Out-of-plane bending of an elliptic plate: ellipticPlate

Prepared by Philip Cardiff and Ivan Batistić

Tutorial Aims

• Demonstrate how to perform a 3D linear-static stress analysis in solids4foam
• Compare the performance of three variants of the same tutorial case: petscSnes, which uses the linearGeometryTotalDisplacement solid model with a PETSc SNES solution algorithm; segregated, which uses the same solid model with an implicit segregated solution algorithm; and unsCoupled, which uses the original coupledUnsLinearGeometryLinearElastic solid model.
• Demonstrates using the solids4foam solid solver based on the PETSc SNES nonlinear solver.

Case Overview

This 3-D test case consists of a thick elliptic plate with a centred elliptic hole (Fig. 1); a constant pressure of \(1\) MPa is applied to the upper surface, and the outer surface is fully clamped. The case has been described by the National Agency for Finite Element Methods and Standards (NAFEMS) [1], and, in the context of the FV method, has been examined and benchmarked in detail by Demirdžić et al. [2]. The assumed mechanical properties are Young's modulus \(E = 210\) GPa and Poisson's ratio \(\nu = 0.3\). Due to double symmetry, only a quarter of the plate is analysed. The thickness of the plate is \(0.6\) m, and the inner and outer ellipses are given by: \(\left(\dfrac{x}{2}\right)^2 + y^2 = 1\)

for the inner ellipse, and

for the outer ellipse.

The problem is solved as quasi-static with one loading increment, neglecting gravitational effects. The mesh is generated using the blockMesh utility and consists of 12 cell layers through the thickness, 24 cells in the circumferential direction, and 16 cells in the radial direction, as shown in Figure 2. The mesh density corresponds to mesh 3 used in [3].

The target stress value given in the NAFEMS benchmark cannot be used directly
for comparison, as the fixed displacement boundary conditions used in the
current study mimic those used in Demirdžić et al. [2] and do not correspond
exactly to those in the NAFEMS benchmark.

Figure 1 - Problem geometry and loading [2]

Figure 2 - Hexahedral mesh (mesh 3 in [2])

Expected Results

Figure 3 shows the stress contours on the \(z=0.3\) m \(x\)–\(y\) plane for the finest mesh. The predictions agree closely with the results of Demirdžić et al. [2]. Further comparisons can be found in [3]: Cardiff et al., 2016, A block-coupled finite volume methodology for linear elasticity and unstructured meshes.

Figure 3 - Stress component distributions on the plane \(z=0.3\) m. The second image is from [2], and the third is generated using finite elements with Abaqus [3].

The wall-clock times and memory requirements for each run are given in Table 1, where the results for the unsCoupled and segregated variants are compared. The unsCoupled solver used a bi-conjugate gradient stabilised linear solver with ILU(0) preconditioner, while the segregated variant used an implicit segregated solve with the same convergence tolerance of \(1 \times 10^{-6}\). In this case, the unsCoupled variant is faster than the segregated method by a factor of 2.5–6. As expected, its memory requirements are greater, by approximately 4.5 times in the largest mesh case. Interestingly, when compared to the FE solution, unsCoupled is faster and considerably more memory efficient in all cases; in the finest mesh case, unsCoupled is almost 6 times faster and uses 8 times less memory.

Table 1: Wall-clock time (in s) and maximum memory usage (in MB) [2].

Running the Case

The tutorial case is located at solids4foam/tutorials/solids/linearElasticity/ellipticPlate. The case can be run using the included Allrun script. The Allrun script optionally takes an argument that specifies the solution approach:

./Allrun                # Defaults to the petscSnes variant
./Allrun segregated     # Segregated variant
./Allrun unsCoupled     # Original foam-extend-only variant

The Allrun script starts by updating the files in the case to match the selected approach; the following files are updated: 0/D, constant/solidProperties, system/fvSchemes, and system/fvSolution. Subsequently, the mesh is created with blockMesh, followed by running the solver solids4Foam.

The default petscSnes variant uses the linearGeometryTotalDisplacement solid model with PETSc SNES. The original unsCoupled variant is also provided through the .unsCoupled file suffixes, and uses the coupledUnsLinearGeometryLinearElastic solid model together with the original block* boundary conditions and coupled discretisation settings. The unsCoupled variant can currently only be run using solids4foam built on foam-extend.

The segregated variant uses the same linearGeometryTotalDisplacement solid model as petscSnes, but with solutionAlgorithm implicitSegregated; in constant/solidProperties. This variant is intended for comparison with the PETSc SNES path using the same geometry and boundary conditions.

References

[1] National Agency for Finite Element Methods and Standards (U.K.). The standard NAFEMS benchmarks. NAFEMS; 1990.

[2] Demirdžić, I., Muzaferija, S., Perić, M.: Benchmark solutions of some structural analysis problems using finite-volume method and multigrid acceleration. International journal for numerical methods in engineering 40(10), 1893–1908 (1997).

[3] P. Cardiff, Ž. Tuković, H. Jasak, A. Ivanković, A block-coupled finite volume methodology for linear elasticity and unstructured meshes. Computers and Structures, 175, 2016, 100–122, 10.1016/j.compstruc.2016.07.004.

[4] P. Cardiff, D. Armfield, Ž. Tuković, I. Batistić, A Jacobian-free Newton-Krylov method for cell-centred finite volume solid mechanics. International Journal for Numerical Methods in Engineering, 127, e70268, 2026, 10.1002/nme.70268.