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

Leung, Shingyu; Lowengrub, John; Zhao, Hongkai

doi:10.1016/j.jcp.2010.12.029

Cited by 55 publications

(51 citation statements)

References 38 publications

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“…26. The third and final broad class of methods are the embedding methods, which solve the surface PDEs in some 3D region encompassing the surface, using standard Cartesian-grid methods (3,(27)(28)(29)(30)(31)(32). These methods do not use the metric tensor on the surface, thus eliminating one of the major technical complexities of working with PDEs on surfaces.…”

Section: Discussionmentioning

confidence: 99%

Simple computation of reaction–diffusion processes on point clouds

Macdonald

Merriman

Ruuth

2013

Proc. Natl. Acad. Sci. U.S.A.

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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...

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Section: Discussionmentioning

confidence: 99%

Simple computation of reaction–diffusion processes on point clouds

Macdonald

Merriman

Ruuth

2013

Proc. Natl. Acad. Sci. U.S.A.

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“…Furthermore, the error made in conserving the enclosed volume converges like O.h=R/ 2 ( Figure 6(b)) because of the second-order polynomials approximation of the interface geometry and the second-order Runge-Kutta algorithm used for the update of the position of the interface. However, because of the presence of fourthorder terms in the membrane bending force, the explicit time evolutive simulations are subjected to a strict Courant-Friedrichs-Lewy (CFL) condition on the time step of the fourth order in mesh size t O.h=R/ 4 [58]. The shapes adopted at equilibrium by the capsule with l C D 0 and for different values of G are shown in Figure 7(a).…”

Section: Remarkmentioning

confidence: 99%

A particle‐based moving interface method (PMIM) for modeling the large deformation of boundaries in soft matter systems

Foucard

2016

Numerical Meth Engineering

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SUMMARYThe mechanics of the interaction between a fluid and a soft interface undergoing large deformations appear in many places, such as in biological systems or industrial processes. We present an Eulerian approach that describes the mechanics of an interface and its interactions with a surrounding fluid via the so-called Navier boundary condition. The interface is modeled as a curvilinear surface with arbitrary mechanical properties across which discontinuities in pressure and tangential fluid velocity can be accounted for using a modified version of the extended finite element method. The coupling between the interface and the fluid is enforced through the use of Lagrange multipliers. The tracking and evolution of the interface are then handled in a Lagrangian step with the grid-based particle method. We show that this method is ideal to describe large membrane deformations and Navier boundary conditions on the interface with velocity/pressure discontinuities. The validity of the model is assessed by evaluating the numerical convergence for a axisymmetrical flow past a spherical capsule with various surface properties. We show the effect of slip length on the shear flow past a two-dimensional capsule and simulate the compression of an elastic membrane lying on a viscous fluid substrate.

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Section: Introductionmentioning

confidence: 99%

“…It has been several decades to develop numerical methods for solving PDEs in surfaces. Many methods have been developed, such as surface finite element method [19], level set method [9,48], grid-based particle method [31,32] and closest point method [35,43].Recently, manifold model attracts more and more attentions in data analysis and image processing [4,11,13,23,26,29,30,36,[40][41][42]47]. In the manifold model, data or images are represented as a point cloud, which is defined as a collection of points that are embedded in a high-dimensional Euclidean space.…”

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confidence: 99%

Convergence of the point integral method for Laplace–Beltrami equation on point cloud

Shi

Sun

2017

Res Math Sci

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The Laplace-Beltrami operator, a fundamental object associated with Riemannian manifolds, encodes all intrinsic geometry of manifolds and has many desirable properties. Recently, we proposed the point integral method (PIM), a novel numerical method for discretizing the Laplace-Beltrami operator on point clouds (Li et al. in Commun Comput Phys 22(1):228-258, 2017). In this paper, we analyze the convergence of PIM for Poisson equation with Neumann boundary condition on submanifolds that are isometrically embedded in Euclidean spaces. Keywords: Laplace-Beltrami operator, Neumann boundary, Point cloud, Point integral method, Convergence analysisMathematics Subject Classification: 65N12, 65N25, 65N75 BackgroundThe partial differential equations on manifolds arise in a wide variety of applications. In many problems, including material science [10,20], fluid flow [22,25], biology and biophysics [2,3,21,37], people need to study the physical process, for instance diffusion and convection, in curved surfaces which introduce different kinds of PDEs in surfaces. It has been several decades to develop numerical methods for solving PDEs in surfaces. Many methods have been developed, such as surface finite element method [19], level set method [9,48], grid-based particle method [31,32] and closest point method [35,43].Recently, manifold model attracts more and more attentions in data analysis and image processing [4,11,13,23,26,29,30,36,[40][41][42]47]. In the manifold model, data or images are represented as a point cloud, which is defined as a collection of points that are embedded in a high-dimensional Euclidean space. One fundamental assumption in the manifold model is that the point cloud samples a smooth manifold. Thus, the information of the manifold is very useful to understand the data or images. PDEs on the manifold, particularly the Laplace-Beltrami equation, encode several intrinsic information of the manifold, thus helping reveal the underlying structures in the data or images. To get the information encoded in PDEs, we need to solve them in the unstructured point cloud. Given that the point cloud is embedded in a high-dimensional space, the traditional methods for PDEs on 2D surfaces do not work.

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

Cited by 55 publications

References 38 publications

Simple computation of reaction–diffusion processes on point clouds

Simple computation of reaction–diffusion processes on point clouds

A particle‐based moving interface method (PMIM) for modeling the large deformation of boundaries in soft matter systems

Convergence of the point integral method for Laplace–Beltrami equation on point cloud

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