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.
