Abstract An approach for reliability-based topology opti-mization of interval parameters structures under dynamicloads is proposed. We modify the equivalent static loadsmethod for non linear static response structural optimiza-tion (ESLSO) to solve the dynamic reliability optimizationproblem. In our modified ESLSO, the equivalent static loads(ESLs) are redefined to consider the uncertainties. The newESLs including all the uncertainties from geometric dimen-sions, material properties and loading conditions generatethe same interval response field as dynamic loads. Basedon the definition of the interval non-probabilistic reliabilityindex, we construct the static reliability topology optimiza-tion model using ESLs. The method of moving asymptotes(MMA) is employed as the optimization problem solver.The applicability and validity of the proposed model andnumerical techniques are demonstrated with three numericalexamples.
Topology optimization is a higher level optimizationmethod than size and shape optimization. Generally, it isused for the preliminary design because optimal resultsonly give a conceptual layout of the structural design.The earliest research on topology optimization is made by
M. Li (�) · W. Tang · M. YuanSchool of Mechanical Engineering, Southeast University,Nanjing 211189, Chinae-mail: liming [email protected]
Michell (1904), but topology optimization is not an activeresearch field until the famous work by Bendoe and Kikuchi(1988).
Taking into account dynamic characteristics of real struc-tures, many researchers have focused their attention ondynamic response topology optimization. In order to easilyobtain sensitivity analysis needed in the process of topologyoptimization, some researchers converted time dependentconstraints into frequency dependent constraints, and stud-ied the dynamic response topology optimization problemin the frequency domain. Ma et al. (1995) proposed themodel of minimizing the dynamic compliance, based onthe homogenization method. Rong et al. (2000) employedESO method to solve the continuum topology optimizationunder mean square random dynamic responses constraints.Jog (2002) adopted the homogenization method to studystructural topology design subjected to periodic loading,and gave a new definition of the global performance indexof dynamic compliance from the viewpoint of reducingthe vibrations and noise. Yoon (2010) investigated struc-tural topology optimization using model reduction schemes.Although dynamic response topology optimization in thefrequency domain is cheaper and easier than that in the timedomain, it is not sufficient to completely include dynamiccharacteristics.
Compared with that in the frequency domain, dynamicresponse topology optimization in the time domain iscostlier and more time-consuming. So the research ofdynamic response topology optimization in the time domainis relatively few. Min and Kikuchi (1999) gave an explicitdirect integration scheme to solve linear transient problems,but if the structure is large or the time duration of loadsis long, the numerical calculation will take an unaccept-able amount of time. In this context, Jang et al. (2012)applied ESLSO into the linear dynamic response topology
optimization, and numerical examples verified the validityof this method.
ESLSO was initially proposed for size and shape opti-mization of structures under dynamic loads by Choi andPark (2002). In the ESLSO, ESLs generate the same dis-placement field as dynamic loads at each time step, and theoriginal dynamic response optimization problem is trans-formed into the linear static response optimization problem.The further research and expansion to ESLSO were con-ducted (Kang et al. 2005; Kim et al. 2009; Jeong et al.2010; Yi et al. 2011; Zhang et al. 2011; Lee and Park2012), and more details can be referred to the review(Park 2011).
In the past three decades, much research was carriedout for topology optimization. However, an overwhelmingmajority of existing studies are based on the determinis-tic assumption. As we know, uncertainties must inevitablyexist in practical engineering designs. Hence, it is quitenecessary to study the reliability-based topology optimiza-tion (RBTO) (Kharmanda et al. 2004; Mogami et al.2006). Some valid RBTO methods have been developedby researchers based on exiting topology optimizationmethods (Kim et al. 2007b; Zhang and Ouyang 2008; Chenet al. 2010; Patel and Choi 2012; Mashayekhi et al. 2012).Maute and Frangopol (2003) and Kim et al. (2007a) dis-cussed the RBTO of microelectromechanical systems. Choet al. (2012) studied the RBTO of electro-thermal-compliantmechanisms. Also, RBTO method was applied to solve themulti-objective topology optimization problem (Cho et al.2011; Greiner and Hajela 2012). Luo et al. (2009) describeda non-probabilistic reliability-based topology optimization(NRBTO) method for continuum structures based on multi-ellipsoid convex model.
Although the potential of the RBTO method is promis-ing, it is still in the infancy. Mainly research on RBTOstill focused on static optimization and random natural fre-quency optimization, but not dynamic response. That isbecause the direct topology optimization considering therandomness of structural parameters and external dynamicloads is very difficult and costly. Ma et al. (2011) investi-gated the reliability-based dimension and shape optimiza-tion of trusses under dynamic loads, but not topologyoptimization.
