Meshless Local Petrov-Galerkin Method for Heat Conduction Problem in an Anisotropic Medium

Meshless methods based on the local Petrov-Galerkin approach are proposed for solution of steady and transient heat conduction problem in a continuously nonhomogeneous anisotropic medium. Fundamental solution of the governing partial differential equations and the Heaviside step function are used as the test functions in the local weak form. It is leading to derive local boundary integral equations which are given in the Laplace transform domain. The analyzed domain is covered by small subdomains with a simple geometry. To eliminate the number of unknowns on artificial boundaries of subdomains the modified fundamental solution and/or the parametrix with a convenient cut-off function are applied. In the formulation with Heaviside step function the final form of local integral equations has a pure contour character even for continuously nonhomogeneous material properties. The moving least square (MLS) method is used for approximation of physical quantities in LBIEs. keyword: meshless method, local weak form, Heaviside step function, fundamental solution, moving least squares interpolation, Laplace transform