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A Stream-Function Formulation for the Surface Stokes-Helfrich Model - Surface Finite Element Discretization and Validation
Enno Igel, Maik Porrmann, Axel Voigt
TL;DR
Fully evolving Surface Stokes-Helfrich surfaces require a stream-function formulation that retains pressure and introduces additional potentials. The paper develops an ALE surface finite-element discretization and finds comparable errors to the velocity-pressure approach, with slightly better inextensibility error but no decisive efficiency advantage.
Problem
Existing stream-function formulations address stationary or prescribed evolving surfaces, while the fully evolving model retains pressure and requires an additional gradient potential.
Method
The paper combines a stream-function formulation with ALE isoparametric surface finite elements and first-order semi-implicit time stepping, comparing it with velocity-pressure discretization.
Results
The stream-function method produces similar errors and is slightly more efficient for inextensibility than the velocity-pressure formulation, though the efficiency difference is not decisive.
Takeaways & Limitations
The formulation is a feasible alternative that avoids the pressure saddle-point structure and requires solving only scalar-valued partial differential equations.
Takeaways & Limitations
The study is restricted to simply-connected surfaces, so harmonic vector fields are not included in the current formulation.
Abstract
from arXiv · showhide
We introduce a stream-function formulation for the Surface Stokes-Helfrich model and study an Arbitrary Lagrangian-Eulerian (ALE) Surface Finite Element (SFEM) discretization in space and a semi-implicit discretization in time of these equations. In contrast with special cases of a surface Stokes model on stationary or prescribed evolving surfaces the surface pressure remains as an unknown and needs to be reconstructed in each time step. As for the surface Stokes model on a prescribed evolving surface the formulation also requires an additional gradient potential. A detailed computational comparison with an ALE-SFEM discretization in space and a similar semi-implicit discretization in time of the corresponding velocity-pressure formulation is explored on a representative problem formulation for simply-connected surfaces, demonstrating the validity of the approach.
1. Introduction.
The introduction situates the work in numerical studies of closed, volume-constrained surface models and identifies standard ALE discretization ingredients. It also notes that stability estimates exist for such schemes, while detailed analysis remains unavailable in the general evolving-surface setting.
- Context: The model concerns closed surfaces with an enclosed-volume constraint, making the problem nonlocal.The full model and variants with symmetry assumptions have been studied numerically in prior work.
- Related numerical methods: Existing general approaches use ALE methods, tangential point redistribution for mesh regularity, stable velocity-pressure elements, and semi-implicit time discretization.The passage specifically mentions Taylor-Hood elements as an example of a stable element pair.
- Open analysis: Stability estimates have been established for these schemes, but detailed analysis is not yet available for the general evolving-surface problem.The situation changes when the surface is stationary or its evolution is prescribed, where the problem becomes a Surface (Navier-)Stokes equation.
2. Modeling.
Section 2 formulates the evolving-surface Stokes-Helfrich problem in velocity-pressure, tangential-normal, and stream-function variables. For simply connected surfaces, the stream-function formulation uses scalar unknowns, eliminates the pressure saddle-point structure, but retains surface pressure through an additional equation.
- Velocity-pressure formulation: The velocity-pressure model determines the surface parametrization, velocity, pressure, and volume-enforcing Lagrange multiplier under force, inextensibility, and volume constraints.Only the normal material velocity updates the parametrization because of reparametrization invariance.
- Tangential-normal formulation: Splitting velocity into tangential and normal components separates force balance and reveals curvature-mediated coupling between tangential motion and shape changes.The parametrization evolves according to ∂tX = uNν.
- Stream-function formulation: On simply connected surfaces, the harmonic component of the tangential Helmholtz decomposition vanishes, enabling representation through a stream-function and gradient potential.The mass-conservation equation becomes ΔSψ − uNH = 0, while curlS and divS operations produce equations for the stream-function, vorticity, and pressure.
- Stream-function formulation: Unlike stream-function formulations for stationary or prescribed evolving surfaces, the evolving-surface system retains surface pressure as an unknown and requires an auxiliary equation to avoid higher-order operators.The inextensibility constraint becomes an equation for the gradient potential, while the volume constraint remains.
- Stream-function formulation: Despite stronger geometric couplings and additional unknowns, the stream-function formulation makes all unknowns scalar-valued and eliminates the pressure saddle-point structure.These advantages persist even though the resulting system is more complicated.
- Numerical discretization: The numerical method uses an ALE Surface Finite Element Method combined with mesh redistribution to solve the nonlinear geometric and surface partial differential equations.The approach exploits surface reparametrization invariance during computation.
3. Numerics.
The numerics use an isoparametric surface finite element discretization with discrete scalar and vector spaces, combined with a first-order implicit-explicit time splitting. The implementation uses AMDiS within the Dune ecosystem and validates the method on a perturbed sphere benchmark.
