DSpace at VNU: Application of two radial basis function-pseudospectral meshfree methods to three-dimensional electromagnetic problems

The least squares RBF-PS method introduced here as a modification of the RBF-PS method allows the authors to work with fewer RBFs while maintaining a high number of collocation points.. T

Published in IET Science, Measurement and Technology

Received on 3rd October 2010

Revised on 14th March 2011

doi: 10.1049/iet-smt.2010.0125

ISSN 1751-8822 Application of two radial basis

function-pseudospectral meshfree methods to

three-dimensional electromagnetic problems

P Vu1 G.E Fasshauer2

1

Department of Power Systems, Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam

2 Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA

E-mail: phantu_vu@hcmut.edu.vn

Abstract: In this study the authors present an application of the radial basis function-pseudospectral (RBF-PS) meshfree method

as well as a least squares variant thereof to a three-dimensional (3D) benchmark engineering problem defined by the Laplace equation To their knowledge this is the first such study The RBF-PS method is a version of the radial basis function (RBF) collocation method formulated in the vein of traditional pseudospectral methods The least squares RBF-PS method introduced here as a modification of the RBF-PS method allows the authors to work with fewer RBFs while maintaining a high number of collocation points In addition, the authors use a leave-one-out cross validation algorithm to choose an

‘optimal’ shape parameter for their basis functions In order to evaluate the accuracy, effectiveness and applicability of their new approach, the authors apply it to a 3D benchmark electromagnetic problem Their numerical results demonstrate that the proposed methods compare favourably to the finite difference and finite element methods

Meshfree methods have been developed and widely used for

solving partial differential equations (PDEs) in science and

engineering in recent years, including electromagnetic [1 –

3] and mechanical[4, 5] applications The use of meshfree

methods for electromagnetic problems encountered in the

literature falls into two groups: (i) weak form formulations

of the problem as in [1, 2], where one uses a local

approximation and numerical integration similar to finite

element method (FEM) to solve Poisson’s equation; and

(ii) strong form or collocation[3]formulations, such as radial

basis function (RBF) methods of the type first introduced by

Kansa[6] in 1990 for solving PDEs The main idea of this

latter method is to use only a set of collocation points to

discretise the domain without having to resort to any mesh or

numerical integration This method is very suitable for

solving problems with more complex geometries, particularly

in engineering applications In the past years, it has also been

shown to perform well with regard to high accuracy

demands Moreover, the RBF collocation method is easy to

implement for the numerical solution of electromagnetic

problems of the type we are interested here

As it is generally formulated, the standard RBF collocation

method requires first the computation of the RBF expansion

coefficients, followed by a second evaluation step Both of

these steps are likely to become ill conditioned In

particular, many choices of RBFs and their shape

parameters lead to collocation matrices that are ill

conditioned and therefore unstable to invert This leads to

potentially inaccurate expansion coefficients that are subsequently multiplied with another ill-conditioned matrix during the evaluation step The RBF-PS method (a reformulation of the RBF collocation method as a pseudospectral method) was developed and used recently (see e.g [4, 5, 7]) to avoid some of these drawbacks One

of its features is given by the fact that the solution at the collocation points can be directly obtained without having

to go through the computation of RBF expansion coefficients Another potential advantage is given by the fact that the basic RBF differentiation matrix (before any boundary conditions are applied) can be shown to be invertible, whereas its polynomial counterpart is generally singular Although this latter fact is not important for the solution of well-posed PDEs (since polynomial differentiation matrices are known to be non-singular once appropriate boundary conditions are added), this feature of the RBF-PS method may be advantageous in the context of ill-posed problems

In this paper we introduce a least squares variant of the RBF-PS method For function approximation problems, numerical studies have shown that least squares formulations may be more stable, more efficient and still yield very high accuracy As is usually the case with least squares methods, we are interested in the case where we use fewer RBFs than collocation conditions – thus arriving at a rectangular differentiation matrix We demonstrate the full flexibility of this least squares approach by picking different sets of points for our RBF centres and the collocation conditions A theoretical analysis of this very general

situation is still open, but our numerical experiments are

encouraging One of the main features of the least squares

RBF-PS method is its efficiency when compared with the

basic RBF-PS method

In the past, RBF-PS methods have been applied mostly to

two-dimensional (2D) problems [4, 5, 7, 8] Therefore

another novelty of this paper is the fact that we apply

perhaps for the first time an RBF-PS method to a 3D

electromagnetic problem defined mathematically in terms of

a Laplace equation Finally, we use a further modification

of the leave-one-out cross validation (LOOCV) algorithm of

[9] which had previously been adapted for the RBF-PS

method in [9] to determine the ‘optimal’ shape parameters

for our meshfree methods We will compare solutions to

our benchmark problem obtained with the two RBF-PS

methods to those obtained with FEM and finite difference

method (FDM)

