Steady Two-Dimensional Problems

Abstract

Two dimensional problems present an additional spatial dimension to one-dimensional problems, and their Green element calculations essentially follow the procedure which we had adopted for 1-D. problems in the earlier chapters. For every differential equation, there has to be found an appropriate complimentary or auxiliary differential equation to which the fundamental solution is obtained. Green’s second identity in two dimensions provides the tool to transform the governing differential equation into an integral one which is discretized by appropriate 2-D. elements such as triangles and rectangles. The resulting discretized integral equation constitutes the element equations which are assembled to form the global matrix equation that is solved to obtain the nodal unknowns. In situations where the differential equation is nonlinear, the global matrix equation has to be linearized and solved by either the Picard or Newton-Raphson algorithm. In the remaining chapters of this text, we shall apply the GEM to steady, transient, linear and nonlinear problems in 2-D. domains which are either homogeneous or heterogeneous.