Journal ArticleDOI
Numerical simulation of two-dimensional combustion using mesh-free methods
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In this article, the authors developed tools for numerical simulations of flame propagation with mesh-free radial basis functions (RBFs), which offer many distinct advantages over traditional finite difference, finite element, and finite volume methods.Abstract:
The purpose of this research was to develop tools for numerical simulations of flame propagation with mesh-free radial basis functions (RBFs). Mesh-free methods offer many distinct advantages over traditional finite difference, finite element, and finite volume methods. Traditional Lagrangian methods with significant swirl require mesh stiffeners and periodic remeshing to avoid excessive mesh distortion; such codes often require user interaction to repair the meshes before the simulation can proceed again. A propagating flame of infinite extent is simulated as a collection of normalized cells with periodic boundary conditions. Rather than capturing the flame front, it is tracked as a discontinuity. The flame front is approximated as a product of a Heaviside function in the normal propagation direction and a piece-wise continuous function represented by RBFs in the tangential direction. The cells are subdivided into the burned and unburned sub-domains approximated by two-dimensional periodic RBFs that are constrained to be strictly conservative. The underlying steady flow is vortical with an input turbulent intensity. The governing equations are rotationally and translationally transformed to produce exact differentials that are integrated exactly in time. In the present paper, the previous results of Aldredge who used a finite-difference level-set method were compared. The physical behavior was remarkably similar, whereas the finite-difference level-set method required 14 h of CPU time, the RBF approach required only 120 CPU seconds on a desktop computer for the case with the largest turbulent intensity. Although there are no other papers that tried to duplicate the original results of Aldredge, the results that are reported here are consistent with the physics observed in other experimental and numerical investigations.read more
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Recent Advances in Radial Basis Function Collocation Methods
TL;DR: Radial basis functions (RBFs) as mentioned in this paper are constructed in terms of one-dimensional distance variable and appear to have certain advantages over the traditional coordinates-based functions, which avoid troublesome mesh generation for high-dimensional problems involving irregular or moving boundary.
Journal ArticleDOI
The use of a meshless technique based on collocation and radial basis functions for solving the time fractional nonlinear Schrödinger equation arising in quantum mechanics
TL;DR: In this paper, the authors proposed a numerical method for the solution of the time-fractional nonlinear Schrodinger equation in one and two dimensions which appear in quantum mechanics.
Journal ArticleDOI
A random variable shape parameter strategy for radial basis function approximation methods
Scott A. Sarra,Derek Sturgill +1 more
TL;DR: This work introduces a new random variable shape parameter strategy and gives numerical results showing that the new random strategy often outperforms both existing variable shape and constant shape strategies.
Multiquadric Radial Basis Function Approximation Methods for the Numerical Solution of Partial Differential Equations
TL;DR: This monograph differs from other recent books on meshfree methods in that it focuses only on the MQ RBF while others have focused on meshless methods in general.
Journal ArticleDOI
The numerical solution of nonlinear high dimensional generalized Benjamin-Bona-Mahony-Burgers equation via the meshless method of radial basis functions
TL;DR: The aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear partial differential equations.
References
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Journal ArticleDOI
Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Stanley Osher,James A. Sethian +1 more
TL;DR: The PSC algorithm as mentioned in this paper approximates the Hamilton-Jacobi equations with parabolic right-hand-sides by using techniques from the hyperbolic conservation laws, which can be used also for more general surface motion problems.
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Multiquadric equations of topography and other irregular surfaces
TL;DR: In this paper, a method of representing irregular surfaces that involves the summation of equations of quadric surfaces having unknown coefficients is described, and procedures are given for solving multiquadric equations of topography that are based on coordinate data.
Journal ArticleDOI
Scattered data interpolation: tests of some methods
TL;DR: In this paper, the evaluation of methods for scattered data interpolation and some of the results of the tests when applied to a number of methods are presented. But the evaluation process involves evaluation of the methods in terms of timing, storage, accuracy, visual pleasantness of the surface, and ease of implementation.
Journal ArticleDOI
Multiquadrics--a scattered data approximation scheme with applications to computational fluid-dynamics-- ii solutions to parabolic, hyperbolic and elliptic partial differential equations
TL;DR: In this paper, the authors used MQ as the spatial approximation scheme for parabolic, hyperbolic and the elliptic Poisson's equation, and showed that MQ is not only exceptionally accurate, but is more efficient than finite difference schemes which require many more operations to achieve the same degree of accuracy.
Approximation scheme with applications to computational fluid-dynamics-- i surface approximations and partial derivative estimates
TL;DR: In this article, the authors presented an enhanced multiquadrics (MQ) scheme for spatial approximations, which is a true scattered data, grid free scheme for representing surfaces and bodies in an arbitrary number of dimensions.