三维弹性接触Taylor级数多极边界元法理论与应用研究
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摘要
从上世纪50年代以来,计算力学就在固体力学的各分支学科和边缘学科中得到了很大发展,它已经成为固体力学除理论研究和实验研究以外的第三种研究手段。如今数值解析方法已经成为现代科学与工程领域中的主要分析工具。数值仿真已成为工程和科学的重要部分,并且强烈地影响到几乎工程与科学的每一个分析领域。
边界元法是一种重要的现代数值解析工具,是继有限元法之后的一种别具特色的数值方法。它是将描述问题的偏微分方程转化为边界积分方程,并吸收了有限元法的离散化技术而发展起来的。由于具有降维和半解析等独特的优势,使其成为有限元法的重要补充,在某些工程领域中甚至成为不可替代的数值方法。由于边界元法最终形成的线性方程组的系数矩阵是非对称满秩矩阵,对于大规模问题的解析时,传统的求解方式(边界积分方程—影响系数数值积分—线性方程组及消去法求解)将导致昂贵的计算代价,这使边界元法仅用于中小规模问题的分析。快速多极算法与边界元法相结合,加速了影响系数的计算,更新了传统边界元法的求解模式,能够有效地提高边界元法的求解效率,使之适应大规模问题的计算和分析。本文主要是对三维Taylor级数多极边界元法的理论进行深入的研究。
多极边界元法在提高求解效率(计算时间和内存使用)的同时,其计算精度与传统边界元法相比精度有所下降。本文首先对径矢函数r的Taylor级数展开性质进行了研究。然后对三维弹性问题Taylor级数多极边界元法的误差进行分析,推导了误差估计公式,给出了远近场的划分准则,分析了影响Taylor级数多极边界元法计算精度的因素。
边界元法中基本解是用张量形式给出的,这种形式的公式具有表示简单,易于编程的特点,但是张量形式的公式中有许多相同的项被重复计算和存储,在一定程度上降低了计算效率。本文根据三维Taylor级数展开项和三维分量对称性的特点,给出了三维弹性问题Taylor级数多极边界元法的矢量化公式。矢量化公式减少了计算量和内存使用量,进一步提高了Taylor级数多极边界元法的求解效率。
对于大规模问题形成的线性方程组,由于条件数的恶化,迭代方法和预条件技术的选择是有效求解的关键。本文对适合求解边界元法形成的非对称稠密线性方程组的常用迭代方法(Krylov subspace methods)进行了研究。对不同Krylov子空间迭代法的求解性能和效率进行了比较分析,给出了不同求解方法的Fortran程序。对常用预条件技术进行了比较分析,提出了一种适合多极边界元法的近场预条件技术,进一步加速收敛,减少迭代次数。
对三维弹性摩擦接触问题的Taylor级数多极边界元法进行了研究,给出了合适的基本解Taylor级数展开策略,提出了基于点—面接触模型多物体摩擦接触问题的数学规划求解方法。给出了大规模数值算例,成功分析了HC轧机辊间接触压力分布和辊系的弹性变形。数值实验证明了该方法的有效性和正确性。
本文对三维问题Taylor级数多极边界元法的基础理论和应用进行了研究,其中涉及弹性问题、弹性接触问题、线性方程组迭代解法和预条件技术等,为进一步提高Taylor级数多极边界元法的计算效率奠定了基础。课题研究属于计算力学的前沿,具有重大的学术意义,工程应用前景广阔。
Since the 1950s, the computational mechanics has developed greatly in various subdiscipline and interdisciplinary subject of solid mechanics, which has become the third means in addition to the theoretical and experimental research. Nowadays, the numerical methods have turned into main analytic tools in the field of modern science and engineering. Numerical simulation is an important component of science and engineering, and it has strong influence on the all analytic domains of science and engineering.