In this paper, an improved ESLSO is proposed forreliability-based topology optimization of interval parame-ters structures under dynamic loads. The ESLs are redefinedto consider uncertainties. In order to avoid interval arith-metic resulting in the expansion of interval range, theinterval is replaced by the set in the process of mathemati-cal operation. In conjunction with the ESLSO and definitionof non-probabilistic reliability index, the reliability-basedtopology optimization model of interval parameters struc-tures under dynamic loads is modeled.
The method of moving asymptotes (MMA; Svanberg1987) is combined with the SIMP model (Bendsoe 1989;Rozvany et al. 1992) to solve the optimization problem.Three standard numerical examples are used to illustratethe applicability and validity of the constructed model andproposed numerical techniques.
2 Interval non-probabilistic reliability index
2.1 Interval vector
An interval vector is defined (Moore 1979) as
a ∈ aI= [a, a
]=(aIi
), ai ∈ aI
i =[ai, ai
](i = 1, 2, · · · , m).
(1)
The midpoint vector or mean value vector aC of intervalvector a is defined as
aC = a + a2
, aCi = ai + ai
2(i = 1, 2, · · · , m) . (2)
The deviation amplitude vector or uncertain radius vectoraR of interval vector a is defined as
aR = a − a2
, aRi = ai − ai
2(i = 1, 2, · · · , m) . (3)
Therefore, the interval vector a can be written as
a = aC + aR ⊗ δ, ai = aCi + aR
i δi ,
δi ∈ [−1, 1] (i = 1, 2, · · · , m) , (4)
where δi ∈ [−1, 1] is called normalized interval number.
2.2 Definition of the interval non-probabilistic reliabilityindex
We assume the limit-state function or performance func-tion as G(a), where a is an interval vector related to thestructure. According to the non-probabilistic reliability con-cept proposed by Ben-Haim (1994), the critical condition ofstructures is defined by the limit-state surface G(a) = 0.For a multi-dimensional interval model C = {
a∣∣a ∈ aI
},
we can easily obtain the normalized interval model E ={δ ||δi | ≤ 1 (i = 1, · · · , m)} through (4). Therefore, thelimit-state surface G(a) = 0 can be transformed intog(δ) = 0 defined in the δ-space, and the δ-space isdivided by the limit-state surface g(δ) = 0 into a reliabledomain (corresponding to g(δ) > 0) and a failure domain(corresponding to g(δ) < 0).
As an illustrative example, the standard δ-space of astructural system with two interval variables is shown inFig. 1. The solid-line square box represents the normalizedinterval model E = {δ ||δi | ≤ 1, i = 1, 2}. Here, we adoptthe infinite norm ||•||∝ to measure the distance from the
Structural dynamic topology optimization based on dynamic reliability
origin to the limit-state curve, so we can reach the min-imal distance dmin as min
δ:g(δ)=0{max (|δ1| , |δ2|)}. dmin can
be obtained by continuously expanding the square box inFig. 1. Obviously, the minimal distance is the distance fromthe origin to the first intersection point the square box andlimit-state curve have.
According to the non-probabilistic reliability conceptproposed by Ben-Haim (1994), it is reasonable to define aquantified measure for the structural reliability index in caseof m-dimensional interval model as
η = sgn (g (0)) · minδ:g(δ)=0
{max (|δ1| , |δ2| , · · ·, |δm|)} , (5)
in which
sgn(g(0)) =⎧⎨
⎩
1 if g(0) > 0,
0 if g(0) = 0,
−1 if g(0) < 0.
(6)
As shown in Fig. 1, if η = 1, the limit-state surface willbe tangent to convex domain E = {δ : |δi | ≤ 1, i = 1, 2},and in this case the structure is in the critical state. If η > 1,the limit-state surface will have no point of intersection withthe convex domain E . In this case, the structure is reliable,and the farther the distance between the failure surface withthe convex domain, the larger the value of η , then the morereliable the structure.
3 Process of ESLSO for dynamic responsereliability-based topology optimization
3.1 Definition of ESLs for interval parameters structures
The traditional equivalent static loads (ESLs) generate thesame displacement field as dynamic loads at each step (Park2011; Choi and Park 2002). The equivalent static loadsmethod has been verified by Park and Kang (2003) that thesolution obtained satisfies the Karush-Kuhn-Tucker neces-sary condition and is an optimum solution of the originaldynamic response optimization. In accordance with interval
Fig. 1 Interval non-probabilistic reliability index
parameters structures, a new definition of ESLs is proposedin this paper. In the new definition, the displacement fieldsrespectively by ESLs and dynamic loads have the samemidpoint vector and deviation amplitude vector.
