改进的无网格局部边界积分方程方法研究
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摘要
无网格方法是目前科学和工程计算方法的研究热点之一，也是科学和工程计算发展的趋势。
无网格局部边界积分方程方法是将局部边界积分方程和移动最小二乘方法相结合而形成的。该方法的优点是无论构造形函数，还是数值积分都不需要网格，是完全的无网格方法。但是移动最小二乘法的使用影响了该方法的计算效率，而且很容易形成病态方程组。本文针对这些问题，将计算量小、精度高以及性能稳定的新的近似函数与局部边界积分方程相结合，建立了改进的无网格局部边界积分方程方法。
径向基函数和多项式基函数耦合可以构造具有插值特性的近似函数。本文将这种近似函数与势问题对应的局部边界积分方程相结合，建立了势问题的径向基函数—局部边界积分方程方法；接着将这种近似函数引入弹性力学的局部边界积分方程方法，提出了弹性力学的径向基函数—局部边界积分方程方法。
为了提高局部边界积分方程方法求解裂纹问题的精度和效率，本文引入考虑裂纹尖端场的扩展的多项式基函数，并将其和径向基函数耦合构造了具有插值特性的近似函数，建立了基于径向基函数—扩展的局部边界积分方程方法。
针对移动最小二乘法存在的计算量大、容易形成病态方程组的问题，本文将改进的移动最小二乘法引入势问题的局部边界积分方程方法，建立了势问题的改进的局部边界积分方程方法；并进一步将改进的移动最小二乘法和弹性力学的局部边界积分方程相结合，提出了弹性力学的改进的局部边界积分方程方法。
现有的局部边界积分方程方法在模拟断裂力学问题时，存在计算量大、精度低以及裂纹尖端的应力数值解容易发生振荡等问题。为了改善这些不足，本文在扩展的多项式基函数的基础上，利用Schmidt正交化方法构造带权的正交基函数，建立了断裂力学的改进的局部边界积分方程方法。
为了证明本文提出的改进的无网格局部边界积分方程方法的有效性，本文编制了MATLAB计算机程序。数值算例说明了本文所建立的方法的正确性和有效性。
The meshless(or meshfree) method has been a hot direction and the development trend of numerical methods for science and engineering problems in recent years.
The local boundary integral equation(LBIE) method is a meshless method, which combines moving least-squares(MLS) approximation with the local boundary integral equation. The advantages of the LBIE method are that the method is a true meshless method, as no mesh is required either for the approximation functions of the variables, or for the numerical integration. Because the MLS is used in this method, the disadvantages of the method are its less efficiency, and can form an ill-conditioned or singular equations sometimes. To these problems, combining the new interpolation function, which has higher accuracy and stablity, with LBIE method, the improved meshless local boundary integral equation methods are presented in this dissertation.
A new approximation function based on the coupling of radial basis functions and polynomial basis functions is obtained, which has the property of Delta function. Combining this new approximate function with LBIE for potential problems, the LBIE method based on radial basis functions is presented for potential problem in the paper. And then combining this new approximate function with LBIE for elasticity problems, the LBIE method based on radial basis functions is proposed for elasticity problem.
To increase the precision and efficiency of the LBIE for fracture problems, by introducing the analysis solution of displacements at the tip of a crack, the approximation function based on the coupling of radial basis functions and enriched polynomial basis functions is obtained, and a new enriched LBIE method based on radial basis functions is proposed in this dissertation. This method used to solve the crack problems has greater precision and computational efficiency.
With the problems of the MLS, such as ill-conditioning equations, precision and efficiency, the improved MLS approximation is combined with LBIE for potential problems, then an improved LBIE method for potential problems is presented. Furthermore, the improved MLS approximation is combined with LBIE for elasticity problems, and an improved LBIE method for elasticity problems is presented too.
When simulating fractures problems with the conventional LBIE method, some problems, such as the computing time, less precision and the vibration of the solution at the tip of the crack, exist. In order to reduce these shortcomings, on the basis of enriched polynomial basis functions, weighted orthogonal basis functions are used to obtain the MLS approximation by using a Schmidt orthogonalization. In the end, the improved LBIE method for elasticity fracture is proposed.
In order to show the validity of the improved LBIE methods in the dissertation, corresponding MATLAB codes of these methods have been written. Some numerical examples are provided, and the validity and efficiency of these methods are demonstrated.
