边界型数学规划非线性多极边界元法
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摘要
近年,四辊轧机支承辊-工作辊及板带耦合轧制模拟,作为轧制理论的前沿课题亟待攻关解决。但是,其任务因微机上难以完成大规模运算而被搁置。本文以研究定量解析轧制工程等的快速数值解析方法为目的,将多极展开法(简称FMM)融合于边界元法(简称BEM),试图建立起适于大规模数值运算的快速多极边界元法(简称FM-BEM)的完整基础理论体系,并将FM-BEM应用到轧制工程等领域。在FM-BEM的具体实施过程中,寻求各种新形式的快速数值计算方法,使FM-BEM不断发展和得到补充,综合改造传统BEM的计算结构。
本文共分六章。第1章绪论部分,分析了科学工程计算与计算方法的研究现状和发展趋势,概述了BEM和FMM的研究背景、发展历史、研究进展及现状,总结了其近年来取得的研究成果和发展方向,论述了FM-BEM的意义、研究进展和应用前景。第2章,构造了边界FM-BEM球面谐函数及数值计算公式,证明了边界FM-BEM基本定理。剖析FM-BEM的实现机制,给出具体的实施算法并开发FORTRAN源程序,建立起快速FM-BEM的理论框架,完善了三维结构体FMM-BEM的理论体系,从而为FM-BEM在轧制工程等领域中的进一步推广提供强有力的数学支撑。第3章,研究FM-BEM的数学机理,剖析了FMM与BEM结合的关键问题。分别推导了弹性问题、弹塑性问题和位势问题的FM-BEM基本解及相关核函数的计算公式,包括推导球坐标系下的偏导公式及与直角坐标的转化公式,并对基本解的等价性给出证明。第4章,研究Krylov子空间广义极小残值法(简称GMRES(m)算法)的机理并作理论推导,对工程用FM-BEM解的存在唯一性问题进行严格的理论分析和数学证明,为FM-BEM理论体系的形成及工程应用奠定数学基础。还提出改进算法收敛性的预条件GMRES(m)算法并对其正确性加以论证,使FM-BEM在轧制工程等领域中的应用前景更加广阔。第5章,对摩擦接触高度非线性问题,提出数学规划法求解的新思路,给出规划型三维弹性摩擦接触FM-BEM。建立了适于大规模快速计算的点-面摩擦接触最优化数学模型,给出优化GMRES(m)算法的求解策略,并开发研制FORTRAN源程序。通过数值实验证明,所给方法可显著提高计算效率,综合改造了传统BEM的计算结构。第6章,针对弹塑性迭代求解的繁杂费时问题,采用截断技术,提出一种基于FMM的规划-迭代型不完全广义极小残值法(简称IGMRES(m)算法)并建立了新算法的收敛性理论。数值实验和截断比较分析
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 for
the 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.
本文共分六章。第1章绪论部分,分析了科学工程计算与计算方法的研究现状和发展趋势,概述了BEM和FMM的研究背景、发展历史、研究进展及现状,总结了其近年来取得的研究成果和发展方向,论述了FM-BEM的意义、研究进展和应用前景。第2章,构造了边界FM-BEM球面谐函数及数值计算公式,证明了边界FM-BEM基本定理。剖析FM-BEM的实现机制,给出具体的实施算法并开发FORTRAN源程序,建立起快速FM-BEM的理论框架,完善了三维结构体FMM-BEM的理论体系,从而为FM-BEM在轧制工程等领域中的进一步推广提供强有力的数学支撑。第3章,研究FM-BEM的数学机理,剖析了FMM与BEM结合的关键问题。分别推导了弹性问题、弹塑性问题和位势问题的FM-BEM基本解及相关核函数的计算公式,包括推导球坐标系下的偏导公式及与直角坐标的转化公式,并对基本解的等价性给出证明。第4章,研究Krylov子空间广义极小残值法(简称GMRES(m)算法)的机理并作理论推导,对工程用FM-BEM解的存在唯一性问题进行严格的理论分析和数学证明,为FM-BEM理论体系的形成及工程应用奠定数学基础。还提出改进算法收敛性的预条件GMRES(m)算法并对其正确性加以论证,使FM-BEM在轧制工程等领域中的应用前景更加广阔。第5章,对摩擦接触高度非线性问题,提出数学规划法求解的新思路,给出规划型三维弹性摩擦接触FM-BEM。建立了适于大规模快速计算的点-面摩擦接触最优化数学模型,给出优化GMRES(m)算法的求解策略,并开发研制FORTRAN源程序。通过数值实验证明,所给方法可显著提高计算效率,综合改造了传统BEM的计算结构。第6章,针对弹塑性迭代求解的繁杂费时问题,采用截断技术,提出一种基于FMM的规划-迭代型不完全广义极小残值法(简称IGMRES(m)算法)并建立了新算法的收敛性理论。数值实验和截断比较分析
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 for
the 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.
引文
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