Numerical simulation of two-dimensional combustion using mesh-free methods

Kansa, Edward J.; Aldredge, R.C.; Ling, Leevan

doi:10.1016/j.enganabound.2009.02.008

Cited by 74 publications

(30 citation statements)

References 45 publications

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“…Later, due to the vortical flow, the flame front becomes significantly curved, and still later, the flame front becomes vertical. This pattern of the flame front curving and flattening continues indefinitely (see [19]). …”

Section: ∞ Rbfs Applied To Pde and Iesmentioning

confidence: 99%

“…Shocks, contact surfaces and material interfaces are surfaces in n−1 that can be represented as a product of a Heaviside function in the local normal direction and a RBF in n−1 in the tangential directions (see [19]). A flame was modelled as a discontinuous surface in an infinitely periodic domain within an inviscid flow field with vortices on either side of the discontinuous flame front.…”

Section: ∞ Rbfs Applied To Pde and Iesmentioning

confidence: 99%

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Radial basis functions: achievements and challenges

Kansa¹

2015

WIT Transactions on Modelling and Simulation

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Traditionally, the various forms of compactly supported polynomial based schemes such as finite difference, element, and volume methods were used in the numerical solution of ordinary and partial differential equations (ODEs and PDEs) as well as integral equations (IEs). The primitive computers had limited memory and processing speed. As time progressed, spectral and pseudo-spectral methods that possess exponential or spectral convergence that use tensor products of global expansions of one-dimensional orthonormal functions were used to solve ODEs, PDEs, and IEs. In the past 25 years, radial basis functions (RBFs) that can be either compactly supported or global gained in importance. However, those global RBFs that are C ∞ with shape parameters enjoy exponential convergence. The solution accuracy can be increased by either the spatial resolution or increasing the shape parameters. The performance of C ∞ RBFs has been demonstrated in a wide variety of linear and nonlinear elliptic, hyperbolic and parabolic PDEs and IE applications. Both well posed and ill-posed problems can be solved as well as with defined arithmetic and fuzzy arithmetic, problems with fractional derivatives, etc. Computers with the standard 32 bit chips limit the inherent power of the global RBFs by giving rise to potentially severely ill-conditioned systems. However, procedures using preconditioning and domain decomposition can be used to overcome this limitation. Recently, fast extended precision software enables the use of large shape parameters outperforming compactly supported finite elements and other methods. Also, the Galperin-Zheng weak formulation allows treatment of both well posed and ill-posed problems because it looks for solutions with global, rather than local minima. Presently, meshfree C ∞ RBFs appear to have the potential of solving many important scientific problems in higher dimensions.

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Section: ∞ Rbfs Applied To Pde and Iesmentioning

confidence: 99%

Section: ∞ Rbfs Applied To Pde and Iesmentioning

confidence: 99%

Radial basis functions: achievements and challenges

Kansa¹

2015

WIT Transactions on Modelling and Simulation

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“…In 2003, Xaio and McCarthy [14] found that β = 1.99 is an "optimal" value. Also, Kansa has recently suggested that the GMQ has desirable outcomes when β = 5/2 [6]. However, the behavior of the GMQ method with various β and ǫ is still not well understood.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…With β > 1, the GMQ is conditionally positive definite of order ⌈β⌉, and to show that the system matrix, B, is invertible, it is necessary to append low order polynomials to the RBF interpolant [1]. Through extensive numerical experiments, researchers have recently suggested, but not proven, that the GMQ has desirable properties for β with non-half-integer powers [6,13,14]. This work will focus on the less understood GMQ.…”

Section: Introductionmentioning

confidence: 99%

A Numerical Study of Generalized Multiquadric Radial Basis Function Interpolation

Chenoweth¹,

Sarra²

2009

SIURO

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This work focuses on the generalized multiquadric (GMQ) radial basis function. The GMQ is derived from the multiquadric (MQ), which is used in radial basis function (RBF) interpolation. This is a relatively new field of research, and many properties of the GMQ are still unknown. Numerical experiments will be performed involving the GMQ, and results will be analyzed to gain further understanding into this type of function.

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“…Its name comes from the use of the so-called multiquadric function. It has been used in various applications [13], from surface interpolation [7] to numerical solutions of differential equations [16,17]. In addition, Micchelli showed that the problem was solvable for any conditionally positive (or negative) definite kernels among which is the mutiquadric function [20].…”

Section: Related Workmentioning

confidence: 99%

Reconstructing faces from their signatures using RBF regression

Mignon¹,

Jurie²

2013

Procedings of the British Machine Vision Conference 2013

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The increasing popularity of social networks raises critical issues regarding privacy and protection of personal information. One typical example -which is the motivation of the paper -is face images and face recognition, face recognition being ubiquitous on the web now. Assuming someone with bad intentions steals a database containing face signatures, would it be possible to reconstruct the corresponding face images, revealing who was in the database? Would these reconstructed images be good enough to allow them to gain access via a face recognition system? This paper brings a contribution to this topic by proposing a face reconstruction algorithm, based on RBF-regression in eigenspace, which is able to reconstruct face images from their signatures (i.e. neither knowing the identity of the persons nor their true facial appearance). We show that in addition to being visually realistic, the images generated by the proposed method can fool a state-of-the-art face recognition algorithm.

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Numerical simulation of two-dimensional combustion using mesh-free methods

Cited by 74 publications

References 45 publications

Radial basis functions: achievements and challenges

Radial basis functions: achievements and challenges

A Numerical Study of Generalized Multiquadric Radial Basis Function Interpolation

Reconstructing faces from their signatures using RBF regression

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