A least squares radial basis function finite difference method with improved stability properties

Tominec, Igor; Larsson, Elisabeth; Heryudono, Alfa

doi:10.48550/arxiv.2003.03132

Cited by 3 publications

(15 citation statements)

References 26 publications

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“…We noted in [17] that the X-points are supposed to be distributed such that the internodal distance is as uniform as possible, since the Lebesgue constants associated with the cardinal functions then stay fairly small. On the other hand the evaluation point set does not influence the magnitude of the Lebesgue constants but is instead important for the implicit integration that occurs when solving a discretized system of equations in the least-squares sense [17]. Thus the constraints for placing the evaluation points are far more forgiving.…”

Section: The Point Setsmentioning

confidence: 99%

“…which are discrete inner products, to continuous inner products plus a first order integration error [17]. Secondly, the factor h −1 is used to impose the Dirichlet condition in a weak sense such that the matrix D h is nonsingular: this is a classical approach in those finite element methods which use a solution space that does not exactly satisfy the Dirichlet condition.…”

Section: The Unfitted Discretization Of a Pdementioning

confidence: 99%

“…A recently introduced RBF-FD method in a least-squares setting (RBF-FD-LS) [17] was proven to be significantly more robust compared to the same method in the collocation setting (RBF-FD-C), especially in the presence of Neumann-type boundary conditions [17]. However the interpolation points are required to conform to Ω.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we use the RBF-FD-LS method [17] and for that method introduce an approach to decouple interpolation nodes from Ω. A computational domain Ω is enclosed in a box that contains a set of regularly spaced interpolation nodes with reasonably good interpolation properties (see Figure 1 and Figure 2).…”

Section: Introductionmentioning

confidence: 99%

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An unfitted RBF-FD method in a least-squares setting for elliptic PDEs on complex geometries

Tominec,

Breznik

2020

Preprint

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Radial basis function generated finite difference (RBF-FD) methods for PDEs require a set of interpolation points which conform to the computational domain Ω. One of the requirements leading to approximation robustness is to place the interpolation points with a locally uniform distance around the boundary of Ω. However generating interpolation points with such properties is a cumbersome problem. Instead, the interpolation points can be extended over the boundary and as such completely decoupled from the shape of Ω. In this paper we present a modification to the least-squares RBF-FD method which allows the interpolation points to be placed in a box that encapsulates Ω. This way, the node placement over a complex domain in 2D and 3D is greatly simplified. Numerical experiments on solving an elliptic model PDE over complex 2D geometries show that our approach is robust. Furthermore it performs better in terms of the approximation error and the runtime vs. error compared with the classic RBF-FD methods. It is also possible to use our approach in 3D, which we indicate by providing convergence results of a solution over a thoracic diaphragm.

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Section: The Point Setsmentioning

confidence: 99%

Section: The Unfitted Discretization Of a Pdementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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An unfitted RBF-FD method in a least-squares setting for elliptic PDEs on complex geometries

Tominec,

Breznik

2020

Preprint

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“…This is an open problem not only for the method of this paper but also for all previous unsymmetric local meshless (RBF-based or else) methods. See [6,42] for recent attempts to tackle a similar problem in least squares settings for the RBF-FD method.…”

Section: Error and Stabilitymentioning

confidence: 99%

The Direct Radial Basis Function Partition of Unity (D-RBF-PU) Method for Solving PDEs

Mirzaei¹

2020

Preprint

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In this paper, a new localized radial basis function (RBF) method based on partition of unity (PU) is proposed for solving boundary and initial-boundary value problems. The new method is benefited from a direct discretization approach and is called the 'direct RBF partition of unity (D-RBF-PU)' method. Thanks to avoiding all derivatives of PU weight functions as well as all lower derivatives of local approximants, the new method is faster and simpler than the standard RBF-PU method. Besides, we are allowed to use discontinuous weight functions for developing other efficient numerical algorithms. Alternatively, the new method is an RBF-generated finite difference (RBF-FD) method in a PU setting which is much faster and in some situations more accurate than the original RBF-FD. The polyharmonic splines are used for local approximations, and the error and stability issues are considered. Some numerical experiments on irregular 2D and 3D domains, as well as cost comparison tests, are performed to support the theoretical analysis and to show the efficiency of the new method.

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An unfitted radial basis function generated finite difference method applied to thoracic diaphragm simulations

Tominec¹,

Villard²,

Larsson³

et al. 2021

Preprint

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The thoracic diaphragm is the muscle that drives the respiratory cycle of a human being. Using a system of partial differential equations (PDEs) that models linear elasticity we compute displacements and stresses in a two-dimensional cross section of the diaphragm in its contracted state. The boundary data consists of a mix of displacement and traction conditions. If these are imposed as they are, and the conditions are not compatible, this leads to reduced smoothness of the solution. Therefore, the boundary data is first smoothed using the least-squares radial basis function generated finite difference (RBF-FD) framework. Then the boundary conditions are reformulated as a Robin boundary condition with smooth coefficients. The same framework is also used to approximate the boundary curve of the diaphragm cross section based on data obtained from a slice of a computed tomography (CT) scan. To solve the PDE we employ the unfitted least-squares RBF-FD method. This makes it easier to handle the geometry of the diaphragm, which is thin and non-convex. We show numerically that our solution converges with high-order towards a finite element solution evaluated on a fine grid. Through this simplified numerical model we also gain an insight into the challenges associated with the diaphragm geometry and the boundary conditions before approaching a more complex three-dimensional model.

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A least squares radial basis function finite difference method with improved stability properties

Cited by 3 publications

References 26 publications

An unfitted RBF-FD method in a least-squares setting for elliptic PDEs on complex geometries

An unfitted RBF-FD method in a least-squares setting for elliptic PDEs on complex geometries

The Direct Radial Basis Function Partition of Unity (D-RBF-PU) Method for Solving PDEs

An unfitted radial basis function generated finite difference method applied to thoracic diaphragm simulations

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