A finite element method with discontinuous rotations for the Mindlin–Reissner plate model

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Abstract

We present a continuous-discontinuous finite element method for the Mindlin–Reissner plate model based on continuous polynomials of degree k  2 for the transverse displacements and discontinuous polynomials of degree k  1 for the rotations. We prove a priori convergence estimates, uniformly in the thickness of the plate, and thus show that locking is avoided. We also derive a posteriori error estimates based on duality, together with corresponding adaptive procedures for controlling linear functionals of the error. Finally, we present some numerical results.

Introduction

Plates are very common simplified models for thin structures in engineering practice. The most basic plate models are the Kirchhoff model, which is a fourth order partial differential equation, and the Mindlin–Reissner (MR) model, which is a system of second order partial differential equations. The Kirchhoff model can be seen as the limit of the MR model as the thickness of the plate tends to zero. Finite element approximations of plate models would seem to be easier to handle with the MR model, since then only C0 continuity is required, as opposed to the C1-continuous elements needed for the Kirchhoff model. However, in order for a finite element method to work asymptotically as t  0 in the MR model, typically rather complicated approximations must be used.

In this paper, we will consider a family of simple continuous-discontinuous Galerkin finite element methods for the MR model, first proposed in [10], based on discontinuous piecewise polynomials for the discretization of the rotations and continuous piecewise polynomials of one degree higher for the transverse displacements.

When the thickness of the plate tends to zero we obtain the Kirchhoff plate and our scheme simplifies to the method proposed in [9]. In this context we also mention the discontinuous Galerkin methods for the Kirchhoff plate developed by Hansbo and Larson [12] and for the Mindlin–Reissner model by Bösing et al. [4].

Section snippets

The continuous problem

The Mindlin–Reissner plate model is described by the following partial differential equations:-·σ(θ)-κt-2(u-θ)=0inΩR2,-κt-2·(u-θ)=ginΩ,where u is the transverse displacement, θ is the rotation of the median surface, t is the thickness, assumed constant, t3g is the transverse surface load, andσ(θ)2με(θ)+λ·θ1is the moment tensor. Here, 1 is the identity tensor and ε is the strain operator with componentsεij(θ)=12θixj+θjxi.The material constants are given by the relationsκEk2(1+ν),μE24

The finite element method

For simplicity, we shall consider the case of clamped boundary conditions. The transverse displacement and rotation vector are solutions to the following variational problem: find θ[H01(Ω)]2 and uH01(Ω) such thata(θ,ϑ)+b(u,θ;v,ϑ)=(g,v)Ωfor all (v,ϑ)H01(Ω)×[H01(Ω)]2. However, as is common in discontinuous Galerkin methods, we need to assume a higher regularity of the solution in order for all terms in our scheme to make sense with the exact solution inserted. In particular, we will need n · σ(θ

Stability estimates

For our analysis, we introduce the following edge norm:ϑE2=EEϑL2(E)2and mesh dependent energy-like normϑ2=TT(σ(ϑ),ε(ϑ))T+12μ+2λhE1/2n·σ(ϑ)E2+(2μ+2λ)hE-1/2[ϑ]E2.The mesh dependent norm ⦀ · ⦀ can be used to bound the broken H1(Ω) norm on Θh, which is the statement of the following lemma.

Lemma 3

There is a constant c, independent of h, μ, and λ such thatTTϑH1(T)2cϑ2forallϑΘh.

Proof

This is a discrete Korn-type inequality that results from the control of the rigid body rotations given by the

A priori error estimates

In this section, we will derive a priori error estimates for CDG methods in the case k = 2, and show that they hold uniformly in t. For higher order methods, edge effects will typically preclude global estimates because of the lack of regularity of the exact solution.

A duality-based posteriori estimate

For duality-based a posteriori error analysis, we consider the following variant of (19): find z and ψ such that-·σ(ψ)-κt-2(z-ψ)=fθinΩ,-κt-2·z-ψ=fuinΩwith zero Dirichlet boundary conditions for z and ψ. With eθ = θ  θh and eu = u  uh we find, using Lemma 2, that(fθ,eθ)Ω+(fu,eu)Ω=ah(ψ,eθ)+b(z,ψ;eu,eθ)=ah(eθ,ψ-πhψ)+beu,eθ;z-π˜hz,ψ-πhψ,where πh and π˜h now denote arbitrary interpolants (or projections) onto the respective subspaces. Using the equilibrium equations we find that(fθ,eθ)Ω+(fu,eu)Ω=(g,z-π

Numerical examples

We apply Algorithm 1 to a set of simple model problems in order to: (1) exemplify the behavior of the adaptive procedure; (2) study convergence rates of the finite element method (6) with respect to meshsize and plate thickness; and (3) study how the choice of stability parameter affects the approximation on different types of meshes.

We remark that the solutions in Sections 7.1 An L-shaped membrane, 7.2 The unit square are not smooth enough to be covered by the a priori error analysis presented

Concluding remarks

We have presented a novel finite element method for the Mindlin–Reissner plate model, based on the discontinuous Galerkin approach. We show that our method does not lock as long as we make a proper choice of a free, but computable, parameter. Our approach avoids the current paradigm of projections of the rotations in the shear energy functional, which, at least from a conceptual point of view, requires a mixed implementation. We pay the price of having to use a higher number of degrees of

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