A polynomial dimensional decomposition framework based on topology derivatives for stochastic topology sensitivity analysis of high-dimensional complex systems and a type of benchmark problems
aa r X i v : . [ c s . C E ] S e p A polynomial dimensional decomposition framework based on topologyderivatives for stochastic topology sensitivity analysis of high-dimensionalcomplex systems and a type of benchmark problems
Xuchun Ren a Mechanical Engineering Department, Georgia Southern University, Statesboro, GA, USA
Abstract
In this paper, a new computational framework based on the topology derivative concept is presented for evaluatingstochastic topological sensitivities of complex systems. The proposed framework, designed for dealing with highdimensional random inputs, dovetails a polynomial dimensional decomposition (PDD) of multivariate stochastic re-sponse functions and deterministic topology derivatives. On one hand, it provides analytical expressions to calculatetopology sensitivities of the first three stochastic moments which are often required in robust topology optimization(RTO). On another hand, it o ff ers embedded Monte Carlo Simulation (MCS) and finite di ff erence formulations to es-timate topology sensitivities of failure probability for reliability-based topology optimization (RBTO). For both cases,the quantification of uncertainties and their topology sensitivities are determined concurrently from a single stochasticanalysis. Moreover, an original example of two random variables is developed for the first time to obtain analyticalsolutions for topology sensitivity of moments and failure probability. Another 53-dimension example is constructedfor analytical solutions of topology sensitivity of moments and semi-analytical solutions of topology sensitivity offailure probabilities in order to verify the accuracy and e ffi ciency of the proposed method for high-dimensional sce-narios. Those examples are new and make it possible for researchers to benchmark stochastic topology sensitivitiesof existing or new algorithms. In addition, it is unveiled that under certain conditions the proposed method achievesbetter accuracies for stochastic topology sensitivities than for the stochastic quantities themselves. Keywords: stochastic topology sensitivity analysis, topology derivatives, polynomial dimensional decomposition,stochastic moments, reliability
1. Introduction
With the rise of additive manufacturing, topology optimization becomes a popular design methodology to deter-mine the optimal distribution of materials in complex engineering structures[1, 2, 3, 4]. Inevitable uncertainties inthe additive manufacturing process and operating environment often undermine the performance of such topology de-signs. Classical deterministic design approaches often lead to unknowingly risky designs due to the underestimation ofuncertainties, or ine ffi cient and conservative designs that overcompensate for uncertainties. In the past decade, robusttopology optimization (RTO) and reliability-based topology optimization (RBTO) are increasingly adopted as an en-abling technology for topology design subject to uncertainty in aerospace, automotive, civil engineering, and additivemanufacturing [5, 6, 7, 8, 9, 10, 11]. The former seeks for insensitive topology design via minimizing the propagationof input uncertainty, whereas the latter delivers reliable topology design by introducing probabilistic characterizationsof response functions into the objective and / or constraints.RTO and RBTO for realistic engineering applications confront two challenges: (1) the theoretically infinite-dimensionaldesign vector; and (2) high-dimensional integration resulted from a large number of random variables. Both lead tothe curse of dimensionality, which hinders or invalidates almost all RTO and RBTO methods. In RTO, the objective or ∗ Corresponding author
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Preprint submitted to Probabilistic Engineering Mechanics September 11, 2020 onstraint functions are usually expressed by first two moment properties, such as means and standard deviations, ofcertain stochastic responses, describing the objective robustness or feasibility robustness of a given topology. RBTOoften contains probabilistic constraint functions, which restrict the probability of failure regarding certain failuremechanisms. Therefore, to solve a practical RTO or RBTO problem using gradient-based algorithm, an e ffi cient andaccurate method for statistical moments, reliability, and their sensitivity analysis of random responses are in demands.The fundamental problem rooted in statistical moment or reliability analyses entails the evaluation of a high-dimensionalintegral in the entire support of random inputs or its unknown subdomain, respectively. In general, such an integralcannot be calculated analytically. Direct numerical quadrature can be applied, but it is computationally prohibitivewhen the number of random inputs exceeds three or four, especially when the evaluation of a response functionis carried out by expensive finite element analyses (FEA). Existing approaches for statistical moment and reliabil-ity analysis include the point estimate method (PEM) [12], Taylor series expansion or perturbation method [12],tensor product quadrature (TPQ) [13], Neumann expansion method [14], the first-order reliability method or FORM-based methods [15, 16, 17, 18, 19, 20], polynomial chaos expansion (PCE) [21], statistically equivalent solution [22],dimension-reduction method [23, 24], and others [25]. Their topology sensitivities have relied mainly on two kinds ofapproaches: SIMP-based approaches (solid isotropic material with penalization) [26] and topology-derivative-basedapproaches [27, 28]. The former is based on a fictitious density field representing a smooth transition between mate-rial and empty, which requires regularization procedures to get a clear topology. The latter introduces the topologicalderivative concept which defines the derivative of functionals whose variable is a geometrical domain with respect tosingular topology perturbation. The topological derivative concept is mathematically rigorous and independent of thefictitious density field.Nonetheless, three major concerns arise when evaluating stochastic quantities and their sensitivity using existingapproaches or techniques. First, when applied to large-scale topology optimization subject to a large number of ran-dom inputs, many of those methods including Taylor series expansions, FORM-based methods, PEM, PCE, TPQ,and dimension-reduction methods, etc. begin to be inapplicable or inadequate. For example, although the Taylorseries expansion, FORM-based methods, and PEM are inexpensive and simple, they deteriorate due to the lack ofaccuracy when the nonlinearity of a response function is high and / or when the input uncertainty is large. PCE ap-proximates the random response via an infinite series of Hermite polynomials of Gaussian variables (or others) andis popular in stochastic mechanics in the last decades. Although truncated forms of PCE were extensively used inpractice [29, 30], it is easily succumbed to the curse of dimensionality due to astronomically large numbers of termsor coe ffi cients required to capture the interaction e ff ects between random inputs when applied to high-dimensionalsystems. Rooted in the referential dimensional decomposition (RDD), the dimension-reduction approximates a highdimensional function via a set of low dimensional components, but it often results in sub-optimal estimations of theoriginal function, and thus its stochastic moments and the associated reliability. Second, to evaluate the topologysensitivity of stochastic quantities, many of the aforementioned methods may not be adequately e ffi cient and accu-rate. Most of those methods rely on a fictitious density field, thus the sensitivities supplied are not the exact topologysensitivity. Furthermore, many of them resort to repetitive stochastic analyses especially for the sensitivity of reli-ability due to employed finite-di ff erence techniques, which restrain their computational e ffi ciency. Although Taylorseries expansions, is able to perform stochastic sensitivities analysis economically, its accuracy is usually deterioratedby inherited errors from the associated stochastic analysis. Third, to the best of the author’s knowledge, in existingliterature, there is no benchmark example that provides analytical or semi-analytical solutions for stochastic topologysensitivity analysis. A successful benchmark example certainly calls for analytical expressions of stress, strain, orother response functions in two domains - an original domain and a perforated domain - subject to the same loadsand supports. These analytical expressions generally are not readily available even for simple domain and ordinaryload cases. Moreover, verifying the performance of a certain method subject to high-dimensional random inputs oftendemands the benchmark example carrying on complex loads to accommodate a large number of random variables,which impede the implementation of analytical solution of stochastic topology sensitivity. These di ffi culties resultin the lack of benchmark examples and make it impossible to verify the accuracy of existing and new algorithms,especially for high-dimensional cases.This paper presents a novel framework for topology sensitivity analysis of statistical moments and reliability forcomplex engineering structures subject to a large number of random inputs. The framework, designed for dealinghigh-dimensional random inputs, is grounded on the polynomial dimensional decomposition (PDD), and thus it is ca-pable of approximating the high-dimensional stochastic responses in an e ffi cient and accurate manner. It also dovetails2he deterministic topology derivatives with PDD and provides stochastic sensitivities in the exactly topological sense.For RTO, the proposed framework is endowed with analytical expressions for topology sensitivities of the first threestochastic moments. For RBTO, it supplies embedded Monte Carlo Simulation (MCS) and a finite di ff erence formula-tion to estimate topology sensitivities of failure probability. Furthermore, the evaluation of moments and / or reliabilityand their topology sensitivity is accomplished concurrently from only a single stochastic analysis. It is noteworthy thattwo benchmark examples, which provide analytical / semi-analytical topology sensitivity of moments and reliability,are developed for the calibration of stochastic topology sensitivity algorithms. The first example contains only tworandom variables but provides analytical expressions for moments, reliability, and their topology sensitivities. Thesecond one accommodates 53 random variables, whereas the analytical expressions provided can be easily expandedto any positive number of random variables. The rest of this paper is organized as follows. Section 2 formally de-fines general RTO and RBTO problems, including a concomitant mathematical statement. Section 3 starts with abrief exposition of the polynomial dimensional decomposition and associated approximations, which result in explicitformulae for the first two moments and an embedded MCS formulation for the reliability of a generic stochastic re-sponse. Section 4 revisits the definition of topology derivative and describes the new framework of stochastic topologysensitivity analysis, which integrates PDD and deterministic topological derivative as well as numerical proceduresfor topology sensitivities of both stochastic moment and reliability. The calculation of PDD expansion coe ffi cients isbriefly described in Section 5. Section 6 presents three numerical examples. Two benchmark examples are developedto probe the accuracy and computational e ff orts of the proposed method. One three-dimensional bracket is used todemonstrate the feasibility of the new method for practical engineering applications. Finally, conclusions are drawnin Section 7.
