The numerical results are compared against Monte–Carlo simulations, and it is shown that the proposed approach provides high-quality approximations for the first two statistical moments at modest computational effort.
The proposed approach is demonstrated for modeling diffusion in a square domain with a rough wall and heat transfer analysis of a three-dimensional gas turbine blade model with uncertainty in the cooling core geometry. The resulting randomly parametrized algebraic equations are solved using stochastic projection schemes to approximate the response statistics. Finite element spatial discretization of the governing equations using a stochastic mesh representation results in a linear random algebraic system of equations in a polynomial chaos basis whose coefficients of expansion can be non-intrusively computed either at the element or the global level. We prove a result to establish the conditions that the input uncertainty model must satisfy to ensure the validity of the stochastic mesh representation and hence the well posedness of the problem. The key idea is to carry out spatial discretization using a combination of finite element methods and stochastic mesh representations. more We present stochastic projection schemes for approximating the solution of a class of deterministic linear elliptic partial differential equations defined on random domains. We present stochastic projection schemes for approximating the solution of a class of determinist. The numerical results are compared against benchmark Monte Carlo simulations, and it is shown that the proposed approach provides good approximations for the response statistics at modest computational effort. Detailed numerical studies are presented for diffusion in a square domain for varying degrees of nonlinearity. This iterative procedure is repeated until an appropriate convergence criterion is met. The approximation to the solution process thus obtained is used as the guess for the next iteration. The linearized stochastic governing equations are then spatially discretized and approximately solved using stochastic reduced basis projection schemes. The key idea is to iteratively solve the nonlinear stochastic governing equations via an inexact Picard iteration scheme, wherein the nonlinear constitutive law is linearized using the current guess of the solution.
more In this paper, we present a numerical scheme for the analysis of steady-state nonlinear diffusion in random heterogeneous media.
In this paper, we present a numerical scheme for the analysis of steady-state nonlinear diffusion.
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