A numerical method called the boundary walk method is described in this paper. The boundary walk method is a local method in the sense that it directly gives the solution at the point of interest. It is based on a global integral representation of the unknown solution in the form of potentials, followed by evaluating the integrals in the resulting series solutions using Monte Carlo simulation. The boundary walk method has been applied to solve interior problems in potential theory with either Dirichlet or Neumann boundary conditions. It has also been applied to solve interior problems in linear elasticity with either displacement or traction boundary conditions. Weakly singular integral formulations in linear elasticity, to which the boundary walk method has been applied, are also derived. Finally, numerical results, which are computed by applying the boundary walk method to solve some two-dimensional problems over convex domains in potential theory and linear elasticity, are presented. These solutions are compared with the known analytical solutions (when available) or with solutions from the standard boundary element method.
Local Solutions in Potential Theory and Linear Elasticity Using Monte Carlo Methods
Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED MECHANICS. Manuscript received by the ASME Applied Mechanics Division, Nov. 30, 2001; final revision, Aug. 20, 2002. Associate Editor: D. A. Kouris. Discussion on the paper should be addressed to the Editor, Prof. Robert M. McMeeking, Department of Mechanical and Environmental Engineering University of California– Santa Barbara, Santa Barbara, CA 93106-5070, and will be accepted until four months after final publication of the paper itself in the ASME JOURNAL OF APPLIED MECHANICS.
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Kulkarni, S. S., Mukherjee, S., and Grigoriu, M. D. (June 11, 2003). "Local Solutions in Potential Theory and Linear Elasticity Using Monte Carlo Methods ." ASME. J. Appl. Mech. May 2003; 70(3): 408–417. https://doi.org/10.1115/1.1558074
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