Simple computation of reaction–diffusion processes on point clouds

Abstract

The study of reaction-diffusion processes is much more complicated on general curved surfaces than on standard Cartesian coordinate spaces. Here we show how to formulate and solve systems of reaction-diffusion equations on surfaces in an extremely simple way, using only the standard Cartesian form of differential operators, and a discrete unorganized point set to represent the surface. Our method decouples surface geometry from the underlying differential operators. As a consequence, it becomes possible to formulate and solve rather general reaction-diffusion equations on general surfaces without having to consider the complexities of differential geometry or sophisticated numerical analysis. To illustrate the generality of the method, computations for surface diffusion, pattern formation, excitable media, and bulksurface coupling are provided for a variety of complex point cloud surfaces.closest point method | embedding method | Laplace-Beltrami P artial differential equations (PDEs) are widely used to describe continuum processes such as diffusion, chemical reactions, fluid flow, or electrodynamics. In standard 3D settings, these take a familiar PDE form, such as a reaction-diffusion equation:and the ways to numerically solve such equations are welldeveloped. The basic approach can be quite simple, such as laying down a uniform Cartesian grid of points, fx i;j;k : 1 ≤ i; j; k ≤ Ng, and using simple, familiar approximations of the differential terms on this grid, such as:where u i;j;k ≈ uðx i;j;k Þ and h is the uniform grid spacing. As a result, it is very easy to implement methods for the numerical solution of such equations to study the phenomena of interest. To achieve efficiency for large-scale computations, more advanced methods are required, such as implicit discretization and solvers for systems of equations. These solvers, while internally complex, have been implemented in standard, well-validated numerical routines and are accessible through numerical subroutine libraries. Important physical processes also arise in complex geometrical settings, such as on complicated surfaces in three dimensions. The abstract form of differential operators on surfaces remains the same as in 3D, however, when explicitly expressed in coordinates, the formulas for the operators and the corresponding discretized expressions are relatively complicated and have received much less attention. Moreover, in practical settings, surfaces are often defined simply as a set of points-a point cloud-sampled from the underlying surface. Because the connectivity of the points is not provided, this adds further complexity to methods that need to reconstruct the geometric properties of the surface, such as the metric distance.Here we present a method for solving reaction-diffusion equations on a point cloud that represents the underlying surface-or any other geometric object-in a way that reduces the problem to working with entirely standard classical 3D discretizations and solver libraries. Our approach is fundamentally different from oth...

Exhibiting cross-diffusion-induced patterns for reaction-diffusion systems on evolving domains and surfaces

“…In particular, an inhibitor and activator or two activators can diffuse at equal rates; however, the product of the rates of the principal diffusion coefficients must be greater than the product of the cross-diffusion rates. For detailed theoretical analytical and computational results on the effects of domain growth on pattern formation, the interested reader is referred to results published in [1,2,4,5,[26][27][28][29]. In this article our focus is to showcase how cross-diffusion induces patterning in the presence of domain and surface evolution.…”

“…Examples include (but are not limited to) the moving grid finite element method [4], the method of lines [30], evolving surface finite element methods on triangulated surfaces [31][32][33], implicit finite element methods using level set descriptions of the surfaces [34][35][36][37], diffuse interface methods of which phase fields are an example [38,39], particle methods using level set descriptions of the surface [40][41][42], and closest-point methods [27,28]. In this article we choose to implement the evolving surface finite element method.…”

Analysis and Simulations of Coupled Bulk-surface Reaction-Diffusion Systems on Exponentially Evolving Volumes

“…h 1 (u, v, r, s) and h 2 (u, v, r, s) are reactions representing the coupling of the internal dynamics in the bulk Ω t to the surface dynamics on the evolving surface Γ t . As a first attempt, we will consider a more generalised form of linear coupling of the following nature [18] …”

Projected Finite Elements for Systems of Reaction-Diffusion Equations on Closed Evolving Spheroidal Surfaces

“…Examples include (but are not limited to) the method of lines [4], evolving surface finite element methods on triangulated surfaces [1,8,9,12,13], implicit finite element methods using level set descriptions of the surfaces [11,12,33,37], diffuse interface methods of which phase-fields are an example [3,6,14], particle methods using level set descriptions of the surface [7,16,18,22] and closest-point methods [24,25]. In all these methods the continuous surface is approximated by a discrete surface thereby committing a geometrical error.…”