2. Stochastic topology design problems
In the presence of uncertainties, a topology optimization problem can include robust, probabilistic, or non-probabilistic constraints. For RTO, both objective and constraint functions may involve the first two moment propertiesfor the assessment of robustness [31]. Whereas for RBTO, probabilistic functions are often embedded as constraintsto restrict the failure probability and achieve a high confidence level on design [32, 16]. Nonetheless, the typicalRTO and RBTO problems interested in this paper are often formulated as the following mathematical programmingproblems min Ω ⊆ D c ( Ω ) : = w E (cid:2) y ( Ω , X ) (cid:3) µ ∗ + w p var (cid:2) y ( Ω , X ) (cid:3) σ ∗ , subject to c k ( Ω ) : = α k q var (cid:2) y k ( Ω , X ) (cid:3) − E (cid:2) y k ( Ω , X ) (cid:3) ≤ k = , · · · , K (1)and min Ω ⊆ D c ( Ω ) : = w E (cid:2) y ( Ω , X ) (cid:3) µ ∗ + w p var (cid:2) y ( Ω , X ) (cid:3) σ ∗ , subject to c k ( Ω ) : = P (cid:2) X ∈ Ω F , k (cid:3) ≤ p k ; k = , · · · , K , (2)respectively, where D ⊂ R is a bounded domain in which all admissible topology design Ω are included; X : = ( X , · · · , X N ) T ∈ R N is an N -dimensional random input vector completely defined by a family of joint probabilitydensity functions { f X ( x ) , x ∈ R N } on the probability triple ( Ω X , F , P ), where Ω X is the sample space; F is the σ -field on Ω X ; P is the probability measure associated with probability density f X ( x ); Ω F , k is the k th failure domaindefined by response function y k ( Ω , X ); 0 < p k < w ∈ R + and w ∈ R + aretwo non-negative, real-valued weights, satisfying w + w = µ ∗ ∈ R \ { } and σ ∗ ∈ R + are two non-zero, real-valued scaling factors; α k ∈ R + , k = , , · · · , K , are positive, real-valued constants associated with the probabilitiesof constraint satisfaction; E and var are expectation operator and variance operator, respectively, with respect to3he probability measure P . The evaluation of both E and var on certain random response demands statistical momentanalysis [33, 34, 35, 36, 37, 14, 22, 23, 24, 25], which is not unduly di ffi cult. By contrast, the evaluation of probabilisticconstraint functions in RBTO, generally more complicated than E and var, is obtained from c k ( Ω ) : = P (cid:2) X ∈ Ω F , k (cid:3) = Z Ω F , k f X ( x ) d x = Z R N I Ω F , k ( Ω , x ) d x : = E h I Ω F , k ( Ω , X ) i (3)which represents a failure probability from reliability analysis [38, 39, 40, 41, 42, 43, 44, 45, 46, 47]. The indicatorfunction I Ω F , k ( Ω , x ) = x ∈ Ω F , k and zero otherwise. For component-level RBTO, the failure domain, oftenadequately described by a single performance function y k ( Ω , x ), and component reliability analysis are relativelysimple. Whereas, interdependent performance functions y ( q ) k ( Ω , x ) , q = , , · · · , are required for a system-level(series, parallel, or general) RBTO, leading to a highly complex failure domain and huge computational demand forsystem reliability analysis.
3. Polynomial dimensional decomposition method and uncertainty quantification
Consider a multivariate stochastic response y ( Ω , X ) of certain topology design Ω subject to random input vector X = { X , · · · , X N } T , representing any of the performance function y k in Eq. (1) or (2). Let L ( Ω X , F , P ) be a Hilbertspace of square-integrable functions y with a probability measure f X ( x ) d x supported on R N . Assuming independentcomponents of X , the PDD expansion of function y generates a hierarchical representation[48, 49] y ( Ω , X ) = y ∅ ( Ω ) + X ∅ , u ⊆{ , ··· , N } X j | u | ∈ N | u | C u j | u | ( Ω ) ψ u j | u | ( X u ; Ω ) , (4)of the original performance function, in terms of an infinite number of multivariate orthonormal basis [48, 49] ψ u j | u | ( X u ; Ω ) : = Q | u | p = ψ i p j p ( X i ; Ω ) in L ( Ω X , F , P ), where j | u | = ( j , · · · , j | u | ) ∈ N | u | is a | u | -dimensional multi-index; y φ ( Ω ) contributes the constant component; for | u | = C u j | u | ( Ω ) ψ u j | u | ( X u ; Ω ) commits all univariate component func-tions representing the individual contribution to y ( Ω , X ) from each single input variable; for | u | =
2, it brings in allbivariate component functions embodying the cooperative influence of any two input variables; and for | u | = S , it ad-mits S -variate component functions quantifying the interactive e ff ects of any S input variables. For most performancefunctions in engineering applications, a truncated version of Eq. (4) is often accurate enough by retaining, at most,the interactive e ff ects of S < N input variables and m th order polynomials,˜ y S , m ( Ω , X ) = y ∅ ( Ω ) + X ∅ , u ⊆{ , ··· , N } ≤| u |≤ S X j | u | ∈ N | u | k j | u | k ∞ ≤ m C u j | u | ( Ω ) ψ u j | u | ( X u ; Ω ) , (5)where y ∅ ( Ω ) = Z R N y ( x , Ω ) f X ( x ) d x (6)and C u j | u | ( Ω ) : = Z R N y ( x , Ω ) ψ u j | u | ( x u ; Ω ) f X ( x ) d x , ∅ , u ⊆ { , · · · , N } , j | u | ∈ N | u | , (7)are referred to as expansion coe ffi cients of PDD expansion (4) or truncated PDD approximation (5). The untruncatedPDD expansion in Eq. (4) employs an orthogonal polynomial basis and exactly represents the response function, itcan be easily refer that it is equivalent to PCE when the basis used is same. However, the PDD expansion provides ahierarchical representation by classifying the interaction between random inputs, which is a key to alleviate the courseof dimensionality when applying its truncated version. For S > m >
