Docs: solidStVK

Mon, 01 Jan 0001 00:00:00 +0000

Overview

solidStVK solves the transient momentum balance for an elastic solid undergoing large deformations. It uses the St. Venant-Kirchhoff hyperelastic law and a total Lagrangian formulation: the unknown is the total displacement D, and every equation is assembled on the undeformed (reference) mesh, which is never moved. The discretisation follows the segregated finite-volume solid-mechanics approach of foam-extend (Cardiff et al.). It uses an implicit “stiffness” Laplacian with explicit deferred correction, and a Rhie-Chow-type momentum stabilisation. The region type is built to act as the structural partner of a fluid region, coupled through fsiInterface. On its own it does not solve anything (see Known Limitations).

Governing Equations

Momentum (total Lagrangian)

The equation assembled in setCoupledEqns() is

$$ \rho \frac{\partial^2 \mathbf{D}}{\partial t^2} = \underbrace{\nabla \cdot \left( K \nabla \mathbf{D} \right)}_{\text{implicit}} - \underbrace{\nabla \cdot \left( K \nabla \mathbf{D} \right)}_{\text{explicit}} + \nabla \cdot \left( J\, \mathbf{F}^{-1} \cdot \boldsymbol{\sigma} \right) + \mathbf{S}_\text{stab} $$

with the implicit stiffness $K = 2\mu + \lambda$. The implicit and explicit Laplacians cancel at convergence, so the converged solution satisfies $\rho\,\ddot{\mathbf{D}} = \nabla_0 \cdot \mathbf{P}^T$ with the first Piola-Kirchhoff stress $\mathbf{P} = J \boldsymbol{\sigma} \mathbf{F}^{-T}$. No body force (gravity) term is included.

Kinematics

$$ \mathbf{F} = \mathbf{I} + (\nabla \mathbf{D})^T, \qquad J = \det \mathbf{F} $$

St. Venant-Kirchhoff constitutive law

$$ \mathbf{C} = \mathbf{F}^T \mathbf{F}, \qquad \mathbf{E} = \tfrac{1}{2}\left( \mathbf{C} - \mathbf{I} \right), \qquad \mathbf{S} = 2\mu\,\mathbf{E} + \lambda\,\mathrm{tr}(\mathbf{E})\,\mathbf{I} $$ $$ \boldsymbol{\sigma} = \frac{1}{J}\, \mathbf{F}\, \mathbf{S}\, \mathbf{F}^T $$

Lamé parameters

$$ \mu = \frac{E}{2(1+\nu)}, \qquad \lambda = \begin{cases} \dfrac{\nu E}{(1+\nu)(1-2\nu)} & \text{plane strain / 3-D} \\[2ex] \dfrac{\nu E}{(1+\nu)(1-\nu)} & \text{plane stress (\texttt{planeStress yes})} \end{cases} $$

Symbol	Description	Unit
$\mathbf{D}$	Total displacement	m
$\rho$	Solid density	kg/m³
$E$	Young’s modulus	Pa
$\nu$	Poisson’s ratio	–
$\mu,\ \lambda$	Lamé parameters	Pa
$K$	Implicit stiffness $2\mu+\lambda$ (`impK`, `impKf`)	Pa
$\mathbf{F}$	Deformation gradient	–
$J$	Jacobian $\det\mathbf{F}$	–
$\mathbf{E}$	Green-Lagrange strain	–
$\mathbf{S}$	2nd Piola-Kirchhoff stress	Pa
$\boldsymbol{\sigma}$	Cauchy stress (`sigma`)	Pa
$\mathbf{S}_\text{stab}$	Momentum stabilisation term	Pa/m

Momentum stabilisation

$\mathbf{S}_\text{stab}$ is evaluated explicitly on the displacement increment DD $= \mathbf{D} - \mathbf{D}^\text{old}$ and its gradient gradDD. Its diffusivity is $\gamma = s\,K$, where $s$ is the scale factor, interpolated to faces as $\gamma_f$:

`type`	Term
`RhieChow` (default)	$\nabla\cdot(\gamma_f \nabla \mathbf{DD}) - \nabla\cdot\left(\gamma_f\, \mathbf{S}_f \cdot (\nabla\mathbf{DD})_f\right)$, i.e. compact minus non-compact Laplacian
`JamesonSchmidtTurkel`	$-\nabla\cdot\left(\lvert\mathbf{S}_f\rvert\, \nabla \left[\nabla\cdot(\gamma_f \nabla \mathbf{DD})\right]\right)$ (fourth-order damping)
`Laplacian`	$\nabla\cdot(\gamma_f \nabla \mathbf{DD})$
`none`	$\mathbf{0}$

On faces where $K$ differs between owner and neighbour (material interfaces), $\gamma_f$ is replaced by interfaceScaleFactor·$\tfrac{1}{2}(K_\text{own}+K_\text{nei})$. On coupled non-processor patches this replacement is applied to every face.