The governing equation of dynamic analysis of the inter-val parameters structure is
M (a, b) x (a, b, t) + C (a, b) x (a, b, t)
+ K (a, b) x (a, b, t) = F (a, t) (t = 1, ..., l) ,
(7)
where the interval vector a ∈ Rm consists of m structuralinterval parameters, the design variable vector b is the den-sity of finite elements, M(a,b) is the mass matrix, C(a,b)is the damping matrix, K(a,b) is the stiffness matrix, F(a,t)is the dynamic loads vector, x (a, b, t) is the displacementvector, x (a, b, t) is the velocity vector, x (a, b, t) is theacceleration vector and l is the number of time steps indynamic analysis.
Firstly, we need to conduct the interval dynamic responseanalysis for (7) before calculating the ESLs.
Here, we employ the non-probabilistic interval analy-sis method (Qiu and Wang 2003) to conduct the intervaldynamic response analysis. But in order to avoid intervalarithmetic resulting in the expansion of interval range in thetheoretical derivation process, the analysis result x(a, b, t)
will be expressed by set form, not interval.Assume that the deviation amplitude vector aR is rela-
tively small compared with the midpoint vector aC. UsingTaylor series, the dynamic response x(a, b, t) aboutaC isdeveloped as
x (a, b, t) = x(
aC + �a, b, t)
≈ x(
aC, b, t)
+m∑
i
∂x(aC, b, t
)
∂ai
�ai, (8)
in which |�a| ≤ aR, �ai ∈ [−aRi , aR
i ], i = 1, · · · , m.Substituting a = aC into (7), then we can reach x(aC,b,t)
by solving the deterministic dynamic response problem (7)using Newmark integration algorithm.
Recall that the mass matrix M(a,b), the damping matrixC(a,b) and the stiffness matrix K(a,b) are symmetric.Assume that all elements in M(a,b), C(a,b), K(a,b) andall components in F(a,t) are continuously differentiablewith respect to every structural interval parameter ai . Basedon the implicit function theorem, the dynamic responsex(a,b,t) from (7) is also continuously differentiable. Differ-entiating both sides of (7) with respect to a yields
M(a, b)∂ x(a, b, t)
∂a+C(a, b)
∂ x(a, b, t)
∂a
+K(a, b)∂x(a, b, t)
∂a= R(a, b, t) (9)
M. Li et al.
in which
R(a, b, t) = ∂F(a, t)
∂a− ∂M(a, b)
∂ax(a, b, t)
−∂C(a, b)
∂ax(a, b, t) − ∂K(a, b)
∂ax(a, b, t).
(10)
Substituting a = aC into (9), then ∂x(aC, b, t)/∂a canbe reached by solving the deterministic (9) using Newmarkintegration algorithm.
Now, substituting x(aC, b, t) and ∂x(aC, b, t)/∂a into(8), thus we can reach a set X(t) consisting of all thepossible values of dynamic response as
X(t) ={
x(t)|x(t) = x(aC, b, t)
+m∑
i
∂x(aC, b, t)
∂ai
�ai, �ai ∈[−aR
i , aRi
]}
(11)
Secondly, calculate the ESLs for interval parametersstructures.
According to the theory of ESLSO, (7) can be modifiedto make the ESLs vector feq(t) as
Here, we propose a numerical processing technique thatK(a,b) in (12) is replaced with deterministic stiffness matrixK0 = K(aC, b). Since x(a, b, t) includes all the uncertain-ties of K(a, b), so the equivalent static loads feq(t), which iscalculated by multiplying the deterministic stiffness matrixK0 and the dynamic response x(a,b,t), can also reflect allthe uncertainties of interval parameters structures. Further-more, the most important thing is that the equivalent staticloads feq(t) can generate the same displacement field as theoriginal dynamic loads.
Replacing the time variable t by s, then the ESLs vectorcan be evaluated by multiplying the deterministic stiffnessmatrix and the dynamic response at each time step as
feq (s) = K0x (s) (s = 1, ..., l) . (13)
Based on the set mapping theory, substituting all the ele-ments of the set X(t) into (13), hence we can reach a setFeq(s) consisting of all the possible ESLs as
Feq(s) ={
feq(s)
∣∣∣feq(s) = K0x(aC, b, s)
+m∑
i
K0∂x(aC, b, s)
∂ai
�ai, �ai, ∈[−aR
i , aRi
]}
.
(14)
From the above theoretical derivation process, it canbe seen that one-to-one mapping is employed to computeESLs, and the uncertainties of interval parameters structuresare successfully and accurately transferred to ESLs.