The Boundary Element Method (BEM) is an important modern numerical analytic tool, and it is a unique numerical method after Finite Element Method (FEM). The basis idea of BEM is to transform the original different equations into the boundary integral equations, and in conjunction with the discrete technologies of FEM. It has become the important complement of FEM owing to the dimensionality reduction and semi analytical features. In some domains, the BEM method is even the irreplaceable numerical method. However, the linear system arising from the BEM is general dense and non-symmetrical matrix. For the large scale problems, the traditional solving scheme which is composed of the boundary integral equation, computing of numerical integration, form the linear system, and solving by elimination method results in expensive computational costs. It limits the BEM only be applied to the small and medium-sized problems. The Multipole-BEMs combine the Fast Multipole Method (FMM) with the BEM, accelerate the computation of influence coefficients, update the solving mode of traditional BEM, improve the efficiency evidently, and be suited to the computing of large scale problems. This dissertation is mainly research the theories and applications of Taylor Series Multipole BEM (TSMBEM).
The Multipole-BEM is capable of improving the solving efficiency of BEM, and meanwhile the precision decreases in comparison with the traditional BEM. This dissertation presents an error analysis with regard to TSMBEM for 3-D elasticity. The error estimating formulas of TSMBEM are derived. The far-field partition criterion is present according to the requirement of precision. The influence factors of the solving accuracy of TSMBEM are specified.
The Taylor series expressions of fundamental solutions are mostly expressed in tensor form. The vectorization expressions of Taylor Series Multipole boundary element formula for 3-D elasticity problems, which take account of the symmetric properties of fundamental solutions and the characteristic of component for 3-D problems. The vectorization expressions reduce the computational costs and storage required, thus improve the computational efficiency.
For the large scale problems, the linear systems arising from BEM are general ill-conditioned, therefore the choice of iterative methods and preconditioning techniques is of great importance to the efficient solution. This dissertation investigates the Krylov subspace methods which are suited to the dense non-symmetric linear system arising from BEM. The comparative performances of different Krylov subspace solvers are studied, and several general preconditionrers are also considered and assessed. A near-field preconditioning techniques are presented, and it can accelerate the convergence, reduce the iterations, and be propitious to the special implicit linear systems arising from Multipole-BEM.
The dissertation researches the TSMBEM for 3-D elastic contact with problem, presents a mathematical programming solution method base on Point-Surface contact model. The source codes are written in FORTRAN. A large scale numerical experiment is presented. The contact pressures of between rolls of HC mill are analyzed. The validity and efficiency of TSMBEM are proved by numerical experiments.
This dissertation is mainly research the fundamental theories of TSMBEM, including the elasticity, contact problems, iterative methods and preconditioning techniques, which lay the foundation for the improvement of computational efficiency of BEM. The research subject is the frontline of computational mechanics, and has great academic value and wide engineering prospects.
边界元法是一种重要的现代数值解析工具,是继有限元法之后的一种别具特色的数值方法。它是将描述问题的偏微分方程转化为边界积分方程,并吸收了有限元法的离散化技术而发展起来的。由于具有降维和半解析等独特的优势,使其成为有限元法的重要补充,在某些工程领域中甚至成为不可替代的数值方法。由于边界元法最终形成的线性方程组的系数矩阵是非对称满秩矩阵,对于大规模问题的解析时,传统的求解方式(边界积分方程—影响系数数值积分—线性方程组及消去法求解)将导致昂贵的计算代价,这使边界元法仅用于中小规模问题的分析。快速多极算法与边界元法相结合,加速了影响系数的计算,更新了传统边界元法的求解模式,能够有效地提高边界元法的求解效率,使之适应大规模问题的计算和分析。本文主要是对三维Taylor级数多极边界元法的理论进行深入的研究。
多极边界元法在提高求解效率(计算时间和内存使用)的同时,其计算精度与传统边界元法相比精度有所下降。本文首先对径矢函数r的Taylor级数展开性质进行了研究。然后对三维弹性问题Taylor级数多极边界元法的误差进行分析,推导了误差估计公式,给出了远近场的划分准则,分析了影响Taylor级数多极边界元法计算精度的因素。
边界元法中基本解是用张量形式给出的,这种形式的公式具有表示简单,易于编程的特点,但是张量形式的公式中有许多相同的项被重复计算和存储,在一定程度上降低了计算效率。本文根据三维Taylor级数展开项和三维分量对称性的特点,给出了三维弹性问题Taylor级数多极边界元法的矢量化公式。矢量化公式减少了计算量和内存使用量,进一步提高了Taylor级数多极边界元法的求解效率。
对于大规模问题形成的线性方程组,由于条件数的恶化,迭代方法和预条件技术的选择是有效求解的关键。本文对适合求解边界元法形成的非对称稠密线性方程组的常用迭代方法(Krylov subspace methods)进行了研究。对不同Krylov子空间迭代法的求解性能和效率进行了比较分析,给出了不同求解方法的Fortran程序。对常用预条件技术进行了比较分析,提出了一种适合多极边界元法的近场预条件技术,进一步加速收敛,减少迭代次数。
