Fonction de base radiale

Résumé

Une fonction de base radiale est une fonction à valeurs réelles dont la valeur ne dépend que de la distance séparant son paramètre d'entrée x à un autre point donné, communément appelé origine ou centre de la fonction. Toute fonction \phi qui vérifie l'égalité \phi(x)=\varphi(\lVert x \rVert) est une fonction de base radiale. La norme \lVert \cdot \rVert utilisée correspond à la distance euclidienne, d'autres métriques peuvent cependant être utilisées. Au cours des dernières décennies, plusieurs méthodes d'approximation et d'interpolation basées sur les fonctions de base radiale ont trouvé leurs utilités dans un large éventail d'applications des sciences de l'ingénieur, allant de l'apprentissage automatique à la résolution numérique des équations différentielles aux dérivées partielles. Définition Une fonction \phi : \mathbb{R}^{d} \rightarrow \mathbb{R} est dite radiale s'il existe une fonction \var

Source officielle

https://fr.wikipedia.org/wiki/Fonction_de_base_radiale

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Essentially nonoscillatory (ENO) and weighted ENO (WENO) methods on equidistant Cartesian grids are widely employed to solve partial differential equations with discontinuous solutions. However, stable ENO/WENO methods on unstructured grids are less well studied. We propose high-order essentially nonoscillatory methods based on radial basis functions (RBFs) to solve hyperbolic conservation laws. We derive a smoothness indicator that guarantees the satisfaction of the sign property of the resulting interpolant on general one-dimensional grids. Based on this algorithm we introduce an entropy stable arbitrary high-order finite difference method (RBF-TeCNOp) and an entropy stable second order finite volume method (RBF-EFV2) for one-dimensional problems. Hence, we show that methods based on radial basis functions are as powerful as methods based on polynomial reconstruction. Next, we propose a high-order ENO method based on radial basis functions to solve hyperbolic conservation laws on general two-dimensional unstructured grids. The radial basis function reconstruction offers a flexible framework to deal with ill-conditioned cell arrangements. We define a smoothness indicator based the one-dimensional version and a stencil selection algorithm suitable for general meshes. Furthermore, we develop a stable method to evaluate the RBF reconstruction in the finite volume setting to circumvent the stagnation of the error and keep the condition number of the reconstruction bounded. To reduce the computational complexity, we develop the RBF-CWENO method. This method exhibits high-order convergence and robustness when solving challenging problems and is considerably faster. However, the resolution close to shocks and turbulent structures is lower than for the RBF-ENO method. Finally, we present a hybrid high-resolution RBF-ENO method which is based on the RBF-ENO method for unstructured patches and the standard WENO method on structured ones. Furthermore, we introduce a positivity preserving limiter for non-polynomial reconstruction methods that stabilizes the hybrid RBF-ENO method for problems with low density or pressure. We show its robustness on the scramjet inflow problem and a conical aerospike nozzle jet simulation.

EPFL2020

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Topology Optimization with Radial Basis Functions

Julien Hess

The purpose of this project is to perform topology optimization on the cantilever beam problem, which is a particular case of the two-dimensional elasticity problem. Based on a finite element discretization, a Matlab implementation of the elasticity problem is first presented. The Young modulus is then utilized in order to model the cavities in the beam. Finally, a topological optimization of the beam is performed by using a radial basis function approximation of the Young modulus.

2012

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RBF Based CWENO Method

Jan Sickmann Hesthaven, Fabian Mönkeberg

Solving hyperbolic conservation laws on general grids can be important to reduce the computational complexity and increase accuracy in many applications. However, the use of non-uniform grids can introduce challenges when using high-order methods. We propose to use a Central WENO (CWENO) scheme based on radial basis function (RBF) interpolation, which is applicable to scattered data. We develop a smoothness indicator, based on RBFs, and CWENO specific weights which depend on the mesh size of the grid to construct an arbitrarily high order RBF-CWENO method. We evaluate the method with multiple examples in one dimension.

2018

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