3.2 ESLSO for dynamic response reliability-basedtopology optimization
The overall process of ESLSO algorithm for dynamicresponse reliability-based topology optimization is illus-trated in Fig. 2 and the steps are as follows:
Step 1. Set initial design variables b(k) = b(0) and thecycle number k = 0;
Step 2. Perform the interval dynamic response analysisof interval parameters structures with b(k). In thisstep, the deterministic stiffness matrix K(k)
0 andset X(k)(t) consisting of all the possible values ofdynamic response are obtained as
K(k)0 = K
(aC, b(k)
), (15)
X(k)(t) ={
x(k)(t)
∣∣∣∣∣x(k)(t) = x(aC, b(k), t) +
m∑
i
× ∂x(aC, b(k), t)
∂ai
�ai , �ai ∈[−aR
i , aRi
]}
.
(16)
Step 3. Calculate the ESLs of interval parameters struc-tures. In this step, the set Feq(s) including all thepossible ESLs is obtained as
F(k)eq (s) =
{
f(k)eq (s)
∣∣∣∣∣f(k)eq (s)=K(k)
0 x(aC, b(k), s)+m∑
i
K(k)
× ∂x(aC, b(k), s)
∂ai
�ai, �ai ∈[−aR
i , aRi
]}
.
(17)
Fig. 2 Process of linear dynamic response topology optimizationusing ESLSO
Structural dynamic topology optimization based on dynamic reliability
Step 4. Based on the definition of non-probabilistic reli-ability index, establish the equivalent linear staticreliability-based topology optimization model(18a, 18b, 18c, 18d, 18e), and solve it.
Find : b(k+1) ∈ Rn (18a)
To maximize : η{g(
b(k+1),δ)
= C∗
−C(
b(k+1),δ)}
(18b)
Subject to. : K(b(k+1)
)u (s) = f(k)
eq (s) , f(k)eq (s)
∈ F(k)eq (s) (s = 1, 2, · · · , l) ,
(18c)
V(
b(k+1),δ)/
V0 ≤ f, (18d)
0.001 ≤ b(k+1)e ≤ 1 (e = 1, · · · , n) , (18e)
where C =1∑
s=1ωs
(f(k)eq (s)
)Tu (s) is the dynamic
compliance (Min et al. 1999; Jang et al. 2012),ωs
is the weighting factor, l is the number of ESLssets, δ is the normalized interval vector of struc-tural interval parameters vector a, C∗ is the ref-erence value of dynamic compliance that is usedto define the limit-state function g(b(k+1), δ), η isthe corresponding interval non-probabilistic reli-ability index, K(b(k+1)) is the deterministic stiff-ness matrix K(aC,b(k+1)) in the kth cycle, u (s) isthe displacement vector, V and V0 are the materialvolume and initial design domain volume, respec-tively, f is the prescribed volume fraction, n is thenumber of design variables. f(k)
eq (s) is the element
of the set F(k)eq (s), and the displacement response
u(s) will is calculated by the set mapping theory.Step 5. When k = 0, go to Step 6. When k > 0, if the con-
vergence criterion is satisfied, terminate the pro-cess. Otherwise, go to Step 6. The convergencecriterion is defined as
countif(∣∣∣b(k+1)
e − b(k)e
∣∣∣ ≥ ε1
)≤ n × ε2 (e = 1, · · · , n) ,
(19)where ε1 and ε2 are small values defined by theuser according to actual problems.
Step 6. Update the design results, set k = k+ 1, and go toStep 2.
It is noted that the SIMP model (Bendsoe 1989; Rozvanyet al. 1992) is employed as the material interpolation scheme
of topology optimization. Therefore, throughout the opti-mization process including analysis domain to obtain theequivalent static loads, the Young’s modulus of the artificialmaterial for each element is calculated by Ee = E0 (be)
p,where E0 is the Young’s modulus for solid material and p
is the penalty exponent.