Numerical Algorithm

usesPIMPLE() returns false. solveRegion(), prePredictor(), momentumPredictor(), pressureCorrector() and meshMotionCorrector() are all no-ops. selfCoupledFieldNames() is not overridden, so it returns an empty list. The displacement equation is solved only when D appears in the DNA (partitioned) coupled field list. In that case each DNA iteration of multiRegionSystem::solve() runs:

solvePIMPLE() on the fluid region(s). D is a special FSI field, and stand-alone PIMPLE is skipped whenever D is DNA-coupled.
assembleAndSolveEqns<fvMatrix, vector>("D"). For solidStVK this is a Picard loop:
1. setCoupledEqns("D") stores D.prevIter() and assembles the momentum equation above. It registers the equation as D<mesh>MeshsolidStVKTypeEqn.
2. The equation is relaxed (relaxationFactors.equations) and solved.
3. postSolve() relaxes the field D (relaxationFactors.fields). It then updates DD, gradD = fvc::grad(D), gradDD, F, Finv, J, sigma, interpolates D to pointD (volPointInterpolation), updates pointDD and sets the solid velocity U = fvc::ddt(D).
4. The loop repeats until the region’s PICARD.D criteria are met (maxCorr, tolerance, relTol). If no PICARD entry is given, it runs one pass per DNA iteration.
meshMotionCorrector() on all regions. The fluid mesh follows pointD of the solid (via interfaceTrackingFvMesh). Then interfaces_->update().
DNA convergence is checked with the fsiInterface residual: the mismatch between the deformed solid interface points (points + pointD) and the fluid interface points.

correct(), which runs from regionTypeList::preSolve() once per time step, only calls D.correctBoundaryConditions().

getMinDeltaT() returns GREAT, so the solid never limits the time step.

Sub-Models

Sub-model	Selection mechanism	Notes
Momentum stabilisation	`stabilisation` sub-dict in `mechanicalProperties`	`none`, `RhieChow`, `JamesonSchmidtTurkel`, `Laplacian`

No fvOptions, thermal-stress, plasticity or other constitutive models are available. The material law is fixed to St. Venant-Kirchhoff.

Required Fields

Field	Type	Description	Unit
`D`	`volVectorField`	Total displacement (must be read)	m

Created at runtime. All of these are read if present and auto-written unless stated otherwise:

Field	Type	Description	Unit	Default
`DD`	`volVectorField`	Displacement increment over the time step	m	0
`gradD`	`volTensorField`	Displacement gradient	–	0
`gradDD`	`volTensorField`	Gradient of displacement increment	–	0
`F`	`volTensorField`	Deformation gradient	–	`I`
`Finv`	`volTensorField`	Inverse deformation gradient	–	`I`
`J`	`volScalarField`	Jacobian	–	1
`sigma`	`volSymmTensorField`	Cauchy stress	Pa	0
`U`	`volVectorField`	Solid velocity $\partial\mathbf{D}/\partial t$	m/s	0
`impK`	`volScalarField`	Implicit stiffness $2\mu+\lambda$	Pa	uniform
`impKf`	`surfaceScalarField`	Face implicit stiffness	Pa	uniform
`rImpK`	`volScalarField`	$1/K$ (read if present, not written)	1/Pa	$1/K$
`pointD`	`pointVectorField`	Displacement interpolated to points	m	from `D`
`pointDD`	`pointVectorField`	Point displacement increment	m	from `DD`

pointD is the field that fsiInterface, interfaceTrackingFvMesh and the solidPointDisplacement function object look up. Its presence is also how fsiInterface tells which side of the interface is the solid.

Case Setup

Required dictionaries

constant/<solid>/mechanicalProperties – rho, E, nu, planeStress, optional stabilisation
constant/<solid>/dynamicMeshDict – staticFvMesh (the solid mesh is never moved)
system/<solid>/fvSolution – D solver, relaxation, PICARD loop for D
system/<solid>/fvSchemes – d2dt2, ddt, grad and the named laplacian schemes

`mechanicalProperties`

planeStress no; // required; yes => plane-stress lambda

rho rho [1 -3 0 0 0 0 0] 1000; // required
nu nu [0 0 0 0 0 0 0] 0.4; // required
E E [1 -1 -2 0 0 0 0] 5.6e6; // required

// Optional. If absent: { type RhieChow; scaleFactor 0.1; }
stabilisation
{
 type RhieChow; // none | RhieChow | JamesonSchmidtTurkel | Laplacian
 scaleFactor 0.1; // required unless type none or useCellZones yes
 useCellZones false; // default false; then <cellZoneName>ScaleFactor per zone
 interfaceScaleFactor 0.01; // default 0.01; used on material-interface faces
}

`multiRegionProperties` entry

regions
(
 ( fluid (pimpleFluid) )
 ( solid (solidStVK) )
);

DNA
{
 D
 {
 maxCoupleIter 100;

 residualControl
 {
 maxIntRes 1e-7;
 outputIntResField yes;
 }
 }
}

solidStVK can be stacked with other region types on the same mesh, e.g. ( solid (solidStVK conductTemperature) ) in heatedBeamInCrossFlow.