0, Eq. (5) retains interactive e ff ects amongat most S input variables X i , · · · , X i S , 1 ≤ i < · · · < i S ≤ N and m th order polynomial nonlinearity in y , thus leadingto the so-called S -variate, m th-order PDD approximation. When S → N and m → ∞ , ˜ y S , m converges to y in the mean-square sense and engenders a sequence of hierarchical and convergent approximations of y . Based on the dimensional4tructure and nonlinearity of a stochastic response, the truncation parameters S and m can be chosen correspondingly.The higher the values of S and m permit the higher the accuracy, but also endow the computational cost of an S th-order polynomial computational complexity [48, 49]. Henceforth, the S -variate, m th-order PDD approximation willbe simply referred to as truncated PDD approximation in this paper. For an arbitrary random response of certain topology design Ω , let m ( r ) ( Ω ) : = E [ y r ( Ω , X )], if it exists, denote the rawmoment of y of order r , where r ∈ N . Let ˜ m ( r ) ( Ω ) : = E [˜ y rS , m ( Ω , X )] denote the raw moment of ˜ y S , m of order r , given an S -variate, m th-order PDD approximation ˜ y S , m ( Ω , X ) of y ( Ω , X ). The analytical expressions or explicit formulae forestimating the moments using PDD approximations are described as follows. Applying the expectation operator on˜ y S , m ( Ω , X ) and ˜ y S , m ( Ω , X ), the first moment or mean [50]˜ m (1) S , m ( Ω ) : = E (cid:2) ˜ y S , m ( Ω , X ) (cid:3) = y ∅ ( Ω ) = E (cid:2) y ( Ω , X ) (cid:3) = : m (1) ( Ω ) (8)of the S -variate, m th-order PDD approximation is simply the constant component in Eq. (5), whereas the secondmoment [50] ˜ m (2) S , m ( Ω ) : = E h ˜ y S , m ( Ω , X ) i = y ∅ ( Ω ) + X ∅ , u ⊆{ , ··· , N } ≤| u |≤ S X j | u | ∈ N | u | k j | u | k ∞ ≤ m C u j | u | ( Ω ) (9)is expressed as the sum of squares of all expansion coe ffi cients of ˜ y S , m ( Ω , X ). It is straightforward that the estimationof the second moment evaluated by Eq. (9) approaches the exact second moment m (2) ( Ω ) : = E h y ( Ω , X ) i = y ∅ ( Ω ) + X ∅ , u ⊆{ , ··· , N } X j | u | ∈ N | u | C u j | u | ( Ω ) (10)of y when S → N and m → ∞ . The mean-square convergence of ˜ y S , m is ensured as its component functions willcontain all required bases of the corresponding Hilbert spaces. Furthermore, the variance of ˜ y S , m ( Ω , X ) is also mean-square convergent. The RBTO problem defined in Eq. (2) requires not only stochastic moment analysis but also evaluations of theprobabilistic constraints, that is, the failure probability P F = P (cid:2) X ∈ Ω F , k (cid:3) = Z Ω F , k f X ( x ) d x = Z R N I Ω F , k ( Ω , x ) d x : = E h I Ω F , k ( Ω , X ) i (11)of a certain topology design Ω with respect to certain failure set Ω F , k . In which, the indicator function I Ω F , k ( Ω , x ) = x ∈ Ω F , k and zero otherwise. For component-level RBTO, the failure set is often adequately characterizedby a single performance function y k ( Ω , x ) as Ω F , k : = { x : y k ( Ω , x ) < } . Whereas for a system-level RBTO, it isusually described by multiple, interdependent performance functions y ( q ) k ( Ω , x ) , q = , , · · · , leading, for example,to Ω F , k : = { x : ∪ q y ( q ) k ( Ω , x ) , < } and Ω F , k : = { x : ∩ q y ( q ) k ( Ω , x ) , < } for series and parallel systems, respectively. Let˜ Ω F , k : = { x : ˜ y S , m ( x ) < } or ˜ Ω F , k : = { x : ∪ q ˜ y ( q ) S , m ( x ) < } or ˜ Ω F , k : = { x : ∩ q ˜ y ( q ) S , m ( x ) < } be an approximate failure set asa result of S -variate, m th-order PDD approximations ˜ y S , m ( X ) of y ( X ) or ˜ y ( q ) S , m ( X ) of y ( q ) ( X ). Then the embedded MCSestimate of the failure probability P F is˜ P F = E d h I ˜ Ω F , k ( Ω , X ) i = lim L →∞ L L X l = I ˜ Ω F , k ( Ω , x ( l ) ) , (12)where L is the sample size, x ( l ) is the l th realization of X , and I ˜ Ω F , k ( Ω , x ), equal to one when x ∈ ˜ Ω F , k and zero otherwise, is an approximation of the indicator function I Ω F , k ( Ω , x ).5 igure 1: A perforated domain Note that the stochastic moment analysis and reliability analysis for RTO and RBTO are quite similar to the onesin a general robust design optimization (RDO) and reliability-based design optimization (RBDO) [51, 52, 53] exceptthat the former is a ffi liated with certain topology designs Ω . However, topology sensitivity analysis of moments andreliability is distinct from sensitivity analysis in RDO and RBDO due to the disparate topology change associated,and is elaborated in the next section.
4. Stochastic topology sensitivity analysis
To evaluate the topology sensitivity of a stochastic response, a new framework is proposed here which dovetailsPDD and deterministic topological derivative. It relies fundamentally on the topology derivative [54, 55, 56, 57, 58,59, 60, 61, 62] of a deterministic objective function y ( Ω ). The new method provides closed-form solutions and anembedded MCS formulation for the topological derivative of stochastic moments and reliability, respectively. Beforepresenting the new framework itself, a brief revisit on the idea of topological derivative appears to be necessary andshould be convenient to those not yet familiar with the concept. Pioneered by Schumacher[63], Sokolowski and Zochowski [64, 65], and Garreau et al. [66], the topologicalderivative measures the change of a performance functional when an infinitesimal hole is introduced in the referencedomain in which a boundary-value problem is defined. For a given reference domain Ω ⊂ R n , a point ξ ∈ Ω , anda hole ω ∈ R n with the radius of 1, a translated and rescaled hole can be defined by ω ρ = ξ + ρω, ∀ ρ > Ω ρ = Ω \ ¯ ω ρ as shown in Fig. 1.For a small ρ >
0, if y ( Ω ρ ) admits the topological asymptotic expansion y ( Ω ρ ) = y ( Ω ) + ρ n D T y ( ξ ) + o ( ρ n ) , (13)then D T y ( ξ ) is called the topological derivative at point ξ and is applicable to general boundary value problemsincluding the linear elastic system ∇ · ( C : ∇ u ) = in Ω u = ¯u on Γ D n · ( C : ∇ u ) = : t = ¯ t on Γ N . (14)where C is the elastic tensor, Γ D and Γ N denote Dirichlet boundary and Neumann boundary of Ω , respectively. Thetopological asymptotic expansion (13) contains two performance functions y ( Ω ) and y ( Ω ρ ). The former is related to6he reference domain Ω and evaluated by solving (14), whereas the latter is a ffi liated with the perforated domain Ω ρ and the associated boundary value problem ∇ · [ C : ∇ ( u + ˆ u )] = in Ω ρ u + ˆ u = ¯ u on Γ D n · [ C : ∇ ( u + ˆ u )] = : t + ˆ t = ¯ t on Γ N t + ˆ t = on − ∂ω ρ (15)where the Neumann type condition is prescribed on − ∂ω ρ , i.e., the boundary ∂ω ρ with the opposite normal vector.Comparing Eq. (14) and Eq. (15), it concludes that ˆ u = on Γ D and ˆ t = on Γ N . Moreover, it was proved thatˆ u + o ( ρ ), where o ( ρ ) is the reminder of higher order compared to ρ , is the solution of the following external problem [66] ∇ · ( C : ∇ ˆ u ) = in R n \ ω ρ n · ( C : ∇ ˆ u ) = : ˆ t = n · C : ∇ u ( ξ ) on − ∂ω ρ , (16)as ρ →
