A boundary-integrated closure for renormalized, least-squares-corrected meshless operators on elliptic boundary-value problems.
FIRM is a meshless discretization framework for incompressible flow that folds the wall (Neumann) and free-surface (Robin/Dirichlet) boundary conditions directly into the differential operators — algebraically, in strong form, on a single code path — instead of padding the domain with auxiliary particles, embedding it in a level set, or enforcing the condition through a weak form. The boundary is detected from the point cloud itself, with no tuning parameters.
The method rests on three principles:
- Trust the neighbors for what they can see; impose physics for what they cannot.
- All projections are performed in physical space on the raw offset vector
oᵢ, before renormalization byBᵢ. - The boundary-condition type — not the physical quantity — decides where the known datum enters: a Neumann flux goes to the right-hand side; a Dirichlet/Robin value goes to the diagonal.
This repository collects the idea, a bottom-up numerical validation suite, and the manuscript describing the closure.
| Path | Contents |
|---|---|
docs/spec.md |
The FIRM method specification (v3) — the single source of truth: operators, boundary closure, free-surface detection, the unified pressure-Poisson equation, validation summary, known limitations, and future directions. |
firm/ |
The validation suite: the operators (firm_core.py), Poisson assembly/solve, the GFDM baseline, the benchmark battery, and ~20 substep tests each checked against an independent analytic ground truth, plus a manufactured-Poisson capstone. See firm/README.md. |
firm/figures/ |
Figures and paper_numbers.json consumed by the manuscript (\graphicspath points here). |
paper/ |
The manuscript firm_boundary_closure.tex, its bibliography firm.bib, and a compiled PDF. |
Renormalized operators. Each particle uses a least-squares-corrected (renormalization
tensor Bᵢ = Mᵢ⁻¹) gradient and Laplacian that are linear-exact on any point cloud —
regular or highly irregular — to round-off.
Boundary closure, organized by condition type:
- A Neumann flux is imposed by a tangential projection performed in physical space before renormalization, entering the right-hand side. A general Gram projector treats non-orthogonal corners exactly.
- A complementary algebraic-ghost closure completes the truncated boundary stencil by reflecting the neighbours and the node itself across the local reflection group — no extra unknowns. It is the LeVeque/ghost-node cure cast algebraically.
- A Dirichlet/Robin value enters the diagonal (self-interaction) term.
- The free surface is detected from the support-deficiency geometry; its enforcement strength cancels to the total kernel weight, removing any penalty or threshold parameter.
What the validation establishes (honest summary):
- Gradient and Laplacian are linear-exact (~1e-14) on any cloud, including truncated
boundary rows; the projector algebra (idempotent, symmetric,
P·nₖ = 0) holds exactly. - The value/free-surface closure supraconverges; the free-surface region is the most accurate part of the capstone.
- The flux-only Neumann wall is the accuracy limiter; the algebraic-ghost closure cuts the wall-limited error by ~an order of magnitude and, with the sum normalization, approaches second-order at a straight Neumann boundary. Non-orthogonal corners cap the attainable order (~1.3).
- Used in a Helmholtz–Hodge decomposition, the renormalized operators are consistent for a single projection — the residual divergence equals the operators' non-adjointness defect and vanishes under refinement.
See docs/spec.md §9 and the manuscript for the quantified results.
Requires Python 3 and NumPy. SciPy (neighbour search + sparse solve) and Matplotlib (plots) are optional — the suite falls back to NumPy without them.
cd firm
python3 run_all.py # full suite: per-check PASS/FAIL + summary (exits nonzero on any fail)
python3 run_all.py --plot # + capstone figures (convergence, error field)
python3 tests/test_03_laplacian.py # any substep standalone
pytest tests/ # the substeps are pytest-discoverable too
# Manuscript artefacts:
python3 paper_benchmarks.py [b1 nb b2 b3 b5] [--full] # FIRM-vs-GFDM benchmark battery
python3 paper_figures.py [--quick] # paper figures -> figures/firm/README.md documents every module, the locked conventions (kernel,
normalization, projection order), and the full findings.
The figures it references are committed under firm/figures/, so it builds out of the box:
cd paper
pdflatex firm_boundary_closure
bibtex firm_boundary_closure
pdflatex firm_boundary_closure
pdflatex firm_boundary_closureA pre-built firm_boundary_closure.pdf is included. To regenerate the figures, run the
paper_figures.py / make_figures.py scripts in firm/.
- Validated: the renormalized operators, the boundary closure (Neumann projection + Gram corner + algebraic ghost), free-surface detection, and the elliptic/Poisson pipeline — verified against analytic ground truths and irregular-domain benchmarks in 2D.
- Not yet validated: the velocity operators near boundaries and the implicit-viscosity Helmholtz system (spec §6–7) are derived but not exercised by the current test suite, and are flagged as such.
FIRM is a research idea under active development — interfaces and results may change.
If you use this work, please cite the manuscript:
J. Bašić and C. Peng, A boundary-integrated closure for renormalised meshless operators on elliptic boundary value problems.
- Josip Bašić — Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Croatia
- Chong Peng — School of Civil Engineering, Southeast University, Nanjing, China
No license has been chosen yet, so all rights are reserved by default. Please contact the authors before reusing or redistributing this work.