无单元法及其面向对象程序实施
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摘要
与有限元法相比,无单元法由于只需节点不需单元而在诸如结构大变形、断裂、裂纹扩展等工程问题的计算分析中具有很多优点,进一步完善无单元法理论,并研究出一套简便快捷和有效的、易于程序修改和扩充的方法,以便编制包含各种无单元法(或无单元法与有限元法相结合)的计算力学软件将是十分有意义的。
本文总结了无单元法的有关理论和该方法在有关领域的应用,并建立了一套完整的面向对象无单元法程序设计体系。在认真回顾与总结各种无单元法理论及其应用的基础上,本文完成了如下工作:
Ⅰ.对几种常见的无单元近似方法如光滑粒子法(SPH)、无单元迦辽金法(EFGM)、再生核点法(RKPM)、滑动最小二乘(MLS)法、单位分解(PU)法等进行了总结,讨论了其相应的列式方法、完备性、一致收敛性及应用情况;研究了它们的共性和特性以及其间的相互关系,为进行面向对象无单元法类定义,并为建立相应的类库建立了理论上的依据。
Ⅱ.对场函数及其导数不连续近似的处理方法如可视性法则、衍射法则和透射法则进行了总结;对无单元法中各种本质边界条件施加方法进行了总结,对罚函数法、Lagrange乘子法、边界奇异核函数法、边界转换法、无单元与有限元耦合法等方法以及其各自的优缺点进行了讨论,为进行面向对象无单元法程序设计时边界条件施加类的定义建立基础;对无单元近似的数值离散方法、核函数紧支半径计算方法、数值积分方法等问题进行了研究,并提出了核函数紧支半径计算的一种新算法,这种新算法可以改进节点随机分布时无单元法的计算效果。
Ⅲ.对小波理论中再生核、再生方程等与无单元法有关的重要概念进行了介绍,并对小波理论在无单元法中的应用,尤其是在多尺度再生核点法中的应用以及列式方法进行了总结和讨论;为进行面向对象无单元法程序设计时小波函数类的定义建立了基础;并在结构动力问题计算中采用小波函数作为核函数;
Ⅳ.基于变分原理,采用边界转换法施加本质边界条件,对弹性力学静力问题和结构动力问题进行了无单元法列式;将无单元法的应用领域扩展到压电陶瓷非线性断裂这一包含多物理场问题的分析中,进行了该问题分析的
Compared with Finite Element Method (FEM), the idea of Element Free Method ( EFM ) or meshless method for numerical analysis (such as large deformation, fracture ,etc.) is very appealing as EFM does not require the generation of a mesh for complicated two- and three-dimensional structures. In order to give a good approach for the development of large scale EFM coupled or connected with FEM analysis software system, the following works have been done in this dissertation:1. The major methodologies of element free approximations including Smoothed Particle Hydrodynamics (SPH) , Partition of Unity (PU), Moving Least Square (MLS) approximation, Element Free Galerkin Method (EFGM), Reproducing Kernel Particle Method (RKPM), have been reviewed. The completeness, consistency, convergence, applications and meshless formulations of these EFM approximations have been discussed. The common features of different EFM approximations have also been studied. According to these features, one can define the classes of object-oriented EFM program.2. The approaches of imposing essential boundary conditions such as Lagrange multiplier method, penalty method, boundary transformation method, boundary singular kernel method, Coupling method with FEM, have been summarized in detail. The advantages and disadvantages of the approaches have also been discussed. In object-oriented EFM programmimg, one can define a set of C++ classes to implement these approaches. The dominant methods of EFM discretization including collocation and Galertkin method, the calculation of compact support radius of kernel (weight or window ) function, the numerical integral methods in EFM, have been described. Considering the issue of functions with discontinuities and discontinuous derivatives, EFM approximations of discontinuous fields play an important role in engineering problems. Several methods (such as visibility criterion, diffraction method , transparency method) for the calculation of the functions with discontinuities have been reviewed. To calculate the compact support radius of the kernel function, an improved method has been introduced.
3. The concepts of reproducing kernel, reproducing conditions, reproducing equations and multiple scale analysis in wavelets theory, have been introduced. The applications of wavelets in RKPM have been described. To calculate wavelet functions, a typical C++ class named as "Wavelet" is defined.4. Based on variational principles, the EFM formulations for elastic static and dynamic analysis of structures have been implemented with the essential boundary conditions imposed by boundary transformation method. Considering the piezoelectric structure involving coupled elastic and electrostatic energy domains, EFM is more appealing for the analysis of piezoelectric ceramics because of the mix energy field. EFM formulation for non-linear fracture analysis of piezoelectric ceramics has also been completed.5. The approaches of defining C++ classes library including weight functions library, basis functions and shape functions library have been discussed. The emphasis is to identify and define the classes of object-oriented element free program, and some typical classes are arranged into hierarchies. An object-oriented element free program code in C++ is named MPM-FEM. Because of the relationship between EFM and FEM, some classes of the existing object-oriented FEM program is reused in MPM-FEM. In MPM-FEM, all the numerical integral approaches, imposing essential boundary conditions approaches of EFM are optional. One can use this program for the EFM and (/or coupled with ) FEM analysis.6. In order to enhance data hiding and decrease data coupling, the EFM data members encapsulated in C++ classes in place of the Fortran COMMON variables have been defined. To decrease data redundancy, static data members are used for implementing common resources that all objects need. A C++ class named "Tree" has been used to manage the objects and data. The extensibility and the extensible approaches of the program code have also been discussed.7. A nonlinear constitutive model for piezoelectric ceramics is proposed, in which the polarization switching and saturation are taken into account. Based on the model, the non-linear fracture analysis is implemented using RKPM that is one of the EFM numerical methods. Using local J-integral as a fracture criterion, a relation curve of fracture loads against electric fields is obtained. Qualitatively, the curve is in agreement with the experimental observations reported by Park &
Sun. The computation is implemented using object-oriented programming method.
