Clamped Circular Plate under Uniform Load (Mindlin-Reissner)

Problem Description

This benchmark tests the Mindlin-Reissner plate model against an analytical solution. The problem is a clamped circular plate of radius \(R\) subject to a uniform out-of-plane load \(F\).

The Mindlin-Reissner model extends Kirchhoff-Love theory by allowing the rotation field \(\boldsymbol{\theta}\) to differ from \(\nabla w\), capturing transverse shear deformation. The equilibrium equations are

\begin{align}\nabla\cdot\mathbf{Q} = F, \qquad -\nabla\cdot\mathbf{M}(\boldsymbol{\theta}) + \mathbf{Q} = \mathbf{0},\end{align}

with the shear resultant \(\mathbf{Q} = \kappa G t\,(\nabla w - \boldsymbol{\theta})\) and the moment tensor \(\mathbf{M}\) from the plane-stress constitutive law. Here \(G = E/(2(1+\nu))\) is the shear modulus and \(\kappa = 5/6\) is the shear correction factor.

The plate is clamped at the boundary:

\begin{align}w = 0, \quad \boldsymbol{\theta} = \mathbf{0} \quad \text{on } \partial\Omega.\end{align}

The system is reformulated using a Helmholtz decomposition of the shear resultant (see the model documentation).

Analytical Solution

For a clamped circular plate of radius \(R\) under uniform load \(F\) the Mindlin-Reissner analytical solution is

\begin{align}w(r) = -\frac{F R^4}{64D}\left(1-\xi^2\right)\left[\left(1-\xi^2\right) + \frac{8}{\kappa}\frac{t^2/R^2}{3(1-\nu)}\right],\end{align}

where \(\xi = r/R\) and \(D = Et^3/(12(1-\nu^2))\). The second term inside the brackets is the shear correction relative to the Kirchhoff-Love solution. As \(t/R \to 0\) the Kirchhoff-Love result is recovered.

Parameters

Parameter	Symbol	Value	Unit
Young's modulus	\(E\)	\(10^6\)	Pa
Poisson ratio	\(\nu\)	0.3	-
Plate thickness	\(t\)	0.05	m
Shear correction factor	\(\kappa\)	5/6	-
Load	\(F\)	1.0	N/m²
Radius	\(R\)	1.0	m

The parameters match the Kirchhoff-Love benchmark. The thickness-to-radius ratio \(t/R = 0.05\) keeps the shear correction term \(8(t/R)^2/[3\kappa(1-\nu)] \approx 1.1\%\) small, so the Mindlin-Reissner and Kirchhoff-Love solutions differ only slightly.

Setup

The domain is a unit disk discretized with an unstructured triangular mesh generated by Gmsh from disk.geo. The mesh size \(h\) controls the resolution. The mixed formulation uses two coupled sub-problems discretized with control-volume finite element (CVFE) methods: the Box scheme (PQ1 CVFE (Box) scheme) for the deformation sub-problem \((\varphi, w, \psi)\) and the PQ1-Bubble scheme (PQ1-B CVFE scheme) for the rotation sub-problem \(\boldsymbol{\theta}\), coupled via the Multidomain framework framework.

Results

To run the convergence test and produce the convergence plot, run:

python3 plot_convergence.py <executable>

The script produces two figures:

convergence.png: relative \(L^2\) error of \(w\) versus mesh size \(h\) on a log-log scale; the expected convergence rate is \(\mathcal{O}(h^2)\).
deformation_3d.png: 3D surface of the deformed plate (requires PyVista).

Convergence plot

Deformation

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