2007, Sabelfeld, Karl, Shalimova, Irina, Levykin, Alexander

We develop a stochastic simulation method for a numerical solution of the Lamé equation with random loads. To treat the general case of large intensity of random loads, we use the Random Walk on Fixed Spheres (RWFS) method described in our paper citesab-lev-shal-2006. The vector random field of loads which stands in the right-hand-side of the system of elasticity equations is simulated by the Randomization Spectral method presented in citesab-1991 and recently revised and generalized in citekurb-sab-2006. Comparative analysis of RWFS method and an alternative direct evaluation of the correlation tensor of the solution is made. We derive also a closed boundary value problem for the correlation tensor of the solution which is applicable in the case of inhomogeneous random loads. Calculations of the longitudinal and transverse correlations are presented for a domain which is a union of two arbitrarily overlapped discs. We also discuss a possibility to solve an inverse problem of determination of the elastic constants from the known longitudinal and transverse correlations of the loads.

2006, Sabelfeld, Karl, Shalimova, Irina, Levykin, Alexander

The Random Walk on Fixed Spheres (RWFS) introduced in our previous paper is presented in details for Laplace and Lamé equations governing static elasticity problems. The approach is based on the Poisson type integral formulae written for each disc of a domain consisting of a family of overlapping discs. The original differential boundary value problem is equivalently reformulated in the form of a system of integral equations defined on the intersection surfaces (arches, in 2D, and caps, if generalized to 3D spheres). To solve the obtained system of integral equations, a Random Walk procedure is constructed where the random walks are living on the intersecting surfaces. Since the spheres are fixed, it is convenient to construct also discrete random walk methods for solving the system of linear equations approximating the system of integral equations. We develop here two classes of special Monte Carlo iterative methods for solving these systems of linear algebraic equations which are constructed as a kind of randomized versions of the Chebyshev iteration method and Successive Over Relaxation (SOR) method. It is found that in this class of randomized SOR methods, the Gauss-Seidel method has a minimal variance ...