本文总结了无单元法的有关理论和该方法在有关领域的应用,并建立了一套完整的面向对象无单元法程序设计体系。在认真回顾与总结各种无单元法理论及其应用的基础上,本文完成了如下工作:
Ⅰ.对几种常见的无单元近似方法如光滑粒子法(SPH)、无单元迦辽金法(EFGM)、再生核点法(RKPM)、滑动最小二乘(MLS)法、单位分解(PU)法等进行了总结,讨论了其相应的列式方法、完备性、一致收敛性及应用情况;研究了它们的共性和特性以及其间的相互关系,为进行面向对象无单元法类定义,并为建立相应的类库建立了理论上的依据。
Ⅱ.对场函数及其导数不连续近似的处理方法如可视性法则、衍射法则和透射法则进行了总结;对无单元法中各种本质边界条件施加方法进行了总结,对罚函数法、Lagrange乘子法、边界奇异核函数法、边界转换法、无单元与有限元耦合法等方法以及其各自的优缺点进行了讨论,为进行面向对象无单元法程序设计时边界条件施加类的定义建立基础;对无单元近似的数值离散方法、核函数紧支半径计算方法、数值积分方法等问题进行了研究,并提出了核函数紧支半径计算的一种新算法,这种新算法可以改进节点随机分布时无单元法的计算效果。
Ⅲ.对小波理论中再生核、再生方程等与无单元法有关的重要概念进行了介绍,并对小波理论在无单元法中的应用,尤其是在多尺度再生核点法中的应用以及列式方法进行了总结和讨论;为进行面向对象无单元法程序设计时小波函数类的定义建立了基础;并在结构动力问题计算中采用小波函数作为核函数;
Ⅳ.基于变分原理,采用边界转换法施加本质边界条件,对弹性力学静力问题和结构动力问题进行了无单元法列式;将无单元法的应用领域扩展到压电陶瓷非线性断裂这一包含多物理场问题的分析中,进行了该问题分析的
Compared with Finite Element Method (FEM), the idea of Element Free Method ( EFM ) or meshless method for numerical analysis (such as large deformation, fracture ,etc.) is very appealing as EFM does not require the generation of a mesh for complicated two- and three-dimensional structures. In order to give a good approach for the development of large scale EFM coupled or connected with FEM analysis software system, the following works have been done in this dissertation:1. The major methodologies of element free approximations including Smoothed Particle Hydrodynamics (SPH) , Partition of Unity (PU), Moving Least Square (MLS) approximation, Element Free Galerkin Method (EFGM), Reproducing Kernel Particle Method (RKPM), have been reviewed. The completeness, consistency, convergence, applications and meshless formulations of these EFM approximations have been discussed. The common features of different EFM approximations have also been studied. According to these features, one can define the classes of object-oriented EFM program.2. The approaches of imposing essential boundary conditions such as Lagrange multiplier method, penalty method, boundary transformation method, boundary singular kernel method, Coupling method with FEM, have been summarized in detail. The advantages and disadvantages of the approaches have also been discussed. In object-oriented EFM programmimg, one can define a set of C++ classes to implement these approaches. The dominant methods of EFM discretization including collocation and Galertkin method, the calculation of compact support radius of kernel (weight or window ) function, the numerical integral methods in EFM, have been described. Considering the issue of functions with discontinuities and discontinuous derivatives, EFM approximations of discontinuous fields play an important role in engineering problems. Several methods (such as visibility criterion, diffraction method , transparency method) for the calculation of the functions with discontinuities have been reviewed. To calculate the compact support radius of the kernel function, an improved method has been introduced.
3. The concepts of reproducing kernel, reproducing conditions, reproducing equations and multiple scale analysis in wavelets theory, have been introduced. The applications of wavelets in RKPM have been described. To calculate wavelet functions, a typical C++ class named as "Wavelet" is defined.4. Based on variational principles, the EFM formulations for elastic static and dynamic analysis of structures have been implemented with the essential boundary conditions imposed by boundary transformation method. Considering the piezoelectric structure involving coupled elastic and electrostatic energy domains, EFM is more appealing for the analysis of piezoelectric ceramics because of the mix energy field. EFM formulation for non-linear fracture analysis of piezoelectric ceramics has also been completed.5. The approaches of defining C++ classes library including weight functions library, basis functions and shape functions library have been discussed. The emphasis is to identify and define the classes of object-oriented element free program, and some typical classes are arranged into hierarchies. An object-oriented element free program code in C++ is named MPM-FEM. Because of the relationship between EFM and FEM, some classes of the existing object-oriented FEM program is reused in MPM-FEM. In MPM-FEM, all the numerical integral approaches, imposing essential boundary conditions approaches of EFM are optional. One can use this program for the EFM and (/or coupled with ) FEM analysis.6. In order to enhance data hiding and decrease data coupling, the EFM data members encapsulated in C++ classes in place of the Fortran COMMON variables have been defined. To decrease data redundancy, static data members are used for implementing common resources that all objects need. A C++ class named "Tree" has been used to manage the objects and data. The extensibility and the extensible approaches of the program code have also been discussed.7. A nonlinear constitutive model for piezoelectric ceramics is proposed, in which the polarization switching and saturation are taken into account. Based on the model, the non-linear fracture analysis is implemented using RKPM that is one of the EFM numerical methods. Using local J-integral as a fracture criterion, a relation curve of fracture loads against electric fields is obtained. Qualitatively, the curve is in agreement with the experimental observations reported by Park &
Sun. The computation is implemented using object-oriented programming method.
引文
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