Cook's tapered panel [26] with a nearly incompressible linear elastic material, a standard test for volumetric locking under combined bending and shear [75]. The tests solve it with the elastic model (Solid mechanics linear elasticity) assembled with local degrees of freedom, with three control-volume finite-element schemes and two Galerkin finite-element schemes.

Problem

The panel has the corners (0, 0), (48, 44), (48, 60) and (0, 44). The left edge \( x = 0 \) is clamped, the right edge \( x = 48 \) carries the vertical traction \( t = 6.25 \) (a total load of 100), and the other edges are traction-free. Plane strain, Young's modulus \( E = 250 \) and Poisson's ratio \( \nu = 0.4999 \), the setup of Ong et al. [75], who compare with the results of Kasper and Taylor [53]. The quantity of interest is the vertical displacement \( u_y \) of the upper right corner \( A = (48, 60) \). For the same load and \( E \) with \( \nu = 0.4999999 \), Chen and Sukumar [22] quote the reference value \( u_y(A) = 7.769 \).

Discretization

Test	Scheme
test_cooks_membrane_box	box (linear Lagrange basis, control volumes for all dofs)
test_cooks_membrane_pq1bubble	linear basis with element bubble, control volumes for all dofs
test_cooks_membrane_pq2	quadratic basis, control volumes for the vertex dofs and Galerkin test functions for the edge dofs
test_cooks_membrane_pq1fe	linear basis, Galerkin finite elements
test_cooks_membrane_pq2fe	quadratic basis, Galerkin finite elements

The problem derives from Experimental::ProblemWithSpatialParams: the clamped edge is imposed as constraints of the local degrees of freedom (constraints()) and the traction with boundaryFluxAtPos. The mesh cooksmembrane.msh has 233 triangles; Grid.Refinement refines it by conforming bisection, which roughly doubles the number of triangles per level.

Results

Vertical displacement \( u_y(A) \) under refinement:

Level	Box: dofs	Box	PQ2: dofs	PQ2 (control volumes)	PQ2 (finite elements)
0	140	4.7086	512	7.5868	7.6491
1	303	7.0334	1140	7.5932	7.7000
2	632	7.4196	2429	7.7115	7.7306
3	1312	7.5569	5102	7.6931	7.7391
4	2709	7.6526	10,633	7.7376	7.7519
5	5577	7.6869	22,016	7.7332	7.7557
6	11,415	7.7204	45,249	7.7558	7.7619
7	23,237	7.7333	92,364	7.7530	7.7637
8	47,092	7.7477

The PQ1-bubble and the PQ1 finite-element schemes give the same tip displacement as the box scheme on every level computed (levels 0 to 4, agreement to \( 10^{-9} \)). The values of the PQ2 control-volume scheme alternate between odd and even levels; those of the PQ2 finite-element scheme increase monotonically. Richardson extrapolation over every second level gives 7.759 (odd levels) and 7.766 (even levels) for the box scheme, 7.772 for the PQ2 control-volume scheme (odd levels) and 7.771 for the PQ2 finite-element scheme (odd and even levels). These agree with the value 7.769 of Chen and Sukumar [22] for \( \nu = 0.4999999 \) to 0.13 %. All schemes approach the limit slowly: on the coarsest mesh the linear displacement elements lock (box: 4.71), and at the upper end of the clamped edge, where clamped and free edge meet at 108°, the displacement behaves like \( r^{0.54} \) in the distance \( r \) from the corner (Williams' eigenvalue for \( \nu = 0.4999 \), plane strain).

The finite-difference Jacobian limits how close \( \nu \) can be to 0.5: at \( \nu = 0.4999999 \) the Newton iteration of the PQ2 control-volume scheme diverges.

The CTests compare the displacement field on the base mesh with a reference.

./test_cooks_membrane_pq2 params.input -Problem.Name test_cooks_membrane_pq2 -Grid.Refinement 4

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