A Monte Carlo solution method for linear elasticity

Abstract

A probabilistic method is proposed and implemented to solve linear elasticity problems. The method, called walk on the boundary method (WBM), uses the same governing equations as the boundary element method. Unlike in finite element and boundary element methods, WBM does not require any meshing. Also, error estimates for WBM are easier to obtain than in finite element and boundary element methods. Furthermore, WBM obtains a point solution at a specific point of interest instead of a full field solution as in finite element and boundary element methods. WBM is developed for general traction boundary value problems in antiplane shear, plane strain, and 3D elasticity. Numerical implementations are performed for three example problems: (1) antiplane shear problem with a centrally located circular hole being loaded by uniformly applied traction, (2) plane strain problem with centrally located circular hole being loaded by uniform tension, and (3) 3D elasticity problem with a centrally located spherical cavity being loaded by uniform tension. Results from the three example problems compared favorably with results from the analytical and finite element solution. Three critical issues associated with the WBM for linear elasticity are pointed out for its further improvement.