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Research Papers: Design Automation

Reliability-Based Topology Optimization Using Mean-Value Second-Order Saddlepoint Approximation

[+] Author and Article Information
Dimitrios I. Papadimitriou

Mechanical Engineering Department,
Oakland University,
2200 N. Squirrel Road,
Rochester 48309, MI
e-mail: dpapadimitriou@oakland.edu

Zissimos P. Mourelatos

Mechanical Engineering Department,
Oakland University,
2200 N. Squirrel Road,
Rochester 48309, MI
e-mail: mourelat@oakland.edu

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received July 5, 2017; final manuscript received November 16, 2017; published online January 10, 2018. Assoc. Editor: Nam H. Kim.

J. Mech. Des 140(3), 031403 (Jan 10, 2018) (11 pages) Paper No: MD-17-1454; doi: 10.1115/1.4038645 History: Received July 05, 2017; Revised November 16, 2017

A reliability-based topology optimization (RBTO) approach is presented using a new mean-value second-order saddlepoint approximation (MVSOSA) method to calculate the probability of failure. The topology optimizer uses a discrete adjoint formulation. MVSOSA is based on a second-order Taylor expansion of the limit state function at the mean values of the random variables. The first- and second-order sensitivity derivatives of the limit state cumulant generating function (CGF), with respect to the random variables in MVSOSA, are computed using direct-differentiation of the structural equations. Third-order sensitivity derivatives, including the sensitivities of the saddlepoint, are calculated using the adjoint approach. The accuracy of the proposed MVSOSA reliability method is demonstrated using a nonlinear mathematical example. Comparison with Monte Carlo simulation (MCS) shows that MVSOSA is more accurate than mean-value first-order saddlepoint approximation (MVFOSA) and more accurate than mean-value second-order second-moment (MVSOSM) method. Finally, the proposed RBTO-MVSOSA method for minimizing a compliance-based probability of failure is demonstrated using two two-dimensional beam structures under random loading. The density-based topology optimization based on the solid isotropic material with penalization (SIMP) method is utilized.

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Figures

Grahic Jump Location
Fig. 1

Comparison of MVFOSA, MVSOSA, MVFOSM, and MVSOSM with MCS for different values of σ—(a) low uncertainty (σ=0.01 or COV = 0.1), (b) average uncertainty (σ=0.05 or COV = 0.5), (c) high uncertainty (σ=0.15 or COV = 1.5)

Grahic Jump Location
Fig. 2

Comparison of CGF among MVFOSA, MVSOSA, and MCS for different values of σ—top: low uncertainty (σ=0.01 or COV = 0.1), mid: average uncertainty (σ=0.05 or COV = 0.5), bottom: high uncertainty (σ=0.15 or COV = 1.5)

Grahic Jump Location
Fig. 3

Schematic of cantilever beam for topology optimization

Grahic Jump Location
Fig. 4

(a) Cantilever beam topology from DTO, (b) cantilever beam topology from RBTO-MVFOSA, (c) cantilever beam topology from RBTO-MVSOSA

Grahic Jump Location
Fig. 5

Cantilever beam convergence history of compliance for deterministic and RBTO cases

Grahic Jump Location
Fig. 6

Cantilever beam convergence history of probability of failure for DTO and RBTO-MVFOSA cases

Grahic Jump Location
Fig. 7

Cantilever beam convergence history of probability of failure for DTO and RBTO-MVSOSA cases

Grahic Jump Location
Fig. 8

Cantilever beam topology from RBTO-MVSOSA for correlated random forces

Grahic Jump Location
Fig. 9

Schematic of simply supported beam for topology optimization

Grahic Jump Location
Fig. 10

(a) Simply supported beam topology from DTO, (b) simply supported beam topology from RBTO-MVFOSA, and (c) simply supported beam topology from RBTO-MVSOSA

Grahic Jump Location
Fig. 11

(a) Simply supported beam convergence history of compliance for DTO and RBTO, (b) simply supported beam convergence history of probability of failure for DTO and RBTO-MVFOSA, and (c) simply supported beam convergence history of probability of failure for DTO and RBTO-MVSOSA

Grahic Jump Location
Fig. 12

Simply supported beam topology from RBTO-MVSOSA for correlated random forces

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