Linear Elastic Problem. The Looking-free Finite Element

The theme of this dissertation is the employment of nonconforming finite element methods based on the minimization of the energy functional to overcome locking problem of linear elasticity associated with a homogeneous isotropic elastic material.For the linear isotropic elasticity,it is well known that the convergence rate for the normal lower order conforming finite elements deteriorates as the Laméconstantλ→∞, i.e.,as the elastic material becomes incompressible.This is known as the phenomenon of locking.Various approaches have been proposed which work uniformly well for allλ,which can be deduce to mixed finite element method based on the Hellinger-Reissner variational principle and standard finite element method based on the minimization of energy functional.In the mixed finite element method,if we could find appropriate displacements and stresses finite element spaces,we would get stably discrete method. However,one drawback of mixed methods is the large number of variables involved. The latter method don't involve more variable than that in the differential equation. Moreover,the the matrix discretized from the finite element is positive definite and easier to be solved.In this paper,we construct nonconforming finite elements with two-order convergence rate for two-dimensional and three-dimensional linear elasticity based on the minimization of the energy functional.For the planar linear isotropic elasticity,we present a 14-freedom triangular element and a 18-freedom rectangular element.We analyze exchangeable property of the divergence operator and the interpolation operator introduced by the finite element schemes in a weaker sense,and the continuous property in integral sense of the functions in the corresponding finite element spaces.Using these properties,we prove the nonconforming elements are locking-free for the pure displacement planar elasticity problem.Convergence rate of the element schemes are uniformly optimal with respect toλ.Error estimates in the energy norm and L~2 norm are O(h~2) and O(h~3),respectively.Numerical tests are carried out,which coincide with our theoretical analysis.In the adaptation of nonconforming finite element methods to the equations of elasticity with traction boundary conditions,the main difficulty stems from the request that the finite element satisfy the discrete version of Korn's second inequality.In virtue of the ideas in[22],we prove the version of Korn's inequality hold in the 14-freedom triangular finite element space.Consequently,we prove the element is locking-free for the planar elasticity with pure traction boundary conditions.Moreover,error estimates in the energy norm and L~2 norm are also O(h~2) and O(h~3) respectively.We extend the results to the three-dimensional linear elasticity.For pure displacement boundary value problem,we present a 39-freedom tetrahedron and a 63-freedom cuboid nonconforming finite element schemes.Then we prove the convergence rates of the elements are uniformly optimal with respect toλ.Finally,we give numerical examples to verify the convergence of the cuboid element.