Hybrid Finite Element Methods For Problems With Multiple Elliptical Holes

Due to the factors such as manufacturing process and mismatch of physical properties of constituents,there are usually micro-pores and micro-cracks in the engineering materials.The interaction between them affects the propagation of cracks and finally affects the strength and safety of materials.Thus,the study on behaviors of holes and cracks is one of key issues in the field of structure safety.However,the distribution of holes or cracks in practical problems is random,so it is difficult to give exact solutions to such problem related to multi-holes or multi-cracks.Alternatively,the numerical methods can be developed for numerical solutions to this problem.Among them,the hybrid element method with fundamental solution kernels(HFS-FEM)attracted much more attention in the past ten years,especially in the construction of special elements.The algorithm has two independent interpolation functions: one is defined inside the element by the linear combination of fundamental solutions of problem considered so that it can exactly satisfy the governing equations of the problem;the other is independently defined along the element boundary to guarantee the continuity between adjacent elements and the boundary conditions imposed on the element edge.Finally,combing these two independent fields into the modified double-variable Hellinger–Reissner variational principle leads to a system of element stiffness equations including element boundary integrals only.Compared to the traditional boundary element and finite element methods,this algorithm has the following advantages:(1)the final stiffness equations include element boundary integrals only,so the integral dimension is reduced by one;(2)the element boundary integration strategy allows that any shaped polygonal element with arbitrary number of sides(n-sided polygonal element)can be constructed in a unified form;(3)the element boundary integration strategy permits great versatility with non-conforming mesh in the pre-processing stage without any difficulty;(4)all elements use the unified internal interpolation function,that is,fundamental solutions of problem;(5)for models containing circular holes,circular inclusions,concentrated loads,the corresponding special elements can be developed to improve the computational efficiency and simplify the mesh division around them.Currently,this method has been extensively developed for many thermal and elastic problems except for the problems on the isotropic/anisotropic plates with randomly distributed elliptic holes and cracks.Therefore,in this thesis,we focus on the numerical solutions of such problems by introducing specially-purposed elliptical hole elements.To do this,the special fundamental solution satisfying the specific boundary condition of a rotated elliptic hole in an infinite plane is first derived by introducing the complex conformal mapping transformation,and then the special polygonal element enclosing the elliptical hole is constructed to calculate the distribution of temperature or displacement field in the computational domain to further investigate the effects of elliptical hole size,distribution and rotation angle on the material properties.This thesis mainly consists of five parts :(1)hybrid finite element formulation for heat conduction in anisotropic plates;(2)hybrid finite element formulation for heat conduction in isotropic plates with multiple elliptical holes;(3)hybrid finite element formulation for heat conduction in anisotropic plates with multiple elliptical holes;(4)hybrid finite element formulation for plane elastic problems in isotropic plates with multiple elliptical holes;(5)hybrid finite element formulation for two-dimensional elastic mixed crack problems.In each part,the solving procedure of the algorithm and the corresponding numerical examples are fully given.The results demonstrate that the special elliptical hole elements constructed in this thesis can achieve good accuracy,and simultaneously can greatly save the computational time and simplify the meshing difficulty around elliptical holes,reduce numbers of elements.So,the present special elliptical hole element can be used for solving problems with multiple elliptical holes.In the future,the more complex crack problems can be analyzed by the special crack element which can be derived by setting the minor axis of elliptical hole into zero.