This benchmark test case implements the Computational Solid Mechanics (CSM) benchmark as proposed by Turek and Hron [100]. It evaluates the performance of the hyperelastic model for large deformations and time-dependent loading.

Problem description

The setup consists of a fixed rigid cylinder with an elastic beam (flag) attached to its rear. The beam is subject to a constant volume force (gravity), which induces oscillation. The benchmark tests the ability of the numerical scheme to handle large displacements and the accuracy of the time integration.

Mathematical model

We use the hyperelastic model Hyperelastic solid mechanics. This requires implementing the first Piola stress tensor \(\mathbf{P}\) as a function of the deformation gradient \(\mathbf{F} = \mathbf{I} + \nabla \mathbf{u}\). \ We use a Saint-Venant Kirchhoff material law similar to the one described in [100], which defines the first Piola-Kirchhoff stress tensor as:

\begin{align}\mathbf{P}(\mathbf{F}) = \mathbf{F} \mathbf{S} \end{align}

with

Lagragian-Green strain tensor: \(\quad\mathbf{E} = 0.5(\mathbf{F}^T\mathbf{F} - \mathbf{I}) = 0.5(\mathbf{C} - \mathbf{I})\)
Second Piola-Kirchhoff stress tensor: \(\quad\mathbf{S}(\mathbf{E}) = \lambda \text{tr}(\mathbf{E})\mathbf{I} + 2\mu\mathbf{E}\),

where \(\lambda\) is the first Lamé parameter and \(\mu\) is the shear modulus.

Numerical Discretization

Implementation	Spatial Discretization	Type	Order	Temporal Discretization	Type	Order
Dumux	Box PQ1 CVFE (Box) scheme	CVFE	1	Newmark- \(\beta\)	Implicit	2
Reference (Featflow)	Galerkin FEM (Q2/P1)	FEM	2	Crank-Nicolson	Implicit	2

Due to the difference in interpolation order (linear vs. quadratic), DuMuX typically requires a higher mesh refinement level to match the peak amplitudes of the reference solution.

Setup & Parameters

The Turek and Hron suite outlines three specific structural sub-benchmarks. While this implementation focuses primarily on CSM3 (the instationary time-dependent case), the domain configuration and parameters across the suite are summarized below for complete context.

Domain Geometry and Boundary Segments

CSM Geometry and Boundary Conditions

The domain isolates the flexible structural appendage from the benchmark layout using key geometric references:

Parameter	Symbol	Value	Unit
Total length of the hyperelastic beam	\(L\)	\(0.35\)	\(\text{m}\)
Total height thickness of the hyperelastic beam	\(H\)	\(0.02\)	\(\text{m}\)
Initial coordinates of monitoring control point \(A\) (beam tip center)	\(A\)	\((0.6, 0.2)\)	\(\text{m}\)
Coordinates of reference boundary point \(B\) (upstream cylinder hull face)	\(B\)	\((0.15, 0.2)\)	\(\text{m}\)
Coordinates of reference point \(C\) (rigid cylinder base center)	\(C\)	\((0.2, 0.2)\)	\(\text{m}\)

Boundary and Initial Conditions

The physical behavior is constrained by prescribing conditions along the closure of the solid boundary \(\partial\Omega = \Gamma_D \cup \Gamma_N\):

**Dirichlet Boundary ( \(\Gamma_D\))**: The left vertical face of the elastic structure is clamped flat against the rigid cylinder profile. The displacement field is strictly fixed:
\begin{align}\mathbf{u} = \mathbf{0} \quad \text{on } \Gamma_D \quad (x = 0.2\text{ m})\end{align}
**Flux Boundary ( \(\Gamma_N\))**: All remaining outer surfaces (top, bottom, and right end) exposed to environmental conditions are completely traction-free. The exact mathematical formulation defines a zero surface traction force vector using the first Piola-Kirchhoff stress tensor \(\mathbf{P}\) and the outward unit normal vector \(\mathbf{n}\):
\begin{align}\mathbf{P} \cdot \mathbf{n} = \mathbf{0} \quad \text{on } \Gamma_N\end{align}
Initial condition: The domain begins in a fully relaxed, non-deformed state at time \(t = 0\). For all steps \(t > 0\), a constant spatial gravity vector ( \(g = 2\text{ m/s}^2\)) acts downward along the negative \(y\)-axis across the solid density.

