An efficient SFE method using Lagrange polynomials: Application to nonlinear mechanical problems with uncertain parameters

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Designing efficient numerical methods for the solution of stochastic partial differential equations with random inputs or random coefficients is meeting growing interest. So far, the stochastic Galerkin method has been successfully used for various problems with small number of independent random variables. The drawback of this method lies in its difficulty of implementation for nonlinear problems. In this paper we propose a high-order stochastic collocation method to solve nonlinear mechanical systems whose uncertain parameters can be modeled as random variables. Similar to the stochastic Galerkin methods, fast convergence can be achieved when the solution in random space is smooth. However, the numerical implementation of stochastic collocation method is as easy as the Monte-Carlo method since it only requires repetitive runs of an existing deterministic solver. We illustrate the efficiency of this method on two nonlinear mechanical problems in which the random parameters are modeled as correlated lognormal random variables.