重构核粒子边界无单元法研究
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摘要
边界积分方程的无网格方法是无网格方法的重要分支。以往的边界积分方程的无网格方法一般是将移动最小二乘法和边界积分方程方法相结合形成的。这类方法的优点是构造形函数不依赖网格，而且具有降维的特点。但是移动最小二乘法容易形成病态方程组而导致求解失效、并且往往给处理边界条件带来困难。本文将具有插值特性的重构核粒子法的边界节点无网格形函数和边界积分方程方法结合，提出了重构核粒子边界无单元法及其与有限元的耦合方法：基于扩展的重构核粒子法形函数，提出了断裂力学的重构核粒子边界无单元法。本文提出的重构核粒子边界无单元法，是解题规模小、节省时间、可直接施加边界条件的无网格方法。
本文研究了重构核粒子法的形函数的插值特性。在有限域上利用重构条件生成的重构核粒子法的形函数，当紧支域半径覆盖的节点数与基函数的单项式个数一致时，其满足点插值性质；改进的重构核粒子法的形函数，是通过简单函数引入插值特性，并利用增强函数构造重构条件，得到一个具有任意离散点插值特性的形函数。改进的重构核粒子法形函数能精确重构插值点多项式的真值、具有不低于核函数的高阶光滑性。本文采用具有点插值特性的重构核粒子法的形函数构造近似函数，克服了多数无网格方法处理本质边界条件的困难。
在改进的的重构核粒子法基础上，结合势问题的边界积分方程方法，本文建立了势问题的重构核粒子边界无单元法，推导了相应的离散化公式，建立了边界节点未知量求解方程。本文提出的重构核粒子边界无单元法是一种直接列式法，具有精度高和求解效率高的优势。
基于改进的重构核粒子法和弹性力学的边界积分方程方法，本文提出了弹性力学的重构核粒子边界无单元法。研究了其数值积分方案，推导了其离散化边界积分方程和边界节点未知量求解方程组，并给出了其离散化内点公式。
为求解具有区域特性的势问题和弹性力学问题，将势问题和弹性力学的重构核粒子边界无单元法与相应问题的有限元法耦合，提出了重构核粒子边界无单元法与有限元耦合的方法。该耦合方法的节点未知量与重构核粒子边界无单元法的节点未知量同样是节点待求真值，所以其同样可以直接施加边界条件。
现有的无网格方法在模拟断裂力学问题时，存在计算量大、精度低等问题。为了克服这些不足，本文引入扩展的重构核粒子法形函数构造具有裂纹尖端场特性的近似函数，建立了断裂力学的重构核粒子边界无单元法。该方法具较好的稳定性。
编制了以上各种方法的FORTRAN计算程序，进行了数值算例分析。采用本文方法，对工程问题进行了分析研究，并和有限元分析结果进行了比较。数值算例和工程应用研究表明，本文方法是稳定性好、具有较高精度和便于应用的有效数值方法。
The meshless boundary integral equation method, which combines moving least-squares approximation with the boundary integral equation method, is an important branch of meshless methods. The advantages of the meshless boundary integral equation method are that no mesh is required for the construction of the shape functions, and it can be applied easily to solve problems. And in the method, nodes are required only on the boundary of the problem domain, then dimension of the problem is decreased. Because the moving least-squares approximation is used in this method, the disadvantages of the method are its less efficiency, and that it can form ill-conditioned or singular equations sometimes, and applied the boundary conditions difficultly. To these problems, combining the reproducing kernel particle method (RKPM) and boundary integral equation method, the reproducing kernel particle boundary element-free (RKP-BEF) method is presented in this dissertation. And the coupled method is also presented by combining th
e RKP-BEF and FEM. Furthermore, the improved RKP-BEF for fracture mechanics is proposed based on the enriched shape functions of RKPM. The RKP-BEF method presented is a meshless method which has advantages of greater precision and computational efficiency. And the RKP-BEF method can apply the boundary conditions directly.
The shape function, which has the property of Delta function, in the RKPM is discussed first. When the shape function covers a number of nodes equal to the number of monomials in its basis, the RKPM shape functions, which satisfy reproducing conditions in a finite region, has an interpolate property. The improved RKPM shape functions which have the interpolate property at arbitrary nodes are obtained by coupling of a simple function and an enrichment function. The smoothness of the shape function of RKPM is no less than that of the kernel function and the values of polynomials at interpolating points can be exactly reconstructed. The difficulties for applying essential boundary conditions can be avoided by using the interpolating shape functions.
On basis of the improved reproducing kernel particle method, combining the boundary integral equation method for potential problems, the reproducing kernel particle boundary element-free (RKP-BEF) method for potential problems is presented in this paper. The formulae of discretization and corresponding discrete equations are also obtained. This RKP-BEF method is a direct meshless method of the boundary integral equation and has the advantages of the higher efficiency and computational precision.
On basis of the improved reproducing kernel particle method and boundary integral equation method, the reproducing kernel particle boundary element-free method for elasticity is presented. The discrete boundary integral equations of the RKP-BEF method are obtained by considering the numerical integral schemes. Then the equations with the variables at boundary nodes are obtained, and the formulae of the displacement and stress at internal points for the RKP-BEF method are given.
By combining the reproducing kernel particle boundary element-free method and finite element method, a coupled RKP-BEF/FE method for elasticity or potential problems is presented. The combined equations on the unknown quantities at nodes for the RKP-BEF/FE method are obtained. As the RKP-BEF and FEM have higher precision, the method presented here is successful.
When simulating fractures problems with the conventional meshless method, some problems, such as the computing time and less precision of the solution at the tip of the crack, are existed. In order to reduce these shortcomings, the improved RKP-BEF method for elasticity fracture is presented based on the enriched shape functions of reproducing kernel particle mothed.
