A low-order nonconforming finite element for Reissner-Mindlin plates

We propose a locking-free element for plate bending problems, based on the use of nonconforming piecewise linear functions for both rotations and deflections. We prove optimal error estimates with respect to both the meshsize and the analytical solution regularity

A subspace correction method for discontinuous Galerkin discretizations of linear elasticity equations

We study preconditioning techniques for discontinuous Galerkin discretizations of isotropic linear elasticity problems in primal (displacement) formulation. We propose subspace correction methods based on a splitting of the vector valued piecewise linear discontinuous finite element space, that are optimal with respect to the mesh size and the Lamé parameters. The pure displacement, the mixed and the traction free problems are discussed in detail. We present a convergence analysis of the proposed preconditioners and include numerical examples that validate the theory and assess the performance of the preconditioners

A discontinuous Galerkin method for nonlinear shear-flexible shells

In this paper, a discontinuous Galerkin method for a nonlinear shear-flexible shell theory is proposed that is suitable for both thick and thin shell analysis. The proposed method extends recent work on Reissner–Mindlin plates to avoid locking without the use of projection operators, such as mixed methods or reduced integration techniques. Instead, the flexibility inherent to discontinuous Galerkin methods in the choice of approximation spaces is exploited to satisfy the thin plate compatibility conditions a priori. A benefit of this approach is that only generalized displacements appear as unknowns. We take advantage of this to craft the method in terms of a discrete energy minimization principle, thereby restoring the Rayleigh–Ritz approach. In addition to providing a straightforward and elegant derivation of the discrete equilibrium equations, the variational character of the method could afford numerous advantages in terms of mesh adaptation and available solution techniques. The proposed method is exercised on a set of benchmarks and example problems to assess its performance numerically, and to test for shear and membrane locking. Keywords: Shells; Discontinuous Galerkin; Locking; VariationalUnited States. Army Research Office (Contract DAAD-19-02-D-0002)United States. Office of Naval Research (Grant N00014-07-1-0764

Development of discontinuous Galerkin method for nonlocal linear elasticity

Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2007.Includes bibliographical references (p. 75-81).A number of constitutive theories have arisen describing materials which, by nature, exhibit a non-local response. The formulation of boundary value problems, in this case, leads to a system of equations involving higher-order derivatives which, in turn, results in requirements of continuity of the solution of higher order. Discontinuous Galerkin methods are particularly attractive toward this end, as they provide a means to naturally enforce higher interelement continuity in a weak manner without the need of modifying the finite element interpolation. In this work, a discontinuous Galerkin formulation for boundary value problems in small strain, non-local linear elasticity is proposed. The underlying theory corresponds to the phenomenological strain-gradient theory developed by Fleck and Hutchinson within the Toupin-Mindlin framework. The single-field displacement method obtained enables the discretization of the boundary value problem with a conventional continuous interpolation inside each finite element, whereas the higher-order interelement continuity is enforced in a weak manner. The proposed method is shown to be consistent and stable both theoretically and with suitable numerical examples.by Ram Bala Chandran.S.M

Multigrid methods for Maxwell\u27s equations

In this work we study finite element methods for two-dimensional Maxwell\u27s equations and their solutions by multigrid algorithms. We begin with a brief survey of finite element methods for Maxwell\u27s equations. Then we review the related fundamentals, such as Sobolev spaces, elliptic regularity results, graded meshes, finite element methods for second order problems, and multigrid algorithms. In Chapter 3, we study two types of nonconforming finite element methods on graded meshes for a two-dimensional curl-curl and grad-div problem that appears in electromagnetics. The first method is based on a discretization using weakly continuous P1 vector fields. The second method uses discontinuous P1 vector fields. Optimal convergence rates (up to an arbitrary positive epsilon) in the energy norm and the L2 norm are established for both methods on graded meshes. In Chapter 4, we consider a class of symmetric discontinuous Galerkin methods for a model Poisson problem on graded meshes that share many techniques with the nonconforming methods in Chapter 3. Optimal order error estimates are derived in both the energy norm and the L2 norm. Then we establish the uniform convergence of W-cycle, V-cycle and F-cycle multigrid algorithms for the resulting discrete problems. In Chapter 5, we propose a new numerical approach for two-dimensional Maxwell\u27s equations that is based on the Hodge decomposition for divergence-free vector fields. In this approach, an approximate solution for Maxwell\u27s equations can be obtained by solving standard second order scalar elliptic boundary value problems. We illustrate this new approach by a P1 finite element method. In Chapter 6, we first report numerical results for multigrid algorithms applied to the discretized curl-curl and grad-div problem using nonconforming finite element methods. Then we present multigrid results for Maxwell\u27s equations based on the approach introduced in Chapter 5. All the theoretical results obtained in this dissertation are confirmed by numerical experiments