A Meshless Boundary Node Method For Signorini Problems

Signorini problem is an important boundary value problem in physics and mathematics. In this problem, the unknown potential and its normal derivative on the boundary are associated with certain inequality constraints, and where the positions change is unknown. Since the Signorini complementary boundary conditions lie on the boundary of the region, it is suitable for boundary type method. Combining the moving least-square(MLS) method and boundary integral equations(BIEs), the boundary node method(BNM) is developed for the numerical solution of Signorini problems.Firstly, according to the projection operator, the Signorini boundary conditions have been transferred into a fixed point equation. Then, projection iterative scheme of Signorini problem has been established. On the one hand, we get the implicit projection iterative scheme, using the fixed point equation directly. On the other hand, we propose the explicit iterative projection scheme to solve the Signorini problem, by introducing the residual function based on the fixed point equation. Thus, we have transferred a special class of elliptic boundary value problem into a normal elliptic boundary value problem. We use the boundary node method to solve numerical solution in each iteration. We proved the convergence of the indirectly method. At last, we give the numerical examples, and they showed high accuracy and high convergence rates of the method. The results have shown the feasibility and effectiveness of the method for Signorini problems, and the precision and convergence rate of the results is better than those of the boundary element method.