Trilinear Interpolation and Particle-Grid Projection

Table of Contents

• 1. Grid -> Particle Interpolation
• 2. Particle -> Grid Scatter and Normalization
• 3. Accuracy and Stability Considerations

Eulerian-Lagrangian coupling in PICurv is built from interpolation (grid -> particle) and scatter/normalization (particle -> grid).

1. Grid -> Particle Interpolation

For a particle inside a host cell with local coordinates \((a_1,a_2,a_3)\in[0,1]^3\), trilinear interpolation uses 8 corner weights:

\[ \phi_p = \sum_{m=1}^{8} w_m(a_1,a_2,a_3)\,\phi_m, \qquad \sum_m w_m = 1. \]

Code touchpoints:

• scalar kernel: TrilinearInterpolation_Scalar
• vector kernel: TrilinearInterpolation_Vector
• field-by-field wrapper: InterpolateEulerFieldToSwarm
• stage wrapper: InterpolateAllFieldsToSwarm

Implementation note: interpolation path includes center-to-corner staging before final particle evaluation.

2. Particle -> Grid Scatter and Normalization

Scatter computes per-cell sums, then normalizes by particle count:

\[ \bar{\phi}_{i,j,k} = \begin{cases} \dfrac{\sum_{p\in cell(i,j,k)} \phi_p}{N_{i,j,k}}, & N_{i,j,k}>0, \\ 0, & N_{i,j,k}=0. \end{cases} \]

Code touchpoints:

• particle count accumulation: CalculateParticleCountPerCell
• scatter orchestrator: ScatterAllParticleFieldsToEulerFields

Current standard scatter path actively maps particle Psi to Eulerian Psi; additional fields are scaffolded and can be enabled with matching vector/DM contracts.

3. Accuracy and Stability Considerations

Coupling quality depends on:

• accurate host-cell IDs and interpolation weights,
• ghost synchronization before interpolation/scatter,
• consistent field DOF/DM association,
• avoiding stale particle-count vectors between scatter calls.

Inconsistency in any of these usually appears as noisy particle statistics or nonphysical reconstructed Eulerian fields.

1. Reference Scales

• Walking Search for Particle Location
• IEM Mixing and Statistical Averaging
• Particle Model and Coupling Overview