Parameter settings for the CSM tests

Parameter	Symbol	Value (CSM1)	Value (CSM2)	Value (CSM3)	Unit
Solid Density	\(\rho^s\)	\(1.0 \times 10^3\)	\(1.0 \times 10^3\)	\(1.0 \times 10^3\)	\(\text{kg/m}^3\)
Poisson's Ratio	\(\nu^s\)	\(0.4\)	\(0.4\)	\(0.4\)	\(-\)
Shear Modulus	\(\mu^s\)	\(0.5 \times 10^6\)	\(2.0 \times 10^6\)	\(0.5 \times 10^6\)	\(\text{kg/(m}\cdot\text{s}^2\text{)}\)
Young's Modulus	\(E\)	\(1.4 \times 10^6\)	\(5.6 \times 10^6\)	\(1.4 \times 10^6\)	\(\text{Pa}\)
Gravity	\(g\)	\(2.0\)	\(2.0\)	\(2.0\)	\(\text{m/s}^2\)
Simulation Type	\(-\)	Steady State	Steady State	Instationary	\(-\)

And the corresponding Lamé parameters are computed as:

\begin{align}\lambda = \frac{\nu^s E}{(1+\nu^s)(1-2\nu^s)} \quad \mu^s = \frac{E}{2(1+\nu^s)} \end{align}

Data creation and comparison

The primary objective is to monitor the displacement fields \(u_x\) and \(u_y\) of control point A (over time). To verify the accuracy of this numerical implementation, the values can be validated against the highly refined grid-independent reference results provided by the TU Dortmund Featflow benchmark suite [1].

Steady State Tests (CSM1 & CSM2)

If utilizing this setup to evaluate steady-state accuracy before running time-dependent loops, compare your final deflection values against the highest mesh-refinement levels:

CSM1 and CSM2 Validation The plots above compare DuMuX (Box scheme) against the Featflow reference data for stationary cases.

Instationary Test (CSM3)

For the dynamic CSM3 case, the beam exhibits large, persistent oscillations due to the sudden application of gravity on the elastic structure starting from an undeformed configuration.

CSM3 Validation

\(\Delta t\)	nel	ndofs	Source	Discretisation	ux of A [mm] (Mean \(\pm\) Amp [Freq])	uy of A [mm] (Mean \(\pm\) Amp [Freq])
0.005	320	6468	FeatFlow	FEM	-14.279 +/- 14.280 [1.0995]	-63.541 +/- 65.094 [1.0995]
0.005	1280	25092	FeatFlow	FEM	-14.299 +/- 14.299 [1.0995]	-63.594 +/- 65.154 [1.0995]
0.005	5120	98820	FeatFlow	FEM	-14.305 +/- 14.305 [1.0995]	-63.607 +/- 65.160 [1.0995]
0.010	320	6468	FeatFlow	FEM	-14.632 +/- 14.636 [1.0978]	-64.744 +/- 64.907 [1.0978]
0.010	1280	25092	FeatFlow	FEM	-14.645 +/- 14.650 [1.0978]	-64.765 +/- 64.946 [1.0978]
0.010	1923	1087	DuMuX	Box	-13.719 +/- 13.720 [1.2295]	-62.448 +/- 63.292 [1.1111]
0.010	5120	98820	FeatFlow	FEM	-14.645 +/- 14.650 [1.0978]	-64.766 +/- 64.948 [1.0978]
0.010	10631	5586	DuMuX	Box	-14.396 +/- 14.402 [1.2097]	-63.970 +/- 64.709 [1.0949]
0.010	49710	25429	DuMuX	Box	-14.590 +/- 14.594 [1.2097]	-64.381 +/- 65.109 [1.0949]
0.010	218076	110216	DuMuX	Box	-14.629 +/- 14.633 [1.2097]	-64.497 +/- 65.178 [1.0989]
0.020	320	6468	FeatFlow	FEM	-14.384 +/- 14.389 [1.0956]	-64.271 +/- 64.595 [1.0956]
0.020	1280	25092	FeatFlow	FEM	-14.402 +/- 14.406 [1.0956]	-64.352 +/- 64.679 [1.0956]
0.020	1923	1087	DuMuX	Box	-13.659 +/- 13.665 [1.2295]	-61.923 +/- 63.544 [1.1194]
0.020	5120	98820	FeatFlow	FEM	-14.404 +/- 14.408 [1.0956]	-64.371 +/- 64.695 [1.0956]
0.020	10631	5586	DuMuX	Box	-14.336 +/- 14.338 [1.2097]	-63.910 +/- 64.510 [1.1029]
0.020	49710	25429	DuMuX	Box	-14.516 +/- 14.521 [1.2097]	-64.315 +/- 64.933 [1.0870]
0.020	218076	110216	DuMuX	Box	-14.580 +/- 14.584 [1.1905]	-64.381 +/- 65.104 [1.0870]

Reproducing our simulation data

To reproduce the benchmark results, navigate to the specific test directory within your build folder and run the benchmark script.

# Example for CSM12
cd build-cmake/test/solidmechanics/turek_hron/csm12
make test_saint_venant_kirchhoff
python3 runBenchmarks.py
# Example for CSM3
cd build-cmake/test/solidmechanics/turek_hron/csm3
make test_saint_venant_kirchhoff_dynamic
python3 runBenchmarks.py

References

[1] Featflow Benchmark Suite, TU Dortmund. CSM Tests Reference Sheet. Available online at: https://wwwold.mathematik.tu-dortmund.de/~featflow/en/benchmarks/cfdbenchmarking/fsi_benchmark/fsi_tests/fsi_csm_tests.html

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