Geometric Flows

Motivation

Computational meshes driving aerospace simulation pipelines are contaminated by non-manifold edges, silver triangles, self-intersections, holes, and noise. Such anomalies break downstream solvers including the immersed-boundary geometries processed by exascale codes such as MFC (Wilfong et al., 2026), PeleLMeX (Esclapez et al., 2023), and CharLES (Cadence).

Geometric flows (mean curvature flow, Willmore flow, and conformal curvature flow) offer a principled, mathematically accurate alternative. Each is a parabolic PDE evolving the mesh embedding $X(t)$ via the discrete Laplace–Beltrami operator $L$. Adoption in scientific computing has been bottlenecked by sparse-solver throughput on irregular meshes. This project investigates whether we can turn geometric-flow mesh repair into a routine, in-situ stage of CFD workflows.

Governing Flows

Each flow reduces to a symmetric positive-definite sparse linear solve per time step.

Conventions: $H = \kappa_1 + \kappa_2$ (sum of principal curvatures), $\mathbf{n}$ is the outward unit normal, and $L$ denotes the (negative) cotangent Laplace–Beltrami matrix so that $\Delta_g X = -H\mathbf{n}$ and $M - \tau L$ is SPD.

Mean curvature flow

$$\partial_t X = -H\,\mathbf{n} = \Delta_g X$$

Willmore flow (variational)

$$\partial_t X = -\nabla_{L^2}\!\int H^2\,dA$$

Willmore flow (explicit, 4th order)

$$\partial_t X = -\bigl[\Delta_g H + 2H(H^2 - K)\bigr]\,\mathbf{n}$$

$K$ is the Gauss curvature; the bracketed term is the Euler–Lagrange expression of the Willmore energy.

Time Stepping

Semi-implicit Euler is used for mean curvature flow, and BDF2 in curvature space (Crane et al., 2013) for the fourth-order Willmore flow.

Semi-implicit Euler

$$(M - \tau L)\,X^{n+1} = M X^{n}$$

BDF2 (curvature space)

$$\tfrac{3}{2}\,\kappa^{n+1} - 2\,\kappa^{n} + \tfrac{1}{2}\,\kappa^{n-1} = \tau\,\Delta_g\,\kappa^{n+1}$$

Approach

Unified differential-geometry filters built around the cotangent Laplace–Beltrami operator with intrinsic-Delaunay edge flips for robustness.
Symmetric positive-definite sparse solves accelerated with preconditioned Conjugate Gradient (Jacobi, block-Jacobi, IC(0)) and algebraic multigrid via Hypre/PETSc.
Hybrid MPI+OpenMP parallelism on a ParMETIS-partitioned dual mesh graph, with greedy graph coloring to vectorize per-vertex Laplacian assembly under SVE predicated gather/scatter.

Software

Libraries: libigl (cotangent Laplacian reference), geometry-central (halfedge data structures, intrinsic-Delaunay flips), CGAL (exact-arithmetic ground truth for self-intersection), PETSc+Hypre (KSP/PC infrastructure), ParMETIS, and Eigen.

Profiling: Nsight Systems, Linux perf, and Google Benchmark.