Application Of IIBEM For Nonlinear Transient Heat Conduction Problems

The boundary element method is very effective in solving many practical engineering problems.In this paper,in order to solve the transient nonlinear problem by using the boundary element method,a single interface integral equation is established for the multi-medium problem,and a new and improved method to transform multi-domain integrals to the boundary and interface is proposed.Firstly,using the boundary element method in the potential problem as a guide,this paper elaborated the process of establishing the boundary-domain integral equation in the single medium,emphasizing the calculation method of various singular integrals and the conversion method from the domain integral to the boundary integral in the single medium—Radial integration method(RIM).For the multi-medium problem,the interface integral equation is derived by the degeneration rule according to the jump effect of the material properties at the interface,and its physical meaning is explained in another derivation.In order to maintain the advantage of BEM that only the boundary is discretized,multi-domain integrals in the interface integral equation need to be transformed to the boundary.However,different from the single-medium problem,the traditional radial integration method cannot be directly used to transform multi-domain integrals.Considering the characteristic that the integrand in the multi-domain integral is not uniformly continuous in the entire calculation domain but continuous in each medium subdomain,the domain integral is transformed on each medium subdomain separately.When the source point is outside the domain,the traditional radial integration method cannot directly transform the domain integration due to the missing part of the definition domain on the integration path.This paper topologically defines the missing definition domains and derives new formulas that can transform domain integrals.Compared with the traditional radial integration method,the application of the new formulas is wider.What's more,the idea of the proposed approach can theoretically prove the correctness of applying RIM to the complex models with multi-connected domain,which has been only confirmed from some numerical examples.In this paper,several numerical examples have been given to verify the correctness,accuracy and applicability of the proposed algorithm.