Rational Finite Element Method For Orthotropic Problems

Due to the complexity of anisotropic elastic problems,it is more difficult to find its analytical solution than isotropic problems.For engineering problems,it is have to use numerical method.According to the characteristics of anisotropic elastic problems,this thesis systematically develops the rational finite element method for anisotropic elasticity problems.The rational finite element method takes the basic solution of the homogeneous differential governing equation in the displacement form as the interpolating function,deduces the FEM formula in the physical domain and makes correction by considering the requirements of patch test in the element level.This method gives a clearer mechanical meaning and avoids the separation between physical and mathematical aspects.This method abandons the isoparametric technique and uses the linearly independent solutions of elastic equations for interpolation to the displacement field and stress field of the element at the same time.Its numerical stability and accuracy of the stress and strain field is significantly increased.This thesis briefly explains the basic idea and procedure of the rational finite element method from the views of functional analysis and approximation theory.Because the characteristic of the rational finite element is using the analytical basic solutions as the interpolation functions,the analytical basic solutions which have clear physical meaning are chosen as the interpolation functions.During the generation process of element stiffness matrix,the integration in the element level is analytical,so the error of the numerical integration method is avoided.As the rational finite element method is non-conforming element method in its nature,it should be pass the patch test.Based on the analysis of several formulations of C0 patch tests,the element level patch test is used to test the element stiffness matrix and then orthogonalized correction is used to the element stiffness matrix according to the test result,while ensures the convergence of the element.This thesis proposes a method for constructing plane anisotropic rational element.The element stiffness matrix is generated rationally from the complete analytical basic solution,and the conforming of element is ensured by patch test and correction,and four elements are realized in this basis.Through the rational selection of the analytical basic solution of the plane anisotropic problem,the interpolation functions are selected with definite mechanical properties.In the generating process of generalized stiffness element matrix,all interpolation functions are integrated analytically.Therefore,the generalized stiffness matrix also has definite mechanical significance.The generation of stiffness matrix adequately considerate the requirement of the patch test in element level.Based on the orthogonalized correction of the element stiffness matrix,the convergence of the plane anisotropic rational element is ensured.Besides,there are two different ways to select the number of the element interpolation functions: one way is to select the number of the analytical basic solution based on the number of freedom degrees of element.For this way,the completeness of the polynomial may not be satisfied and the element is direction dependent.Another way is to consider the completeness of the polynomial and the number of analytical basic solution is selected so that it is greater than the number of degrees of freedom of the element.For this way,it is necessary to use the static condensation method.Numerical results of these two ways show that the first one results in the absence of deformation modes while the second one has better element property and gives more stable numerical result.This thesis also develops the 3D anisotropic rational element.A method is proposed to construct the complete analytical solutions and a detailed process for generating the analytical stiffness matrix is given.Two specific 3D elements which satisfy the patch test are constructed.To ensure the mechanical significance of the analytical basic solutions,this thesis uses two different methods to determine the analytical basic solutions.For a low order displacement solution,a stress solution is assumed and the displacement solution is derived inversely.For a displacement solution with higher order,the method of undetermined coefficients is used to find the solution.An analytical method is used to integrate the generalized stiffness matrix,which are the function of the geometric properties and physical parameters of the element.As a result,the characteristic of rationality of this method is sufficiently reflected.In the process of patch test and orthogonalized correction,this thesis studied the generation of the node force vector.By using the virtual displacement principle and isoparametric idea,the node force vector of a hexahedron element is obtained.This thesis establishes four plane anisotropic rational quadrilateral elements which have four nodes,five nodes,eight nodes and nine nodes,respectively,and two 3D anisotropic rational hexahedron elements which have eight node and twenty nodes,respectively.Numerical experiment provided by this thesis show that the above anisotropic rational finite elements have higher accuracy,better numerical stability,and satisfactory adaptability to distortion mesh.It is an effective method in numerical analysis to anisotropic elastic problems.