Local mesh refinement scheme for finite difference solution of mixed boundary-value elastic problems

Abstract

Some numerical simulations of multi-scale physical phenomena consume a significant amount of computational resources, since their domains are discretized on high resolution meshes. An enormous wastage of these resources occurs in refinement of sections of the domain where computation of the solution does not require high resolutions. This problem is effectively addressed by mesh refinement (MR) technique, a technique of local refinement of mesh only in sections where needed, thus allowing concentration of effort where it is required. Sections of the domain needing high resolution are generally determined by means of a criterion which may vary depending on the nature of the problem. Fairly straightforward criteria could include comparing the solution to a threshold or the gradient of a solution, that is, its local rate of change to a threshold or the presence of stress concentrator or stress riser, sharp change in cross section, void in material, cracks, holes etc. While the comparing of solution to a threshold is not particularly rigorous and hardly ever represents a physical phenomenon of interest, it is simple to implement. However, the gradient criterion is not as simple to implement as a direct comparison of values, but it is still quick and a good indicator of the effectiveness of the MR technique. The MR technique can be classified into two categories. One is h-refinement, where either the existing mesh is split into several smaller cells or additional nodes are inserted locally and the other one is r-refinement in which move the mesh points inside the domain in order to better capture the dynamic changes of solution. The objective of this thesis is to develop a MR algorithm for the solution of fourth order bi-harmonic equation using FDM. In the MR algorithm developed, a mesh of increasingly fine resolution permits high resolution computation in sub-domains of interest and low resolution in others. In this thesis work, the gradient of the solution has been considered as region selecting criteria and existing mesh is split into smaller meshes to achieve refine mesh. The developed MR algorithm has been applied for the solution of an embedded crack problem. The validity, effectiveness, soundness and superiority of this MR algorithm has been verified by the comparing of obtained solutions with uniform mesh results, FEM results and also with the well known published results of the same embedded crack problem having same material, geometry and loading conditions.