Least Squares for the Perturbed Stokes Equations and the Reissner--Mindlin Plate (original) (raw)

Analysis of a Linear–Linear Finite Element for the Reissner–Mindlin Plate Model

Mathematical Models and Methods in Applied Sciences, 1997

An analysis is presented for a recently proposed nite element method for the Reissner{ Mindlin plate problem. The method is based on the standard variational principle, uses nonconforming linear elements to approximate the rotations and conforming linear elements to approximate the transverse displacements, and avoids the usual \locking problem" by interpolating the shear stress into a rotated space of lowest order Raviart-Thomas elements. When the plate thickness t = O(h), it is proved that the method gives optimal order error estimates uniform in t. However, the analysis suggests and numerical calculations con rm that the method can produce poor approximations for moderate sized values of the plate thickness. Indeed, for t xed, the method does not converge as the mesh size h tends to zero.

Multigrid Methods for a Stabilized Reissner–Mindlin Plate Formulation

SIAM Journal on Numerical Analysis, 2009

We consider a stabilized finite element formulation for the Reissner-Mindlin plate bending model. The method uses standard bases functions for the deflection and the rotation vector. We apply a standard multigrid algorithm to obtain a preconditioner. We prove that the condition number of the preconditioned system is uniformly bounded with respect to the multigrid level and the thickness parameter. The abstract multigrid theory is applied for carefully chosen norms. We have to prove also some new finite element error estimates. Numerical results confirm the analysis.

An implementation of thehp-version of the finite element method for Reissner-Mindlin plate problems

International Journal for Numerical Methods in Engineering, 1990

Reissner-Mindlin plate theory is still a topic of research in finite element analysis. One reason for the continuous development of new plate elements is that it is still difficult to construct elements which are accurate and stable against the well-known shear locking effect. In this paper we suggest an approach which allows high order polynomial degrees of the shape functions for deflection and rotations. A balanced adaptive mesh-refinement and increase of the polynomial degree in an hp-version finite element program is presented and it is shown in numerical examples that the results are highly accurate and that high order elements show virtually no shear locking even for very small plate thickness.

FIRST-ORDER SYSTEM LEAST SQUARES FOR THE STOKES AND LINEAR ELASTICITY EQUATIONS: FURTHER RESULTS

First-order system least squares (FOSLS) was developed in for Stokes and elasticity equations. Several new results for these methods are obtained here. First, the inverse-norm FOSLS scheme that was introduced but not analyzed in [SIAM J. Numer. Anal., 34 (1997), pp. 1727-1741 is shown to be continuous and coercive in the L 2 norm. This result is shown to hold for pure displacement or pure traction boundary conditions in two or three dimensions, and for mixed boundary conditions in two dimensions. Next, the FOSLS schemes developed in [SIAM J. Numer. Anal., 35 (1998), pp. 320-335] are applied to the pure displacement problem in planar and spatial linear elasticity by eliminating the pressure variable in the FOSLS formulations of [SIAM J. Numer. Anal., 34 (1997), pp. 1727-1741]. The idea of two-dimensional variable rotation is then extended to three dimensions to make the intervariable coupling subdominant (uniformly so in the Poisson ratio for elasticity). This decoupling ensures optimal (uniform) performance of finite element discretization and multigrid solution methods. It also allows special treatment of the new trace variable, which corresponds to the divergence of velocity in the case of Stokes, so that conservation can be easily imposed, for example. Numerical results for various boundary value problems of planar linear elasticity are studied in a companion paper [SIAM

A negative-norm least squares method for Reissner-Mindlin plates

Mathematics of Computation, 1998

In this paper a least squares method, using the minus one norm developed by Bramble, Lazarov, and Pasciak, is introduced to approximate the solution of the Reissner-Mindlin plate problem with small parameter t t , the thickness of the plate. The reformulation of Brezzi and Fortin is employed to prevent locking. Taking advantage of the least squares approach, we use only continuous finite elements for all the unknowns. In particular, we may use continuous linear finite elements. The difficulty of satisfying the inf-sup condition is overcome by the introduction of a stabilization term into the least squares bilinear form, which is very cheap computationally. It is proved that the error of the discrete solution is optimal with respect to regularity and uniform with respect to the parameter t t . Apart from the simplicity of the elements, the stability theorem gives a natural block diagonal preconditioner of the resulting least squares system. For each diagonal block, one only needs a p...

First-Order System Least Squares for the Stokes Equations, with Application to Linear Elasticity

SIAM Journal on Numerical Analysis, 1997

Following our earlier work on general second-order scalar equations, here we develop a least-squares functional for the two-and three-dimensional Stokes equations, generalized slightly by allowing a pressure term in the continuity equation. By introducing a velocity flux variable and associated curl and trace equations, we are able to establish ellipticity in an H 1 product norm appropriately weighted by the Reynolds number. This immediately yields optimal discretization error estimates for finite element spaces in this norm and optimal algebraic convergence estimates for multiplicative and additive multigrid methods applied to the resulting discrete systems. Both estimates are naturally uniform in the Reynolds number. Moreover, our pressure-perturbed form of the generalized Stokes equations allows us to develop an analogous result for the Dirichlet problem for linear elasticity, where we obtain the more substantive result that the estimates are uniform in the Poisson ratio.

On the uniform approximation of the Reissner-Mindlin plate model by p/hp finite element methods

Proceedings of the Fifth World Congress on Computational Mechanics (WCCM V), 2002

We study the approximation of the Reissner-Mindlin plate using the p/hp version of the finite element method (FEM). Our goal is to identify a method that: (i) is free of shear locking, (ii) approximates the boundary layer independently of the thickness of the plate and (iii) converges exponentially with respect to the number of degrees of freedom. We will consider both standard and reduced constraint/mixed formulations, in the context of the p/hp version of the FEM, and we will give guidelines for the construction of appropriate mesh-degree combinations that accomplish the above three goals, using straight as well as curved sided elements.

A least squares finite element formulation for elastodynamic problems

International Journal for Numerical Methods in Engineering, 1988

A residual finite element formulation is developed in this paper to solve elastodynamic problems in which body wave potentials are primary unknowns. The formulation is based on minimizing the square of the residuals of governing equations as well as all boundary conditions. Since the boundary conditions in terms of wave potentials are neither Dirichlet nor Neumann type it is difficult to construct a functional to satisfy all governing equations and boundary conditions following the variational principle designed for conventional finite element formulation. That is why the least squares technique is sought. AH boundary conditions are included in the functional expression so that the satisfaction of any boundary condition does not become a requirement of the trial functions, but they should satisfy some continuity conditions across the interelement boundary to guarantee proper convergence. In this paper it is demonstrated that the technique works well for elastodynamic problems; however, it is equally applicable to any other field problem.