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Research Papers: Design Automation

Augmented Single Loop Single Vector Algorithm Using Nonlinear Approximations of Constraints in Reliability-Based Design Optimization

[+] Author and Article Information
Petter N. Lind

Department of Solid Mechanics,
Royal Institute of Technology (KTH),
SE-100 44, Stockholm, Sweden
e-mail: pettlind@kth.se

Mårten Olsson

Professor
Department of Solid Mechanics,
Royal Institute of Technology (KTH),
SE-100 44, Stockholm, Sweden
e-mail: mart@kth.se

Contributed by the Design Automation Committee of ASME for publication in the Journal of Mechanical Design. Manuscript received December 18, 2018; final manuscript received April 23, 2019; published online May 23, 2019. Assoc. Editor: Ping Zhu.

J. Mech. Des 141(10), 101403 (May 23, 2019) (9 pages) Paper No: MD-18-1904; doi: 10.1115/1.4043679 History: Received December 18, 2018; Accepted April 29, 2019

Reliability-based design optimization (RBDO) aims at minimizing a function of probabilistic design variables, given a maximum allowed probability of failure. The most efficient methods available for solving moderately nonlinear problems are single loop single vector (SLSV) algorithms that use a first-order approximation of the probability of failure in order to rewrite the inherently nested structure of the loop into a more efficient single loop algorithm. The research presented in this paper takes off from the fundamental idea of this algorithm. An augmented SLSV algorithm is proposed that increases the rate of convergence by making nonlinear approximations of the constraints. The nonlinear approximations are constructed in the following way: first, the SLSV experiments are performed. The gradient of the performance function is known, as well as an estimate of the most probable failure point (MPP). Then, one extra experiment, a probe point, per performance function is conducted at the first estimate of the MPP. The gradient of each performance function is not updated but the probe point facilitates the use of a natural cubic spline as an approximation of an augmented MPP estimate. The SLSV algorithm using probing (SLSVP) also incorporates a simple and effective move limit (ML) strategy that also minimizes the heuristics needed for initiating the optimization algorithm. The size of the forward finite difference design of experiment (DOE) is scaled proportionally with the change of the ML and so is the relative position of the MPP estimate at the current iteration. Benchmark comparisons against results taken from the literature show that the SLSVP algorithm is more efficient than other established RBDO algorithms and converge in situations where the SLSV algorithm fails.

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Figures

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Fig. 1

Comparison of performance between established RBDO algorithms

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Fig. 2

Illustration of probe MPP estimate. Experiments corresponding to coordinates ux* and uxp have been marked accordingly.

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Fig. 3

Flowchart of the SLSVP algorithm

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Fig. 4

Reinforced concrete cross section subjected to pure bending

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Fig. 5

Relative error versus the number of experiments for the reinforced bar example

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Fig. 6

Convergence comparison for Example 1 with βT = 3 using SLSVP, SLSV, and SAP (result from Ref. [15]). The results from the SLSVCG and SLSV algorithms are for this example identical.

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Fig. 7

Convergence of design variables for Example 2 with βT = 3 using SLSVP and SLSVCG (result from Ref. [10]). Note that both algorithms get close to the optimal solution after just a few iterations. The following slow convergence is due to the nonlinearity of the second performance function.

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Fig. 8

The 10-bar truss setup used in Example 3

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Fig. 9

Rate of convergence comparison for design variables in Example 3 using SLSVP and SAP (result from Ref. [15]). The strengths of the SLSVP method are shown here; probing quickly reduces a set of bars to the lower bound and the ML strategy damps out the oscillations enforced from the remaining under constrained optimization problem.

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Fig. 10

Convergence comparison for objective and constraints for Example 3 using SLSVP and SAP (result from Ref. [15])

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Fig. 11

The FE-problem setup with the design variables (h,d,w), applied forces (F), and the edges of the rigidly supported surface marked with thick lines

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Fig. 12

Optimum design with the nonaveraged element nodal stresses that are the same as used in the optimization algorithm

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