4 Solution strategy
4.1 Transformation of the optimization problem
According to the definition of the non-probabilistic relia-bility index, the optimization problem (18a, 18b, 18c, 18d,18e) can be converted to an equivalent problem (Luo et al.2012) that finds the maximization of the limit-state functionvalue under the worst condition. Actually, the equivalentproblem is a nested optimization problem, which can beexpressed as follows:
maxb(k+1)
g(
b(k+1),δ)
(20a)
s.t.V (b(k+1),δ)/
V0 ≤ f, (20b)
0.001 ≤ b(k+1)e ≤ 1(e = 1, · · · , n), (20c)
and
minδ
g(b(k+1),δ) (21a)
s.t. δ2i ≤ η2(i = 1, · · · , m), (21b)
where η is the specified non-probabilistic reliability index.In the nested optimization problem, the inner optimiza-
tion process shown in (21a, 21b) is to find the worst
condition δ of E ={δ∣∣∣|δi | ≤ η , i = 1, 2
}that minimizes
the limit-state function value g(b(k+1),δ), the outer opti-mization process shown in (20a, 20b, 20c) is to find theoptimum design that maximizes the limit-state functionvalue g(b(k+1),δ) under the volume constraint. Obviously,if g(b(k+1),δ) ≥ 0, the reliability index of the optimumdesign is η ≥ η. In contrast, the reliability index of the
optimum design is η < η while g(b(k+1),δ) < 0.It should be pointed out that structural uncertainties have
been fully considered while η = 1, and that the optimum
design in this case is reliable if g(b(k+1),δ) ≥ 0, theoreti-cally. But for some special structures, enough safety marginis usually required. Thus, η > 1 indicates that the optimumstructure retains enough safety margin.
Substituting g(b(k+1),δ) = C∗ −C(b(k+1),δ) into (20a,20b, 20c) and (21a, 21b), the optimization problem can besimplified as
minb(k+1)
maxδ
C(b(k+1),δ) (22a)
M. Li et al.
(a) (b)
Fig. 3 a Design domain of short cantilever b unit dynamic load
s.t. δ2i ≤ η2(i = 1, · · · , m), (22b)
V (b(k+1),δ)/
V0 ≤ f, (22c)
0.001 ≤ b(k+1)e ≤ 1(e = 1, · · · , n). (22d)
In (22a, 22b, 22c, 22d), we do not need to predefine theupper bound of the dynamic compliance.
4.2 Sensitivity analysis
The method of moving asymptotes (MMA; Svanberg 1987)is employed as the solver of the optimization problem (22a,22b, 22c, 22d), thus the sensitivity analysis is necessitated.
Firstly, the derivative of the dynamic complianceC(b(k+1),δ) with respect to uncertain parameters can bereached by
∂C
∂δi
=l∑
s=1
ωs
⎛
⎜⎝
∂(
f(k)eq (s)
)T
∂δi
u(s) +(
f(k)eq (s)
)T ∂u(s)
∂δi
⎞
⎟⎠.
(23)
Taking the derivative of the equilibrium (18c), we obtain
∂u (s)
∂δi
= K−1(b(k+1))
⎛
⎜⎝
∂(
f(k)eq (s)
)T
∂δi
− ∂K(b(k+1))
∂δi
u (s)
⎞
⎟⎠ .
(24)
(a) Deterministic optimization (b) NRBTO
Fig. 4 Comparison of optimal layouts. a Deterministic optimizationb NRBTO
Table 1 Comparison of optimal results for the cantilever plate
Volume Objective Uncertain No. of
fraction function(J) interval(J) cycles
DO 0.4 0.3392 [0.2654,0.5132] 3
NRBTO 0.4 0.4428 [0.2372,0.4436] 3
Equation (24) can be simplified as (25) becauseK(b(k+1)) is the deterministic stiffness matrix.
∂u (s)
∂δi
= K−1(b(k+1))∂(f(k)
eq (s))T
∂δi
. (25)
Substituting (25) and (18c) into (23), we can obtain (26)after rearranging.
∂C
∂δi
=l∑
s=1
2ωs
∂(f(k)eq (s))T
∂δi
u(s). (26)
Secondly, the derivative of the dynamic complianceC(b(k+1), δ) with respect to design variables can be reachedby
∂C
∂be
= −l∑
s=1
ωs
(
uT(s)∂K(b(k+1))
∂be
u(s)
)
. (27)
5 Numerical examples
Three standard examples are used to numerically validatethe proposed method. The first example only considersthe uncertainties from geometric dimensions. The secondexample only considers the uncertainties from externalloads. The third example considers all the possible uncer-tainties from geometric dimensions, material properties andexternal loads. It is noted that the safety margin is not con-sidered in the numerical examples, so that the specifiednon-probabilistic reliability index η is set to be 1.
5.1 Short cantilever beam with a single load
The design domain of the short cantilever beam is illustratedin Fig. 3a. The length of the beam is a ∈[0.15,0.17]m, thewidth is b ∈[0.09,0.11]m and the thickness is 0.002 m. The
Fig. 5 Design domain and loading condition of planar simply-supported structure
Structural dynamic topology optimization based on dynamic reliability
Fig. 6 Comparison of optimallayouts. a Deterministicoptimization(DO) b NRBTO
(a) Deterministic optimization (DO) (b) NRBTO
Poisson’s ratio of the material is v = 0.3. The Young’s mod-ulus and density are E = 2.0e11 Pa and ρ = 3000 kg/m3,respectively. An dynamic load f (t) = 2000 f0(t) is appliedat the bottom-right corner of the design domain in down-ward direction, where f0(t) is illustrated in Fig. 3b. Thevolume should be less than 40 % of the initial volume.
