The Analysis Of Two-dimensional Elasticity Fast Multipole Boundary Element Method Algorithm And Truncation Error

The boundary element method (BEM) can be proved to be a kind of efficient andprecise method in engineering numerical analysis. After recent decades of research anddevelopment, not only to a certain extent solved the difficulties caused by integralsingularity, and the convergence and error analysis theory of the boundary elementmethod has made the further analysis. Boundary element method, the mathematicalfoundation is a boundary integral equation(BIE), is developed under the condition of fastdevelopment of computer technology. Coefficient matrix of algebraic equations which aregenerated by boundary element method is symmetry full matrix, so the conventionalboundary element method is not suitable for large-scale problem in practical engineering.But the fast multipole method (FMM) combined with boundary element method for thefast multipole boundary element method (FM-BEM), can make the boundary elementmethod have exponentially increase in scale of the problem solving and computationalefficiency. Now a regular computer can quickly finish large-scale problem of hundreds ofthousands or even millions of degrees of freedom.The main content of the thesis are as follows: first of all, summarizes the wholearticle writing background and shows that BEM development history and research status.Then, it inttroduces the basic ideas of BEM to solve the actual problem, the advantagesand disadvantages of algorithm exists. In order to overcome the disadvantages of BEMand meet the need of processing large-scale problems, we combine BEM with FMM, alsointroduces the development of FMM and the research achievements of FM-BEM in recentyears. Some basic knowledge of conventional BEM is presented, which includes theestablishment of BIEs for elastic mechanics problem and three ways of discretization ofBIEs. And the basic theory of generalized minimal residual algorithm to solve thealgebraic equations that generated by discretization of BIEs was given. Then, it was on thebasic thoughts of the BEM and expansions and translations for basic solution ofdisplacement and traction. In order to improve solving efficiency, it introduced the processof exponential expansion in moment-to-local translation, which changes translation matrix of the coefficient of two kinds to diagonal form, reducing the amount of calculation.FM-BEM computation process is given. And the algorithm is improved by theimprovement of two concepts of neighbors and list of interaction. The last through theanalysis of operations complexity of FM-BEM, the relationship of operations and degreesof freedom is linear. The analysis of operations complexity of preprocessing matrix isgiven. At the same time, it gives a multipole expansion truncation error analysis. Then, itstates how to select expression of the truncated number. Thus, it reaches a conclusion thatthe truncation error is closely related to the number of the expansion terms.