A grid based particle method for solving partial differential equations on evolving surfaces and modeling high order geometrical motion

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Abstract

We develop numerical methods for solving partial differential equations (PDE) defined on an evolving interface represented by the grid based particle method (GBPM) recently proposed in [S. Leung, H.K. Zhao, A grid based particle method for moving interface problems, J. Comput. Phys. 228 (2009) 7706–7728]. In particular, we develop implicit time discretization methods for the advection–diffusion equation where the time step is restricted solely by the advection part of the equation. We also generalize the GBPM to solve high order geometrical flows including surface diffusion and Willmore-type flows. The resulting algorithm can be easily implemented since the method is based on meshless particles quasi-uniformly sampled on the interface. Furthermore, without any computational mesh or triangulation defined on the interface, we do not require remeshing or reparametrization in the case of highly distorted motion or when there are topological changes. As an interesting application, we study locally inextensible flows governed by energy minimization. We introduce tension force via a Lagrange multiplier determined by the solution to a Helmholtz equation defined on the evolving interface. Extensive numerical examples are also given to demonstrate the efficiency of the proposed approach.

Introduction

The numerical solution of partial differential equations (PDEs) on evolving surfaces is a challenging problem as is simulating the motion of high order geometric evolution equations. The interface may stretch, deform and break apart. In the case of geometric motion, the velocity of the interface depends on high order derivatives of the interface position which poses challenges for accuracy and stability of the method. Such problems have application in the biological, physical and engineering sciences.

Various numerical methods have been proposed to solve PDEs on surfaces and geometric flows. One popular approach is based on explicit representation of surfaces by parametrizing or laying down a mesh on the surface (e.g. [13], [25], [26], [5], [2], [10], [9], [3]). When the surface is evolving, this type of approach is also called Lagrangian formulation since parametrization or meshes on the surface is tracked along the Lagrangian trajectory. One major issue of this approach is that it is difficult (especially in 3 dimensions), to maintain a smooth global parametrization or a quasi-uniform mesh for a moving interface with complicated geometry and dynamics involving large deformations or topological changes. As a result, the resulting algorithm usually requires reparametrization/remeshing during the evolution, although a recent algorithm developed for geometric motion of surfaces has the potential to overcome this problem [3]. In addition, if the topology of the interface changes, then local interface surgery is required which can be highly problematic to perform in three dimensions. However, these Lagrangian type methods are relatively easy to implement and are efficient and accurate numerically.

An alternative approach is based on an implicit representation of surfaces using an auxiliary function on a fixed Eulerian mesh. For example, the level set method can be used for implicit representation while the PDEs on the surface, which is the zero level set of the level set function, are extended off the interface onto neighboring level sets (e.g. [4], [1], [44], [15], [43], [11]). There are three main advantages using an implicit formulation. First, there is no need to parametrize or triangulate the surface. Second, it is easy to handle topological changes. Third, the surface PDEs is extended off the surface to a corresponding PDE in a neighborhood of the zero level set on a fixed Eulerian mesh for which many standard numerical methods for PDEs can be applied. Examples of other recent Eulerian approaches include [31], [23], which are based on the closest point method, [17], which is based on volume of fluid method, [32] which is based on a grid-free particle level set method, and [28], [12], [37], which are based on diffuse interface method. Typically the Eulerian formulation has an increased computation cost compared to Lagrangian formulation due to embedding of the equation into one higher dimension. Also it is difficult to deal with open surfaces. For applications of Eulerian methods to geometric evolution equations, we refer the reader to the recent review [22] for references.

In [20], we have recently proposed a novel approach, the grid based particle method (GBPM), to represent and model an interface and its motion. The idea is to sample the interface by meshless and non-parametrized Lagrangian particles according to an underlying uniform or adaptive Eulerian mesh. This results in a quasi-uniform sampling of the interface. As the interface moves, we continuously update the location of these particles by solving ordinary differential equations (ODEs), rather than partial differential equations (PDEs). Using extra Lagrangian information defined on these sampling points, we can naturally capture topological changes such as merging or breakup of surfaces.

Unlike usual Lagrangian methods depending on surface parametrizations or meshes, the GBPM does not require any connectivity information among the particles. This feature allows one to easily add or delete particles, which is important for maintaining a consistent resolution of the surface as well as dealing with topological changes. Moreover, the sampling of the particles has a one-to-one reference to the underlying grid points which are in the neighborhood of the interface. This Eulerian reference provides both a quasi-uniform sampling of the interface and neighborhood information among meshless particles. This is useful for local construction of the surface and detection of interface collision or self-intersection. The Eulerian reference is continuously updated without using a PDEs approach. Even though it is not demonstrated in the current paper, an adaptive sampling of the interface can be achieved easily through local grid refinement of the underlying grid taking advantage of the fact that no PDE is solved on the grid. For a more detailed description, we again refer interested readers to some recent publications [20], [19].