Key `fvSolution` settings

solvers
{
 "D|DD"
 {
 solver PCG;
 preconditioner FDIC;
 tolerance 1e-09;
 relTol 0.1;
 }
}

relaxationFactors
{
 fields
 {
 "DD|D" 0.9; // applied to D in postSolve()
 }
}

PICARD
{
 D
 {
 maxCorr 1000; // Picard (outer) corrections per DNA iteration
 tolerance 1e-09;
 relTol 1e-09;
 infoFrequency 100;
 }
}

Key `fvSchemes` settings

The momentum equation and the RhieChow stabilisation use the named Laplacian scheme laplacian(DD,D):

d2dt2Schemes { default backward; }
ddtSchemes { default backward; }
gradSchemes { default leastSquares; }
divSchemes { default Gauss linear; }

laplacianSchemes
{
 default none;
 laplacian(DD,D) Gauss linear corrected;
}

interpolationSchemes { default linear; } // also used for interpolate(impK)

With default none, the JamesonSchmidtTurkel stabilisation also needs JSTinner and JSTouter Laplacian entries. The Laplacian stabilisation needs an entry for its default scheme name.

Boundary conditions for `D`

Patch role	BC type
FSI interface (solid side)	`regionCoupledTraction` (with `interfaceType (fsiInterface)`)
Clamped	`fixedValue`
Symmetry plane	`solidSymmetry`
Prescribed traction / pressure (uncoupled)	`solidTraction`

interface
{
 type regionCoupledTraction;
 neighbourFieldName sigma;
 interfaceType (fsiInterface);
 secondOrder false;
 nonOrthCorr false;
 value uniform (0 0 0);
}

regionCoupledTraction and solidTraction look up impK, rImpK, gradD, sigma and Finv from the solid region. All of these are provided by solidStVK.

Coupling

Interface type	Exchanged fields	Coupling modes
`fsiInterface`	Fluid traction `sigma·n` → solid (`regionCoupledTraction` on `D`); solid `pointD` → fluid mesh motion (`interfaceTrackingFvMesh`)	partitioned (DNA)
`heatTransferInterface`	`T`, heat flux (through a stacked `conductTemperature`, not `solidStVK` itself)	partitioned (DNA)

Notes on fsiInterface with this region type:

The interface mapping is built with the solid mesh temporarily moved to its deformed configuration (points + pointD). The mesh is then moved back, so the solid stays total Lagrangian.
The fluid traction is mapped to the solid patch and its sign is flipped. The deformed normal is computed as $\mathbf{n} = \mathbf{F}^{-T}\mathbf{n}_0 / |\cdot|$.
The interface residual for DNA convergence is the point-position mismatch between the deformed solid patch and the fluid patch.

Known Limitations

Requires DNA coupling of D — solveRegion() is a no-op and selfCoupledFieldNames() returns an empty list. If D is not listed in a partitionedCoupledPatches coupledFields entry (and in the DNA dict), the displacement equation is never solved. Single-region (stand-alone solid) runs are therefore not supported.
Partitioned only — there is no monolithic FSI path (monolithic coupling is disabled in multiRegionSystem).
St. Venant-Kirchhoff only — linear elastic in Green strain. It is not suited to large compressive strains. No other material laws, thermal expansion, or body forces (gravity) are implemented.
Constant, uniform material properties — rho, E, nu are single dimensionedScalars read from mechanicalProperties.
No time-step control — getMinDeltaT() returns GREAT.
Static solid mesh — meshMotionCorrector() is a no-op. The staticFvMesh reference configuration is used throughout, and deformation is represented only through D/pointD.
Unused code — solidStVKDdtCoeffs.H (ddt-scheme coefficients) and the threeK member are present but not used anywhere.

Tutorials

Tutorial path	Description
`tutorials/fluidStructureInteraction/HronTurekFsi3`	Hron-Turek FSI3 benchmark (pimpleFluid + solidStVK)
`tutorials/fluidStructureInteraction/beamInCrossFlow`	Elastic beam in cross-flow (pimpleFluid + solidStVK)
`tutorials/fluidStructureInteraction/heatedBeamInCrossFlow`	Heated elastic beam in cross-flow; FSI plus conjugate heat transfer (pimpleFluid + transportTemperature / solidStVK + conductTemperature)

References

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

multiPhysicsFoam – Nonlinear-Elasticity