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Solving the Incompressible Navier-Stokes Equations on Oriented Curved Surfaces Discretized by Point Clouds
Alejandra Foggia, Ivo F. Sbalzarini
TL;DR
Curved-surface incompressible Navier–Stokes simulation is difficult because geometry changes the differential operators and pressure–velocity coupling. The paper combines meshfree surface DC-PSE with EDAC to solve these equations locally and explicitly on oriented point-cloud surfaces, demonstrating consistent high-order approximations and simulations on general curved geometries. The method is bounded by a non-intersecting tubular-neighborhood requirement and homogeneous point-density implementation.
Problem
Curved surfaces complicate differential operators and incompressibility enforcement, motivating numerical methods that handle geometry and pressure–velocity coupling without cumbersome global solves.
Method
The method combines meshfree surface DC-PSE for intrinsic differential operators with EDAC for local artificial-compressibility pressure–velocity coupling on oriented surface point clouds.
Results
The combined method solves INS equations on general oriented curved surfaces using explicit time stepping, with improved accuracy from consistency orders higher than two and demonstrated robustness on asymmetric surfaces.
Takeaways & Limitations
Surface DC-PSE combined with EDAC provides a meshfree approach for curved-surface fluid simulations, including geometries represented directly by surface point clouds.
Takeaways & Limitations
The solver is limited to oriented surfaces with a non-intersecting tubular neighborhood, and homogeneous density forces resolution to meet the highest-curvature region's requirement.
Abstract
from arXiv · showhide
We present a meshfree numerical solver for the incompressible Navier-Stokes equations on oriented curved surfaces that are represented by surface point clouds. On curved surfaces, numerical challenges pertaining to stiffness and pressure-velocity coupling are exacerbated. Moreover, vector calculus on curved surfaces differs from its Euclidean counterpart. The presented method operates on surface point clouds in an Eulerian frame of reference without requiring a computational grid or mesh. It achieves consistent approximation in space and time with high order of accuracy; we demonstrate up to order six. The incompressibility constraint is locally imposed as a weak artificial compressibility approximation, avoiding global matrix inversion. We show that the method provides consistent and convergent approximations of surface vector fields and differential operators. We study the relationship between error, spatial resolution, and artificial Mach number and characterize the frequency spectrum of the artificial oscillations. We provide numerical solutions of the incompressible Navier-Stokes equations on symmetric surfaces, such as the sphere and torus, and on parametric and non-parametric asymmetric surfaces. Since the proposed method works directly on unstructured surface point clouds, it provides a promising approach for simulations on image-derived geometries, such as in biological morphogenesis from microscopy videos.
1 Introduction
Surface-fluid dynamics couple curved geometry with incompressible Navier–Stokes modeling, creating challenges for differential operators, geometry representation, and pressure–velocity coupling. The paper proposes a meshfree Eulerian method combining surface DC-PSE with EDAC for oriented surface point clouds.
- Motivation: Curved-surface fluid models require differential-geometric operators because surface fields live in tangent spaces and depend on local curvature.Covariant derivatives differ from Euclidean operators, while curvature couples geometry and dynamics.
- Challenges: Non-parametric surfaces complicate local coordinate, metric, and normal-field construction, while embedding methods add numerical errors, extrapolation, or computational cost.These issues are especially relevant for irregular or dynamically deforming geometries.
- Challenges: Pressure projection imposes incompressibility through non-local implicit solves, hindering parallelization and requiring matrix inversions or preconditioners.Artificial-compressibility methods instead enable fully explicit schemes by evolving pressure.
- Contribution: The proposed solver combines meshfree surface DC-PSE operators with EDAC to approximate surface INS equations on oriented in-surface point clouds.The method uses local collocation rather than a computational grid or mesh.
- Evaluation: The paper evaluates operator consistency and numerical solutions across symmetric, parametric, and non-parametric curved surfaces.The study includes verification of surface operators and full surface INS simulations.
2 Covariant derivatives in the embedding space
Vector and tensor differentiation on curved surfaces requires comparing quantities in different tangent spaces. The paper handles this through parallel transport or embedding-space projection onto the tangent space.
- Covariant derivatives: Curved-surface derivatives require vectors or tensors at nearby points to be compared within a common tangent space.Parallel transport performs this comparison and is nontrivial on curved surfaces.
- Covariant derivatives: A local surface frame represents tangent-space coordinates, and covariant derivatives describe vector fields using that geometry.Christoffel symbols encode how basis vectors change along the surface.
- Embedding-space formulation: For embedded surfaces, surface derivatives can be computed by extending fields smoothly into the ambient Euclidean space and projecting Euclidean derivatives tangentially.This provides an embedding-space route to surface differential operators.
