Application Of Fast Multipole Boundary Element Method For Two Dimensional Elasticity

Fast multopole boundary element method is a new algorithm developedrecently for numerical solution of boundary element method (BEM). Thisalgorithm solves equation system formed by BEM with higher speed and lowerstorage than that of traditional solvers. Fast multipole BEM is applicable forcertain large scale problems. It has caused much interest in the area of BEM.O(NlogN) and O(N) fast multipole BEM are studied and applied for twodimensional elastostatics in this thesis. Numerical tests are carried out to verifythe efficiency of fast multipole BEM.Two dimensional complex Taylor expansion and adaptive tree structure areused to obtain a new shift of multipole expansion for two dimensionalelastostatics. An O(NlogN) fast multipole BEM is formed. a new shift ofmultipole to local expansion and a new shift of local expansion are obtainedfurther to form an O(N) fast multipole BEM. These two algorithms areapplicable for numerical simulation of single-and multi-domain problems.Fast multipole BEM is compared with traditional solvers such as Gausselimination. The precision, efficiency of computation and storage of fastmultipole BEM are tested. The effective properties of two dimensionalcomposite materials with large number of randomly distributed inclusions aresimulated by fast multipole BEM. Numerical results show that fast multipoleBEM is capable of solving large scale problems without losing precision.When applied for some special structures such as elastic body with randomlydistributed inclusions or cracks, fast multipole has unique advantages.Sparse approximate inverse preconditioner is modified according to newcharacteristics of fast multipole BEM and can be greatly simplified whendealing with some special structures such as elastic body with randomlydistributed inclusions or cracks. Since fast multipole BEM is based on iterativesolvers and the coefficient matrix formed by BEM is always ill-conditioned,preconditioning is inevitable. Numerical results show that modified sparseapproximate inverse technique is stable and achieves high convergence ratewhen dealing with even large scale problems.