Adaptive Unstructured Grid Generation Scheme for Solution of the Heat Equation

Abstract

An adaptive unstructured grid generation scheme is introduced to use finite volume (FV) and finite element (FE) formulation to solve the heat equation with singular boundary conditions. Regular grids could not acheive accurate solution to this problem. The grid generation scheme uses an optimal time complexity frontal method for the automatic generation and delaunay triangulation of the grid points. The algorithm is incremental, so it is the most appropriate for an adaptive solver. Using adaptive grids, the solution is refined to get enough accuracy in all grid points. Two schemes are applied for the solution of the equations to show the flexibility of the adaptive grid scheme. First a cell-vertex finite volume formulation is used. Then, for the FE scheme, using linear shape functions, a set of linear equations are solved explicitly, with overrelaxation. A sequence of adaptation is applied and appropriate number of grid points are introduced in finite predetermined formats to the existent elements, till convergence in the solution is observed. A postprocess is used to smooth the distribution of the set of nodes. This procedure is applied to a few study cases to show that the method is convergent, and produces accurate solution even in the case of singular boundary conditions.