应变重构光滑有限元法及其在反问题中的应用

发布时间：2018-04-03 20:06

本文选题：有限元法　切入点：无网格法　出处：《吉林大学》2017年硕士论文

【摘要】：有限元法作为一种重要的数值方法,在工程、科学等各个领域有着广泛的应用。随着有限元法应用的不断深入,学者们发现有限元法对某些问题的分析具有局限性。比如:在求解大变形问题中,有限元法往往会导致网格产生较大的畸变,影响求解的精度。另外有限元法得到的刚度矩阵过硬,导致其在应力分析和锁定问题上,解的精度较低。为了解决这些问题,学者们提出了许多类型的无网格法。由于无网格法中的形函数大多数不具备?函数性质,施加本质边界条件需要进行一些特殊的处理。Liu和他的团队,结合两种方法的优点,将有限元法与无网格法中的光滑技术相结合提出了光滑有限元法(S-FEM)。本文绪论部分对有限元法,无网格法及光滑有限元法进行了简要的介绍。之后分别介绍了光滑节点域有限元法(NS-FEM)和光滑边域有限元法(ES-FEM)。对这两种方法光滑区域的构造和如何计算光滑应变矩阵进行了说明,并对体积锁定问题的处理方法进行了介绍。NS-FEM获得的刚度矩阵偏软,从而使求得的应变能要比实际的应变能偏大,即可以获得其应变能的上界。ES-FEM缓解了NS-FEM刚度矩阵过软的缺陷,使得解的精度得到提高,更接近于精确解。本文将这两种方法的应变做一个线性组合,引入可调参数?,得到一种新的应变重构数值方法。文中理论证明了应变重构光滑有限元法的收敛性,分析了刚度矩阵与NS-FEM和ES-FEM刚度矩阵之间的关系,简化了计算。文中尝试给出确定最优参数?的方法,确保得到精度较高的收敛解。通过对刚度矩阵的合适构造,不仅具有免于体积锁定的优点,还可以提高解的精度。此外,将应变重构光滑有限元法应用到反问题中,通过对可压的悬臂梁问题和不可压的软组织病变问题的研究,验证了该方法的可行性。
[Abstract]:As an important numerical method, finite element method is widely used in engineering, science and other fields.With the further application of finite element method, scholars find that finite element method has some limitations on some problems.For example, in solving large deformation problems, the finite element method often leads to a large distortion of the mesh, which affects the accuracy of the solution.In addition, the stiffness matrix obtained by the finite element method is very hard, which leads to the low accuracy of the solution in the stress analysis and locking problem.In order to solve these problems, many kinds of meshless methods have been proposed by scholars.Because most of the shape functions in the meshless method do not have?The application of essential boundary conditions requires some special treatment. Liu and his team combine the advantages of the two methods and combine the finite element method with the smooth technique in meshless method to propose a smooth finite element method (S-FEMN).In the introduction part, the finite element method, meshless method and smooth finite element method are briefly introduced.Then the smooth node domain finite element method (NS-FEMM) and smooth edge domain finite element method (ES-FEMN) are introduced respectively.The construction of the smooth region of these two methods and how to calculate the smooth strain matrix are described, and the method to deal with the volume locking problem is introduced. The stiffness matrix obtained by NS-FEM is soft.The obtained strain energy is larger than the actual strain energy, that is, the upper bound of the strain energy can be obtained. ES-FEM alleviates the weakness of the NS-FEM stiffness matrix, improves the accuracy of the solution and is closer to the exact solution.In this paper, the strain of these two methods is used as a linear combination, and the adjustable parameters are introduced to obtain a new strain reconstruction numerical method.In this paper, the convergence of the strain reconstruction smooth finite element method is proved theoretically, the relationship between the stiffness matrix and the NS-FEM and ES-FEM stiffness matrices is analyzed, and the calculation is simplified.This paper attempts to determine the optimal parameters?The convergence solution with higher accuracy can be obtained by the method of the proposed method.By constructing the stiffness matrix properly, it not only has the advantage of avoiding volume locking, but also improves the accuracy of the solution.In addition, the strain reconstruction smooth finite element method is applied to the inverse problem. The feasibility of the method is verified by studying the compressible cantilever beam problem and the incompressible soft tissue lesion problem.
【学位授予单位】：吉林大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O241.82

【相似文献】