0. Solutions ˆ u for various cases of isotropic elasticity are summarized in Table 1, for more details and an easysolution utilizing the Eshelby tensor , refer to Appendix A.Both y ( Ω ) and y ( Ω ρ ) admit a general class of performance functions. Consider the compliance of the structure asthe performance functional, y ( Ω ) : = R Γ D ∪ Γ N u · t d Γ , which can be augmented by a Lagrange multiplier λ to introducethe governing equation as follows, y ( Ω ) : = Z Γ D ∪ Γ N u · t d Γ = Z Γ D ∪ Γ N u · t d Γ + Z Ω λ · [ ∇ · ( C : ∇ u )] d Ω , (17)by noticing u being the solution of Eq. (14) in advance, where λ can be any kinematically admissible field that meetsappropriate smoothness requirements. Similarly for the perforated domain, y ( Ω ρ ) : = Z Γ D ∪ Γ N ( u + ˆ u ) · (cid:16) t + ˆ t (cid:17) d Γ + Z Ω ρ λ · [ ∇ · ( C : ∇ ( u + ˆ u ))] d Ω . (18)The change of compliance after perforation y ( Ω ρ ) − y ( Ω ) = Z Γ D ∪ Γ N (cid:16) u · ˆ t + ˆ u · t + ˆ u · ˆ t (cid:17) d Γ + Z Ω ρ λ · [ ∇ · ( C : ∇ ˆ u )] d Ω − Z ω ρ λ · [ ∇ · ( C : ∇ u )] d Ω= Z Γ D ∪ Γ N (cid:16) u · ˆ t + ˆ u · t (cid:17) d Γ + Z Ω ρ λ · [ ∇ · ( C : ∇ ˆ u )] d Ω − Z ω ρ λ · [ ∇ · ( C : ∇ u )] d Ω , (19)employing ˆ u → on Γ D and ˆ t → on Γ N as ρ →
0. Integrate the second term of the above equation by parts twice7nd the third term one time, meanwhile applying divergence theorem, y ( Ω ρ ) − y ( Ω ) = Z Γ D ∪ Γ N (cid:16) u · ˆ t + ˆ u · t (cid:17) d Γ + Z Γ D ∪ Γ N ∪− ∂ω ρ λ · ˆ t d Γ − Z Ω ρ ∇ λ : C : ∇ ˆ u d Ω − Z ∂ω ρ λ · t d Γ + Z ω ρ ∇ λ : C : ∇ u d Ω= Z Γ D ∪ Γ N (cid:16) u · ˆ t + ˆ u · t + λ · ˆ t (cid:17) d Γ − Z ∂ω ρ λ · ˆ t d Γ − Z Γ D ∪ Γ N ∪− ∂ω ρ ˆ u · ( n · C : ∇ λ ) d Γ + Z Ω ρ ˆ u · [ ∇ · ( C : ∇ λ )] d Ω − Z ∂ω ρ λ · t d Γ + Z ω ρ ∇ λ : C : ∇ u d Ω= Z Γ D ( u + λ ) · ˆ t d Γ + Z Γ N ˆ u · t d Γ − Z ∂ω ρ λ · (cid:16) t + ˆ t (cid:17) d Γ − Z Γ N ˆ u · ( n · C : ∇ λ ) d Γ + Z Ω ρ ˆ u · [ ∇ · ( C : ∇ λ )] d Ω+ Z ∂ω ρ ˆ u · ( n · C : ∇ λ ) d Γ + Z ω ρ ∇ λ : C : ∇ u d Ω= Z Γ D ( u + λ ) · ˆ t d Γ + Z Γ N ˆ u · ( t − n · C : ∇ λ ) d Γ + Z Ω ρ ˆ u · [ ∇ · ( C : ∇ λ )] d Ω + Z ∂ω ρ ˆ u · ( n · C : ∇ λ ) d Γ+ Z ω ρ ∇ λ : C : ∇ u d Ω , (20)noticing ˆ u = on Γ D , ˆ t = on Γ N , t + ˆ t = on ∂ω ρ , and n is always the normal of the current integration surfaceduring the above derivation. Take λ as the displacement solution of the following adjoint problem ∇ · ( C : ∇ λ ) = Ω λ = − ¯ u on Γ D n · ( C : ∇ λ ) = ¯ t on Γ N (21)and apply the solution ˆ u on ∂ω ρ of the external problem for the three-dimensional case in Table 1, we have y ( Ω ρ ) − y ( Ω ) = Z ω ρ ∇ λ : C : ∇ u d Ω + Z ∂ω ρ ˆ u · ( n · C : ∇ λ ) d Γ= πρ (cid:16) C − : ˜ σ (cid:17) : σ + ρ Z ∂ω ρ a − b σ ) n + b n · σ ! · ( n · ˜ σ ) d Γ= πρ σ : C − : σ + ρ b ( ˜ σ · σ ) : Z ∂ω ρ nn d Γ + a − b σ ) ˜ σ : Z ∂ω ρ nn d Γ = πρ " ˜ σ : C − : σ + " b ˜ σ : I : σ + a − b ˜ σ : δδ : σ , (22)identifying R ∂ω ρ nn = πρ δ for the three-dimensional case, where δ is the second-order unit tensor, I is the fourth-orderidentity tensor, and ˜ σ = C : ∇ λ is the stress solution at ξ of the adjoint problem. Further calculations lead to y ( Ω ρ ) − y ( Ω ) = πρ σ : " (1 + ν ) E + b ! I + a − b − ν E ! δδ : σ = πρ − ν E (7 − ν ) ˜ σ : [10(1 + ν ) I − (5 ν + δδ ] : σ : = ρ ˜ σ : A : σ (23)noticing C − = + ν E I − ν E δδ for this case. Therefore the corresponding topological derivative D T y ( Ω , ξ ) has a concreteform D T y ( Ω , ξ ) = ˜ σ ( ξ ) : A : σ ( ξ ) , (24)8 able 1: Displacement solutions on ∂ω ρ of (16) and tensor A for various cases Isotropic Displacement on ∂ω ρ of Eq. (16) A Plane stress ρ h ν − E tr (cid:0) σ (cid:0) ξ (cid:1)(cid:1) n + − ν E n · σ (cid:0) ξ (cid:1)i π E [4 I − δδ ]Plane strain ρ (1 + ν ) E (cid:2) (2 ν −
1) tr (cid:0) σ (cid:0) ξ (cid:1)(cid:1) n + (3 − ν ) n · σ (cid:0) ξ (cid:1)(cid:3) π ( − ν ) E [4 I − δδ ]3D ρ h a − b tr (cid:0) σ (cid:0) ξ (cid:1)(cid:1) n + b n · σ (cid:0) ξ (cid:1)i † π (1 − ν ) E (7 − ν ) [10(1 + ν ) I − (5 ν + δδ ] † a = + ν E , b = − ν − ν ) E (7 − ν ) , σ (cid:0) ξ (cid:1) = C : ǫ ( ξ ), where n is the normal of ∂ω ρ where the fourth-order tensor A = π (1 − ν ) E (7 − ν ) [10(1 + ν ) I − (5 ν + δδ ]. The evaluation of D T y ( Ω , ξ ) requires the stresssolution at ξ from both the original problem and the adjoint problem. In the case that ¯ u =
0, the latter becomesself-adjoint and only the solution of Eq. (14) is needed. The expressions of A for various cases are summarized inTable 1. Let y ( Ω , X ) be a response function of the linear system (14) subject to random input X , which can be uncertainloads, geometry, or material properties. For a point ξ ∈ Ω , taking topology derivative of r th moments of the responsefunction y ( Ω , X ) and applying the Lebesgue dominated convergence theorem, which permits the interchange of thedi ff erential and integral operators, yields D T m ( r ) ( Ω , ξ ) : = D T E (cid:2) y r ( Ω , X ) (cid:3) | ξ = Z R N ry r − ( Ω , x ) D T y ( Ω , x , ξ ) f X ( x ) d x = E h ry r − ( Ω , X ) D T y ( Ω , X , ξ ) i , (25)that is, the topology derivative is obtained from the expectation of a product comprised of the response function andits topology derivative.For simplicity, we denote D T y ( Ω , X , ξ ) by z ( Ω , X , ξ ), and construct its S -variate, m th-order PDD approximation˜ z S , m as ˜ z S , m ( Ω , X , ξ ) : = z ∅ ( Ω , ξ ) + X ∅ , u ⊆{ , ··· , N } ≤| u |≤ S X j | u | ∈ N | u | k j | u | k ∞ ≤ m D u j | u | ( Ω , ξ ) ψ u j | u | ( X u ; Ω ) , (26)Replacing y and D T y of Eq. (25) with their S -variate, m th-order PDD approximations ˜ y S , m and ˜ z S , m , respectively,we have D T ˜ m ( r ) S , m ( Ω , ξ ) = E h r ˜ y r − S , m ( Ω , X )˜ z S , m ( Ω , X , ξ ) i (27)For r = , ,
