A finite cell approach for discretization of random fields

A new method for discretization of Gaussian random fields with only a small number of random variables in the representation is introduced. The method is based on the Karhunen-Loève (KL) expansion, which is optimal among series expansion methods with respect to the global mean square truncation error. The resulting integral eigenvalue problem in the KL-expansion is discretized using a finite cell (FC) approach; i.e. the domain of computation is extended beyond the physical domain up to the boundaries of an embedding domain with a primitive geometrical shape. Higher order polynomials are used as FC shape functions. The approach is useful for random fields defined on domains with complex geometries since it shifts the problem from the mesh generation to the integration of discontinuous functions defined over a fictitious domain. A suitable approach for numerical integration is described. The presented method is compared to the Expansion Optimal Linear Estimation (EOLE) method and to the finite element discretization of the KL-expansion with respect to the mean error variance and in terms of computational costs. On the one hand, the proposed approach shows an exponential rate of convergence in terms of the dimension of the matrix eigenvalue problem to solve for a fixed number of random variables. On the other hand, obtaining a solution for the random field approximation takes considerably longer than with the EOLE method. However, the generation of a realization of the random field representation with the finite cell approach is computationally more efficient than with EOLE.     «

A new method for discretization of Gaussian random fields with only a small number of random variables in the representation is introduced. The method is based on the Karhunen-Loève (KL) expansion, which is optimal among series expansion methods with respect to the global mean square truncation error. The resulting integral eigenvalue problem in the KL-expansion is discretized using a finite cell (FC) approach; i.e. the domain of computation is extended beyond the physical domain up to th...     »