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A meshfree solver for coupled bulk-surface problems with self-organizing surface geometry
Lennart J. Schulze, Alejandra Foggia, Ivo F. Sbalzarini
TL;DR
Coupled bulk–surface problems require simultaneous treatment of deforming geometry, surface PDEs, and surrounding fluid dynamics. The paper presents a fully meshfree solver using implicit geometric tracking and high-order particle operators, demonstrating convergence, oscillating-droplet reaction–diffusion, and two-way-coupled morphogenesis. Large-deformation simulations produce dynamically evolving organic shapes, while the method’s geometry requires a non-intersecting tubular neighborhood.
Problem
Deformable surfaces with intrinsic dynamics and surrounding bulk-fluid coupling require numerical methods that resolve both evolving geometry and coupled surface and bulk equations.
Method
The solver combines implicit level-set surface tracking, PCP geometry, surface DC-PSE operators, particle regularization, and coupled bulk-fluid discretization in a meshfree framework.
Results
The solver shows convergence for spherical-harmonic interpolation and mass conservation on a growing sphere, reproduces reaction–diffusion patterns on an oscillating droplet, and generates evolving organic shapes in a two-way-coupled morphogenetic model.
Takeaways & Limitations
The approach supports high-order Lagrangian simulation of nonlinear surface dynamics on deforming surfaces coupled with bulk fluids.
Takeaways & Limitations
Surface DC-PSE requires a non-intersecting tubular neighborhood at least as wide as the narrow band used for surface representation.
Abstract
from arXiv · showhide
In many systems, the interaction between a deformable surface or interface and the surrounding bulk fluid is coupled with intrinsic spatiotemporal dynamics within the moving surface. Examples include tumor growth, biological tissue morphogenesis, cardiac mechanics, multi-phase surfactant chemistry, additive manufacturing, clothing wear-and-tear, and reactive combustion flows. Solving such problems requires both geometric computing algorithms to track and resolve the surface and numerical methods to solve the coupled governing equations in the surface and the surrounding bulk phase. Here, we present a fully meshfree numerical solver for such coupled bulk-surface problems with deformable interfaces. The presented solver tracks the surface implicitly, solving for the dynamic surface geometry based on stress balance coupled to surrounding fluid phases. We show convergence for a mass-conserving case on a growing sphere and solve bulk-surface problems with incompressible Navier-Stokes fluids coupled to in-surface nonlinear reaction-diffusion dynamics. Finally, we show a model of biological morphogenesis, solving simultaneously for the dynamic surface shape and the fields on the curved surface with two-way coupling.
1 Introduction
The paper targets coupled bulk–surface systems in which deformable surfaces co-evolve with intrinsic surface dynamics and surrounding fluids. It develops a meshfree solver combining implicit geometric tracking with high-order surface operators and demonstrates convergence and coupled applications.
- Deformable surfaces can evolve from coupled in-surface and bulk stress balances rather than imposed boundary conditions.
- Applications include melt-pool Marangoni flows, tissue morphogenesis, cardiac development, and deformable thin-shell objects.Surface tension gradients can drive in-surface flows and alter melt-pool shape, while tissue processes can produce buckling and folding.
- Accurate solution of coupled bulk–surface PDEs requires high-order approximation of surface locations, normals, curvature or metric tensors, and spatial resolution.These geometric quantities vary in space and time and can otherwise dominate PDE discretization error.
- Existing approaches include mesh-based finite elements and meshfree collocation with explicit or implicit surface representations.Moving meshes support deformations but introduce mesh-generation and maintenance challenges in dynamic domains.
- The proposed solver combines particle closest-point geometry, surface DC-PSE operators, and periodic SAISS resampling in an entirely meshfree framework.The implementation uses OpenFPM for scalable parallel scientific computing.
- Demonstrations progress from convergence and conservation benchmarks to reaction–diffusion on an oscillating droplet and two-way-coupled morphogenesis.The morphogenetic model couples evolving surface shape with fields on the curved surface.
2 Physical model and governing equations
The model describes nonlinear reaction–diffusion on a moving closed surface coupled either to constitutive deformation laws or to surrounding bulk hydrodynamics. Surface geometry and concentration dynamics exchange information through transport, curvature, and stress-mediated deformation.
- The surface domain is a closed dynamic 2D manifold Γ(t) embedded in a 3D bulk domain Ω, carrying multiple reacting and diffusing species.Each species has its own diffusion constant, while reactions are generally nonlinear in the concentrations.
- Surface concentration evolves through diffusion, reaction, and a deformation term involving c[s]∇Γ·u.The material derivative is taken with respect to the moving and deforming surface.
- Tangential deformation advects surface concentration, whereas normal deformation contributes through local surface divergence and curvature.The formulation separates the tangential velocity uΓ from the normal component u⊥.
- Surface deformation is obtained either from a constitutive law or by solving coupled PDEs for incompressible bulk fluid phases.The bulk model includes viscosity, density, pressure, and surface tension acting at the interface.
- Surface tension is modeled as a curvature-dependent volumetric force using a smoothed surface delta function and constant tension coefficient.The force acts orthogonally to the surface and is proportional to the sum of principal curvatures.
- The coupling is two-way when surface variables affect deformation and deformation feeds back into surface dynamics; bulk hydrodynamics can instead provide one-way geometric forcing.The morphogenetic case uses concentration-driven shape change, while the hydrodynamic case determines the embedding space for surface dynamics.
3 Numerical method
The numerical method uses separate Lagrangian particle sets for the surface and bulk, with a semi-implicit geometric representation supporting meshfree PDE discretization. High-order geometry and surface operators are combined with particle regularization to preserve stability during deformation.