The corresponding FORTRAN codes of the above methods have been written and the numerical examples are given in the corresponding chapters. Some engineering examples are solved by the methods in this paper, and compared with the FEM simulating results. The numerical results show that the methods in this paper have higher stability and efficiency, and can be applied easily.
本文研究了重构核粒子法的形函数的插值特性。在有限域上利用重构条件生成的重构核粒子法的形函数，当紧支域半径覆盖的节点数与基函数的单项式个数一致时，其满足点插值性质；改进的重构核粒子法的形函数，是通过简单函数引入插值特性，并利用增强函数构造重构条件，得到一个具有任意离散点插值特性的形函数。改进的重构核粒子法形函数能精确重构插值点多项式的真值、具有不低于核函数的高阶光滑性。本文采用具有点插值特性的重构核粒子法的形函数构造近似函数，克服了多数无网格方法处理本质边界条件的困难。
在改进的的重构核粒子法基础上，结合势问题的边界积分方程方法，本文建立了势问题的重构核粒子边界无单元法，推导了相应的离散化公式，建立了边界节点未知量求解方程。本文提出的重构核粒子边界无单元法是一种直接列式法，具有精度高和求解效率高的优势。
基于改进的重构核粒子法和弹性力学的边界积分方程方法，本文提出了弹性力学的重构核粒子边界无单元法。研究了其数值积分方案，推导了其离散化边界积分方程和边界节点未知量求解方程组，并给出了其离散化内点公式。
为求解具有区域特性的势问题和弹性力学问题，将势问题和弹性力学的重构核粒子边界无单元法与相应问题的有限元法耦合，提出了重构核粒子边界无单元法与有限元耦合的方法。该耦合方法的节点未知量与重构核粒子边界无单元法的节点未知量同样是节点待求真值，所以其同样可以直接施加边界条件。
现有的无网格方法在模拟断裂力学问题时，存在计算量大、精度低等问题。为了克服这些不足，本文引入扩展的重构核粒子法形函数构造具有裂纹尖端场特性的近似函数，建立了断裂力学的重构核粒子边界无单元法。该方法具较好的稳定性。
编制了以上各种方法的FORTRAN计算程序，进行了数值算例分析。采用本文方法，对工程问题进行了分析研究，并和有限元分析结果进行了比较。数值算例和工程应用研究表明，本文方法是稳定性好、具有较高精度和便于应用的有效数值方法。
The meshless boundary integral equation method, which combines moving least-squares approximation with the boundary integral equation method, is an important branch of meshless methods. The advantages of the meshless boundary integral equation method are that no mesh is required for the construction of the shape functions, and it can be applied easily to solve problems. And in the method, nodes are required only on the boundary of the problem domain, then dimension of the problem is decreased. Because the moving least-squares approximation is used in this method, the disadvantages of the method are its less efficiency, and that it can form ill-conditioned or singular equations sometimes, and applied the boundary conditions difficultly. To these problems, combining the reproducing kernel particle method (RKPM) and boundary integral equation method, the reproducing kernel particle boundary element-free (RKP-BEF) method is presented in this dissertation. And the coupled method is also presented by combining th
e RKP-BEF and FEM. Furthermore, the improved RKP-BEF for fracture mechanics is proposed based on the enriched shape functions of RKPM. The RKP-BEF method presented is a meshless method which has advantages of greater precision and computational efficiency. And the RKP-BEF method can apply the boundary conditions directly.
The shape function, which has the property of Delta function, in the RKPM is discussed first. When the shape function covers a number of nodes equal to the number of monomials in its basis, the RKPM shape functions, which satisfy reproducing conditions in a finite region, has an interpolate property. The improved RKPM shape functions which have the interpolate property at arbitrary nodes are obtained by coupling of a simple function and an enrichment function. The smoothness of the shape function of RKPM is no less than that of the kernel function and the values of polynomials at interpolating points can be exactly reconstructed. The difficulties for applying essential boundary conditions can be avoided by using the interpolating shape functions.
On basis of the improved reproducing kernel particle method, combining the boundary integral equation method for potential problems, the reproducing kernel particle boundary element-free (RKP-BEF) method for potential problems is presented in this paper. The formulae of discretization and corresponding discrete equations are also obtained. This RKP-BEF method is a direct meshless method of the boundary integral equation and has the advantages of the higher efficiency and computational precision.
On basis of the improved reproducing kernel particle method and boundary integral equation method, the reproducing kernel particle boundary element-free method for elasticity is presented. The discrete boundary integral equations of the RKP-BEF method are obtained by considering the numerical integral schemes. Then the equations with the variables at boundary nodes are obtained, and the formulae of the displacement and stress at internal points for the RKP-BEF method are given.
By combining the reproducing kernel particle boundary element-free method and finite element method, a coupled RKP-BEF/FE method for elasticity or potential problems is presented. The combined equations on the unknown quantities at nodes for the RKP-BEF/FE method are obtained. As the RKP-BEF and FEM have higher precision, the method presented here is successful.
When simulating fractures problems with the conventional meshless method, some problems, such as the computing time and less precision of the solution at the tip of the crack, are existed. In order to reduce these shortcomings, the improved RKP-BEF method for elasticity fracture is presented based on the enriched shape functions of reproducing kernel particle mothed.
The corresponding FORTRAN codes of the above methods have been written and the numerical examples are given in the corresponding chapters. Some engineering examples are solved by the methods in this paper, and compared with the FEM simulating results. The numerical results show that the methods in this paper have higher stability and efficiency, and can be applied easily.
引文
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