Stochastic simulation method for a 2D elasticity problem with random loads

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 [Sabelfeld KK, Shalimova IA, Levykin AI. Discrete random walk over large spherical grids generated by spherical means for PDEs. Monte Carlo Methods and Applications 2006; 12(1): 55–93]. The vector random field of loads which stands on the right-hand side of the system of elasticity equations is simulated by the Randomization Spectral method presented in [Sabelfeld KK. Monte Carlo methods in boundary value problems. Berlin (Heidelberg, New York): Springer-Verlag; 1991] and recently revised and generalized in [Kurbanmuradov O, Sabelfeld KK. Stochastic spectral and Fourier-wavelet methods for vector Gaussian random field. Monte Carlo Methods and Applications 2006; 12(5–6): 395–445]. Comparative analysis of the 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 the determination of the elastic constants from the known longitudinal and transverse correlations of the loads, and give some relevant numerical illustrations.