对三维弹性摩擦接触问题的Taylor级数多极边界元法进行了研究,给出了合适的基本解Taylor级数展开策略,提出了基于点—面接触模型多物体摩擦接触问题的数学规划求解方法。给出了大规模数值算例,成功分析了HC轧机辊间接触压力分布和辊系的弹性变形。数值实验证明了该方法的有效性和正确性。
本文对三维问题Taylor级数多极边界元法的基础理论和应用进行了研究,其中涉及弹性问题、弹性接触问题、线性方程组迭代解法和预条件技术等,为进一步提高Taylor级数多极边界元法的计算效率奠定了基础。课题研究属于计算力学的前沿,具有重大的学术意义,工程应用前景广阔。
Since the 1950s, the computational mechanics has developed greatly in various subdiscipline and interdisciplinary subject of solid mechanics, which has become the third means in addition to the theoretical and experimental research. Nowadays, the numerical methods have turned into main analytic tools in the field of modern science and engineering. Numerical simulation is an important component of science and engineering, and it has strong influence on the all analytic domains of science and engineering.
The Boundary Element Method (BEM) is an important modern numerical analytic tool, and it is a unique numerical method after Finite Element Method (FEM). The basis idea of BEM is to transform the original different equations into the boundary integral equations, and in conjunction with the discrete technologies of FEM. It has become the important complement of FEM owing to the dimensionality reduction and semi analytical features. In some domains, the BEM method is even the irreplaceable numerical method. However, the linear system arising from the BEM is general dense and non-symmetrical matrix. For the large scale problems, the traditional solving scheme which is composed of the boundary integral equation, computing of numerical integration, form the linear system, and solving by elimination method results in expensive computational costs. It limits the BEM only be applied to the small and medium-sized problems. The Multipole-BEMs combine the Fast Multipole Method (FMM) with the BEM, accelerate the computation of influence coefficients, update the solving mode of traditional BEM, improve the efficiency evidently, and be suited to the computing of large scale problems. This dissertation is mainly research the theories and applications of Taylor Series Multipole BEM (TSMBEM).
The Multipole-BEM is capable of improving the solving efficiency of BEM, and meanwhile the precision decreases in comparison with the traditional BEM. This dissertation presents an error analysis with regard to TSMBEM for 3-D elasticity. The error estimating formulas of TSMBEM are derived. The far-field partition criterion is present according to the requirement of precision. The influence factors of the solving accuracy of TSMBEM are specified.
The Taylor series expressions of fundamental solutions are mostly expressed in tensor form. The vectorization expressions of Taylor Series Multipole boundary element formula for 3-D elasticity problems, which take account of the symmetric properties of fundamental solutions and the characteristic of component for 3-D problems. The vectorization expressions reduce the computational costs and storage required, thus improve the computational efficiency.
For the large scale problems, the linear systems arising from BEM are general ill-conditioned, therefore the choice of iterative methods and preconditioning techniques is of great importance to the efficient solution. This dissertation investigates the Krylov subspace methods which are suited to the dense non-symmetric linear system arising from BEM. The comparative performances of different Krylov subspace solvers are studied, and several general preconditionrers are also considered and assessed. A near-field preconditioning techniques are presented, and it can accelerate the convergence, reduce the iterations, and be propitious to the special implicit linear systems arising from Multipole-BEM.
The dissertation researches the TSMBEM for 3-D elastic contact with problem, presents a mathematical programming solution method base on Point-Surface contact model. The source codes are written in FORTRAN. A large scale numerical experiment is presented. The contact pressures of between rolls of HC mill are analyzed. The validity and efficiency of TSMBEM are proved by numerical experiments.
This dissertation is mainly research the fundamental theories of TSMBEM, including the elasticity, contact problems, iterative methods and preconditioning techniques, which lay the foundation for the improvement of computational efficiency of BEM. The research subject is the frontline of computational mechanics, and has great academic value and wide engineering prospects.
引文
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