- Spatial discretization: The spatial discretization uses an isoparametric k-th order surface approximation with continuous piecewise-polynomial scalar spaces and corresponding three-dimensional vector spaces.The formulation takes X_h in V_3(S_h) and represents the other discrete unknowns in the vector-valued space.
- Time discretization: A first-order implicit-explicit time discretization solves a linear system at each time step, treating linear terms implicitly and geometric nonlinearities using previous-step or semi-implicit evaluations.Geometric properties are evaluated at the previous time step, while the mean curvature is treated semi-implicitly.
- Comparison method: The velocity-pressure comparison discretization uses a (P3, P2) Taylor-Hood velocity-pressure pair, k = 3 isoparametric treatment for remaining unknowns, and a similar time discretization.The comparison approach is described as following the discretization in [41].
- Implementation: The discretizations are implemented in AMDiS, using Dune-alugrid for a sphere-topology grid and Dune-curvedgrid to map it onto the surface through the parametrization X_h.The parametrization, discrete functions, and bases are represented through Dune-function-related modules.
- Benchmark problem: The benchmark is a perturbed sphere with radius r(θ, ϑ) = 1 + r0 cos θ sin 3ϑ and r0 = 2 5.This surface was previously used to benchmark a Surface Navier-Stokes-Helfrich model.
4. Results.
The stream-function and velocity-pressure formulations produce comparable solution properties and refinement errors, while the stream-function formulation shows higher experimental convergence order for the tested conservation and inextensibility measures. The simulations reproduce relaxation through an oblate shape toward an equilibrium prolate configuration, with differences attributed to volume and area constraints.
- Evolution: The stream-function simulation evolves through an oblate shape before reaching the equilibrium prolate configuration with zero tangential velocity.Nonzero tangential velocity helps avoid the oblate local minimum of bending energy.
- Comparison: Both formulations show decaying bending energy and nearly coinciding dissipations, with deviations during strong shape transitions.The transition times and final oblate and prolate shapes differ slightly because of different volume and area constraints; the reached accuracy is considered sufficient.
- Convergence: Errors for the two methods are comparable in convergence order and absolute value, consistent with previous velocity-pressure results.The study uses experimental convergence toward the finest-resolution numerical solution because no nontrivial analytical solution or numerical analysis results are available.
- Convergence: The optimal order 3 for k = 3 is not achieved; results indicate order 1 for velocity-pressure and order 2 for stream-function formulations.The reduced order is attributed to additional unknowns and their explicit coupling with geometric quantities.
5. Conclusions.
The stream-function formulation and proposed discretization are feasible for the Surface Stokes-Helfrich model, providing a viable alternative to the classical velocity-pressure approach. For the considered test case, it achieves similar errors and slightly better inextensibility efficiency.
- The stream-function method yields similar errors and is slightly more efficient regarding inextensibility error than the velocity-pressure formulation.This conclusion applies to the considered test case.
- The results demonstrate that the stream-function formulation and proposed discretization are feasible for the Surface Stokes-Helfrich model.
- Despite requiring additional geometric quantities and retaining pressure, the formulation is a viable alternative that circumvents the saddle-point structure.
Appendix A. Derivation of stream-function formulation.
The appendix derives the stream-function formulation from the surface Stokes-Helfrich equations by applying surface identities, testing with stream-function-based fields, and obtaining equivalent strong formulations. It concludes that the resulting equations, excluding the volume constraint, are those of Problem 2.3.
- Surface identities: Surface product-rule and Binet-Cauchy identities convert tangential velocity terms into stream functions and simplify covariant-divergence expressions.The derivation uses identities for smooth surface scalar fields and obtains divS uT = ∆S ψ.
- Starting equations: The derivation starts from the surface Stokes-Helfrich equations, temporarily excluding volume conservation because it can be reintroduced after reformulation.The appendix identifies the repeated equations as the starting point for the proposed formulation.
- Tangential equation: Testing the tangential equation with gradient and curl components yields coupled weak relations involving ψ, ϕ, pressure, curvature K, and normal velocity uN.The tested expressions include Laplace–Beltrami terms, curvature couplings, and gradient and curl test fields.
- Resulting formulation: Equations (A.6a), (A.6b), (A.7), (A.6c), (A.5), and (A.3) represent the stream-function formulation in Problem 2.3, neglecting volume conservation.This is the appendix’s explicit summary of the derived formulation.
- Normal equation: Testing the normal equation with c uN and substituting uT = ∇Sψ + curlSϕ produces the strong normal-force formulation.The calculation uses shape-operator symmetry and surface identities before obtaining equation (A.7).