The Research Of Virtual Boundary Element Method For A Boundary Value Problem About Elliptic Differential Equation

Engineering problems can be divided into two kinds of parts contain potentialproblems and structure problems. Actually, many problems can be translated intoboundary value problems of elliptic differential equations. This paper mainly discussed thepotential problems. Boundary element method is an effective numerical method forsolving potential problem, but it needs to calculate singular integrals in the numericalcalculation process, if use virtual boundary element method can avoid the weakness, whenvirtual boundary method is used, we set a closed virtual surface, so called virtual boundaryoutside the domain under consideration. So the unknown field source function distributedon the virtual boundary can be determined by the boundary conditions given on thephysical boundary via nonsingular integral equation, since the source points are on thephysical boundary, and the integral points on virtual boundary respectively, the integral isno longer singular.The paper mainly discusses boundary value problems of laplace equation,helmholtz equation and poisson equation, through translating the boundary integralequation into the virtual boundary integral equation based on single layer potential, andusing least square method and point collocation to calculate the virtual boundary integralequation, and by using the least squares method and match points calculation method withvirtual boundary integral equation, in specific combined other relevant knowledge solvingprocess, and gives corresponding numerical examples verify the feasibility and validity ofthe method.This paper includes five chapters. In the first chapter, the development of boundaryelement method and virtual boundary element method are mainly introduced, and showingthe source and significance of the subject.In the second chapter, some of the important basic concepts and conclusions havelaid a theoretical foundation for the studied contents in the paper are described.In the third chapter, the laplace equation that is the most typical elliptic differentialequation is taken as the study object, given the virtual boundary integral equation contains field source function combining the traditional potential theory, then using least squaremethod to solve, and the unknowable field source function distributed on the virtualboundary is discretized into constant unit. Finally the numerical example shows feasibilityof the method.In the fourth chapter, the helmholtz equation that is applied widely is taken as thestudy object. The numerical solution is presented using virtual boundary element method.Using series development to approximate the helmholtz equation basic solution in solvingprocess, the boundary discrete processing used constant unit interpolation form, simpleand effective, and through numerical example validated the conclusion.In the last chapter, taken poisson equation of paper to discuss, through virtualboundary element method based on single layer potential and radial basis functioncombination of methods to solve it, and through the example proved the feasibility of themethod.