On Approximations of the Curve Shortening Flow and of the Mean Curvature Flow based on the DeTurck trick

In this paper we discuss novel numerical schemes for the computation of the curve shortening and mean curvature flows that are based on special reparametrizations. The main idea is to use special solutions to the harmonic map heat flow in order to reparametrize the equations of motion. This idea is widely known from the Ricci flow as the DeTurck trick. By introducing a variable time scale for the harmonic map heat flow, we obtain families of numerical schemes for the reparametrized flows. For the curve shortening flow this family unveils a surprising geometric connection between the numerical schemes in [5] and [9]. For the mean curvature flow we obtain families of schemes with good mesh properties similar to those in [3]. We prove error estimates for the semi-discrete scheme of the curve shortening flow. The behaviour of the fully-discrete schemes with respect to the redistribution of mesh points is studied in numerical experiments. We also discuss possible generalizations of our ideas to other extrinsic flows

Geometry-induced patterns through mechanochemical coupling

Intracellular protein patterns regulate a variety of vital cellular processes such as cell division and motility, which often involve dynamic changes of cell shape. These changes in cell shape may in turn affect the dynamics of pattern-forming proteins, hence leading to an intricate feedback loop between cell shape and chemical dynamics. While several computational studies have examined the resulting rich dynamics, the underlying mechanisms are not yet fully understood. To elucidate some of these mechanisms, we explore a conceptual model for cell polarity on a dynamic one-dimensional manifold. Using concepts from differential geometry, we derive the equations governing mass-conserving reaction-diffusion systems on time-evolving manifolds. Analyzing these equations mathematically, we show that dynamic shape changes of the membrane can induce pattern-forming instabilities in parts of the membrane, which we refer to as regional instabilities. Deformations of the local membrane geometry can also (regionally) suppress pattern formation and spatially shift already existing patterns. We explain our findings by applying and generalizing the local equilibria theory of mass-conserving reaction-diffusion systems. This allows us to determine a simple onset criterion for geometry-induced pattern-forming instabilities, which is linked to the phase-space structure of the reaction-diffusion system. The feedback loop between membrane shape deformations and reaction-diffusion dynamics then leads to a surprisingly rich phenomenology of patterns, including oscillations, traveling waves, and standing waves that do not occur in systems with a fixed membrane shape. Our work reveals that the local conformation of the membrane geometry acts as an important dynamical control parameter for pattern formation in mass-conserving reaction-diffusion systems

A computational method for the coupled solution of reaction–diffusion equations on evolving domains and manifolds: application to a model of cell migration and chemotaxis

In this paper, we devise a moving mesh finite element method for the approximate solution of coupled bulk–surface reaction–diffusion equations on an evolving two dimensional domain. Fundamental to the success of the method is the robust generation of bulk and surface meshes. For this purpose, we use a novel moving mesh partial differential equation (MMPDE) approach. The developed method is applied to model problems with known analytical solutions; these experiments indicate second-order spatial and temporal accuracy. Coupled bulk–surface problems occur frequently in many areas; in particular, in the modelling of eukaryotic cell migration and chemotaxis. We apply the method to a model of the two-way interaction of a migrating cell in a chemotactic field, where the bulk region corresponds to the extracellular region and the surface to the cell membrane

Simulating incompressible thin-film fluid with a Moving Eulerian-Lagrangian Particle method

In this thesis, we introduce a Moving Eulerian-Lagrangian Particle (MELP) method, a mesh-free method to simulate incompressible thin-film fluid systems: soap bubbles, bubble clusters, and foams. The realistic simulation of such systems depends upon the successful treatment of three aspects: (1) the soap film\u27s deformation due to the tendency to minimize the surface energy, giving rise to the bouncy characteristics of soap bubbles, (2) the tangential fluid flow on the thin film, causing the thickness to vary spatially, which in conjunction with thin-film interference creates evolving and highly sophisticated iridescent color patterns, (3) the topological changes due to collision, separation, and fragmentation, which may create partition surfaces and non-manifold junctions that spontaneously settle into honeycomb structures due to force balance. The interleaving complexities from all three fronts render the task of accurately and affordably simulating thin-film fluid an open problem for the Computational Physics and Computer Graphics community. The proposed MELP method tackles these multifaceted challenges by employing a novel, bi-layer particle structure: a sparse set of Eulerian particles for dynamic interface tracking and PDE solving, and a fine set of Lagrangian particles for material and momentum transport. Such a design provides crucially advantageous numerical traits compared to existing frameworks. Compared to mesh-based methods, MELP\u27s particle-based nature makes it topologically agnostic, which allows it to conveniently simulate topological changes such as bubble-cluster formation and thin-film rupture. Furthermore, these Lagrangian structures carry out fluid advection naturally, conserve mass by design, and track sub-grid flow details. Compared to existing particle methods, our bi-layer design improves drastically on the computational performance in terms of both stability and efficiency. The advantage of this design will manifest in a wide range of experiments, including dynamic foam formation, Rayleigh-Taylor instability, Newton Black Films, and bubble bursting, showing an increased level of flow detail, increased number of regions in bubble clusters, and increased flexibility to recreate multi-junction formation on-the-fly. Furthermore, we validate its physical correctness against a variety of analytical baselines, by successfully recovering the equilibrium dihedral and tetrahedral angles, the exponential thickness profile of drainage under gravity, the curvature of partition surfaces, and the minimum surface area of double-bubbles