- Embedding-space formulation: The projection operator P = I − n ⊗ n extracts tangent components using the surface normal n.A closest-point extension constant along the normal satisfies ∇v · n = 0 at the surface.
3 Vector surface DC-PSE
Surface DC-PSE extends discretization-corrected particle operators to curved surfaces and vector or tensor fields. It uses virtual normal points, local moment conditions, and tangent projection to obtain consistent high-order surface derivatives.
- Method overview: Surface DC-PSE generalizes DC-PSE to curved surfaces, removing quadrature error on irregular point distributions through discrete reproducing moment conditions.The method also handles boundaries transparently and extends to vector and higher-rank tensor fields.
- Operator evaluation: After embedding-space differentiation, tangent projection yields order-r approximations of surface differential operators.For a three-dimensional surface gradient, nine Euclidean partial derivatives are approximated before projection.
- Surface extension: Virtual points placed along surface normals create a smooth closest-point extension within a non-intersecting tubular neighborhood.The extension copies each surface quantity componentwise without interpolation.
- Kernel construction: Surface operator evaluation uses neighboring surface and virtual points, with the surface kernel encoding curvature through local Taylor-based conditions.The neighborhood radius is defined from the average distance between a point and its neighbors.
- Kernel construction: Local linear systems determine kernel coefficients so discrete moments satisfy consistency conditions for the requested differential operator and accuracy order.The number of equations depends on desired accuracy, operator order, and embedding dimension.
4 EDAC
EDAC replaces the incompressibility continuity equation with a pressure evolution equation that locally couples pressure and velocity. Combined with surface DC-PSE, it provides an explicit method for low-Reynolds-number flows on point-cloud surfaces.
- EDAC formulation: EDAC solves incompressible Navier–Stokes equations without a global inversion by damping acoustic pressure waves through entropy generation.It replaces the continuity equation with a pressure evolution equation.
- EDAC formulation: The artificial Mach number Ma controls pressure response to incompressibility loss, and EDAC converges to INS with O(Ma2) errors.The stated regime is Ma < 0.8 for both high and low Reynolds numbers.
- Prior combinations: EDAC has previously been combined with SPH and DC-PSE for incompressible bulk flows and low-Reynolds-number porous-media and meniscus simulations.These prior combinations motivate its use with surface DC-PSE.
- Surface application: The paper combines EDAC with surface DC-PSE to solve surface INS equations on smooth orientable surfaces discretized by surface point clouds.The combination is studied at low Reynolds numbers, approximately Re ≈ 1.
5 Results
The method is evaluated through convergence tests for surface differential operators and incompressible Navier–Stokes solutions on flat, spherical, and general curved surfaces. Results show high-order spatial convergence, artificial-compressibility effects controlled by Mach number, and applicability to asymmetric geometries.
- 5.1 Convergence of the vector Bochner Laplacian: Surface DC-PSE vector operators converge at orders r = 2, 3, 4, 5, 6, with slopes 2–15% below theoretical bounds.For r = 5 and 6, errors increase at the smallest resolutions because of numerical round-off.
- 5.1 Convergence of the vector Bochner Laplacian: The vector-field solution on the sphere converges toward v(θ, t) = Ψ10 e−t + Φ30 e−11t as spatial resolution increases.Antipodal error cancellation on the symmetric sphere likely explains slightly better empirical convergence than theoretically expected.
- 5.3 Traveling wave on a two-dimensional plane: EDAC-generated oscillations change frequency with artificial Mach number, scaling consistently with acoustic wave speed c ∼ 1/Ma, while spatial resolution leaves their frequency and behavior largely unchanged.The reported frequency ratios between Ma = 0.005 and Ma = 0.01, 0.05, 0.1 are 2, 10, and 20, respectively.
- 5.3 Traveling wave on a two-dimensional plane: Velocity achieves expected second-order spatial convergence only at Ma = 0.005 and 0.01, where discretization error dominates incompressibility error.Pressure convergence is reported as O(h1/2) and O(Ma−1), with decreasing Mach number amplifying sensitivity to velocity-divergence residuals.
- 5.3 Traveling wave on a two-dimensional plane: Reducing artificial Mach number is cheaper for viscous-dominated systems, whereas increasing spatial resolution is cheaper when inertial effects constrain stability.The trade-off depends on the physical regime and available computational effort.
- 5.4 Convergence of Navier–Stokes on the unit sphere: On the sphere, both velocity and pressure converge with spatial discretization, while orders r = 3 and 4 reduce error relative to r = 2.At the highest tested resolution, the r = 2 solution becomes unstable for times t > 0.1.