3, employing the zero mean property and orthonormal property of the PDD basis ψ u j | u | ( X u ; Ω ) yieldsanalytical formulations for topology sensitivity of first three moments D T ˜ m (1) S , m ( Ω , ξ ) = z ∅ ( Ω , ξ ) , (28) D T ˜ m (2) S , m ( Ω , ξ ) = × y ∅ ( Ω ) z ∅ ( Ω , ξ ) + X ∅ , u ⊆{ , ··· , N } ≤| u |≤ S X j | u | ∈ N | u | || j | u | || ∞ ≤ m C u j | u | ( Ω ) D u j | u | ( Ω , ξ ) , (29) D T ˜ m (3) S , m ( Ω , ξ ) = × z ∅ ( Ω , ξ ) ˜ m (2) S , m ( Ω ) + y ∅ ( Ω ) X ∅ , u ⊆{ , ··· , N } ≤| u |≤ S X j | u | ∈ N | u | || j | u | || ∞ ≤ m C u j | u | ( Ω ) D u j | u | ( Ω , ξ ) + T k , (30)9 k = X ∅ , u , v , w ⊆{ , ··· , N } ≤| u | , | v | , | w |≤ S X j | u | , j | v | , j | w | ∈ N | u | || j | u | || ∞ , || j | v | || ∞ , || j | w | || ∞ ≤ m C u j | u | ( Ω ) C v j | v | ( Ω ) D w j | w | ( Ω , ξ ) × E d h ψ u j | u | ( X u ; Ω ) ψ v j | v | ( X v ; Ω ) ψ w j | w | ( X w ; Ω ) i , (31)which requires expectations of various products of three random orthonormal polynomials [51]. However, if X fol-lows classical distributions such as Gaussian, Exponential, and Uniform distribution, then the expectations are easilydetermined from the properties of univariate Hermite, Laguerre, and Legendre polynomials [67, 68, 52]. For generaldistributions, numerical integration methods will apply. Using PDD to approximate the performance function y , the Monte Carlo estimate for topology sensitivity of failureprobability is D T P (cid:2) X ∈ Ω F , k (cid:3) (cid:27) lim ρ → ρ n lim L →∞ L L X l = h I ˜ Ω F , k ,ρ ( x ( l ) ) − I ˜ Ω F , k ( x ( l ) ) i , (32)where L is the sample size; x ( l ) is the l th realization of X ; I ˜ Ω F , k and I ˜ Ω F , k ,ρ are the indicator functions for failure domains˜ Ω F , k : = { x : ˜ y k ( Ω , x ) < } and ˜ Ω F , k ,ρ : = { x : ˜ y k ( Ω ρ , x ) < } , respectively. The PDD approximation of the responsefunction of the current topology design Ω is ˜ y k ( Ω , x ), while at perturbed design Ω ρ , it is ˜ y k ( Ω ρ , x ). When ρ takes finitevalues, Equation (32) leads to a finite-di ff erence approximation D T P (cid:2) X ∈ Ω F , k (cid:3) (cid:27) ρ n lim L →∞ L L X l = h I ˜ Ω F , k ,ρ ( x ( l ) ) − I ˜ Ω F , k ( x ( l ) ) i (33)of the topology derivative for reliability. It requires ˜ y k ( Ω ρ , X ), which is simply obtained from˜ y k ( Ω ρ , X ) (cid:27) ˜ y k ( Ω , X ) + ρ n D T ˜ y k ( Ω , X ) , (34)without additional PDD expansion or FEA involved. This Monte Carlo estimation entails only two PDD approxima-tions, Eq. (5) for the response function itself and Eq. (26) for its deterministic topology derivative, both of whichare generated from the same stochastic analysis. Therefore little additional computational cost is needed to evalu-ate the topology sensitivity of reliability once the stochastic analysis is done, facilitating a novel and highly e ffi cientsensitivity analysis approach for RBTO.
5. Calculation of PDD Coe ffi cients The expansion coe ffi cients in Eq. (5) and Eq. (26) are defined by N -dimensional integrations y ∅ ( Ω ) : = R R N y ( x ) f X ( x ) d x and C u j | u | ( Ω ) : = R R N y ( x ) ψ u j | u | ( X u ; Ω ) f X ( x ) d x etc. For large N , direct numerical integration is often prohibitive, es-pecially when FEA is involved in the Gauss point evaluation. Instead, we will use the dimension-reduction method[23, 69, 24], which entails multiple low-dimensional integrations as an e ff ective replacement of a single N -dimensionalintegration.Let c = ( c , · · · , c N ) T ∈ R N , which is commonly adopted as the mean of X , be a reference point, and y ( x v , c − v ) rep-resent an | v | -variate referential dimensional decomposition (RDD) component function of y ( X ), where v ⊆ { , · · · , N } and − v = { , · · · , N } \ v . Given a positive integer S ≤ R ≤ N , when y ( x ) in the above N -dimensional integration isreplaced by its R -variate RDD approximation, the coe ffi cients are estimated from[23] y ∅ ( Ω ) (cid:27) R X i = ( − i N − R + i − i ! X v ⊆{ , ··· , N }| v | = R − i Z R | v | y ( x v , c − v ) f X v ( x v ) d x v (35)10 igure 2: A round disk subject to a uniform pressure C u j | u | ( Ω ) (cid:27) R X i = ( − i N − R + i − i ! X v ⊆{ , ··· , N }| v | = R − i , u ⊆ v Z R | v | y ( x v , c − v ) ψ u j | u | ( x u ; Ω ) f X v ( x v ) d x v (36)entailing at most R -dimensional integrations. For each integration involved, the Gauss quadrature rule applies. Forengineering problems, the evaluation of Gauss points often relies on FEA. For instance, each FEA with X realized atcertain gauss point supplies response function value for that Gauss point. Whereas to approximate the coe ffi cients forthe topology sensitivity D T y ( Ω , X , ξ ) or z ( Ω , X , ξ ) in section 4.2, each FEA provides stress results for Eq. 24 andfurther produces z values at the corresponding Gauss point. Nonetheless the reduced integration is significantly moree ffi cient than performing one N -dimensional integration owing to a much fewer number of Gauss points required bythe former, particularly when R ≪ N . Moreover, it facilitates the calculation of coe ffi cients approaching their exactvalue as R → N . In addition, the same set of Gauss points thus the same set of FEAs will be reused for the evaluationof coe ffi cients in Eq. (26), rendering a significantly e ffi cient framework for stochastic topology sensitivity analysis.
6. Numerical Examples
In this section, two new benchmark examples are developed for the analytical or semi-analytical solution of mo-ments and reliability and their topology sensitivities. The first one involves two random variables and renders analyt-ical expression for both stochastic quantities and their topology sensitivities of compliance. The second one contains53 random variables to test the accuracy and e ffi ciency of the proposed method for high dimensional problems bydeveloping corresponding analytical and semi-analytical solutions. The third example is a three-dimensional bracket,whose topology has already been optimized, illustrating a practical application of the proposed method. In all ex-amples, orthonormal polynomials and associated Gauss quadrature rules consistent with the probability distributionsof input variables, including classical forms, if they exist, were employed. No unit for length, force, and Young’smodulus is specified in all examples for simplicity while permitting any consistent unit system for the results. Assuming the plane stress state, consider a round disk
Ω = { ( r , θ ) : r ≤ , θ ∈ [0 , π ) } subject to a uniform pressure p as shown in Fig. 2, where ( r , θ ) is the polar coordinate system with its origin locating at the center of the disk. TheYoung’s module E and pressure p are random variables. The Poisson’s ratio ν = .
