Research On The Fast Multipole Boundary Element Method And Its Application For Simulation Of Composite Materials

Boundary Element Method (BEM) is a numerical scheme along with thedevelopment of Finite Element Method (FEM). The advantage of boundarydiscretization makes BEM suitable for the simulation of certain structuresthat have complicated surfaces or interfaces, such as particle-reinforced orfiber-reinforced composite materials.However, the coefficient matrix arisingfrom BEM is always dense and asymmetric, and traditional solutiontechniques are not efficient for such matrix, making BEM not available forlarge scale problems.In order to overcome the above-mentioned difficulty, Fast MultipoleMethod (FMM) is applied to accerate the numerical solutions of BEM. In thefield of solid mechanics, the study of fast multipole BEM is still focused onfinding appropriate expanding and shifting formulations for foundamentalsolutions because of their complex expressions. In addition, the research ofthe new version algorithm is limited.In this paper, an original scheme of fast multipole BEM fortwo-dimensional elasticity is presented, reducing the complexities of bothmemory and operation to O(N). Thus the computational efficiency is greatlyimproved. The numerical results show that, the computational cost andmemory requirement of fast multipole BEM is more than an order ofmagnitude lower than that of traditional BEM solvers in two dimensionswithout losing accuracy, making fast multipole BEM applicable for thesimulation of large scale problems on an ordinary desktop computer.Based on the original scheme, a new version fast multipole BEM ispresented for two-dimensional elasticity. The differences of the original andnew version schemes are studied theoretically and numerically. Comparedwith the original scheme, the new version scheme further increases thecomputational speed with only a little more memory appended.Fast multipole BEM is extended from two dimensions to threedimensions and a uniform expansion formulation of fundamental solutionsfor three-dimensional elasticity is established in order to deal with mixedboundary conditions which are commonly encountered in solid mechanics.Unlike two-dimensional problems, the original scheme of fast multipoleBEM for three-dimensional problems achieves little improvement comparedwith traditional solvers for moderate scale problems, while the new versionscheme still achieves much improvement.Composite materials are used increasingly in various areas of industrialapplications. Fast multipole BEM schemes presented in this paper are used tosimulate both two-dimensional composite materials containing up to 1000randomly distributed inclusions and three-dimensional composite materialscontaining up to 100 randomly distributed particles or fibers. The effect ofmicro-structural parameters on the macro-properties of composite materialsis studied and compared with several classical theoretically approximatemethods. Some important referential conclusions have been obtained.In general, the advantages of fast multipole BEM for large scalecomputations make BEM applicable for many potential applications. In theresearch field of composite materials, fast multipole BEM has an extensiveapplied prospect.