A thesis submitted in partial fulfillment of the requirements for the degree of
Master of Science in Engineering Mathematics
Faculty of Engineering -Shobra
This Thesis presents a new fast multi-pole boundary element formulation for the solution of Reissner’s plate bending problems. The solution of Reissner’s plate bending problems using the conventional direct boundary element method leads to a non-symmetric fully populated system of matrices. The complexity of the solution then becomes of the order O(N3) mathematical operations, where N is the total number of problem unknowns. Hence, the use of fast multi-pole technique becomes practically essential in case of solving large-scale problems by the direct boundary element method.
The suggested formulation is based on representing the fundamental solutions as function of potentials. These potentials and their relevant fundamental solutions are expanded by means of Taylor series expansions. In the present formulation, equivalent collocations are based on the first shift expansion of kernels. This is achieved by representation of far field integrations by series expansions and carrying out summations of far clusters, whereas the near field integrations are kept to be computed directly.
In the presented implementation, the fast multi-pole boundary element method is coupled with the iterative solver: Generalized Minimal Residual System (GMRES). The computational complexity is rapidly reduced to be O(N log N). Numerical examples are presented to demonstrate the efficiency, time saving, and accuracy of the formulation against the conventional direct BEM. The accuracy of the results is traced by cutting Taylor series to few terms. It was proven via numerical examples that three terms are enough to produce sufficient accuracy with substantial reduction of solution time.