Clamped Circular Membrane under Uniform Load

Problem Description

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

\begin{align}T\,\nabla^2 w = F \quad \text{in } \Omega,\end{align}

where \(T\) is the membrane tension and \(w\) is the vertical deformation. The plate is clamped at the boundary:

\begin{align}w = 0 \quad \text{on } \partial\Omega.\end{align}

Analytical Solution

For a circular domain of radius \(R\) the analytical solution is

\begin{align}w(r) = -\frac{F}{4T}\left(R^2 - r^2\right),\end{align}

where \(r = \sqrt{x^2 + y^2}\) is the radial coordinate.

Parameters

Parameter	Symbol	Value	Unit
Load	\(F\)	1.0	N/m²
Tension	\(T\)	10.0	N/m
Radius	\(R\)	1.0	m

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 equation is discretized with the Box scheme (PQ1 CVFE (Box) scheme).

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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