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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

## Abstract

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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## References

Ghanem, R., and Spanos, P. D., 1991, Stochastic Finite Elements: A Spectral Approach, Dover, New York.
Silva, C. R. A., Jr., Beck, A. T., and Rosa, E., 2009, “Solution of the Stochastic Beam Bending Problem by Galerkin Method and the Askey-Wiener Scheme,” Lat. Am. J. Solids Struct., 6, pp. 51–72.
Silva, C. R. A., Jr., Beck, A. T., Mantovani, G., and Azikri, H., 2010, “Galerkin Solution of Stochastic Beam Bending on Winkler Foundations,” Comput. Model. Eng. Sci., 1729, pp. 1–31.
Silva, C. R. A., Jr., and Beck, A. T., 2010, “Bending of Stochastic Kirchhoff Plates on Winkler Foundations via the Galerkin Method and the Askey-Wiener Scheme,” Probab. Eng. Mech., 25, pp. 172–182.
Beck, A. T., and Silva, C. R. A., Jr., 2011, “Timoshenko Versus Euler Beam Theory: Pitfalls of a Deterministic Approach,” Struct. Saf., 33, pp. 19–25.
Silva, C. R. A., Jr., and Beck, A. T., 2011, “Chaos-Galerkin Solution of Stochastic Timoshenko Bending Problems,” Comput. Struct., 89, pp. 599–611.
Yamazaki, F., Shinozuka, M., and Dasgupta, G., 1989, “Neumann Expansion for Stochastic Finite Element Analysis,” J. Eng. Mech. ASCE, 114, pp. 1335–1354.
Araújo, J. M., and Awruch, A. M., 1994, “On Stochastic Finite Elements for Structural Analysis,” Comput. Struct., 52, pp. 461–469.
Chakraborty, S., and Dey, S. S., 1998, “A Stochastic Finite Element Dynamic Analysis of Structures with Uncertain Parameters,” Int. J. Mech. Sci., 40, pp. 1071–1087.
Chakraborty, S., and Dey, S. S., 1996, “Stochastic Finite Element Simulation of Random Structure on Uncertain Foundation Under Random Loading,” Int. J. Mech. Sci., 38, pp. 1209–1218.
Chakraborty, S., and Dey, S. S., 1995, “Stochastic Finite Element Method for Spatial Distribution of Material Properties and External Loading,” Comput. Struct., 55, pp. 41–45.
Lei, Z., and Qiu, C., 2000, “Neumann Dynamic Stochastic Finite Element Method of Vibration for Structures with Stochastic Parameters to Random Excitation,” Comput. Struct., 77, pp. 651–657.
Chakraborty, S., and Sarkar, S. K., 2000, “Analysis of a Curved Beam on Uncertain Elastic foundation,” Finite Elem. Anal. Des., 36, pp. 73–82.
Chakraborty, S., and Bhattacharyya, B., 2002, “An Efficient 3D Stochastic Finite Element Method,” Int. J. Solids Struct., 39, pp. 2465–2475.
Li, C. F., Feng, Y. T., and Owen, D. R. J., 2006, “Explicit Solution to the Stochastic System of Linear Algebraic Equations,” Comput. Methods Appl. Mech. Eng., 195, pp. 6560–6576.
Schevenels, M., Lombaert, G., Degrande, G., and Clouteau, D., 2007, “The Wave Propagation in a Beam on a Random Elastic Foundation,” Probab. Eng. Mech., 22, pp. 150–158.
Babuska, I., Tempone, R., and Zouraris, G. E., 2005, “Solving Elliptic Boundary Value Problems with Uncertain Coefficients by the Finite Element Method: The Stochastic Formulation,” Comput. Methods Appl. Mech. Eng., 194, pp. 1251–1294.
Silva, C. R. A., Jr., Beck, A. T., Franco, A. T., and Suarez, O. A., 2013, “Galerkin Solution of Stochastic Reaction-Diffusion Problems,” J. Heat Transfer, 135, pp. 071201-1–071201-12.
Kincaid, D., and Cheney, W., 2002, Numerical Analysis: Mathematics of Scientific Computing, 3rd ed., AMS, Boston.
Golub, G. H., and Van Loan, C. F., 2012, Matrix Computations (Johns Hopkins Studies in the Mathematical Sciences), JHU Press, Johns Hopkins University, Baltimore.
Nocedal, J., and Wright, S., 2006, Numerical Optimization, 2nd ed., Springer, New York.
Olsson, A. M. J., and Sandberg, G. E., 2002, “Latin Hypercube Sampling for Stochastic Finite Element Analysis,” J. Eng. Mech., 128, pp. 121–125.

## Figures

Fig. 1

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

Fig. 7

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

Fig. 8

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

Fig. 2

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

Fig. 3

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

Fig. 4

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

Fig. 5

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

Fig. 6

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

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