for calculating the mean and deviationvalues of the performance function. The results of MonteCarlo simulations in Table 1 show that the proposed robustapproach possesses improved efficiency without losing muchaccuracy.

From the comparison of the results, with the improve-ment of level 𝑙 accuracy, the number of collocation points intensor product grid and sparse grid corresponding increases.As the results listed in Table 1 show, the mean and standard



1.0e + 008.9e − 017.8e − 016.7e − 015.6e − 014.5e − 013.4e − 012.3e − 011.2e − 011.0e − 02

Max = 1.000E + 002D 2371

Min = 1.000E − 022D 2375

No result

(a) Tensor product formula (𝑙 = 1)


1.0e + 008.9e − 017.8e − 016.7e − 015.6e − 014.5e − 013.4e − 012.3e − 011.2e − 011.0e − 02

Max = 1.000E + 002D 2371

Min = 1.000E − 022D 2375

No result

(b) Sparse grid method (𝑙 = 1)


1.0e + 008.9e − 017.8e − 016.7e − 015.6e − 014.5e − 013.4e − 012.3e − 011.2e − 011.0e − 02

Max = 1.000E + 002D 2371

Min = 1.000E − 022D 2375

No result

(c) Tensor product formula (𝑙 = 2)


1.0e + 008.9e − 017.8e − 016.7e − 015.6e − 014.5e − 013.4e − 012.3e − 011.2e − 011.0e − 02

Max = 1.000E + 002D 2371

Min = 1.000E − 022D 2375

No result

(d) Sparse grid method (𝑙 = 2)


1.0e + 008.9e − 017.8e − 016.7e − 015.6e − 014.5e − 013.4e − 012.3e − 011.2e − 011.0e − 02

Max = 1.000E + 002D 2371

Min = 1.000E − 022D 2375

No result

(e) Tensor product formula (𝑙 = 3)


1.0e + 008.9e − 017.8e − 016.7e − 015.6e − 014.5e − 013.4e − 012.3e − 011.2e − 011.0e − 02

Max = 1.000E + 002D 2371

Min = 1.000E − 022D 2375

No result

(f) Sparse grid method (𝑙 = 3)

Figure 6: Robust designs of the simple supported beam.

deviation of the compliance for sparse grid and tensorproduct grid are gradually close to those for Monte Carlosimulation under the increase of level 𝑙. This indicates thatmore robust designs are clearly obtained under higher level𝑙. Meanwhile, the higher computation cost is required dueto the increase in the number of evaluations of performancefunction responses. This means robustness enhancement isinevitably accompanied with the sacrifice of convergencetime. It can also be seen from the results listed in Table 1that, under the same level 𝑙, sparse grid uses fewer collocationpoints than those of tensor product grid to produce similareven more robust results. At the same time, compared withsparse grid, the computation burden for the exponentialdependent calculation of tensor product grid can be over-come to a certain extent.Thismeans sparse grid, in amanner,

performs better than the tensor product grid in compromisebetween robustness and cost in robust design.

4.2.2. A 3D L-Shaped Structure. The design domain andloading conditions of this example are illustrated in Figure 7.The geometrical dimensions of the structure are 𝐿 = 120mmand 𝑇 = 30mm. Young’s modulus and Poisson’s ratio of thefully solid material are 2.1 × 105MPa and 0.30, respectively.Two concentrated forces, 𝐹

𝑌and 𝐹

𝑍, are applied at central

point of right bottom surface simultaneously.Themagnitudesof the forces are characterized by two uncorrelated randomvariables, which follow Gaussian distributions with meanof 1000N and 30% variance from the mean values. Theoptimization problem is still to minimize the weightedsummation of the mean 𝜇


𝑓of


L

L/2

L

YXO

Z

T

FY

FZL/2

Figure 7: Design domain of the 3D L-shaped structure.


1.0e + 00

8.9e − 01

7.8e − 01

6.7e − 01

5.6e − 01

4.5e − 01

3.4e − 01

2.3e − 01

1.2e − 01

1.0e

Y

XZ

− 02

Max = 1.000E + 003D 49394

Min = 1.000E − 023D 47433

No result

(a) Deterministic design


1.0e + 00

8.9e − 01

7.8e − 01

6.7e − 01

5.6e − 01

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Max = 1.000E + 003D 49394

Min = 1.000E − 023D 47433

No resultY

XZ

(b) Monte Carlo simulation design

Figure 8: Optimal topologies of the L-shaped structure.

the compliance of the structure under the volume constraint.The design domain is divided into 1.2 × 104 eight-nodehexahedron elements.

The optimal topologies obtained by deterministic, MonteCarlo simulation and the robust approaches with densityvalue above 0.3 thresholds are presented in Figures 8 and9, respectively. The corresponding optimization results aresummarized in Table 2, including the required number ofpoints, statisticalmoments, volume constraint, iterations, andcomputing time. Meanwhile, full tensor product grid andsparse grid are investigated under level 𝑙 = 1, 2, and 3.

