Trilinear Interpolation and Particle-Grid Projection

Table of Contents

• 1. Grid -> Particle Interpolation
  • 1.1 Trilinear (Direct Cell-Center) — Default
  • 1.2 Corner-Averaged (Legacy)
  • 1.3 Shared Kernels
  • 1.4 Configuration
• 2. Particle -> Grid Scatter and Normalization
• 3. Accuracy and Stability Considerations
• 4. Related Pages
  • CFD Reader Guidance and Practical Use
    • What To Extract Before Changing A Case
    • Practical CFD Troubleshooting Pattern

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 source-point weights:

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

PICurv supports two grid-to-particle interpolation methods, selectable at runtime via the interpolation_method YAML key (or -interpolation_method flag).

1.1 Trilinear (Direct Cell-Center) — Default

The recommended method. For each particle, the 8 nearest cell centers are identified via octant detection (based on existing host-cell weights), forming a "dual cell." Trilinear interpolation is performed directly from these cell-centered field values to the particle position.

Key properties:

• Second-order accurate on both uniform and curvilinear grids.
• No intermediate staging: operates directly on the ghosted cell-centered field (lUcat, lDiffusivity, etc.) and ghosted cell center coordinates (lCent). No extra ghost exchange is needed beyond what the solver already provides.
• Boundary handling: at non-periodic boundaries, the dual-cell octant is clamped to use only physical cell centers, and trilinear weights are left unclamped (may exceed [0,1]) to provide second-order linear extrapolation. Weights still sum to 1.0.
• Periodic boundaries: ghost cell centers and field values are properly set by ApplyPeriodicCorrectionsToCellCentersAndSpacing() and DMGlobalToLocal with DM_BOUNDARY_PERIODIC, so no special handling is needed.

Code path: InterpolateEulerFieldToSwarm dispatches to InterpolateEulerFieldFromCenterToSwarm.

1.2 Corner-Averaged (Legacy)

The original two-stage method:

1. Center -> Corner: average surrounding cell-center values to each grid node via InterpolateFieldFromCenterToCorner (unweighted sum/count).
2. Corner -> Particle: standard trilinear interpolation from the 8 corner values of the host cell.

This is second-order on uniform Cartesian grids but degrades to first-order on curvilinear grids because the unweighted averaging does not account for the asymmetric placement of cell centers around nodes.

Code path: InterpolateEulerFieldToSwarm dispatches to InterpolateEulerFieldFromCornerToSwarm.

1.3 Shared Kernels

Both methods share the same underlying trilinear kernels:

• scalar kernel: TrilinearInterpolation_Scalar
• vector kernel: TrilinearInterpolation_Vector
• weight computation: ComputeTrilinearWeights
• field-by-field dispatcher: InterpolateEulerFieldToSwarm
• stage wrapper: InterpolateAllFieldsToSwarm

1.4 Configuration

YAML (solver.yml):

interpolation:
  method: "Trilinear"     # default; or "CornerAveraged"

C flag: -interpolation_method 0 (Trilinear) or -interpolation_method 1 (CornerAveraged).

Enum: InterpolationMethod in include/variables.h. Stored in SimCtx.interpolationMethod.

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.

4. Related Pages

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

CFD Reader Guidance and Practical Use

This page describes Trilinear Interpolation and Particle-Grid Projection within the PICurv workflow. For CFD users, the most reliable reading strategy is to map the page content to a concrete run decision: what is configured, what runtime stage it influences, and which diagnostics should confirm expected behavior.

Treat this page as both a conceptual reference and a runbook. If you are debugging, pair the method/procedure described here with monitor output, generated runtime artifacts under runs/<run_id>/config, and the associated solver/post logs so numerical intent and implementation behavior stay aligned.

What To Extract Before Changing A Case

• Identify which YAML role or runtime stage this page governs.
• List the primary control knobs (tolerances, cadence, paths, selectors, or mode flags).
• Record expected success indicators (convergence trend, artifact presence, or stable derived metrics).
• Record failure signals that require rollback or parameter isolation.

Practical CFD Troubleshooting Pattern

1. Reproduce the issue on a tiny case or narrow timestep window.
2. Change one control at a time and keep all other roles/configs fixed.
3. Validate generated artifacts and logs after each change before scaling up.
4. If behavior remains inconsistent, compare against a known-good baseline example and re-check grid/BC consistency.