We have successfully applied this technique to various velocity models in [20], including motion by mean curvature. We have also demonstrated that the method can be used to capture the viscosity solution or the multivalued solution when two interfaces come across each other. Numerous test examples have been shown in [20] that demonstrate the flexibility of the approach. We can also deal with curves/surfaces with high codimension as well as open curves/surfaces easily with this method [19], [21].

In this paper, we extend our previous work and use the GBPM to solve advection–diffusion equations on surfaces. In particular, we develop implicit methods that remove time step restrictions due to diffusion. We then further extend these techniques and consider high-order geometric flows including surface diffusion and Willmore flow [42]. As an interesting application of these techniques, we further impose a local inextensibility condition on an evolving interface. This constraint is important in many fields including lipid vesicle modeling in mathematical biology (e.g. [30], [8], [36], [40], [39], [35]), flexible fiber interactions (e.g. [38]) and robotics (e.g. [6]). To enforce local inextensibility, we incorporate a Lagrange multiplier which acts as the local tension of the evolving interface. This Lagrange multiplier satisfies a surface Helmholtz equation relating the geometry of the evolving interface with the tangential and the normal velocities of the unconstrained flow.

The paper is organized as follows. In Section 2, we first summarize the grid based particle method we have introduced in [20]. We explain how we generalize the method for solving PDEs on evolving surface in Sections 3.1 Advection equation, 3.2 Surface Laplacian, 3.3 Advection–diffusion equation. With these techniques, we then model higher order motions of the interface including the motion by surface diffusion and the Willmore flow in Sections 3.4 Smoothing, 3.5 Surface diffusion flow and Willmore flow. In Section 4, we apply the techniques we have developed to simulate local inextensible flows. Section 5 shows examples to demonstrate the performance of our method.

Section snippets

Grid based particle method (GBPM)

In this section, we give a brief review of the grid based particle method. For a complete description of the algorithm, we refer interested readers to [20]. The interface is represented by meshless particles which are associated to an underlying Eulerian mesh. In our current algorithm, each sampling particle on the interface is chosen to be the closest point from each underlying grid point in a small neighborhood of the interface. This one to one correspondence gives each particle an Eulerian

Time and spatial derivatives

In the first part of this section, we discuss how we apply the GBPM to solve a PDE on an evolving surface. Then in the second part we apply the GBPM to compute high order flow including motion by surface diffusion and Willmore flow.

Application to flows with local inextensibility

In this section, we will consider modeling only a two dimensional flow with a local inextensibility constraint given by [18], [30](vt)s-κvn=0,where s is the arclength parametrization, κ is the signed curvature and vn and vt are the normal and the tangential velocities, respectively. It is possible to generalize this approach to three dimensions. For instance, preserving the surface area, we havedivsu=0,where divs = tr s is the surface divergence. This generalizes the local inextensibility

Advection equation on a circle

The first example is a two-dimensional case where a circle centered at (0.5, 0.75) with radius 0.15 is rotated by the velocityu=2(0.5-y),v=2(x-0.5).The circle rotates about (0.5, 0.5) with period T = π. On the circle, we solve the advection equationDfDt=0,whereDDt=t+ux+vyis the material derivative along the trajectory of a particle on the interface. The initial condition defined on the circle is given byf(x,y,0)=x.The exact solution to this interface problem is given byf(x(T),y(T),T)=x0,where

Conclusion and future work

In this work, we have developed numerical algorithms for solving advection–diffusion PDEs on moving interfaces using the grid based particle method (GBPM). We also demonstrated the ability of the GBPM to simulate high order geometric flows including changes in the interface topology. To solve the equations efficiently, we developed semi-implicit methods to overcome the severe time step constraints required for the stability of explicit methods.

In the future, we plan to extend the algorithms to

Acknowledgments

S.L. acknowledges the hospitality of the University of California, Irvine where preliminary work was performed. S.L. gratefully acknowledges partial support from the Hong Kong RGC under Grant GRF602210. J.L. acknowledges partial support from the National Science Foundation (NSF), Division of Mathematical Sciences (DMS). H.Z. was partially supported by NSF Grant DMS0811254.

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