Comparing the deterministic design with the proposedrobust designs, obvious distinctions can be clearly observedin optimal topologies as shown in Figures 8 and 9. Fromthe viewpoint of geometry, the robust designs representmore rational material layouts, which lead to more robuststructural configuration. According to Table 2, the finalstatistical moments of the robust designs are superior tothat of deterministic design. This indicates that the robust

design is less sensitive to the variations of load magnitude.It can also be seen from the results listed in Table 2 that asthe level 𝑙 is increasing, the mean and standard deviation ofthe structure compliance for tensor product grid and sparsegrid are closer to those for Monte Carlo simulation in whichmore robust designs are obtained, but it is also revealed thatthe corresponding computing time is increasing. This meansrobustness and cost are conflicting criteria in robust topologyoptimization design. For comparison of the results of tensorproduct grid and sparse grid, these indicate that sparse grid,in a manner, performs better than the tensor product gridcomprehensively considering the accuracy and efficiency.

5. Conclusion

This paper integrates the stochastic collocation methodsinto structural topology optimization for solving the RTOproblem with random loading uncertainty. Specially, tensor



1.0e + 00

8.9e − 01

7.8e − 01

6.7e − 01

5.6e − 01

4.5e − 01

3.4e − 01

2.3e − 01

1.2e − 01

1.0e − 02

Max = 1.000E + 003D 50394

Min = 1.000E − 023D 47433

No resultY

XZ

(a) Tensor product grid (𝑙 = 1)


1.0e + 00

8.9e − 01

7.8e − 01

6.7e − 01

5.6e − 01

4.5e − 01

3.4e − 01

2.3e − 01

1.2e − 01

1.0e − 02

Max = 1.000E + 003D 48164

Min = 1.000E − 023D 47433

No resultY

XZ

(b) Sparse grid (𝑙 = 1)


1.0e + 00

8.9e − 01

7.8e − 01

6.7e − 01

5.6e − 01

4.5e − 01

3.4e − 01

2.3e − 01

1.2e − 01

1.0e − 02

Max = 1.000E + 003D 49394

Min = 1.000E − 023D 47433

No resultY

XZ

(c) Tensor product grid (𝑙 = 2)


1.0e + 00

8.9e − 01

7.8e − 01

6.7e − 01

5.6e − 01

4.5e − 01

3.4e − 01

2.3e − 01

1.2e − 01

1.0e − 02

Max = 1.000E + 003D 49394

Min = 1.000E − 023D 47433

No resultY

XZ

(d) Sparse grid (𝑙 = 2)


1.0e + 00

8.9e − 01

7.8e − 01

6.7e − 01

5.6e − 01

4.5e − 01

3.4e − 01

2.3e − 01

1.2e − 01

1.0e − 02

Max = 1.000E + 003D 49394

Min = 1.000E − 023D 47433

No resultY

XZ

(e) Tensor product grid (𝑙 = 3)


1.0e + 00

8.9e − 01

7.8e − 01

6.7e − 01

5.6e − 01

4.5e − 01

3.4e − 01

2.3e − 01

1.2e − 01

1.0e − 02

Max = 1.000E + 003D 49394

Min = 1.000E − 023D 47433

No resultY

XZ

(f) Sparse grid (𝑙 = 3)

Figure 9: Optimal designs of the L-shaped structure by robust topology optimization.


product grid and sparse grid are employed to transformthe RTO into a simplified weighted multiple loading deter-ministic problem at the collocation points. The proposedrobust approaches are implemented, discussed, and testedon two numerical examples by implementation in existingcommercial topology optimization software package. Anextensive Monte Carlo simulation is also performed toverify the accuracy and validation of the proposed approach.From the comparison of the results, the following conclusionsare summarized:

(1) The proposed RTO approach possesses more robusttopologies thanDTOandmay exhibitmanifest distinct topol-ogy configuration. It is revealed that robust topology opti-mization is essential when involving the uncertainty param-eters in optimization design. The stochastic collocationmethods are compatible with RTO for calculating statisticalmoments of the objective performance response and arereadily implemented in existing commercial topology opti-mization software package.

(2) For tensor product grid and sparse grid, more robustresults are clearly obtained under higher level setting. Mean-while, the required computation cost is more expensive.This indicates that robustness enhancement is inevitablyaccompanied with the sacrifice of computational expense.Sparse grid demands fewer collocation points than tensorproduct grid to generate similar even more robust resultsunder the same level. At the same time, the computationburden for tensor product grid can be overcome to a certainextent. It is demonstrated that sparse grid, in a manner,performs better than the tensor product grid in compromisebetween robustness and computation cost.

(3) It is important to notice that, compared to deter-ministic topology optimization, the application of robusttopology optimization to practical structural engineering isstill a challenge task because of the high RTO computationcost. Future research is still recommended to implement theproposed RTO approach for optimization problems subjectto non-Gaussian type random loading uncertainty.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

This research is supported by National Natural ScienceFoundation of China (Grants no. 51275040 and no. 50905017)and the Programme of Introducing Talents of Discipline toUniversities (no. B12022).

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