Solid–Solid Interaction

Overview

The InteractionSolidSolid class models short-range repulsive forces between solid particles (grains) immersed in a fluid. Without such forces, solid particles can overlap during simulation, causing numerical artifacts in sintering and sedimentation simulations.

Defined in openphase/include/FluidDynamics/InteractionSolidSolid.h.

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Class declaration

class InteractionSolidSolid : public OPObject
{
public:
    InteractionSolidSolid(Settings& locSettings,
                          std::string InputFileName = DefaultInputFileName);

    void Initialize(Settings& locSettings,
                    std::string ObjectNameSuffix = "") override;

    void Calculate(PhaseField& Phase,
                   const BoundaryConditions& BC,
                   const std::function<double(int,int,int)>& MassDensity,
                   const double dt) const;
};

The Calculate method iterates over all solid grain pairs and applies a repulsive force contribution to the grain velocity via PhaseField grain data.

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Interaction models

Two models are available, selected by the Model parameter:

enum class SolidSolidInteractionModel
{
    Standard,  ///< ICAMS-developed model
    Wang,      ///< Wang (2006) sintering model
};

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Standard model

The Standard model computes a repulsive force between overlapping solid interfaces based on the local phase-field gradient. The force is proportional to the overlap (measured by the phase-field values) and the user-defined interaction strength:

$$\begin{array}{}\text{(1)}& {\mathbf{F}}_{ij}(\mathbf{x})=\text{strength}\cdot {\varphi }_i(\mathbf{x}){\varphi }_j(\mathbf{x})\mathrm{\nabla }{\varphi }_i(\mathbf{x})\end{array}$$

where ${\varphi }_i$, ${\varphi }_j$ are the phase-field order parameters of grains $i$ and $j$.

The order parameter controls the polynomial power of the overlap term (default 2). The cutoff parameter limits the interaction to a finite interface layer.

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Wang model

The Wang model is based on the formulation from:

M. Wang (2006), Computer modeling and simulation of solid-state sintering: A phase field approach. Acta Materialia.

It computes an elastic contact force proportional to the grain-pair overlap, weighted by the local mass density:

$$\begin{array}{}\text{(2)}& {\mathbf{F}}_{ij}\propto \text{strength\_wang}\cdot {\varphi }_i{\varphi }_j\cdot \mathrm{\nabla }({\varphi }_i+{\varphi }_j)\end{array}$$

The MassDensity callable passed to Calculate provides the local density at each node, enabling density-weighted force computation.

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Parameters

Parameter	Type	Description
Model	SolidSolidInteractionModel	Standard or Wang
strength	double	Interaction strength (Standard model) [Pa]
strength_wang	double	Interaction strength (Wang model) [Pa]
order	int	Polynomial order of overlap (Standard model)
cutoff	int	Cutoff distance in grid nodes
elastic	vector<vector<double>>	Per-phase elastic moduli
PhaseAggregateStates	vector<AggregateStates>	Identifies which phases are solid

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Applicability check

The class applies interactions only to grains that satisfy all of:

• grain.Exist == true
• grain.State == AggregateStates::Solid
• grain.Volume > epsilon

Grains that are too small or not in solid state are skipped.

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Typical usage

InteractionSolidSolid ISS(OPSettings, InputFile);

// Provide a density function (e.g., from FlowSolverLBM)
auto massDensity = [&](int i, int j, int k) {
    return FL.Density(Phase, i, j, k);
};

// each timestep, after FlowSolverLBM::Solve():
ISS.Calculate(Phase, BC, massDensity, dt);

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Numerical considerations

• The interaction strength must be tuned relative to the fluid forces and viscosity to prevent unphysical overlap without over-constraining the dynamics.
• Very high strength values combined with large time steps can cause instability.
• The Wang model is better suited for sintering simulations where grain contact mechanics follow an elastic-contact law.
• For fluid-only simulations without solid particle collision, InteractionSolidSolid is not needed and should not be instantiated.