- 5.5 Incompressible fluids on general surfaces: The method also handles nonzero initial divergence on decic and peanut-shaped surfaces, where EDAC relaxes the continuity mismatch.The broader tests cover symmetric and asymmetric, analytical and non-analytical surfaces with non-uniform curvature and different topology.
6 Conclusion
The paper presents a meshfree surface INS solver that combines surface DC-PSE with EDAC for explicit, local computation on oriented curved point clouds. Verification and simulations support convergence, geometric robustness, and applicability across several curved surfaces, while identifying scope limitations.
- 6 Conclusion: Surface DC-PSE combined with EDAC provides a local, explicit-in-time solver for INS equations on oriented curved surface point clouds.The method uses unconnected points, local pressure-velocity coupling, and explicit time stepping.
- 6 Conclusion: The solver is limited to oriented surfaces with a non-intersecting tubular neighborhood and currently assumes homogeneous point density and codimension one.These constraints can increase computational effort in high-curvature regions.
- 6 Conclusion: The solver exhibits expected convergence for surface differential operators and diffusion equations, including the connection/Bochner Laplacian on the sphere.Diffusion tests also showed qualitatively correct behavior on the torus and decic surface.
- 6 Conclusion: Artificial oscillation frequency is proportional to 1/Ma and independent of spatial resolution, while excessively low Mach numbers pollute the solution with oscillations.The study also found that lower Mach numbers do not necessarily improve incompressibility at sufficiently high spatial resolution.
- 6 Conclusion: Higher-than-second-order surface DC-PSE operators consistently improve accuracy in full surface INS convergence tests on the unit sphere.Spatial discretization dominated the tested curved cases, while lower Mach numbers did not affect the solution or incompressibility.
- 6 Conclusion: Simulations on torus, decic, and peanut-shaped surfaces demonstrate qualitative agreement, geometric robustness, and handling of initially non-divergence-free velocity fields.The examples use explicit time stepping and include long simulation times.
A Covariant operators in the embedding space
This section formulates surface differential operators through smoothly extended scalar, vector, and tensor fields in the embedding space. It organizes gradients, divergences, and Laplacians using covariant projections and gives sphere-specific expressions for manufactured solutions.
- A Covariant operators in the embedding space: Surface operators are computed for smoothly extended scalar, vector, and tensor fields using indices in the embedding dimension.The construction covers scalar gradients, vector and tensor gradients, divergences, and Laplacians.
- A Covariant operators in the embedding space: The divergence of a tensor is obtained by tracing its covariant gradient, with the gradient itself computed from the preceding tensor-gradient relation.The section explicitly connects the divergence formula to the tensor gradient operator.
- A Covariant operators in the embedding space: The operator list includes scalar and vector Laplacians, including the Laplace–Beltrami and connection/Bochner Laplacians.These operators are presented alongside covariant gradients and divergences.
- A Covariant operators in the embedding space: For the sphere, metric components, Christoffel symbols, and covariant-operator expressions are specified for constructing manufactured solutions.The sphere is parametrized in R3 using angular coordinates θ and ϕ.
C Torus
The torus and asymmetric decic surfaces are defined through parametric embeddings and discretized with specified point-cloud resolutions and initial conditions. The setup supplies the geometric and field data used in the simulations.
- C Torus: The torus is an axis-aligned surface embedded in R3 with major radius R and minor tube radius r.Its angular coordinates describe rotation around the tube and around the symmetry axis.
- C Torus: The torus point cloud contains Np = NinNout points, with main-text parameters R = 2, r = 0.5, Nin = 100, and Nout = 400.The angular sampling uses the specified Nout-based discretization.
- C Torus: The torus velocity initial condition is a linear combination of divergence-free and curl-free harmonic functions.The same class of initial condition is used for the vector-valued equation and surface INS velocity.
- C Torus: The decic surface is parametrized by controls for stretching, compression, and asymmetry, with main-text values C = 1.1, r = 0.55, and B = 7/20C.The resulting surface lacks rotational symmetry and is symmetric only with respect to the xz-plane.
E Error at final time in the convergence of the surface INS equations
Final-time errors are compared using an extrapolated envelope because EDAC solutions oscillate in time. The procedure samples error peaks and fits their envelope to estimate a meaningful value at the final time.
- E Error at final time in the convergence of the surface INS equations: EDAC time oscillations make the L2 error phase-dependent at t = 0.1, so direct final-time comparisons can be misleading.The issue varies across Mach numbers, spatial resolutions, and operator orders.
- E Error at final time in the convergence of the surface INS equations: The analysis samples error peaks and uses a log-linear fit to extrapolate the oscillation envelope to the final time.For higher Mach numbers, oscillations are nearly absent.