2, and is deterministic.Assume E follows inverse uniform distribution on [2 ,
4] with the probability density function (PDF) f E ( x E ) = x − E (37)11 able 2: Analytical solutions, numerical results, and relative errors: moments m (1) m (2) m (3) values RelativeError (%) values RelativeError (%) values RelativeError (%)PDD S = , m = S = , m = S = , m = S = , m = S = , m = S = , m = . × − . × − π (1 − ν ) π (1 − ν ) π (1 − ν ) and P follows uniform distribution on [1 , y ( Ω ) = π − ν E p . (38)The exact PDF of the compliance for this particular problem is found as f Y ( y ) = π (1 − ν ) (cid:18) − (cid:16) y π (1 − ν ) (cid:17) − (cid:19) π (1 − ν )2 ≤ y < π (1 − ν ) π (1 − ν ) (cid:16) y π (1 − ν ) (cid:17) − (cid:16) √ − (cid:17) π (1 − ν ) ≤ y < π (1 − ν ) π (1 − ν ) (cid:18) √ (cid:16) y π (1 − ν ) (cid:17) − − (cid:19) π (1 − ν ) ≤ y < π (1 − ν ) . (39)Moreover, the analytical expression of the first three moments of compliance are summarized in the Table 2To calculate the analytical topology sensitivity of moments and failure probability at the center ξ , another analyt-ical solution for the perforated domain with a tiny hole at the center is needed. It reads y (cid:16) Ω ρ (cid:17) = π p o E (cid:0) − ρ (cid:1) h (1 + ν ) ρ + (1 − ν ) i , (40)which can be derived based on the Lam´e’s strain potential C ln rK with undetermined constants C and K via thedisplacement method. The deterministic topology derivative D T y by definition is D T y (cid:0) Ω , ξ (cid:1) = lim ǫ → y (cid:16) Ω ρ (cid:17) − y ( Ω ) ρ = π p E . (41)Together with Eqs. (25) and (39), the analytical expressions of topology sensitivity for the first three moments can bedetermined, and are listed in Table 3.The finite element model employed in the proposed method consists of 404800 quadrilateral and 1600 triangularelements. The displacement u θ at (1 , (cid:16) , π (cid:17) , and (cid:16) , π (cid:17) , are specified as zero to make the FEA model well-posedand keep the same solution of stress, strain, and compliance in Fig. 2. Table 2 displays the approximate moments ofthe compliance, committed by the proposed univariate ( S =
1) and bivariate ( S =
2) PDD for m = , ,
3. Relativeerrors, defined as the ratio of the absolute error to the exact value, are also presented. For the first moments, the errorsrange from 3 . × − to 0 .
252 percent. When the order of moments increases, the errors show an uptrend as expecteddue to the accumulation of approximation errors, but still maintains good levels, 1 . × − to 2 .
308 percent for thesecond moments and 0 .
983 to 7 .
523 percent for the third moments.Table 3 presents the approximate topology sensitivity of the center point and their relative errors for the first threemoments. For the same set of S and m values, the relative errors of topology sensitivity are almost identical withthe ones of moments in Table 2. It seems unusual since for many methods the numerical estimation of stochasticsensitivity is often less accurate than the estimation of the function itself. However, the proposed method dovetailsthe deterministic topology derivative D T y as shown in Eq. (25) and the nonlinearity and interactive e ff ects in D T y are12 able 3: Analytical solutions, numerical results, and relative errors: sensitivity of moments at ξ = (0 , D T m (1) ( Ω , ξ ) D T m (2) ( Ω , ξ ) D T m (3) ( Ω , ξ )values RelativeError (%) values RelativeError (%) values RelativeError (%)PDD S = , m = S = , m = S = , m = S = , m = S = , m = S = , m = . × − . × − π π (1 − ν ) π (1 − ν ) Table 4: comparison between analytical solution and numerical results: reliability and its sensitivity for ρ = . , ¯ y = . ξ = (0 , P F : = P ( y ≥ ¯ y ), ¯ y = . D T P F ( Ω , ξ )values RelativeError (%) values RelativeError (%)PDD S = , m = . × − . S = , m = . × − . S = , m = . × − . S = , m = . . S = , m = . S = , m = . ρ →
0) 1 − √ . π − . π − . π √ . π − . π often similar with the response y as shown in Eqs (38) and (41), which lead to similar or identical relative errors in thesensitivity of moments. The errors from the propose method drop as m and S increase as expected for both momentsand their topology sensitivities.Analytical expressions and numerical results of failure probabilities and their topology sensitivity are presentedin Tables 4 and 5 for two limit-state values 7 . .
5, respectively. The numerical estimations of failure probabilityby the proposed method are evaluated via Eq. (12) using the embedded MCS, whereas their topology sensitivities arecalculated based on Eqs. (33) and (34) with a finite ρ value of 0 .
05. The sample size for both is L = . The totalnumber of FEA simulations for various combinations of the truncation parameters S = , m = , , m = S =
1) PDD are relatively large, but it requires only 5 FEAs. But the errors drop significantly as S and / or m increases. For instance, the errors of failure probability become less than one percent for S = , m = ,
3, requiring15 and 25 FEAs respectively. Similar trends are observed in their topology sensitivity. Comparing results for ¯ y = . y = .
5, the errors of failure probability increase as expected when the limit state values move away from themean of the response function. Further developments address this problem in our future work.
Consider the same round disk in last example but subject to a more complex pressure as shown in Fig. 3, wherethe pressure function f ( θ ) = D + K X k = ( D k cos ( k + θ + E k sin ( k + θ ) (42)accommodating 2 K + D k , k = , · · · , K and E k , k = , · · · , K .13 able 5: comparison between analytical solution and numerical results: reliability and its sensitivity for ρ = . , ¯ y = . ξ = (0 , P F : = P ( y ≥ ¯ y ), ¯ y = . D T P F ( Ω , ξ ) S = , m = . × − S = , m = . × − S = , m = . × − S = , m = . × − S = , m = . × − S = , m = . × − ρ →
0) 1 − √ π − . π − . . π √ π − . π NA Figure 3: A round disk subject to a complex pressure .2.1. Analytical solutions Employing the
Taylor series expansion of holomorphic functions in a simply-connected domain and
Goursatformula [70], the analytical solution for compliance of the disk subject to the above pressure is found in the form of y ( Ω ) = D π (1 − ν ) E + K X k = (cid:16) D k + E k (cid:17) π ( ν + k + k ( k + E . (43)The solution (43) is general and applicable for the pressure function (42) for any positive integer K .Now consider perforating a tiny hole of radius ρ in the center of the disk. Its compliance, subject to the samepressure function (42), is found as follows y (cid:16) Ω ρ (cid:17) = D π h ρ (1 + ν ) + (1 − ν ) i E (cid:0) − ρ (cid:1) + K X k = A k B k C k F k , (44)where A k = (cid:16) D k + E k (cid:17) π B k = ρ k ( k + h k ν − (3 k + − ( k ν + k + ρ i + h ( ν − k − ρ k + − ( ν + k + i k − X j = ρ j C k = k ( k + EF k = k ( k + ρ k (cid:16) − ρ (cid:17) + (cid:16) ρ k + − (cid:17) k − X j = ρ j . which requires Laurent series expansion of holomorphic functions in a double-connected region.Employing Eqs. (43) and (44), the analytical expression of the deterministic topology derivative at the center reads D T y ( Ω , ξ ) = lim ρ → y (cid:16) Ω ρ (cid:17) − y ( Ω ) ρ = π (cid:16) D + D + E (cid:17) E , (45)indicating that at the center of the disk the topology derivative of compliance is merely related to Young’s modulus E and three parameters D , D , E in the pressure function.The exact topological sensitivities of moments at the center are derived from D T m ( r ) ( Ω , ξ ) = Z R N ry r − ( Ω , X ) D T y ( Ω , X , ξ ) f X ( x ) d x , (46)employing Eqs. (43) and (45). Generally, Eq. (46) admits any proper distributions for the 2 K + Let K =
25, random variables D k , k = , · · · ,
25 and E k , k = , · · · ,
25 follow four-parameter Beta distributionswith mean value µ D k = k + µ E k = k +
1, and coe ffi cient of variance (CV) be 0 . D k and E k . Two isotropicelastic material constants also follow four-parameter Beta distributions, where Young’s modulus E has a mean valueof 10 and CV of 0 .
1, the Poisson’s ratio ν has a mean value of 0.2 and CV of 0 .
01. The support of each Beta variableis (cid:2) µ − σ, µ + σ (cid:3) , where µ and σ here denote mean and standard deviation of the corresponding variable.The exact solutions of the first three moments of the compliance, obtained based on the analytical solution (43),are exhibited in Table 6. For the finite element model used in the proposed method, two types of mesh are adopted:1) coarse mesh (24800 quadrilateral and 400 triangular elements), and 2) fine mesh (404800 quadrilateral and 1600triangular elements), as shown in Tables 6 and 7. The displacement u θ at (1 , (cid:16) , π (cid:17) , and (cid:16) , π (cid:17) , are specified as zero to make the FEA model well-posed and meanwhile keep the compliance unchanged. For the results by the coarsemesh, the relative errors of the first moment by the proposed method with various truncations range from 1 .
056 to15 able 6: Exact solutions, numerical results, and relative errors for moments - coarse mesh m (1) m (2) m (3) values RelativeError (%) values RelativeError (%) values RelativeError (%)PDD S = , m = S = , m = S = , m = S = , m = S = , m = S = , m = .