- Separate surface and bulk particle sets respectively solve surface reaction–diffusion and store level-set or bulk hydrodynamic fields.Both sets move with the material, and differential operators are evaluated in a Lagrangian frame.
- The semi-implicit representation uses explicit surface particles for PDEs and a local level-set description for geometry and bulk coupling.This combines explicit surface sampling with implicit geometric information.
- Bulk SPH advances density, velocity, pressure, and surface-tension interactions on particles initialized on a Cartesian grid.Bulk particles subsequently move with the computed fluid flow.
- The PCP method computes closest points, normals, and curvatures from local regression polynomials in surface patches.It requires bulk particles within a narrow tubular band around the surface for geometric reconstruction.
- Surface DC-PSE approximates intrinsic Laplace–Beltrami and divergence operators by orthogonally extending fields along surface normals.Virtual particles copy surface values, allowing bulk DC-PSE operators to produce consistent surface derivatives at desired convergence order.
- Particle stretching and clustering can destabilize surface operators, so SAISS periodically adds, removes, and redistributes particles toward the target resolution.Rapid deformation requires more frequent regularization, followed by interpolation of fields onto the new particle set.
- The predictor step can cause surface particles to drift from the zero level-set, although closest-point projection provides a no-additional-cost correction.The reported simulations did not require this correction.
4 Results
The solver is evaluated from operator convergence and mass conservation through coupled droplet dynamics and fully coupled morphogenesis. Results show accurate surface discretization, reaction-diffusion patterns on deforming surfaces, and the importance of particle regularity for stability and runtime.
- Overview: The C++ OpenFPM implementation combines convergence benchmarks with oscillating-droplet and two-way coupled morphogenesis simulations.The study proceeds from analytically tractable tests to nonlinear reaction-diffusion and emergent interface deformation.
- Operator convergence: 1.13, 2.06, and 3.12 are the empirical convergence orders for surface DC-PSE operators of orders 1, 2, and 3.The unit-sphere test uses Fibonacci-distributed particles and particle-to-particle interpolation to 256 target particles.
- Growing-sphere convergence: 1.26 and 1.45 are the empirical convergence rates for first- and second-order operators in the mass-conserved growing-sphere test.The maximum absolute errors are measured after 1000 time steps; second-order operators consistently produce smaller maximum errors.
- Oscillating droplet: Gray–Scott dynamics on the oscillating droplet produce lines, evenly distributed dots, or indefinitely moving, fusing, and splitting dots depending on the parameter pattern.The γ pattern reaches a line-dominated steady state, the λ pattern forms slower and yields dots, while the α pattern remains dynamic.
- Fully coupled morphogenesis: The morphogenesis model increases surface particles to 9766 for the α pattern and 8430 for the γ pattern as the surface evolves.The γ pattern develops roughly spherical lobes, whereas the α pattern produces a more irregular asymmetric shape; both differ qualitatively from the oscillating-droplet patterns.
- Particle regularity: Without periodic surface re-regularization, the simulation develops numerical instability and underestimates final shape growth by over 40%.Irregular particles produce non-smooth concentrations, PCP instability, and slower computation; with SAISS, PCP calls require only 26% of the time despite 16% more particles at t = 0.3.
5 Conclusions and outlook
The paper presents a meshfree solver for dynamically deforming bulk-surface systems, validates its convergence, and demonstrates one- and two-way coupled applications. Large deformations require periodic particle re-regularization, while tubular-neighborhood constraints and missing feedback to bulk hydrodynamics remain limitations.
- Conclusions and outlook: The solver combines semi-implicit surface geometry, intrinsic surface particles, and predictor-corrector time integration for nonlinear coupled bulk-surface dynamics.It computes normals and curvatures at high order and periodically regularizes particle distributions.
- Conclusions and outlook: Convergence was verified for surface DC-PSE interpolation of spherical harmonics through order 5 and for mass conservation on an isotropically growing sphere.The mass-conservation test achieved the theoretically expected convergence order.
- Conclusions and outlook: Gray–Scott dynamics on an oscillating 3D droplet reached steady patterns, but irregular surface particles caused instability or non-physical checkerboard patterns.The result demonstrates sensitivity of the surface PDE solution to particle-distribution quality.
- Conclusions and outlook: Two-way morphogenesis coupling produced dynamically evolving organic shapes, with periodic particle re-regularization mandatory under large deformations and associated with faster runtimes.Chemical reaction-diffusion fields induced local growth while surface deformation fed back into the surface dynamics.
- Conclusions and outlook: The morphogenesis simulations were limited by the need for a non-intersecting tubular neighborhood and by absent feedback from surface dynamics to the embedding hydrodynamic phases.Extending the method would require adaptive-resolution operators and surfactant interpolation to bulk-particle closest points.
- Conclusions and outlook: The solver offers meshfree, high-order Lagrangian solutions for nonlinear surface dynamics on deforming surfaces coupled with bulk fluids.The authors identify applications in biological morphogenesis, additive manufacturing, and computer graphics.
CRediT authorship contribution statement
The contribution statement assigns work across conceptualization, methodology, software, validation, analysis, investigation, writing, and project responsibilities.
- CRediT authorship contribution statement: LS contributed to conceptualization, methodology, software, validation, analysis, investigation, data curation, writing, visualization, and related project activities.The statement lists both original-draft and review-and-editing contributions.
- CRediT authorship contribution statement: AF contributed to conceptualization, methodology, software, validation, analysis, investigation, and writing.The statement includes original-draft and review-and-editing contributions.
- CRediT authorship contribution statement: IFS contributed to conceptualization, methodology, analysis, investigation, resources, writing, supervision, project administration, and funding acquisition.The passage lists original-draft and review-and-editing contributions.