Boundary Element Method For Heat Conduction Problems Of Thin Structures

As the application of anisotropic materials becomes more and more widely, researchers not only limit their research to isotropic medium problems, but also expand their work to the anisotropic media. Compared to isotropic problems, the parameters of material properties in the anisotropic heat conduction problems has increased a lot, which makes the establishment of the fundamental solution used in the boundary element method more difficult. This thesis presents a new method for establishing fundamental solutions of general variable coefficient heat conduction problems in anisotropic media, and a pure boundary integral equation is derived for solving general two-and three-dimensional steady heat conduction problems of anisotropic media. The established fundamental solutions are suitable for the situation that the thermal conductivity is a function of spatial coordinates, and therefore the developed integral equation can be used to solve the heterogeneous material heat transfer problems. The domain integrals induced by heat sources are transformed into boundary integrals using the radial integral method, so a pure boundary element method which does not need any interior cells is formed.The rapid development of modern science and technology has greatly promoted the engineering applications of thin structures. Due to the special geometric shapes of thin structures, the numerical analysis of temperature distribution using boundary element method is one of difficult problems, mainly due to the nearly-singular integral problem. In recent years, numerous research works have already been published on the treatment of nearly singular integrals, such as the virtual boundary element method、the interval segmentation method、the precise integration method、various transformation methods, and the special Gaussian integration formula. This thesis presents a new method for the evaluation of nearly singular boundary integrals based on3D non-homogeneous heat conduction problems. In the proposed method, the Newton-Raphson iteration algorithm is adopted to determine the point on the boundary element which is closest to the source point, and then the distance from the source point to any point on the integral element is calculated by expanding the coordinates of the point as Taylor’s series at the closet point. Finally, the formulation for the evaluation of nearly singular boundary integrals is derived by substituting the distance function into the nearly singular boundary integral and using the exponential transform method. The resulting expression for the nearly singular boundary integrals can be calculated directly by using Gaussian integral formula.This thesis also presents a dual boundary integral equation method for isotropic heat conduction problems. Based on the fundamental solution of three-dimensional isotropic heat conduction problems, through deriving derivatives of the temperature boundary integral equation with respect to the spatial coordinates, the boundary integral equation for the heat flux is obtained. The flux boundary integral equation is collocated at nodes on one face of the nearest two faces in the thin structure, while the temperature boundary integral equation is collocated at other remaining nodes. This collocation scheme can generate a closed independent system of algebra equations.Based on the presented theory, a Fortran code has been written. Related numerical examples are given to verify the correctness, effectiveness and stability of the presented method.