Development of a boundary management technique in finite difference method of solution

Abstract

This thesis develops a new boundary management technique for finite-difference analysis of any arbitrarily bounded two-dimensional bodies. Management of irregular boundary shapes was the prime obstacle in the finite difference method of solutions. The technique developed here uses a uniform rectangular grid-system for arbitrary shaped two-dimensional bodies with desired grid size. Boundary values at the boundary points, not matching with the nodal points of the generated mesh network, is expressed as the linear interpolated value of the four neighboring nodal points of that boundary point. The local mesh coordinate parameters (a, s) locates a boundary point within a mesh of the grid network. Boundary conditions, expressed in terms of normal and tangential components, are applied at these boundary points. From a set of different discretized boundary equations, depending upon the direction cosines of the boundary points, an appropriate equation is applied at that point. Because of the application of governing equation at the interior nodal points, some nodal points outside the boundary are also includcd in the mcthod of solution. It is ensured that the total number of unknowns (number of nodal points involved in the solution) is equal to the number of equations (summation of the available boundary conditions and the number of interior nodal points) to get a unique solution. The present finite-difference approach is verified by applying it to three different fields of engineering, namely, an one-dimensional beam problem, a two-dimensional heat conduction problem, and a two-dimensional elasticity problem. The results are compared with analytical and other available solutions in the literature. The results are found appropriate and accurate, and thus establish the reliability and appropriateness of the present methodology in solving arbitrarily bounded boundary value problem of engineering.