SUMMARYThis paper describes the application of the so-called hybridizable discontinuous Galerkin (HDG) method to linear elasticity problems. The method has three significant features. The first is that the only globally coupled degrees of freedom are those of an approximation of the displacement defined solely on the faces of the elements. The corresponding stiffness matrix is symmetric, positive definite, and possesses a block-wise sparse structure that allows for a very efficient implementation of the method. The second feature is that, when polynomials of degree k are used to approximate the displacement and the stress, both variables converge with the optimal order of k+1 for any k 0. The third feature is that, by using an element-by-element post-processing, a new approximate displacement can be obtained that converges at the order of k +2, whenever k 2. Numerical experiments are provided to compare the performance of the HDG method with that of the continuous Galerkin (CG) method for problems with smooth solutions, and to assess its performance in situations where the CG method is not adequate, that is, when the material is nearly incompressible and when there is a crack.
L1nii)ersity of llfinois at Chicago, Chicago, I L 60680, U.S.A.
SUMMARYAn assumed strain (strain interpolation) method is used to construct a stabilization matrix for the 9-node shell element. The stabilization procedure can bc justified based on the Hellinger-Reissner variational method. It involves a projection vcctor which is orthogonal to both linear and quadratic fields in the local co-ordinate system of cach quadrature point. All terms in the development involve 2 x 2 quadrature in the 9-node element. Example problems show good accuracy and an almost optimal rate of convergence.
The behavior of a curved beam element is studied by comparison to an analytic solution. In the curved beam element, curvature effects are incorporated through a “shallow-shell” type theory. It is shown that when low-order, inplane displacement fields are used for the element, the curvature terms increase the bending stiffness due to contributions from the membrane strains; this is called “membrane locking.” Reduced integration yields a bending stiffness, which is in better agreement with the analytic value, and yet it retains the bending-membrane coupling, which is characteristic of curved elements. The results of the analysis are verified by several numerical examples.
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