Boundary-Pattern Mathematical Programming Nonlinear Multipole Boundary Element Method

Of late years, the coupling rolling simulation of the back-up roll, working roll andslat of the four-high mill has been an urgent problem to challenge and solve, which wasthe frontal subject of rolling theory. However, this task has been leaving aside because thelarge-scale computing was difficult to finish. This paper focused on fast numericalanalytic solutions that were used in quantitative analyzing the rolling engineering andother problems. The Fast Multipole Method (FMM) was introduced and integrated withthe Boundary Element Method (BEM). Then a complete fundamental theory system wastried to establish for the FM-BEM, which was suitable for the large-scale numericalcomputing. In addition, the FM-BEM was attempted to apply into the rolling field andother fields. During the specific implementation of the FM-BEM, some new fastnumerical computation methods were studied, which would further develop and optimizethe FM-BEM. The presented FM-BEM could synthetically reform the computationstructure of the traditional BEM.The paper included six chapters. The first chapter was introduction. The presentresearch state and the development trend of the scientific & engineering computing andcomputational methods were analyzed. For the BEM and the FMM, the researchbackground, development history, research progress and present state of were overviewed.Its achievements of later years and development direction were summarized. Then themeaning, research progress and application prospect of the FM-BEM were discussed. Inthe second chapter, a spherical harmonic function and some related FM-BEM numericalformulas were constructed for the boundary. Fundamental theorems were presented andproved. The implementation mechanism of the FM-BEM was analyzed, and the specificimplementation algorithm was given. Then the theoretical frame of FM-BEM waspreliminarily established and a complete FMM-BEM theoretical system was optimizedfor 3-D structural objects, which provided strong mathematical support for furtherpromotion of the FM-BEM in rolling engineering field and other fields. In the thirdchapter, the mathematical mechanism of the FM-BEM was studied, and the key problemwas analyzed for the combination of the FMM and the BEM. The computational formulasof the FM-BEM fundamental solutions and its related kernel functions were derived forthe elasticity, the elasto-plasticity and the potential problems, respectively. Also the partialderivative formulas were obtained under the spherical coordinate system and wereconverted into those under rectangular coordinate system. The FM-BEM fundamentalsolutions were proved to be equivalent with the traditional ones. In the fourth chapter, themechanism of GMRES(m) algorithm in Krylov subspace was studied and derived intheory. The existence and uniqueness of the solution was strictly analyzed in theory andproved in mathematics for the FM-BEM used in engineering, which established themathematical foundation for the formation of the FM-BEM theoretical architecture and itsengineering application. In addition, a preconditioning GMRES(m) algorithm wasproposed to improve the convergence and its correctness was argued, which made theapplication prospect of the FM-BEM more extensive in rolling engineering field andother fields. In the fifth chapter, to solve the highly nonlinear problems with frictionalcontact, a new mathematical programming method was proposed and a program-patternFM-BEM was developed for 3-D elastic contact with friction. An optimizationmathematical model suitable for large-scale fast computing was built for thenode-to-surface contact. Then an optimization GMRES(m) algorithm was presented asthe solution strategy. Also the FORTRAN source program was developed. The numericalresults showed that the presented method could significantly improve the computationalefficiency and synthetically reform the structure of traditional BEM. In the sixth chapter,a new program-iteration pattern IGMRES(m) algorithm was presented using truncationtechnique to solve the complicated and time-consuming problem in the iterative solutionof elasto-plasticity. Its convergence theory was established and analyzed. The numericalresults and truncation comparison analysis showed that the new algorithm wasparticularly efficient for the complicated and time-consuming problem in theelasto-plastic contact with friction. If the truncation ratio was selected properly, thecomputation and memory requirement could be greatly reduced.