Pure Displacement Of The Plane Elasticity Problem, The Quadratic Convergence Locking-free Finite Element

The topic of this paper is to construct locking-free finite elements with two-order of convergence for planar elasticity problem with pure displacement conditions, and to analyze anisotropic property of three-dimensional quasi-Wilson finite element.For the planar linear isotropic elasticity, it is well known that the performance of some finite elements may suffer deteriorations as the Lame constantλ→∞, i.e., as the material becomes incompressible. This is the so-called locking phenomenon. In order to overcome this phenomenon, several approaches are developed recent years, such as mixed finite element method, p-version and hp-version method of higher order finite element scheme, macro-element method, etc. In this paper, we present four locking-free finite elements basing on the minimization of the energy functional.To construct locking-free element with two-order of convergence, the conforming element needs more degrees of freedom and more complex process. So we focus on non-conforming element. Nonconforming element have the virtue of lesser global degrees of freedom, and it can get optimal convergence rate. In this paper all elements we construct are nonconforming.In the space of incomplete cubic polynomial, we present two kinds of 14-freedom triangular elements by restricting the operator div in P₁. We name them 14-freedom triangular element I and 14-freedom triangular element II, respectively. In chapter 5, by restricting the operator div in Q₁ or P₁, we present a 20-freedom rectangular element and a 18-freedom rectangular element. These elements are all locking-free. In chapter 4, the convergence properties of the four nonconforming elements are proved in detail and uniform optimal error estimates with respect toλ∈(0,∞) are obtained.In chapter 6, by constructing a trilinear interpolation operator, we present anisotropic analysis of quasi-Wilson element in three-dimensional space. In chapter 7, by analyzing the forced boundary conditions, we present an efficient algorithm for the system of elliptic boundary value equations and carry out the numerical experiments.Finally, we carry out two kinds of numerical experiments to assess the validity of our proposed elements. The first one has a analytic solution, so we can-give error estimates of L²-norm and energy norm. The second one the analytic solution is unknown. We use a estimator to verify the convergence rate. From the numerical results, we can see that optimal error estimates which are uniform with respect toλ∈(0,∞) are obtained for nearly incompressible planar elasticity problem. As a remark, we also list some numerical results of two kinds of locking phenomena lastly.