Fast, adaptive summation of point forces in the two-dimensional Poisson equation

Dommelen, Leon van; Rundensteiner, Elke A.

doi:10.1016/0021-9991(89)90225-8

Cited by 69 publications

(47 citation statements)

References 10 publications

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“…One of the simplest is using quad tree structures (in 2D) and has been used in the context of finite volume method in [14][15][16], for FEM in [17], and with the meshfree element-free Galerkin method [18], where level-1 quad-tree structures are employed even though this is not explicitly mentioned in the paper. The level-1 (or 1-irregular) grid refinement has been originally introduced in [1].…”

Section: Literature Reviewmentioning

confidence: 99%

Convergence, adaptive refinement, and scaling in 1D peridynamics

Bobaru

Yang

Alves

et al. 2008

Numerical Meth Engineering

460

331

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SUMMARYWe introduce here adaptive refinement algorithms for the non-local method peridynamics, which was proposed in (J. Mech. Phys. Solids 2000; 48:175-209) as a reformulation of classical elasticity for discontinuities and long-range forces. We use scaling of the micromodulus and horizon and discuss the particular features of adaptivity in peridynamics for which multiscale modeling and grid refinement are closely connected. We discuss three types of numerical convergence for peridynamics and obtain uniform convergence to the classical solutions of static and dynamic elasticity problems in 1D in the limit of the horizon going to zero. Continuous micromoduli lead to optimal rates of convergence independent of the grid used, while discontinuous micromoduli produce optimal rates of convergence only for uniform grids. Examples for static and dynamic elasticity problems in 1D are shown. The relative error for the static and dynamic solutions obtained using adaptive refinement are significantly lower than those obtained using uniform refinement, for the same number of nodes.

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Section: Literature Reviewmentioning

confidence: 99%

Convergence, adaptive refinement, and scaling in 1D peridynamics

Bobaru

Yang

Alves

et al. 2008

Numerical Meth Engineering

460

331

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“…The methods based on multipole expansions due to Greengard and Rokhlin [7] and the adaptive FMM due to Carrier, Greengard, and Rokhlin [4] are probably the most efficient in two dimensions. The adaptation of the FMM using Poisson's formula due to Anderson [2] and the FMM based on the Laurent series due to van Dommelen and Rundensteiner [21] are other FMM based schemes. Lustig, Rastogi, and Wagner [12] detail a technique that modifies and speeds up the FMM by telescoping the multipole method by using Chebyshev economization.…”

Section: Fast Summation Techniquesmentioning

confidence: 99%

A Fast, Two-Dimensional Panel Method

Ramachandran¹,

Rajan²,

Ramakrishna³

2003

SIAM J. Sci. Comput.

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Abstract.A new two-dimensional panel technique has been developed to solve Laplacian flows, which eliminates the edge effect present in traditional panel methods. Such a method is very useful for applications where the velocity induced by the panels is required at arbitrary locations. Particle based flow solvers are a prime example. The method, however, requires considerably more computational effort. In this paper the method is modified to improve computational efficiency by adapting the fast multipole algorithm for the panel method. Significant improvement in computational efficiency is obtained while ensuring that the edge effects are eliminated.Key words. panel method, edge effect, fast multipole method AMS subject classifications. 31A99, 34B60, 35J05, 65E05, 65Y99, 76M15 PII. S1064827500374662 1. Introduction. Panel methods provide an elegant methodology for solving a class of flows past arbitrarily shaped bodies in both two and three dimensions. The basic idea is to discretize the body in terms of a singularity distribution on the body surface, satisfy the necessary boundary conditions, find the resulting distribution of singularity on the surface, and thereby obtain fluid dynamic properties of the flow. The body geometry is represented in terms of smaller subunits called panels, hence the name "panel" method. In two dimensions the panels are usually straight lines, and in three dimensions planar elements are used. The singularities used can be either sources, doublets, or vortices. Each panel is constructed to have some kind of singularity distribution. Depending on the accuracy, computational speed and other factors one can use constant, linear, parabolic, or even higher orders of distribution of the singularity on each panel. The number of panels that represent the body can also be varied. The actual singularity distribution is initially unknown, but by enforcing the boundary conditions on the body, it is possible to solve for them. The boundary conditions can be represented in terms of the velocity field, called the Neumann condition, or in terms of the potential inside the body, called the Dirichlet condition. Hess and Smith [8] laid the foundation for the source panel method. The idea of the vortex panel method is due to Martensen [14] and is extended by Lewis [10]. Katz [9] gives an excellent, comprehensive overview of panel methods in general (in both two and three dimensions).The main advantage of the panel method is that it is relatively easy to formulate and compute. The panel methods are also "grid free" and represent the geometry as accurately as possible. Rajan [16] also shows that the vortex based panel methods are capable of explaining both the kinematic motion of the rigid body and the fluid flow. Unlike finite difference methods panel methods are, however, a little restrictive in their applicability. Typically panel methods are used to solve Laplacian flows, which they solve very accurately and efficiently. There are techniques to handle compressible flows past thin bodies by using t...

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“…This cost is prohibitive for large-scale, three-dimensional computations. To alleviate this difficulty, fast summation algorithms have been proposed [2,3,8,19]. In many of these approaches, particles are divided into a nested set of clusters, and particle-particle interactions are replaced by particle-cluster interactions, which can be efficiently evaluated by using an expansion.…”

Section: Introductionmentioning

confidence: 99%

Convergence Characteristics and Computational Cost of Two Algebraic Kernels in Vortex Methods with a Tree-Code Algorithm

Wee¹,

Marzouk²,

Schlegel³

et al. 2009

SIAM J. Sci. Comput.

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Abstract. We study the convergence characteristics of two algebraic kernels used in vortex calculations: the Rosenhead-Moore kernel, which is a low-order kernel, and the Winckelmans-Leonard kernel, which is a high-order kernel. To facilitate the study, a method of evaluating particle-cluster interactions is introduced for the Winckelmans-Leonard kernel. The method is based on Taylor series expansion in Cartesian coordinates, as initially proposed by Lindsay and Krasny [J. Comput. Phys., 172 (2001), pp. 879-907] for the Rosenhead-Moore kernel. A recurrence relation for the Taylor coefficients of the Winckelmans-Leonard kernel is derived by separating the kernel into two parts, and an error estimate is obtained to ensure adaptive error control. The recurrence relation is incorporated into a tree-code to evaluate vorticity-induced velocity. Next, comparison of convergence is made while utilizing the tree-code. Both algebraic kernels lead to convergence, but the Winckelmans-Leonard kernel exhibits a superior convergence rate. The combined desingularization and discretization error from the Winckelmans-Leonard kernel is an order of magnitude smaller than that from the Rosenhead-Moore kernel at a typical resolution. Simulations of vortex rings are performed using the two algebraic kernels in order to compare their performance in a practical setting. In particular, numerical simulations of the side-by-side collision of two identical vortex rings suggest that the three-dimensional evolution of vorticity at finite resolution can be greatly affected by the choice of the kernel. We find that the Winckelmans-Leonard kernel is able to perform the same task with a much smaller number of vortex elements than the Rosenhead-Moore kernel, greatly reducing the overall computational cost.

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Fast, adaptive summation of point forces in the two-dimensional Poisson equation

Cited by 69 publications

References 10 publications

Convergence, adaptive refinement, and scaling in 1D peridynamics

Convergence, adaptive refinement, and scaling in 1D peridynamics

A Fast, Two-Dimensional Panel Method

Convergence Characteristics and Computational Cost of Two Algebraic Kernels in Vortex Methods with a Tree-Code Algorithm

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