Methods and Models Overview

Table of Contents

• 1. Governing Model Snapshot
• 2. Runtime Execution Order
• 3. Method Map
• 4. External Method References
• 5. How To Read These Pages
  • CFD Reader Guidance and Practical Use
    • What To Extract Before Changing A Case
    • Practical CFD Troubleshooting Pattern

This section maps PICurv's numerical methods to the code paths that execute each step. It is intended as the bridge between theory-level terminology and what the current codebase actually does.

1. Governing Model Snapshot

PICurv advances incompressible flow in non-dimensional form on curvilinear grids:

\[ \frac{\partial \mathbf{u}}{\partial t} + \nabla\cdot(\mathbf{u}\mathbf{u}) = -\nabla p + \frac{1}{Re}\nabla^2\mathbf{u} + \mathbf{f}, \qquad \nabla\cdot\mathbf{u}=0. \]

Operationally, the solver uses a projection workflow:

1. build momentum residuals,
2. advance momentum with selected strategy,
3. solve Poisson equation for pressure correction,
4. project velocity to divergence-free space,
5. execute particle coupling (if enabled).

2. Runtime Execution Order

At runtime, the top-level sequence is:

1. CreateSimulationContext parses generated control files.
2. InitializeEulerianState sets initial Eulerian fields.
3. FlowSolver advances one fluid step (or AnalyticalSolutionEngine for analytical mode).
4. Particle stage (when active): interpolation -> motion -> relocation -> physics -> scatter.

The method pages below document each major stage in detail.

3. Method Map

• CurvIB Method Overview: curvilinear grid/metric framework and immersed-boundary context.
• Fractional-Step (Projection) Method: predictor/projection incompressible update.
• Dual-Time Picard RK4 Momentum Solver: implicit-in-physical-time momentum iteration.
• Pressure-Poisson, GMRES, and Multigrid: Poisson assembly and PETSc multigrid/KSP path.
• Walking Search for Particle Location: particle cell location and migration orchestration.
• Trilinear Interpolation and Particle-Grid Projection: Eulerian-Lagrangian field exchange.
• IEM Mixing and Statistical Averaging: particle micromixing and post statistics kernels.
• Momentum Solver Implementations: supported momentum-solver options and dispatch status.
• Analytical Solution Modes: analytical Eulerian modes and geometry policies.
• Initial Condition Modes: Eulerian and particle initialization modes.
• Particle Model and Coupling Overview: end-to-end particle model pipeline.
• Boundary Conditions Guide: boundary handler combinations and runtime enforcement path.
• Particle Initialization and Restart Guide: detailed particle seeding/restart/migration behavior.
• C Runtime Execution Map: code-level startup and timestep execution map.

4. External Method References

For readers connecting PICurv implementation to the CURVIB literature, these are the primary starting references:

• Borazjani I, Ge L, Sotiropoulos F. "Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies." Journal of Computational Physics 227(16), 7587-7620 (2008). DOI: 10.1016/j.jcp.2008.04.024.
• Borazjani I, Di Achille P, D'Souza RM, et al. "The functional role of left atrial flow in ventricular filling and flow evolution in the left ventricle." Annals of Biomedical Engineering 41(6), 1265-1275 (2013). DOI: 10.1007/s10439-013-0758-9.

Project context:

• PICurv project page: https://vishalkandala.me/projects/Picurv/

5. How To Read These Pages

Use each page with two goals:

1. theory alignment (which equation/model is being approximated),
2. implementation alignment (which function actually executes it now).

When behavior differs from classical textbook formulations, the implementation notes take precedence for this repository.

CFD Reader Guidance and Practical Use

This page describes Methods and Models Overview 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.