# Radial Basis Functions Theory And Implementations Pdf

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Published: 07.05.2021  Networks with kernel functions ; Radial basis function approximation ; Radial basis function neural networks ; Regularization networks. Radial basis function networks are a means of approximation by algorithms using linear combinations of translates of a rotationally invariant function, called the radial basis function. The coefficients of these approximations usually solve a minimization problem and can also be computed by interpolation processes.

## Radial Basis Functions - Theory and Implementations

Radial basis functions RBFs based mesh morphing allows to adapt the shape of a computational grid onto a new one by updating the position of all its nodes. Usually nodes on surfaces are used as sources to define the interpolation field that is propagated into the volume mesh by the RBF. The method comes with two distinctive advantages that makes it very flexible: it is mesh independent and it allows a node wise precision.

There are however two major drawbacks: large data set management and excessive distortion of the morphed mesh that may occur. The BHS minimizes the mesh distortion but it is computational intense as a dense linear system has to be solved whilist the WC2 leads to a sparse system easier to solve but which can lack in smoothness. In this paper we compare these two radial kernels with a specific focus on mesh distortion.

A detailed insight about RBF fields resulting from BHS and WC2 is first provided by inspecting the intensity and the distribution of the strain for a very simple shape: a square plate with a central circular hole. An aeronautical example, the ice formation onto the leading edge of a wing, is then exposed adopting an industrial software implementation based on the state of the art of RBF solvers. Radial basis functions RBFs are a powerful mathematical tool introduced by Hardy [ 1 ] in the late sixties for the interpolation of scattered data in the field of surveying and mapping.

A review of the multiquadric MQ approach was published by Hardy himself [ 2 ] twenty years later; here the author explains that the MQ is bi-harmonic for 3D problems. Since their inception, RBF where adopted in many different fields and their mathematical framework developed [ 3 ] by exploring a variety of RBF kernels; among them the compact supported ones were introduced by Wendland [ 4 ]. An RBF interpolation at a generic point see Eq.

The weights are computed so that the RBF gets at the sources the known input values to be interpolated. RBF are nowadays adopted in several engineering applications [ 5 ] and are well accepted as one of the most powerful and versatile mathematical approach to manage mesh morphing.

The RBF allows creating a scalar field that interpolates a function defined on a set of source points: in the case of 3d mesh morphing a vector field is defined by individually interpolating the three components of the displacement known at source points.

The RBF vector field is a point function independent from the mesh itself, all the points in the space receiving the field are called targets. A typical mesh morphing problem that can be faced using RBF consists in a three-dimensional mesh to be adapted according to a known displacement of the surface; surface nodes are extracted from the surface mesh as sources and all the nodes of the volume mesh receive the morphing field as targets.

Computer Aided Engineering CAE is more and more demanding for advanced methods capable to generate and adapt computational grids for multi-physics models. High fidelity models are widely employed, for instance, in computational fluid dynamics CFD and computational structural mechanics CSM. The size of the grids daily adopted for CFD according to industrial best practices can be comprised of many millions of cells and, in some situation, close to one billion [ 6 ]; structured meshes of hexahedrons, hybrid meshes of tetrahedrons with prisms layers at the wall and meshes of Cartesian polyhedrons with inflation of prisms at surfaces are common adopted options for CFD.

For CSM applications the mesh size is about one-two order lower than CFD ranging from hundred thousand up to some million of nodes for the most complex cases [ 7 ]; parabolic tetrahedrons are in this case widely adopted and the extra complexity of mid-side nodes management is added.

In view of mesh morphing we have to consider that to the complexity of the mesh typology i. This is typical of design optimization which requires the automatic update of the CAE grid onto new design configurations to be explored: the baseline mesh is updated onto the new configuration instead of generating a new mesh onto the updated geometry [ 8 , 9 ].

Morphing becomes very useful for optimization where the new shape is not known in advance but predicted by automatic sculpting methods as the adjoint based [ 10 , 11 ] and the biological growth method [ 12 ]. Mesh morphing is also a key enabler toward the creation of reduced order models ROM that are adopted for the creation of digital twins [ 15 ]. Whatever is the need for mesh morphing, there are common requirements to fulfill to set-up an effective mesh morphing approach.

Among these there are the ability to:. RBF mesh morphing satisfactorily fits above mentioned requirements because it comes with two distinctive advantages that makes it very flexible: it is mesh independent and it allows a node wise precision.

