Study On The Steady Flow Problems By Boundary Element Method

The most engineering problems can come down to the boundary value problemof partial differential equations. However, the analytical solution can not beobtained generally due to the complex boundary conditions, uneven properties ofobject and irregular geometry etc. Therefore, people begin to pay attention to theapproximation algorithm-numerical analysis method to solve the problems. As oneof numerical analysis methods, the boundary element method has developed rapidlyin the past50years with its own unique advantages, although it is later than finiteelement method. Nowadays, more and more fields have adopted the boundaryelement method, such as potential problem, elasticity problem, elastoplastic problem,dynamic problem etc. This paper uses direct boundary element method to study fourseepage problems separately. According to the seepage medium, i.e. the seepagecoefficient, the four seepage problems are as follows: homogeneous isotropicseepage problem (k x ky?and both are constants), orthotropic seepage problem(k x ky?but both are constants), unsaturated soil seepage problem (k k e hs) andlayered soil seepage problem. In addition that the process of solving the fourseepage problems is described in this article, matlab program is also written for eachtype of problem and verified with examples. The detail of the study is as follows:Firstly, deduction of the boundary integral equation for the each kind ofproblem. Based on the governing equation for each problem, this paper converts thepartial differential equation to boundary integral equation by using the weightedresidual approach and Green's formula. Among the four problems, unsaturated soilseepage problem is quite special. Its governing equation is nonlinear, so there arealso integral items in domain in the boundary integral equation.Secondly, study on the boundary divergence method. The boundary is divergedwith constant element and linear element for both of the first two seepage problems.And the accuracy of final computing results for the two divergence methods isjudged by comparing.Thirdly, study on the discontinuous normal flow problem at the corner joints.When the boundary is diverged with linear element and the two end points of theunit are served as joints, the corner joint problem will appear. This paper adopts themultiple joints method and single-unknown multiple joints mthod to solve the problem, and concludes the advantages and disadvantages of the two methods fordifferent seepage problems as well as the their own applicable condition.Finally, handling of singular integral. As for the one step singularity in theboundary integral equation, the paper uses analysis algorithm to solve the problem.As for the singularity in the integral equation in domain, the paper transforms theintegral coordinates first, i.e. the rectangular coordinates integral is transformed tothe polar coordinates integral, and then uses the semi-analytic and semi-numericalmethod to handle the problem.