Element Free Method And Its Object-Oriented Programming

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.