The meshless nature is a distinctive feature of another widely adopted mesh morphing method: the free form deformation FFD [ 16 ]. One of the major drawbacks of FFD is that a point-wise precision cannot be achieved. Mesh based methods, as for instance the use of an auxiliary FEM solution [ 17 , 18 ], allow the user to have pointwise control; despite this benefit their mesh based nature makes complex the management of arbitrary elements typology and interfaces.

Considering the great advantages of RBF, it is clear why a large research effort has been invested over the last decades toward their effective implementation. There are however two major drawbacks to be handled: numerical complexity and deformed mesh quality.

RBFs require the solution of large linear systems whose size is equal to the number of source points. The source points count grows fast especially if a node-wise control is needed because large portions of the nodes on the surface mesh are used as sources.

Most of the implementations that are considered for research purposes exploits direct linear solvers because a great flexibility is possible and different radial function kernels can be seamlessly adopted. Unfortunately the numerical complexity scales up in this case with a cubic law so the maximum size of the RBF problem number of sources is limited to about 10 points.

To overtake such limit there are different strategies: replace the original cloud with a smaller one point decimation fine enough to guarantee the desired precision [ 19 ], adopt iterative [ 20 ] solvers, use of the partition of unity [ 21 ] and use of fast multipole expansion [ 22 ]. The quality of the morphed mesh is of paramount importance and the effectiveness of deformation depends on the radial kernel adopted. Keeping a good quality usually is not enough; a proper spacing of the cells close to the surfaces is required, as example, for CFD meshes in order to preserve the capability to correctly solve the boundary layer.

The same spacing is expected in the deformed mesh. This means that while the shape of the surface is changed, the deformation orthogonal to the surface itself has to be minimum so that the boundary layers are solved keeping a similar overall thickness and spacing. The aim of this paper is to provide useful insights on two specific kernels that are widely adopted for mesh morphing: the bi-harmonic spline BHS and the Wendland C2 WC2.

The WC2 is compactly supported and so a sparse system has to be solved; however it may lack in smoothness and the behavior close to the surfaces is strongly dependent on the radius of the support. An overview of the studied problem is given in the first section this introduction , a refresh of the math of RBF, including first derivatives calculation, is provided in the second section; the third section demonstrates how various RBFs perform for a square plate with a central circular hole, the fourth section deals with an aeronautical example faced with an industrial software and, finally, the fifth section wraps the study with the concluding remarks.

The RBF interpolation of a generic scalar field can be adopted to represent a variety of quantities. A generic component of a displacement field can be interpolated as. In this study we consider the case where the quantity to be interpolated is given at the source point locations.

The unknown coefficient vector can be computed by solving the linear system. Where the interpolation matrix is. Different radial functions can be adopted. In the case of three-dimensional spaces the interpolated function can be rewritten as. And can be differentiated, for instance, with respect to.

The full gradient of the interpolated function can be computed accordingly. When RBFs are used for mesh morphing, the three components of a displacement field Eq. The local deformation due to the morphing field can be inspected by computing the derivatives of the three components of displacement thus obtaining the strains as follows.

The definition of the RBF problem, i. An overview of RBF mesh morphing strategies is given in [ 23 , 24 ], while a deeper presentation about the use of RBF mesh morphing in industrial applications can be found in [ 25 ]. Among such a variety of mesh morphing paradigms the simpler one, which is the easiest to be automated, consists in the usage of all the nodes or a subset onto the surfaces as sources [ 26 ].

For sake of simplicity the study herein presented is based on such approach. The example of Fig. An uniform spacing is imposed to all curves of the original geometry. The deformed positions of sources are then computed onto the updated CAD by keeping the same parametric distribution along the curve.

The RBF field produced by source points is then used to update all the nodes of the mesh. The notched bar is reshaped adopting RBF mesh morphing. The nodes on all the curves are updated onto the new geometrical model keeping the same parametric spacing along the curves.

The first application faced to understand how the different radial functions are capable to handle mesh morphing consists in a simple geometry: the square plate 1. Square plate with a circular hole: the nodes on the curves are used to define the RBF sources, the nodes of the mesh are the morphing targets. The implementation of this RBF mesh morphing demonstrator is divided in two parts.

The mesh is exported in a readable ASCII format Nastran data deck in this example so that the full mesh nodes and elements together with boundary conditions constrained nodes with prescribed displacements can be translated. The second part is a Mathcad application that implements basic RBF according to [ 5 ] and provides a fast bench to play around with the math and with obtainable morphed mesh. This implementation is good for investigate the method thanks to great flexibility provided, but it is not intended to be used for the assessment of numerical performances.

