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Single-loop system reliability-based topology optimization considering statistical dependence between limit-states

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Abstract

This paper presents a single-loop algorithm for system reliability-based topology optimization (SRBTO) that can account for statistical dependence between multiple limit-states, and its applications to computationally demanding topology optimization (TO) problems. A single-loop reliability-based design optimization (RBDO) algorithm replaces the inner-loop iterations to evaluate probabilistic constraints by a non-iterative approximation. The proposed single-loop SRBTO algorithm accounts for the statistical dependence between the limit-states by using the matrix-based system reliability (MSR) method to compute the system failure probability and its parameter sensitivities. The SRBTO/MSR approach is applicable to general system events including series, parallel, cut-set and link-set systems and provides the gradients of the system failure probability to facilitate gradient-based optimization. In most RBTO applications, probabilistic constraints are evaluated by use of the first-order reliability method for efficiency. In order to improve the accuracy of the reliability calculations for RBDO or RBTO problems with high nonlinearity, we introduce a new single-loop RBDO scheme utilizing the second-order reliability method and implement it to the proposed SRBTO algorithm. Moreover, in order to overcome challenges in applying the proposed algorithm to computationally demanding topology optimization problems, we utilize the multiresolution topology optimization (MTOP) method, which achieves computational efficiency in topology optimization by assigning different levels of resolutions to three meshes representing finite element analysis, design variables and material density distribution respectively. The paper provides numerical examples of two- and three-dimensional topology optimization problems to demonstrate the proposed SRBTO algorithm and its applications. The optimal topologies from deterministic, component and system RBTOs are compared with one another to investigate the impact of optimization schemes on final topologies. Monte Carlo simulations are also performed to verify the accuracy of the failure probabilities computed by the proposed approach.

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Acknowledgments

This research was funded in part by a grant from the Vietnam Education Foundation (VEF) and the National Science Foundation. The supports are gratefully acknowledged. The opinions, findings, and conclusions stated herein are those of the authors and do not necessarily reflect those of sponsors.

Nomenclature

B :

strain-displacement matrix of shape function derivatives

c :

“event” vector

d :

vector of design variables

D(·):

constitutive matrix determined by material density function

d n :

the n-th design variable

D 0 :

constitutive matrix corresponding to the solid material

E i :

the i-th failure event

E sys :

the system failure event

E 0 :

Young’s modulus corresponding to the solid material

f(·):

objective function

f :

global load vector

\(g_{i}({\uprho}, {\rm \textbf{X}}\)):

limit-state (or performance) function of the i-th failure mode

\(g_{P_i^t}\) :

performance function of the i-th failure mode

Jx,u :

Jacobian matrix of the transformation x = x(u)

K :

global stiffness matrix

K e :

element stiffness matrix

p :

penalization parameter

p :

“probability” vector

P i :

actual failure probability of the i-th mode

\(P_i^t\) :

target failure probability of the i-th mode

P sys :

actual system failure probability

\(P_{sys}^t\) :

target system failure probability

r ik :

Dunnett-Sobel correlation parameter

r min :

minimum length scale

r ni :

distance from the point associated design variable to the centroid

S :

common source random variables

u d :

displacement vector

\({\boldsymbol{\tilde u}}_i^t\) :

approximate location for the performance function value for the i-th failure mode

\({\boldsymbol{\tilde u}}_i\) :

approximate location for the performance function value by the KKT condition

volfrac :

volume fraction

V s :

prescribed volume constraint

X :

random variables

\(\hat{\pmb{\upalpha}}_i\) :

negative normalized gradient vector

β i :

reliability index

\({\upbeta}_i^t\) :

target reliability index

\({\upbeta} _i^{t(k)}\) :

updated target reliability index at the k-th design iteration

\({\upbeta}_i^{t(k)(SORM)}\) :

improved reliability index using Breitung’s formula at the k-th design iteration

\(\pmb{\rm \mu}_{\bf x}\) :

vector of means of x

\({\uprho} (\cdot )\) :

material density function

ρ i :

density of element i

\(\pmb{\uppsi}\) :

position vector

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Authors and Affiliations

  1. Dept. of Civil and Environmental Engineering, Univ. of Illinois, Urbana, IL, 61801, USA

    Tam H. Nguyen, Junho Song & Glaucio H. Paulino

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  1. Tam H. Nguyen
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  2. Junho Song
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  3. Glaucio H. Paulino
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Correspondence to Glaucio H. Paulino.

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Nguyen, T.H., Song, J. & Paulino, G.H. Single-loop system reliability-based topology optimization considering statistical dependence between limit-states. Struct Multidisc Optim 44, 593–611 (2011). https://doi.org/10.1007/s00158-011-0669-0

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