The design domain is discretized into 40 × 25 four-nodeplane stress elements. The weighting factors of the dynamiccompliance are set to be ωs = 1/l (s = 1, · · · , l). In thiscase, the dynamic compliance C means the average value ofstrain energy within the given time.
The optimal layout of non-probabilistic reliability-basedtopology optimization (NRBTO) is shown in Fig. 4b. Forthe comparison purpose, the optimal layout of determin-istic optimization (DO), which takes the midpoint valuesof structural interval parameters as deterministic parame-ter values, is given in Fig. 4a. We conduct the reanalysisfor both layouts, respectively, then uncertain intervals ofthe dynamic compliance for both layouts are obtained. Thecorresponding results are listed in Table 1.
From the result of DO shown in Table 1, it can be seenthat uncertainties from geometric dimensions have a largeeffect on the dynamic compliance. Thus, it is necessary forrational topology optimization designs to take uncertain-ties into account. From Fig. 4, we can find that NRBTOsuggests a different layout. From Table 1, it can be foundthat the upper bound of dynamic compliance of the differ-ent layout obtained by NRBTO is smaller than that of DO.Thus, the design reached by NRBTO is more reasonableand more reliable than that by DO. And if we predefinethe reference value of dynamic compliance C* = 0.5, wecan reach the non-probabilistic reliability index of the opti-mum design by NRBTO η > 1 according to the definition ofnon-probabilistic reliability index. Obviously, in this case,the optimum design of NRBTO is considered to be reliable,
Table 2 Comparison of optimal results for the simply-supportedstructure
Volume Objective Uncertain No. of
fraction function(J) interval(J) cycles
DO 0.5 728.5 [630.6,848.2] 4
NRBTO 0.5 817.6 [621.7,814.9] 4
but the optimum design of DO is not reliable. In addition,from the result of NRBTO shown in Table 1, we find thatthe objective function value is very similar to the upperbound of the dynamic compliance interval obtained throughreanalysis, which proves the validity of the proposed modeland numerical techniques. It is worth to note that NRBTOonly needs the same number of cycles as DO while theconvergence criterion is satisfied.
5.2 Planar simply-supported structure with multiple loads
The design domain and loading condition of the simply-supported structure are shown in Fig. 5. It has dimensionsof 1.0 m × 0.2 m × 0.005 m and is discretized into100 × 20 four nodes plane stress elements. The Pois-son’s ratio, Young’s modulus and density of the materialare v = 0.3, E = 2.0e11 Pa and ρ = 7800 kg/m3, res-pectively. The values of dynamic loads f1 − f3 aref1 = A1sin(5πt)[kN], f2 = A2sin(5πt)[kN] and f3 =A3sin(5πt)[kN], where A1 ∈[95,105], A2 ∈[0,40] andA3 ∈[95,105], respectively. The specified volume fractionis 0.5. The weighting factors of the dynamic complianceare also set to be ωs = 1/l (s = 1, · · · , l).
For the comparison purpose, the optimal topologies ofmidpoint-value-based deterministic optimization (DO) andNRBTO are shown in Fig. 6. We can see that NRBTO pro-duces a different layout from DO. This is mainly becausethe larger disturbance of external loads f2 makes the effectof f2 on the optimum design become larger. From theresults of uncertainty analysis for optimum layouts shown inTable 2, it can be found that the design reached by NRBTOis more reasonable and more reliable than that of midpoint-value-based deterministic optimization. Meanwhile, the twooptimizations need the same number of cycles. In otherwords, NRBTO does not cost more than DO, but it reaches
Fig. 7 Design domain of the short cantilever
M. Li et al.
(a) Deterministic optimization (b) NRBTO
Fig. 8 Comparison of optimal layouts. a Deterministic optimizationb NRBTO
a better design than DO. As we expect, the objective func-tion value of NRBTO is very similar to the upper boundof dynamic compliance interval reached by reanalysis. Thatproves the validity of the proposed method again.
