Numerical Simulations Of Signorini Problems And The Seepage Problems

This dissertation first reviews the recent developments of the Signorini problemsand the Seepage problems in the Chapter one. Then, based on the large amount ofreferences, projection iterative algorithms based on the boundary element methods arepresented for the Signorini problems of Laplacian equation and unilateral elastic contactproblems in Chapter two and three, respectively. In Chapter four, projection iterativealgorithms are proposed for solving the simplified Signorini problems. Finally, wedevelop linear complementarity-projection iterative algorithms based on the finitedifference methods for a seepage problem with free boundary in Chapter five.Signorini problems form a special class of elliptic boundary value problems sincethe potential and its normal derivative on the boundary are associated with certaininequality constraints, and where the positions change are unknown. As the Signoriniconditions are given on the boundary of the domain, the boundary element methods aresuitable for solving such problems. Projection iterative algorithms based on fixed pointequations are proposed for solving Signorini problems for the Laplacian equation andthe frictionless unilateral elastic contact problem of an elastic body with a rigidfoundation, which are suitable for any domain. The satisfaction of the Signoriniboundary conditions is verified in a projection iterative manner, and at each iterativestep, an elliptic mixed boundary value problem is solved by the boundary elementmethods. The convergence of the algorithms is proved by the property of projection.The advantages of these algorithms are that it is easy to be implemented and convergequickly. For Signorini problems of Poisson equation, domain integrals are involved.While a particular solution of the Poisson equation can be obtained the problem couldbe transformed into a problem of the Laplace equation. If the inhomogeneous term isharmonic in the whole domain, then we can transform the domain integrals into theboundary integrals by using the dual reciprocity methods.For simplified Signorini problems, the projection iterative algorithms based onfixed point equations are proposed, and convergence of the algorithms is proved by theproperty of projection. We then perform the numerical experiment for simplifiedSignorini problems in square domain with the finite defference methods.For solving a kind of seepage problem with free boundary, the projection iterativealgorithms based on linear complementarity problems are presented. As the differential inequalities describing the seepage problem are given in the domain, domain numericalmethods are suitable for solving such problem. Applying the finite difference methodsto the partial differential operator, we obtain a finite-dimensional linearcomplementarity problem which can be transformed into an equivalent fixed pointproblem. Then projection iterative algorithms are proposed for the linearcomplementarity problem. Using the property of projection and positive definiteness,we prove the convergence of the algorithms. Similarly, these algorithms are easy to beimplemented and converge quickly.Finally, some numerical results show the accuracy and effectiveness of thealgorithms and we also compare our results with the known numerical results in otherreferences.