Stable and Convergent Unsymmetric Meshless Collocation Methods

Ling, Leevan; Schaback, Robert

doi:10.1137/06067300x

Cited by 71 publications

(52 citation statements)

References 14 publications

(30 reference statements)

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“…Mesh motion techniques that exploit RBF's are efficient and inexpensive since the coupling matrix is calculated only once, with no further modifications needed thereafter, and all other operations being simple matrix multiplications. An optimal selection of the reference points or centres can be obtained effectively through a greedy method [16][17][18] which improves greatly the efficiency by finding redundant centres. Finally, Laplacian smoothing attempts to fix the mesh and recover the original grid quality via finite differences of Laplace's equation [19] when large displacements of boundary nodes affect the quality of cells nearby.…”

Section: Background a Existing Unsteady Cfd Meshing Methodsmentioning

confidence: 99%

23rd AIAA Computational Fluid Dynamics Conference

Ederra¹,

Rendall²,

Gaitonde³

et al. 2017

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-terms A spacetime framework is presented to solve unsteady aerodynamics problems as an alternative to conventional approaches for complex problems involving large deformation or topological change such as store separation, slat and flap deployment or spoiler deflection. It avoids complex CFD meshing methods, such as Chimera, by the use of a finite-volume approach both in space and time. The use of a central-difference scheme in the time direction yields non-physical transient solutions as a consequence of pressure waves travelling backwards in time. Therefore, an upwind formulation is provided and validated against one-dimensional test cases: a semi-infinite piston and a finite piston with a sharp change in direction. Also a hybrid formulation is given and several two-dimensional unsteady aerodynamics cases are computed with all three formulations and compared, demonstrating that the use of an upwind time stencil yields more representative physical solutions and improves the rate of convergence. In particular, the following problems are presented: a pitching NACA-0012 (periodic); a simple flap deflection; a spoiler deployment; a full landing case with a combination of slat, flap and spoiler deployments along with ground effect; and a case where aerofoils fly in opposite directions at subsonic and supersonic speeds.

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Section: Background a Existing Unsteady Cfd Meshing Methodsmentioning

confidence: 99%

23rd AIAA Computational Fluid Dynamics Conference

Ederra¹,

Rendall²,

Gaitonde³

et al. 2017

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“…[4]). Complex analysis techniques in the form of a Contour-Padé algorithm were suggested in [21], and numerical linear techniques based on the QR or singular value decompositions have also been proposed [9,20,24,27].…”

Section: Conditioning Of Rbf Interpolationmentioning

confidence: 99%

Preconditioning of Radial Basis Function Interpolation Systems via Accelerated Iterated Approximate Moving Least Squares Approximation

Fasshauer

Zhang

2009

Progress on Meshless Methods

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The standard approach to the solution of the radial basis function interpolation problem has been recognized as an ill-conditioned problem for many years. This is especially true when infinitely smooth basic functions such as multiquadrics or Gaussians are used with extreme values of their associated shape parameters. Various approaches have been described to deal with this phenomenon. These techniques include applying specialized preconditioners to the system matrix, changing the basis of the approximation space or using techniques from complex analysis. In this paper we present a preconditioning technique based on residual iteration of an approximate moving least squares quasi-interpolant that can be interpreted as a change of basis. In the limit our algorithm will produce the perfectly conditioned cardinal basis of the underlying radial basis function approximation space. Although our method is motivated by radial basis function interpolation problems, it can also be adapted for similar problems when the solution of a linear system is involved such as collocation methods for solving differential equations.

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“…One solution is to look for a well-behaved subspace of the trial space so that the condition of the reduced linear system can be controlled. We proposed various subspace selection algorithms [13,14] for such a purpose. The previously proposed sequential versions of adaptive greedy algorithms have a potentially high computational cost when the number of selections is large.…”

Section: Introductionmentioning

confidence: 99%

Convergence Studies for an Adaptive Meshless Least-Squares Collocation Method

Cheung¹,

Ling²

2017

Int. J. CMEM

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In this paper, we apply the recently proposed fast block-greedy algorithm to a convergent kernel-based collocation method. In particular, we discretize three-dimensional second-order elliptic differential equations by the meshless asymmetric collocation method with over-sampling. Approximated solutions are obtained by solving the resulting weighted least squares problem. Such formulation has been proven to have optimal convergence in H 2 . Our aim is to investigate the convergence behaviour of some three dimensional test problems. We also study the low-rank solution by restricting the approximation in some smaller trial subspaces. A block-greedy algorithm, which costs at most O(NK 2 ) to select K columns (or trial centers) out of an M × N overdetermined matrix, is employed for such an adaptivity. Numerical simulations are provided to justify these reductions. Keywords: ansa method, kernel-based collocation, adaptive greedy algorithm, elliptic equation1 INTRODUCTION Unsymmetric meshless kernel-based collocation methods, a.k.a. Kansa methods, have been used to solve various problems in science and engineering [1][2][3][4][5]. In 1990, Kansa solved time-dependent partial differential equation (PDEs) for the first time by such formulation using multiquadric [6,7]. Because of the ease of implementation and (potentially) high accuracy, the methods were widely adopted for over 10 years without any theoretical backup. In 2001, Hon and Schaback [8] showed that linear systems for Kansa methods could be singular. In order to obtain a non-singular linear system, the symmetric collocation method has been proposed [9], which requires a set of collocation-dependent basis functions. Until 2006, the convergence of the unsymmetric collocation method [10] has been proven provided that the collocation points are dense enough relative to the trial centers. The resulting linear system becomes over-determined, and it is natural to solve the resulting system in a least square sense. In [11], we investigate the convergence of the meshless collocation method for solving second-order elliptic equations in some bounded domain ll. The theories showed that we have to impose certain weighting for the boundary part to obtain the stability estimate. Using reproducing kernels of the Sobolev space H m (Ω), we proved that the weighted least square solution converges to the analytical solution in an optimal rate h m-2 in H 2 -norm. Numerical evidence in two dimensions also showed that the weighted least square problem gives more accurate results than the unweighted one [12].So far, we did not address the problem of ill-conditioning, which depends on the choice of the kernel, its shape parameter, data points distribution, and literally everything in the partial differential equation. One solution is to look for a well-behaved subspace of the trial space so that the condition of the reduced linear system can be controlled. We proposed various subspace selection algorithms [13,14] for such a purpose. The previously proposed sequential versions...

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Stable and Convergent Unsymmetric Meshless Collocation Methods

Cited by 71 publications

References 14 publications

23rd AIAA Computational Fluid Dynamics Conference

23rd AIAA Computational Fluid Dynamics Conference

Preconditioning of Radial Basis Function Interpolation Systems via Accelerated Iterated Approximate Moving Least Squares Approximation

Convergence Studies for an Adaptive Meshless Least-Squares Collocation Method

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