The Research On Some Lower-order Noncomforming Elements And Efficient Solving Algorithms For Nearly Incompressible Problems In Three Dimensions

Nearly incompressible problem in three dimensions is a class of important issues in actual engineering calculation,such as the common rubber,plastic and etc,are part of the nearly incompressible material which is characterized by Poisson's ratio is close to 0.5 or Lame constant tends to infinity.By using the usual finite element method to solve this problem,volume locking phenomenon appears.There are many ways to overcome this volume locking,for example the mixed finite element,higher-order conforming finite element,nonconforming finite element,reduced integration method and so on.Taking into account the computational complexity for three-dimensional problem,nonconforming element(e.g.,Wilson element)is often being used to solve the volume locking whose computing scale is too large because it takes the advantage of less freedom and high precision.In this paper,two kinds of calculation format for low-order Wilson element are firstly established based on hexahedral element and then applied to two types nearly incompressible problems (i.e.cantilever and Cook membrane problems) with mixture boundary conditions.Nonconforming element can overcome the volume locking and has a higher accuracy comparing with the coordination element under the same size.However,this method relies on meshing quality,the convergence will greatly deteriorated or converge no more since the element can not pass the patch test when there is a large deformation or distortion unit.Refinement element method is obtained by modifying the constant strain item based on Wilson element,which can improve the speed of convergence and accuracy on a large extent.Aiming cantilever and Cook membrane problems,numerical testing and results comparing are being taken for 8-nodes and 20-nodes refinement elements by using the large deformation semi-structured and unstructured grids.Refinement element has higher accuracy and better anti-distortion ability to grid than Wilson with the same scale.In order to improve the overall efficiency of the nonconforming element analysis,designing a fast algorithm for the corresponding discrete system is necessary.Thus,the discrete system is a highly ill conditioned positive definite system when came to the nearly incompressible problem and PCG method is one of the most effective methods to solve these equations.Moreover,anisotropic grid is often produced with a large aspect ratio(i.e.,the size of the units vary greatly in three directions)due to the special of structure,which will greatly affect the convergence of PCG method.A kind of PCG method is designed based on DAMG method in this thesis,then applied to solve the discrete systems of 8-nodes Wilson element and 8-nodes refinement element for nearly incompressible problem.The results indicated that the algebraic multigrid method(DAMG)based on the "distance matrix" can solve the problem with anisotropy grid more effectively.And by combining with effective polished operator again,the corresponding PCG method has a good robustness and efficiency for the nearly incompressible problems.