A New Interior Penalty Discontinuous Galerkin Method for the Reissner–mindlin Model

Abstract

We introduce an interior penalty discontinuous Galerkin¯nite element method for the ReissnerÀ Mindlin plate model that, as the plate's half-thickness tends to zero, recovers a hp interior penalty discontinuous Galerkin¯nite element methods for biharmonic equation. Our method does not introduce shear as an extra unknown, and does not need reduced integration techniques. We develop the a priori error analysis of these methods and prove error bounds that are optimal in h and uniform in . Numerical tests, that con¯rm our predictions, are provided. Math. Models Methods Appl. Sci. 2010Sci. .20:1343Sci. -1361. Downloaded from www.worldscientific.com by UNIVERSITY OF PITTSBURGH on 02/05/15. For personal use only.require ingenious numerical methods. The matter is, despite being of second-order and elliptic, the system depends in a nontrivial manner on , the half-thickness of the plate. As goes to zero, the ReissnerÀMindlin solution approaches the Kirchho®ÀLove solution, which comes from a fourth-order partial di®erential equation. Thus, for positive but quite small, naive numerical schemes designed to solve ReissnerÀMindlin fail, since in general they do not approximate well solutions of fourth-order problems. This is described as a locking problem. This is all well known and sharply described in the introductory section of Ref. 7. There are in the literature¯nite element schemes that avoid locking altogether, for instance, see Refs. 2, 3, 8, 13, 18À20, 26À28 and 33; for a comprehensive review, see Ref. 24 and references therein. More recently some authors started to take advantage of the°exibility of discontinuous Galerkin (DG)¯nite element methods 5,6 to design new, locking free, plate models 4,14,16,29 ; after the completion of our work, we also learned about Ref. 25. Our work¯ts in this realm.Discontinuous Galerkin methods admit discontinuities of the elements in the discrete space, allowing the use of non-matching grids, approximations with varying polynomial order, and o®er the possibility to implement weakly the desired smoothness of elements of the approximation space. This is particularly important when designing¯nite element methods for the biharmonic equation, since one can use highorder polynomial approximations without the necessity of fairly involved construction of C 1 continuous¯nite element spaces. Furthermore, the local elementwise conservation property of the DG methods is often desired in applications. The sub-stantial°exibility of DG methods makes this approach quite suitable for a large range of computational problems such as linear and nonlinear hyperbolic PDEs, convection dominated di®usion PDEs and elliptic problems in general. A uni¯ed analysis of DG methods for the second-order elliptic equations can be found in Ref. 5, while a uni¯ed analysis encompassing both elliptic and hyperbolic equations in the framework of Friedrich's system can be found in Refs. 21À23.The discontinuous Galerkin method for fourth-order elliptic equation was introduced and analyzed by Baker. 10 A hp version of interior ...

The TDNNS method for Reissner–Mindlin plates

“…Deflections, rotations and the shear are approximated using different continuity assumptions in [5]. In [14,27] DG methods for deflection and rotation without reduced integration techniques are presented. We also mention [19] for a discontinuous Petrov-Galerkin method, where optimal test functions of higher polynomial order are chosen to suit the trial functions.…”

Locking free quadrilateral continuous/discontinuous finite element methods for the Reissner–Mindlin plate

“…This approach was first proposed by Hansbo and Larson [20], where continuous piecewise quadratics for the displacements and discontinuous piecewise linears for the rotations in a discontinuous Galerkin formulation. Further developments, still using simplicial elements, were given by Arnold et al [4], Heintz et al [18], and Bösing et al [8]. When the thickness of the plate tends to zero we obtain the Kirchhoff plate and our scheme can be seen as a version of the method proposed in [16], see [18].…”

Developments of Mindlin-Reissner Plate Elements

“…Recently, some scholars, mostly among mathematicians, began to employ the discontinuous Galerkin (DG) finite element methods [17] to design plate elements [120][121][122][123][124][125][126]. This DG method admitted the discontinuities in the element discrete space, leading to new types of conforming or nonconforming elements.…”