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Research Papers

A Fast Convergence Parameter for Monte Carlo–Neumann Solution of Linear Stochastic Systems

[+] Author and Article Information
Cláudio R. Ávila da Silva, Jr.

Department of Mechanical Engineering, Federal University of Technology of Parana, Av. 7 de setembro, Curitiba - PR 80230-901, Brazile-mail: avila@utfpr.edu.br

André Teófilo Beck

Structural Engineering Department, EESC, University of São Paulo, Av. Trabalhador Sancarlense, 400, 13566-590 São Carlos, São Paulo, Brazile-mail: atbeck@sc.usp.br

Manuscript received July 23, 2014; final manuscript received February 2, 2015; published online April 20, 2015. Assoc. Editor: Ioannis Kougioumtzoglou.

ASME J. Risk Uncertainty Part B 1(2), 021002 (Apr 20, 2015) (8 pages) Paper No: RISK-14-1031; doi: 10.1115/1.4029741 History: Received July 23, 2014; Accepted February 05, 2015; Online April 20, 2015

The Neumann series is a well-known technique to aid the solution of uncertainty propagation problems. However, convergence of the Neumann series can be very slow, often making its use highly inefficient. In this article, a fast convergence parameter (λ) convergence parameter is introduced, which yields accurate and efficient Monte Carlo–Neumann (MC-N) solutions of linear stochastic systems using first-order Neumann expansions. The λ convergence parameter is found as a solution to the distance minimization problem, for an approximation of the inverse of the system matrix using the Neumann series. The method presented herein is called Monte Carlo–Neumann with λ convergence, or simply the MC-N λ method. The accuracy and efficiency of the MC-N λ method are demonstrated in application to stochastic beam-bending problems.

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Figures

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Fig. 1

Convergence of Monte Carlo simulation results for mean displacement at x=3l5

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Fig. 2

Convergence of Monte Carlo simulation results for variance of displacements at x=3l5

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Fig. 3

Deviation in expected value (εμ̑um2) for random membrane stiffness, δEA=110

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Fig. 4

Deviation in variance (εσ̑um2) for random membrane stiffness, δEA=110

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Fig. 5

Deviation in expected value (εμ̑um) for random membrane stiffness, δEA=310

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Fig. 6

Deviation in variance (εσ̑um2) for random membrane stiffness, δEA=310

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Fig. 7

Deviation in expected value (εμ̑um) for random reaction stiffness, δκ=310

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Fig. 8

Deviation in variance (εσ̑um2) for random reaction stiffness, δκ=310

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