无网格局部边界积分方程方法是将局部边界积分方程和移动最小二乘方法相结合而形成的。该方法的优点是无论构造形函数，还是数值积分都不需要网格，是完全的无网格方法。但是移动最小二乘法的使用影响了该方法的计算效率，而且很容易形成病态方程组。本文针对这些问题，将计算量小、精度高以及性能稳定的新的近似函数与局部边界积分方程相结合，建立了改进的无网格局部边界积分方程方法。
径向基函数和多项式基函数耦合可以构造具有插值特性的近似函数。本文将这种近似函数与势问题对应的局部边界积分方程相结合，建立了势问题的径向基函数—局部边界积分方程方法；接着将这种近似函数引入弹性力学的局部边界积分方程方法，提出了弹性力学的径向基函数—局部边界积分方程方法。
为了提高局部边界积分方程方法求解裂纹问题的精度和效率，本文引入考虑裂纹尖端场的扩展的多项式基函数，并将其和径向基函数耦合构造了具有插值特性的近似函数，建立了基于径向基函数—扩展的局部边界积分方程方法。
针对移动最小二乘法存在的计算量大、容易形成病态方程组的问题，本文将改进的移动最小二乘法引入势问题的局部边界积分方程方法，建立了势问题的改进的局部边界积分方程方法；并进一步将改进的移动最小二乘法和弹性力学的局部边界积分方程相结合，提出了弹性力学的改进的局部边界积分方程方法。
现有的局部边界积分方程方法在模拟断裂力学问题时，存在计算量大、精度低以及裂纹尖端的应力数值解容易发生振荡等问题。为了改善这些不足，本文在扩展的多项式基函数的基础上，利用Schmidt正交化方法构造带权的正交基函数，建立了断裂力学的改进的局部边界积分方程方法。
为了证明本文提出的改进的无网格局部边界积分方程方法的有效性，本文编制了MATLAB计算机程序。数值算例说明了本文所建立的方法的正确性和有效性。
The meshless(or meshfree) method has been a hot direction and the development trend of numerical methods for science and engineering problems in recent years.
The local boundary integral equation(LBIE) method is a meshless method, which combines moving least-squares(MLS) approximation with the local boundary integral equation. The advantages of the LBIE method are that the method is a true meshless method, as no mesh is required either for the approximation functions of the variables, or for the numerical integration. Because the MLS is used in this method, the disadvantages of the method are its less efficiency, and can form an ill-conditioned or singular equations sometimes. To these problems, combining the new interpolation function, which has higher accuracy and stablity, with LBIE method, the improved meshless local boundary integral equation methods are presented in this dissertation.
A new approximation function based on the coupling of radial basis functions and polynomial basis functions is obtained, which has the property of Delta function. Combining this new approximate function with LBIE for potential problems, the LBIE method based on radial basis functions is presented for potential problem in the paper. And then combining this new approximate function with LBIE for elasticity problems, the LBIE method based on radial basis functions is proposed for elasticity problem.
To increase the precision and efficiency of the LBIE for fracture problems, by introducing the analysis solution of displacements at the tip of a crack, the approximation function based on the coupling of radial basis functions and enriched polynomial basis functions is obtained, and a new enriched LBIE method based on radial basis functions is proposed in this dissertation. This method used to solve the crack problems has greater precision and computational efficiency.
With the problems of the MLS, such as ill-conditioning equations, precision and efficiency, the improved MLS approximation is combined with LBIE for potential problems, then an improved LBIE method for potential problems is presented. Furthermore, the improved MLS approximation is combined with LBIE for elasticity problems, and an improved LBIE method for elasticity problems is presented too.
When simulating fractures problems with the conventional LBIE method, some problems, such as the computing time, less precision and the vibration of the solution at the tip of the crack, exist. In order to reduce these shortcomings, on the basis of enriched polynomial basis functions, weighted orthogonal basis functions are used to obtain the MLS approximation by using a Schmidt orthogonalization. In the end, the improved LBIE method for elasticity fracture is proposed.
In order to show the validity of the improved LBIE methods in the dissertation, corresponding MATLAB codes of these methods have been written. Some numerical examples are provided, and the validity and efficiency of these methods are demonstrated.
引文
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