5.3 Short cantilever beam with two loads
The design domain is illustrated in Fig. 7. The length ofthe beam is a ∈[0.19,0.21]m, the width is b ∈[0.09,0.11]mand the thickness is 0.002 m. The cantilever beam is dis-cretized into 50 × 25 four-node plane stress elements.The Poisson’s ratio, the Young’s modulus and the den-sity of the material are v = 0.3, E ∈[1.8e11,2.2e11]Paand ρ ∈[7.02e3,8.58e3]kg/m3, respectively. As shown inFig. 7, the plate is subjected to two dynamic loads, respec-tively f1(t) = Asin(5πt)[N] at the point P1 in down-ward direction and f2(t) = Bcos(5πt)[N] at the bottom-center P2 in upward direction, where A∈[9000,11000] andB∈[19000,21000]. The volume should be less than 40 %of the initial volume. The weighting factors of the dynamiccompliance are also set to be ωs = 1/l (s = 1, · · · , l).
Figure 8a is the layout of midpoint-value-based deter-ministic optimization and Fig. 8b is the layout of NRBTO.The corresponding optimal results and reanalysis results aredetailed in Table 3. As the previous examples, the NRBTOalso suggests a more reliable and more reasonable struc-tural topology, and the iterations needed by DO and RBTOare still the same when convergence criterion is satisfied.Besides, the objective function value is also similar to themaximization of the dynamic compliance interval obtainedby reanalysis in the NRBTO.
Table 3 Comparison of optimal results for the cantilever plate
Volume Objective Uncertain No. of
fraction function(J) interval(J) cycles
DO 0.4 22.76 [13.64,46.10] 4
NRBTO 0.4 39.52 [ 9.54, 39.17] 4
6 Conclusions
In practical applications, it is very difficult to obtain theprobability distribution of structural uncertainties, but some-times we can easily obtain the bounds of uncertainties.In this context, an effective approach for reliability-basedtopology optimization of interval parameters structuresunder dynamic loads is proposed. A modified ESLSO, theESLs of which are redefined to consider uncertainties, isemployed to convert the dynamic response optimizationproblem to the static response optimization problem. In con-junction with the definition of the interval non-probabilisticindex, the reliability-based topology optimization modelconsidering dynamic characteristics is constructed. In orderto avoid interval arithmetic resulting in the expansion ofinterval range, intervals are replaced by sets in the processof mathematical operation.
The applicability and validity of the constructed modeland proposed numerical techniques are verified by threestandard examples. From the results of topology optimiza-tion, we can find that uncertainties have a large effecton the dynamic response structural optimization, and thatreliability-based topology optimization usually suggestsmore reasonable and more reliable structural topologiesdifferent from deterministic optimization.
Acknowledgments The authors would like to thank the anonymousreviewers for their critical reviews and valuable suggestions.
References
Bendsoe MP (1989) Optimal shape design as a material distributionproblem. Struct Optim 1(4):193–202
Bendoe MP, Kikuchi N (1988) Generating optimal topologies in struc-tural design using a homogenization method. Comput MethodsAppl Mech Eng 71(1):197–224
Ben-Haim Y (1994) A non-probabilistic concept of reliability. StructSaf 14(4):227–245
Chen SK, Chen W, Lee SH (2010) Level set based robust shape andtopology optimization under random field uncertainties. StructMultidiscip Optim 41(4):507–524
Cho KH, Park JY, Im MG, Han SY (2012) Reliability-based topologyoptimization of electro-thermal-compliant mechanisms with a newmaterial mixing method. Int J Prec Eng Manuf 13(5):693–699
Cho KH, Park JY, Ryu SP, Han SY (2011) Reliability-based topologyoptimization based on bidirectional evolutionary structural opti-mization using multi-objective sensitivity numbers. Int J AutomotTechnol 12(6):849–856
Choi WS, Park GJ (2002) Structural optimization using equivalentstatic loads at all time intervals. Comput Methods Appl Mech Eng191(19–20):2077–2094
Greiner D, Hajela P (2012) Truss topology optimization for mass andreliability considerations-co-evolutionary multiobjective formula-tions. Struct Multidiscip Optim 45(4):589–613
Jang HH, Lee HA et al (2012) Dynamic response topology opti-mization in the time domain using equivalent static loads. Aiaa J50(1):226–234
Structural dynamic topology optimization based on dynamic reliability
Jeong SB, Yoon S, Xu S, Park GJ (2010) Non-linear dynamic responsestructural optimization of an automobile frontal structure usingequivalent static loads. Proc Inst Mech Eng D J Automob Eng224(D4):489–501
