Research Papers

Time-Dependent Reliability Estimation for Dynamic Problems Using a Niching Genetic Algorithm

[+] Author and Article Information
Jing Li

Department of Mechanical Engineering, Oakland University, Rochester, MI 48309li2@oakland.edu

Zissimos P. Mourelatos1

Department of Mechanical Engineering, Oakland University, Rochester, MI 48309mourelat@oakland.edu


Corresponding author.

J. Mech. Des 131(7), 071009 (Jun 25, 2009) (13 pages) doi:10.1115/1.3149842 History: Received November 25, 2007; Revised April 01, 2009; Published June 25, 2009

A time-dependent reliability analysis method is presented for dynamic systems under uncertainty using a niching genetic algorithm (GA). The system response is modeled as a parametric random process. A double-loop optimization algorithm is used. The inner loop calculates the maximum response in time, using a hybrid (global-local) optimization algorithm. A global GA quickly locates the vicinity of the global maximum, and a gradient-based optimizer subsequently refines its location. A time-dependent problem is, therefore, transformed into a time-independent one. The outer loop calculates multiple most probable points (MPPs), which are commonly encountered in vibration problems. The dominant MPPs with the highest contribution to the probability of failure are identified. A niching GA is used because of its ability to simultaneously identify multiple solutions. All potential MPPs are initially identified approximately, and their location is efficiently refined using a gradient-based optimizer with local metamodels. For computational efficiency, the local metamodels are built using mostly available sample points from the niching GA. Among all MPPs, the significant and independent ones are identified using a correlation analysis. Approximate limit states are built at the identified MPPs, and the system failure probability is estimated using bimodal bounds. The vibration response of a cantilever plate under a random oscillating pressure load and a point load is used to illustrate the present method and demonstrate its robustness and efficiency. A finite-element model is used to calculate the plate response.

Copyright © 2009 by American Society of Mechanical Engineers
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Figure 1

A sample function of random process S=S(X,t)

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Figure 2

MMP, reliability index, and linearization of a limit state in the standard normal space

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Figure 3

Schematic of niches versus MPPs. A (fitness) function is built considering the limit state as a “mountain ridge” with multiple peaks.

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Figure 4

Linearized limit states at potential MPPs

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Figure 5

Linearized limit states at actual MPPs

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Figure 6

A ridge-shaped fitness function using a penalty function

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Figure 7

Cantilevered plate under uniform pressure p(t) and point load P(t)

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Figure 8

(a) Multimodal maximum displacement in standard normal space. (b) Limit state g(U)=wmax−3=0 and MPPs for the cantilever plate.

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Figure 9

Approximate maximum displacement using the GA

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Figure 10

Four niches after generation 5

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Figure 11

Four potential MMPs and all 78 samples from niching GA

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Figure 12

The 78 niching GA sample points and additional 8 Latin hypercube sample points

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Figure 13

Refined locations of potential MPPs

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Figure 14

Location of refined MPPs

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Figure 15

Linearized convex safe domain and largest β-circle

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Figure 16

Himmelblau’s function (63): refined niches (○) and identified niches ( ∗) after generation 100

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Figure 17

Himmelblau’s function (63): 4 niches (○) after generation 100 with final population (×)

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Figure 18

Himmelblau’s function (63): percentage of found peaks versus requested number of peaks

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Figure 19

Six-hump camel back function: percentage of found peaks versus requested number of peaks




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