146 percent. When the order of moments increases, the relative errors rise, for instance, to 2 . .
532 percentfor the second moment and to 3 . .
183 percent for the third moment. This trend is foreseeable since the momentcalculation accumulates the error of the approximated response function when its order increases. Checking anyparticular moment in Table 6, the prevailing trend of the relative errors is down when increasing truncation parameters S and m , but it is insignificant. The reason as disclosed in the later discussion is that the error introduced by FEAapproximations is dominant comparing to the error of the PDD approximation. Nonetheless, roughly 1 . m (1) , 2 . m (2) , and 4 . m (3) are highly satisfactory for stochastic momentanalysis using the coarse mesh. When employing the fine mesh, the relative errors of all three moments plummetapproximately by half for every combination of truncation parameters as shown in Table 7, which indicates the errorfrom FEA may dominate the error of PDD approximations. The relative errors for m (1) , m (2) , and m (3) by the proposedmethod using the fine mesh are merely 0 .
4, 1 .
1, and 2 . ξ = (0 , ξ . Their exact solutions are unveiledin Tables 8 and 9. The proposed method is implemented in all combinations of S = , m = , , . , . . , . . , . . , . . , . . , . D T m (2) ( Ω , ξ ) and D T m (3) ( Ω , ξ ). Tables 6-9 demonstrate thatthe proposed method is capable of performing highly accurate moment analysis as well as their topology sensitivityanalysis. By comparing results from two mesh cases, it can be inferred that a significant portion of errors comefrom FEA, conjointly evincing the accuracy of the proposed method. Moreover, sensitivity analyses not limited totopology sensitivity analyses of a generic response function are often less accurate than the evaluation of the functionitself. However, comparing Table 6 with Table 8, or Table 7 with Table 9, it shows that for the same mesh case andthe same set of S and m the topology sensitivity is surprisingly more accurate than the moment analysis itself. Forinstance,1 .
800 percent error for D T m (2) ( Ω , ξ ) is less than 2 .
532 percent error for m (2) itself in the case of coarse mesh, S =
1, and m =
1. The remarkable more accuracy of sensitivity seems occasional and rare, however, it is reasonablefor the proposed framework due to the deterministic topology embedded in Eqs. (25)-(30). Scrutinizing the definitionof the r th moment m ( r ) ( Ω ) : = E [ y r ( Ω , X )] and its topology sensitivity Eq. (25), a major di ff erence between them isthe replacement of y by D T y in the topology sensitivity. When the nonlinearity and interaction structure of D T y isequal or simpler than ones of y , for the same set of truncation parameter S and m , the topology sensitivity of momentscalculated by the proposed method is bound to be equally or more accurate than the moments itself. The deterministictopology derivative at the center for this example is shown in Eq. (45), which is obviously simpler than the complianceitself as shown in Eq. (43). The structure of the proposed method in Eqs. (25)-(30) well explains the observationthat topology sensitivity is more accurate than the moment itself and also demonstrates another advantage of the newmethod.For failure probability and its topology sensitivity, analytical expressions or exact values are not readily available16 able 7: Exact solutions, numerical results, and relative errors for moments - fine mesh m (1) m (2) m (3) values RelativeError (%) values RelativeError (%) values RelativeError (%)PDD S = , m = S = , m = S = , m = S = , m = S = , m = S = , m = Table 8: Exact solutions, numerical results, and relative errors for sensitivities of moments at ξ = (0 ,
0) - coarse mesh D T m (1) ( Ω , ξ ) D T m (2) ( Ω , ξ ) D T m (3) ( Ω , ξ )values RelativeError(%) values RelativeError(%) values RelativeError(%)PDD S = , m = S = , m = S = , m = S = , m = S = , m = S = , m = Table 9: Exact solutions, numerical results, and relative errors for sensitivities of moments at ξ = (0 ,
0) - fine mesh D T m (1) ( Ω , ξ ) D T m (2) ( Ω , ξ ) D T m (3) ( Ω , ξ )values RelativeError(%) values RelativeError(%) values RelativeError(%)PDD S = , m = S = , m = S = , m = S = , m = S = , m = S = , m = Table 10: Benchmark of reliability and its sensitivity for ρ = . , ¯ y = . ξ = (0 ,
0) - coarse mesh P F : = P ( y ≤ ¯ y ), ¯ y = . D T P F ( Ω , ξ ) S = , m = † S = , m = S = , m = S = , m = S = , m = S = , m = ‡ † The sample size for results by proposed method is L = ‡ The sample size for the Crude MCS-FD is L = .4 3.5 3.6 3.7 3.8 3.9 410 -3 -3 -2 -1 Figure 4: CDF of the complianceTable 11: Benchmark of reliability and its sensitivity for ρ = . , ¯ y = . ξ = (0 ,
0) - fine mesh P F : = P ( y ≤ ¯ y ), ¯ y = . D T P F ( Ω , ξ ) S = , m = † S = , m = S = , m = S = , m = S = , m = S = , m = ‡ † The sample size for results by the proposed method is L = ‡ The sample size for the Crude MCS-FD is L = L = is taken as the benchmark solution of failure probability. Meanwhile, a finite di ff erence formulationembedded the crude MCS (Crude MCS-FD) D T P [ X ∈ Ω F ] (cid:27) ρ n lim L →∞ L L X l = h I Ω F ,ρ ( x ( l ) ) − I Ω F ( x ( l ) ) i (47)is adopted as the benchmark solution of topology sensitivity of failure probability, where the radius of the perforatedhole takes a finite value ρ = .
05, the sample size L = , I Ω F and I Ω F ,ρ are the indicator functions of the exact failuredomains Ω F : = { x : y ( Ω , x ) < ¯ y } and Ω F ,ρ : = { x : y ( Ω ρ , x ) < ¯ y } with y ( Ω , x ) taking the exact compliance function ofthe disk as shown in Eq. (43) and y ( Ω ρ , x ) taking the exact compliance function of the perforated disk as shown inEq. (44). These benchmark solutions, involving analytical expressions of compliance, MCS, and the finite-di ff erencemethod, is also referred to as semi-analytical solutions in this paper.The cumulative distribution function (CDF) of the compliance by crude MCS as well as ones by the proposedmethod employing two mesh cases and various PDD truncations are plotted in Fig. 4. An identical sample size L = is used for all plots in this figure. All the CDF curves spontaneously group into two bundles. The first bundleconsists of all linear ( m =
1) approximations whether univariate ( S =
1) or bivariate ( S = m ≥ S = , m = S = , m =
3, and their curves are almost coincide with the one by the crude MCS. Nonetheless, an overall trendof convergence can be roughly observed in Fig. 4 as increasing S and m and adopting finer mesh. More quantitativeverifications of failure probability and its topology sensitivity are displayed in Tables 10 and 11, in which the failureprobability at 0 . m =
1) carries the largest errors among their same-variate and same-meshcounterparts, specifically 127 .
015 and 122 .
761 percent for coarse mesh S = ,
2, 96 .
831 and 95 .
086 percent for finemesh S = ,
2. After increasing m , the errors plummet dramatically to about 19-41 percent for coarse mesh cases and2-14 percent for fine mesh cases. The significant di ff erences in error levels of two mesh types imply that the errorfrom FEA predominates in those cases. Similar behaviors are observed in the results of its topology sensitivity butthe level of errors have slight or moderate drops for most of m ≥ m ≥
2) provides satisfactory evaluation for the topology sensitivity of failure probability, merely 5-8percent for univariate and 1-3 percent for bivariate as shown in Table 11. For both failure probability and its sensitivity,Table 10-11 show that the error level roughly drops when increasing S and m , but the trend is not monotonic becauseof the synthetic e ff ect of four kinds of error sources - finite di ff erence, MCS, PDD, and FEA. The number of FEAsrequired by the proposed method for each PDD truncation is also listed in Table 10-11. Univariate cases are muchmore e ffi cient than bivariate ones as expected, involving only 107 and 213 FEAs to level down the errors to 1 .
536 and14 .
732 percent in failure probability and 4 .
574 and 7 .
