solidSymmetry

Overview

solidSymmetry is a symmetry-plane condition ported from the foam-extend solid-mechanics toolbox (Cardiff, Tuković), where it is also known as symmetryDisplacement. It exists for two field types, both registered under the same type name:

• solidSymmetryFvPatchVectorField, intended for the total displacement D of a solid region (solidStVK). It mirrors the near-wall displacement across the plane, so the normal displacement on the patch is zero and the tangential displacement is free. An optional second-order correction is available.
• solidSymmetryFvPatchScalarField, the scalar counterpart. It sets the patch value to the corrected near-wall value (zero normal gradient).

Both classes derive from the stock symmetry constraint (symmetryFvPatchField). They only replace snGrad() and evaluate(), so the patch must be of mesh type symmetry. The class keeps the matrix coefficients (implicit part) of the stock condition.

Mathematical Formulation

Let $\mathbf{n}$ be the patch unit normal, $\boldsymbol{\Delta}$ the patch delta vector (face centre minus cell centre), $\delta = 1/\texttt{deltaCoeffs}$ the cell-to-face distance, and $\mathbf{R} = \mathbf{I} - 2\,\mathbf{n}\mathbf{n}$ the reflection tensor. The non-orthogonal correction vector and the corrected near-wall value are

Vector field, first order (default)

Vector field, second order (secondOrder yes)

The tangential part of the value is extrapolated over half a cell with the mean of the zero face gradient and the cell gradient. The normal part of the gradient is the one-sided second-order estimate $2(\mathbf{D}_b - \mathbf{D}_P)\cdot\mathbf{n}/\delta - (\partial_n D_n)_P$.

Scalar field

A scalar is invariant under reflection, so $\mathbf{R}\,T = T$ and the formulas above reduce to

Note on OpenFOAM.com. In OpenFOAM.com (v2512 and v2606), fvPatch::delta() returns the patch-normal delta $\mathbf{n}(\mathbf{n}\cdot(\mathbf{C}_f-\mathbf{C}_P))$ for all non-coupled patches. Then $\mathbf{k} \equiv \mathbf{0}$, $\mathbf{D}_P^{*} = \mathbf{D}_P$, and the non-orthogonal correction does nothing. It was written for foam-extend, where delta() is the full $\mathbf{C}_f - \mathbf{C}_P$. With the default first-order setting, the vector condition therefore gives exactly the values and normal gradient of the stock symmetry condition. The scalar condition is equivalent to stock symmetry / zeroGradient for scalars.

Implementation

Construction. The dictionary constructor calls the stock symmetry constructor. That constructor ignores value and evaluates the field once with the stock symmetry formula. The constructor then checks that the patch is exactly a symmetryFvPatch (isType). Any other patch type, including symmetryPlane, stops the run with a FatalIOError. The vector version reads the optional secondOrder switch and prints it. Both versions print “Symmetry boundary condition with non-orthogonal correction”.

evaluate() (called from correctBoundaryConditions()):

1. Calls updateCoeffs() if needed. This is the base no-op.

2. Looks up the cell gradient on the patch.

   • Vector: the volTensorField named gradD, whatever the name of the field the BC is attached to.
   • Scalar: the volVectorField named grad(<fieldName>), e.g. grad(T).

   The lookup is mandatory. If the field is not registered on the mesh, the run stops with a fatal error.

3. Builds $\mathbf{D}_P^{*}$ and assigns the patch value with the formula above.

4. Vector only: calls the base evaluate(), which resets the updated flag. The scalar version does not call it.

snGrad() returns the normal gradient above. It is used wherever the patch gradient is requested explicitly. The matrix coefficients are inherited from the stock transform/basicSymmetry implementation: for vectors the implicit diagonal is $|n_i|$ per component. Because gradientBoundaryCoeffs() is snGrad() - gradientInternalCoeffs()·D_P, the overridden snGrad() (including the second-order term) enters fvm::laplacian as an explicit source. For scalars, the inherited coefficients are those of a zero-gradient condition.

In solidStVK. The momentum equation uses fvm::laplacian(impKf, D, "laplacian(DD,D)"), which picks up the coefficients above. gradD = fvc::grad(D) is recomputed in postSolve() from the patch values set in evaluate(). solidStVK registers gradD, so the vector BC works on its D field. It also works on any other vector field of that region, but it always uses gradD.

Dictionary Entries

Vector version:

Scalar version:

The scalar version does not read secondOrder.

Required registered fields:

Usage

The mesh patch must be of type symmetry (e.g. symmetry symmetry (...) in blockMeshDict). From tutorials/fluidStructureInteraction/beamInCrossFlow/0/solid/orig/partitioned/D:

symmetry
{
    type            solidSymmetry;
    patchType       symmetry;
    value           uniform (0 0 0);
}

With the second-order correction:

symmetry
{
    type            solidSymmetry;
    secondOrder     yes;
    value           uniform (0 0 0);
}

For other fields of the solid region on the same patch, the tutorials use the stock condition (e.g. T in heatedBeamInCrossFlow: type symmetry;).

Known Limitations

• Non-orthogonal correction has no effect in OpenFOAM.com. The patch delta is patch-normal, so $\mathbf{k} = \mathbf{0}$ (see the note above). With secondOrder no, the vector BC is numerically identical to stock symmetry. It still needs gradD.
• Hard-coded gradient name (vector). The vector version always reads gradD. It is only usable in a region that registers a volTensorField named gradD, i.e. solidStVK.
• Scalar version needs grad(<field>). No compiled region type registers a volVectorField with that name. Unless something else in the case provides it, the scalar BC fails on its first evaluation. Because it is equivalent to the stock condition in OpenFOAM.com, use symmetry for scalar fields instead.
• Patch type symmetry only. symmetryPlane and other patch types are rejected.
• value is ignored on read. The initial patch value is always the stock symmetry evaluation of the internal field.

Tutorials

No tutorial uses the scalar version or secondOrder yes.

References

• Cardiff, P., Karač, A., De Jaeger, P., Jasak, H., Nagy, J., Ivanković, A., Tuković, Ž., An open-source finite volume toolbox for solid mechanics and fluid-solid interaction simulations, arXiv:1808.10736, 2018.
• Alkfri, A., Habes, C., Marschall, H., multiRegionFoam: A Unified Multiphysics Framework for Multi-Region Coupled Continuum-Physical Problems, Engineering with Computers, 2023. DOI: 10.1007/s00366-024-01974-4