Jog CS (2002) Topology design of structures subjected to periodicloading. J Sound Vib 253(3):687–709
Kang BS, Park GJ, Arora JS (2005) Optimization of flexible multibodydynamic systems using the equivalent static load method. Aiaa J43(4):846–852
Kharmanda G, Olhoff N, Mohamed A, Lemaire M (2004) Reliability-based topology optimization. Struct Multidiscip Optim 26(5):295–307
Kim C, Wang S, Hwang I, Lee J (2007a) Application of reliability-based topology optimization for microelectromechanical systems.Aiaa J 45(12):2926–2934
Kim SR, Park JY, Lee WG, Yu JS, Han SY (2007b) Reliability-basedtopology optimization based on evolutionary structural optimiza-tion. In: Ardil C (ed) Proceedings of world academy of science,engineering and technology, Parts 1 and 2, December 2007, vol26. Proceedings of World Academy of Science Engineering andTechnology, pp 168–172
Kim YI, Park GJ, Kolonay RM, Blair M, Canfield RA (2009)Nonlinear dynamic response structural optimization of ajoined-wing using equivalent static loads. J Aircr 46(3):821–831
Lee H-A, Park G-J (2012) Topology optimization for structures withnonlinear behavior using the equivalent static loads method. JMech Des 134(3)
Luo YJ, Kang Z, Luo Z, Li A (2009) Continuum topology optimiza-tion with non-probabilistic reliability constraints based on multi-ellipsoid convex model. Struct Multidiscip Optim 39(3):297–310
Luo Y, Kang Z, Yue Z (2012) Maximal stiffness design of two-material structures by topology optimization with nonprobabilisticreliability. Aiaa J 50(9):1993–2003
Ma ZD, Kikuchi N, Cheng HC (1995) Topological design for vibrat-ing structures. Comput Method Appl Mech Eng 121(3):259–280
Ma J, Wriggers P, Gao W, Chen JJ, Sahraee S (2011) Reliability-basedoptimization of trusses with random parameters under dynamicloads. Comput Mech 47(6):627–640
Mashayekhi M, Salajegheh E, Salajegheh J, Fadaee MJ (2012)Reliability-based topology optimization of double layer gridsusing a two-stage optimization method. Struct Multidiscip Optim45(6):815–833
Maute K, Frangopol DM (2003) Reliability-based design of MEMSmechanisms by topology optimization. Comput Struct 81(8–11):813–824
Michell AGM (1904) The limits of economy of material in framestructures. Phil Mag 8:589–597
Min SJ, Kikuchi N (1999) Optimal topology design of structures underdynamic loads. Struct Optim 17(2):208–218
Min S, Kikuchi N, Park YC, Kim S, Chang S (1999) Optimal topol-ogy design of structures under dynamic loads. Struct Optim17(2–3):208–218
Mogami K, Nishiwaki S, Izui K, Yoshimura M, Kogiso N (2006)Reliability-based structural optimization of frame structures formultiple failure criteria using topology optimization techniques.Struct Multidiscip Optim 32(4):299–311
Moore RE (1979) Methods and applications of interval analysis.Prentice-Hall Inc., London
Park GJ (2011) Technical overview of the equivalent static loadsmethod for non-linear static response structural optimization.Struct Multidiscip Optim 43(3):319–337
Park GJ, Kang BS (2003) Validation of a structural optimization algo-rithm transforming dynamic loads into equivalent static loads. JOptim Theory Appl 118(1):191–200
Patel J, Choi SK (2012) Classification approach for reliability-basedtopology optimization using probabilistic neural networks. StructMultidiscip Optim 45(4):529–543
Qiu ZP, Wang XJ (2003) Comparison of dynamic response ofstructures with uncertain-but-bounded parameters using non-probabilistic interval analysis method and probabilistic approach.Int J Solids Struct 40(20):5423–5439
Rong JH, Xie YM, Yang XY (2000) Topology optimization ofstructures under dynamic response constraints. J Sound Vib234(2):177–189
Rozvany GIN, Zhou M, Birker T (1992) Generalized shape opti-mization without homogenization. Struct Optim 4(3–4):250–252
Svanberg K (1987) The method of moving asymptotes—a new methodfor structural optimization. Int J Numer Methods Eng 24(2):359–373
Yi S-I, Lee H-A, Park G-J (2011) Optimization of a structure withcontact conditions using equivalent loads. J Mech Sci Technol25(3):773–782
Yoon GH (2010) Structural topology optimization for frequencyresponse problem using model reduction schemes. ComputMethod Appl Mech Eng 199:1744–1763
Zhang X, Ouyang G (2008) A level set method for reliability-basedtopology optimization of compliant mechanisms. Sci China ETechnol Sci 51(4):443–455
Zhang W, Sun C, Wang H (2011) Equivalent static loads method forflexible structural shape optimization. In: Gao J (ed) Advanceddesign technology, Pts 1-3, vol 308–310. Advanced MaterialsResearch, pp 2364–2367