864 in its topology sensitivity for fine mesh and m = ,
3. Itis noteworthy that the same set of FEAs can be used to generate estimations for not only failure probability and itssensitivity but also moments and their sensitivity in preceding tables.To sum up, this example is constructed to gauge the accuracy of new or existing methods for stochastic analysesand their topology sensitivities by analytical or semi-analytical solutions developed. Although K =
25 is specified,the analytical and semi-analytical solutions developed can be directly used or easily expanded for any positive K toaccommodate even more random variables. Nonetheless, the proposed method is capable of evaluating moments andtheir sensitivities in a highly accurate manner even using low-variate low-order approximation. For failure probabilityand its sensitivity, it is also feasible to provide satisfactory evaluations using low-variate but nonlinear approxima-tion. The least number of FEAs required for those fine approximations is 107 for this 53 random variable example,demonstrating the high e ffi ciency of the proposed method for high-dimensional stochastic topology sensitivity analy-sis. Another advantage of the proposed method observed in this example is its capability of providing higher accuracyin topology sensitivity than in stochastic quantities themselves.19 igure 5: Geometry and mesh of the bracket Last, the proposed method is applied to a three-dimensional engineering bracket [71] shown in Fig. 5. Withthe fixed support at the middle hole, the bracket is subject to nine random tractions along x , y , or z -direction onthe surfaces of one top hole and two bottom holes as shown in Fig. 5. Their mean values are (cid:0) µ F , µ F , · · · , µ F (cid:1) = (2500 . , . , − . , . , − . , − . , − . , . , − . , respectively. The Young’s modulusand Poisson’s ratio are also random with mean values µ E = . × and µ ν = .
3. The CV for all 11 randomvariables is 0 .
1. In this example, all 11 random variables follow truncated Gaussian distribution, which has thefollowing PDF in general f X ( x ) = Φ ( D ) − Φ ( − D ) φ (cid:16) x − µσ (cid:17) α ≤ x ≤ β, , (48)where µ and σ denote the mean and standard deviation of each variable before the truncation and α = µ − D , β = µ + D .For nine random tractions and Young’s modulus, D takes 10 times of the corresponding standard deviation, thatis, D = σ . For the random Poisson’s ratio, D takes six times of the corresponding standard deviation to avoidunrealistic materials.The second-order univariate PDD ( S = , m =
2) is used to perform stochastic topology sensitivity analysis.The finite element model required contains 182540 quadratic tetrahedron elements. Compliance is selected as theperformance function y and failure criteria for the reliability is defined as P F : = P (cid:16) y < . × (cid:17) . Contours ofstochastic topology sensitivities for compliance are plotted in Fig. 6. The contours for sensitivities of the threemoments follow similar patterns but di ff erent value ranges as expected since the sensitivity is eventually related tothe stress field. The contour for the sensitivity of failure probability is also similar due to the same reason althoughdistinct colors manifest the value di ff erence. Only 23 FEAs are needed to evaluate the first three moments, probabilityof failure, and their sensitivities for this 11-dimensional example, illustrating the e ff ectiveness of the proposed methodfor high-dimensional engineering problems.
7. Conclusions
A new framework for stochastic topology sensitivity analysis was developed for solving RTO and RBTO problemscommonly encountered in engineering. The framework is grounded on the polynomial dimensional decompositionand the concept of topology derivative. Comparing with previous developments, the new method is capable of pro-viding accurate evaluations of stochastic topology sensitivity owing to the dovetailed topology derivative concept.Furthermore, the new method can e ffi ciently tackle high-dimensional stochastic response functions and their topologysensitivities as a result of the hierarchical structure of PDD which decomposes a high-dimensional function in termsof lower-variate component functions. With these two intrinsic advantages, the new method endows the first three mo-ments and their topology sensitivities with analytical expressions. And it also provides embedded MCS for reliability20 a) (b) (c) (d) Figure 6: Stochastic topology sensitivity of compliance: (a)-(c) topology sensitivity of 1st moment, 2nd moments, and 3rd moments; (d) topologysensitivity of failure probability analysis and finite di ff erence formulations for topology sensitivity of reliability. In the finite di ff erence formulations,the definition of topology derivative is utilized as a callback to evaluate the perturbed performance function requiringno additional function evaluations and thus results in a self-consistent framework. It is noteworthy that the evalua-tion of moments, reliability, and their topology sensitivity is acquired from a sing stochastic analysis. In addition,the adjoint method inherited from deterministic topology sensitivity analysis, together with PDD, grant the proposedframework a significantly high e ffi ciency for solving high-dimensional engineering problems especially when FEA isinvolved.Two new benchmark examples were developed to address the issue of lacking analytical solutions of stochastictopology sensitivity for verification. The first example provides not only the analytical expression for the first threemoments of compliance and their topology sensitivities but also the analytical expression for the failure probabilityand its topology sensitivity. Aided by this example, the accuracy and e ffi ciency of the proposed method are examinedand demonstrated. The second example, accommodating 53 random variables via applying an intricate pressure,supplies analytical solutions for compliance of both the original domain and the perforated domain. These analyticalcompliances generate exact solutions for the moments and their sensitivities, and also o ff er a precise evaluation offailure probability via crude Monte Carlo simulation as well as an accurate assessment for its topology sensitivity byvirtue of finite di ff erence method. The e ff ectiveness of the proposed method is thus verified and the advantages of thedovetailed decomposition are illustrated by this 53-dimension example. It also demonstrates that topology sensitivitiesof moments by the proposed method possess higher accuracies than moments themselves when the function structureof deterministic topology derivative is simpler than the response itself. A similar advantage is also observed in thetopology sensitivity of failure probability in this example. The proposed method is finally applied to a three-dimensionbracket with 11 random variables, by which the application to complex engineering problems is examined.In summary, the introduction of the topology derivative concept enables a rigorous description of stochastic topol-ogy sensitivity and permits the development of new benchmark examples for this research field. The grounded poly-nomial dimensional decomposition empowers its high e ffi ciency to solve stochastic topology sensitivity for high-dimensional complex engineering problems. In addition, when the deterministic topology derivative of response takesa simpler form than the response itself, the proposed method often supplies better accuracies on stochastic topologysensitivities than on the stochastic analysis. Acknowledgments
The authors acknowledge financial support from the U.S. National Science Foundation under Grant No. CMMI-1635167 and the startup funding of Georgia Southern University. Also to commemorate Niels Henrik Abel.21 ppendix A. The solutions for the external problem
The solution of Eq. (16) were well studied by mathematicians in early research [72, 66]. However, topologicalderivatives require only the solution on the boundary ∂ω ρ , which can be obtained in an easier approach comparingto those in literature [72, 66]. In this appendix, an approach based on Eshelby tensor [73] and solutions for planestress, plane strain, and three-dimensional cases are compiled for easy accessibility of researchers in mechanics andengineering field. When the elastic medium in Eshelby phase-transition strain problem is isotropic and the inclusiondomain Ω is a sphere, the Eshelby tensor is isotropic S = ( α − β ) 13 δδ + β I (A.1)where α = K K + G , β = K + G )5 (3 K + G ) , and G and K are shear modulus and bulk modulus, respectively. The real strain on the boundary of the inclusion readsˆ ǫ = (cid:16) S − − I (cid:17) − C − : ˆ σ (A.2)where ˆ σ is the stress on the surface of the inclusion. To utilize it for the solution on ∂ω ρ of Eq. (16), letˆ σ = σ (cid:0) ξ (cid:1) (A.3)where σ (cid:0) ξ (cid:1) is the stress at ξ in Eq. (14). Therefore the strain solution for Eq. (16)ˆ ǫ = a − b δδ + b I ! : σ (cid:0) ξ (cid:1) (A.4)where a = G = + ν E , b = K + G ) G (9 K + G ) = − ν − ν ) E (7 − ν ) . The corresponding displacement solution on ∂ω ρ readsˆ u = a − b δδ + b I ! : σ (cid:0) ξ (cid:1) · n ρ = ρ a − b (cid:0) σ (cid:0) ξ (cid:1)(cid:1) n + b n · σ (cid:0) ξ (cid:1)! (A.5)For plane strain cases, the Eshelby tensor becomes S = ( α − β ) 12 δδ + β I (A.6)with α = − ν ) , β = − ν − ν ) , and the displacement solution on ∂ω ρ becomesˆ u = (1 + ν ) E [(2 ν − δδ + (3 − ν ) I ] : σ (cid:0) ξ (cid:1) · n ρ = ρ (1 + ν ) E (cid:2) (2 ν −
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