Multi-scale goal-oriented adaptive modeling of random heterogeneous materials

This paper addresses the general problem of modeling local features of the response of highly heterogeneous elastic materials with random distributions of the material constituents. The theory and methodologies of goal-oriented adaptive modeling of heterogeneous materials [Oden, J.T., Vemaganti, K., 2000a. Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. Part I: Error estimates and adaptive algorithms. J. Comp. Phys. 164, 22–47; Oden, J.T., Vemaganti, K., 2000b. Adaptive modeling of composite structures: modeling error estimation. Int. J. Comput. Civil Str. Eng. 1, 1–16; Vemaganti, K., Oden, J.T., 2001. Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. Part II: A computational environment for adaptive modeling of heterogeneous elastic solids. Comput. Methods Appl. Mech. Eng. 190, 6089–6124; Romkes, A., Vemaganti, K., Oden, J.T., 2004. The extension of the GOALS algorithm to the analysis of elastostatics problems of random heterogeneous materials. ICES Report 04-45, The University of Texas at Austin] are extended to incorporate uncertainty in the material data by using classical Monte Carlo methods for calculating local quantities of interest. Techniques for estimating modeling error are extended to cases in which the material data are random variables. Several numerical examples involving two-phase composites with random material properties are given.