A posteriori analysis of classical plate elements

Summary. We outline the results of our recent article on the a posteriori error analysis of C 1 ﬁnite elements for the classical Kirchhoﬀ plate model with general boundary conditions. Numerical examples are given.


Introduction
The purpose of our work is to fill a gap in the literature.Surprisingly, the a posteriori error analysis for classical plate finite elements has so far only been given for the fully clamped case and a load in L 2 , cf. [2].In our recent work [1], we treated a combination of all common boundary conditions (clamped, simply supported and free).In addition, we considered the cases of point and line loads.

The Kirchhoff plate problem
We denote the deflection of the plate's midsurface by u, the curvature by K and the moment by M , and we assume isotropic linear elasticity.Hence, it holds with where d denotes the thickness of the plate.E and ν are the Young's modulus and Poisson ratio, respectively.The strain energy for an admissible deflection v is then1 2 a(v, v), with The potential energy l(v) stems from the loading, which we assume to consist of a distributed load f ∈ L 2 (Ω), a load g ∈ L 2 (S) along the line S ⊂ Ω, and of a point load F at the point x 0 , so that The total energy is thus 1 2 a(v, v) − l(v), and its minimisation leads to the variational form: with We assume that the plate is clamped on the boundary part Γ c , simply supported on Γ s , and free on By the well-known integration by parts, we get the boundary value problem.To this end we have to recall the following quantities for a admissible displacement v; the normal shear force Q n (v), the normal and twisting moments M nn (v), M ns (v), and the effective shear force With the constitutive relationship (2), an elimination yields the plate equation for the deflection u: where the so-called bending stiffness D is defined as The boundary value problem is the following.
• In the domain we have the distributional differential equation where l is the distribution defined by (4).
• On the clamped part we have the conditions: u = 0 and ∂u ∂n = 0 on Γ c .• On the simply supported part it holds: u = 0 and M nn (u) = 0 on Γ s .
• On the free part it holds: M nn (u) = 0 and V n (u) = 0 on Γ f .
• At the corners on the free part we have the jump condition on the twisting moment [[M ns (u)(c)]] = 0 for all corners c ∈ Γ f .

Here and below [[•]] denotes the jump.
We consider conforming finite element methods: The finite element partitioning is denoted by C h .We assume that mesh is such that the point load is a vertex and the line load is along edges.The edges are divided into interior edges E i h , edges on S, E S h , edges on the free boundary E f h , and edges on the simply supported boundary E s h .The local error indicators are then the following.
• The residual on each element • The jump residuals of the normal moment along interior edges • The jump residuals in the effective shear force along interior edges • The normal moment along edges on the free and simply supported boundaries • The effective shear force along edges on the free boundary The error estimator is defined through Our a posteriori estimate is the following, where the energy norm is defined as Theorem 1 There exists positive constants C 1 , C 2 , such that

Numerical examples
In the examples, we have used the Argyris triangle.In the figures, we give the meshes for the adaptive solution of a square plate with a point and line load, and for a L-shaped domain with a free boundary for the edges sharing the re-entrant corner and simply supported along the rest of the boundary.

Figure 1 .
Figure 1.The adaptive meshes for the point and line loads.

Figure 2 .
Figure 2. The adaptive mesh for the L-shaped domain.