The method of fundamental solutions for solving direct and inverse Signorini problems

Signorini problems model phenomena in which a known or unknown portion of the boundary is subjected to alternating Dirichlet and Neumann boundary conditions. In this paper, we apply the method of fundamental solutions (MFS) for the solution of two-dimensional both direct and inverse Signorini problems for the Laplace equation. In this meshless and integration-free method, the harmonic solution representing the steady-state temperature or the electric potential is approximated by a linear combination of non-singular fundamental solutions with sources located outside the closure of the solution domain. The unknown coefficients in this expansion, the points of separation of the Signorini boundary conditions and possibly the unknown Signorini boundary (in the inverse problem) are determined by imposing/collocating the boundary conditions which can be of Dirichlet, Neumann, Cauchy or Signorini type. This results in a constrained minimization problem which is solved using the MATLAB© toolbox routine fmincon. Several numerical examples involving both direct and inverse problems are presented and discussed in order to illustrate the accuracy and stability of the numerical method employed.