All the nodes belonging to the curves on the boundary are in this case used as RBF sources. The ones along the square perimeter are kept fixed while the ones on the hole are moved radially of 0. The surface plot of the strain is then generated for all the points with the exception of the ones inside the circle for which a zero value is imposed.

All surface plots of Fig. We can clearly notice that, as expected, the maximum strain is 0. As far as the minimum strain, this is related to the compression of the cell required to accommodate morphing, and it has a peak along the y direction pure radial on the circle.

The minimum value represents the severity of element compression due to morphing. A better insight can be gained by plotting the strain along the vertical segment that starts from the midpoint of the bottom side of the square and ends at the intersection with the circle Fig. In this comparison also the WC0 and WC4 functions are included. The ability of the RBF to succesfully adapt the mesh is investigated by plotting the minimum quality of the triangles i.

The WC2 with radius 0. Chart of minimum mesh quality as a function of the deformation intensity for the BHS left and for the WC2 with support radius 0. Chart of minimum mesh quality as a function of the deformation intensity for the WC2 with support radius 0.

To better understand the effect of the support radius for the WC2 a larger value of 0. The case of 0. It is worth to notice that such large support radius allows to add the long distance interactions; the benefit of generating sparse system will be in this case reduced or lost. RBF Morph allows to set-up the RBF mesh morphing problems with a variety of methods including the simple ones relevant for this study.

A library of Wendland functions is available as well. The study presented in this section refers to the problem studied in [ 27 , 28 ]. Mesh morphing is adopted to simulate the growth of ice; advanced CFD simulations coupled with icing models allow to compute the distribution of ice thickness onto the surfaces. The ice profile has an aerodynamic impact which cannot be neglected if the shape change is relevant. It is worth to notice that, as anticipated in the introduction, the spacing of the cells close to the wall is defined to solve the boundary layer.

A key feature of the mesh adapted onto new shapes is to preserve such spacing similar to the one of the baseline mesh. Lateral view of the CFD mesh left with a detail of the mesh around the airfoil profile right. Mesh vertexes at the walls are used as RBF sources red points.

Color figure online. The mesh morphing set-up can be appreciated in Fig. KOHN, M. The series publishes expositions on all aspects of applicable and numerical mathematics, with an emphasis on new developments in this fast-moving area of research. State-of-the-art methods and algorithms as well as modern mathematical descriptions of physical and mechanical ideas are presented in a manner suited to graduate research students and professionals alike. Sound pedagogical presentation is a prerequisite. It is intended that books in the series will serve to inform a new generation of researchers. Also in this series: 1. Dynamical Systems and Numerical Analysis, A.

Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. DOI: Buhmann Published in Cambridge monographs on…. Preface 1. Introduction 2. Summary of methods and applications 3. Preface 1. Introduction 2. Summary of methods and applications 3. General methods for approximation and interpolation 4. Radial basis function approximation.

In the field of mathematical modeling , a radial basis function network is an artificial neural network that uses radial basis functions as activation functions. The output of the network is a linear combination of radial basis functions of the inputs and neuron parameters. Radial basis function networks have many uses, including function approximation , time series prediction , classification , and system control. They were first formulated in a paper by Broomhead and Lowe, both researchers at the Royal Signals and Radar Establishment. Radial basis function RBF networks typically have three layers: an input layer, a hidden layer with a non-linear RBF activation function and a linear output layer.

The distance is usually Euclidean distance , although other metrics are sometimes used. Sums of radial basis functions are typically used to approximate given functions. This approximation process can also be interpreted as a simple kind of neural network ; this was the context in which they were originally applied to machine learning, in work by David Broomhead and David Lowe in ,   which stemmed from Michael J. Powell 's seminal research from Approximation schemes of this kind have been particularly used [ citation needed ] in time series prediction and control of nonlinear systems exhibiting sufficiently simple chaotic behaviour and 3D reconstruction in computer graphics for example, hierarchical RBF and Pose Space Deformation.

Radial basis functions provide powerful meshfree method for multivariate interpolation for scattered data. But both the approximation quality and stability depend on the distribution of the center set. Many methods have been constructed to select optimal center sets for radial basis function interpolation. A review of these methods is given. Four kinds of center choosing algorithms which are thinning algorithm, greedy algorithm, arclength equipartition like algorithm and k-means clustering algorithm are introduced with some algorithmic analysis.

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Radial basis functions RBFs based mesh morphing allows to adapt the shape of a computational grid onto a new one by updating the position of all its nodes. Edilio M.

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