The organisation of this paper is as follows: we will first

present the RBF-PS and least squares RBF-PS meshfree

methods by introducing the strong form for a general

elliptic boundary value problem in Section 2 Section 3

contains a brief introduction to the LOOCV algorithm In

Section 4 we apply the proposed methods and other

standard numerical methods to a 3D benchmark

electromagnetic problem Conclusions will be presented in

Section 5

2.1 RBF-pseudospectral method

In a scattered data interpolation problem we are given a set of

data sites X ¼{x1, , xN} , V and associated function

values u(xi), i ¼ 1, , N, where V is a bounded domain in

Rs, and we want to find a function uhwhich interpolates (or

more generally approximates) the function u on the set of

given data, that is

uh(xi)= u(xi) i= 1, , N (1)

The approximate RBF solution is formulated as a linear

combination of RBFs as

uh(x)=N

j =1

cjw(||x − xj||2) (2)

where the basis functions w(† 2 xj2) depend only on the

distance from the centre xj and w is usually assumed to be

strictly (conditionally) positive definite The unknown

coefficients c ¼ [c1, , cN]T in (2) are determined using

the interpolation conditions (1)

We can formulate this problem in matrix – vector form, that

is

u= Ac (3) where u ¼ [uh(x1), , uh(xN)]T results from applying the

interpolation conditions (1) to (2) so that the N× N

interpolation matrix A has entries Therefore c is given by

c= A−1u (4) The interpolation matrix A in (3) is usually symmetric and

positive definite (or becomes positive definite when it is

augmented by appropriate polynomial blocks) However, it may be ill conditioned In particular, the matrix A will become badly ill conditioned for infinitely smooth RBFs that are made increasingly flat by the choice of their shape parameters Increasing problem size, N, of course also has a detrimental effect on the condition number of A As a consequence, solution of the linear system (3) – or inversion of the matrix A – may be unstable A similar problem arises in the collocation solution of PDEs as first suggested in[6]

In order to introduce the RBF-PS method, we will consider

a general linear elliptic boundary value problem with linear boundary condition given by

Lu= f

Bu= g (5) The approximate solution uhis assumed to be of the form (2)

as before and we can apply the differential operators L and B

to this expansion This results in

Luh(x)=N

j =1

cjLw(||x − xj||2) (6a)

Buh(x)=N

j=1

cjBw(||x − xj||2) (6b)

An important characteristic of the RBF-PS method compared with the RBF collocation method of [6] is that it directly evaluates the approximate solution at the collocation points

xi Therefore, similarly as in (3), we can express the problem in matrix – vector form (see[4, 5, 7])

uL= ALc (7) where uL¼ [Luh

(x1), , Luh(xNI), Buh(xNI +1), , Buh(xN)]T, the N× N matrix ALhas entries

(AL)ij = Lw(||xi− xj||2) for i= 1, , NI

Bw(||xi− xj||2) for i= NI + 1, , N



j ¼ 1, , N and NI denotes the number of collocation conditions applied on the interior of the domain V

By substituting the representation (4) for c into (7) we end

up with

uL= ALA−1u (8)

Finally, the approximate solution of the boundary value problem (5) at the collocation points of the RBF-PS method

is given by

u= L−1 G

f g

 

(9)

in which

LG= ALA−1 (10)

is referred to as a differentiation matrix, and ALis as above

As mentioned earlier, using the traditional RBF collocation method of[6]one first computes the coefficients c by solving

(7) and then uses those coefficients to evaluate the expansion

(2) at a set of desired evaluation points With the RBF-PS

method, on the other hand, we compute the matrix LG of

(10) and then obtain the approximate solution at the

collocation points via (9) For most commonly used RBFs

the matrix A is invertible and therefore the differentiation

matrix LGis well-defined Note that this does not imply that

LG is always guaranteed to be invertible (for more details

see[7])

2.2 Least squares RBF-PS method

In order to obtain a numerical method that is more efficient

than the RBF-PS (or basic RBF collocation) method of the

previous section while retaining similar accuracy we

propose the use of a least squares approach This means

that we use a smaller number, M, of RBF centres than

collocation points, N, and then enforce the collocation

conditions (6) only in the discrete least squares sense

instead of satisfying them exactly

The formulation of the previous section changes only in

two small – but important – places The matrix AL is now

rectangular, N× M, and so is the differentiation matrix LG

This means that we obtain the approximate solution at the

RBF centres as the least squares solution of the rectangular

system

LGu= fg

 

(11)

with LGas in (10), but with rectangular matrix AL

The second modification is given by the fact that we allow

the RBFs centres to be different from (in fact, not even a

subset of) the collocation points This means that the

theoretical foundation of this method is not at all clear

However, by choosing the RBF centres uniformly randomly

distributed in the domain they approximately represent

clusters of collocation points (which we take equally spaced

in our experiments) This is in the spirit of the only existing

work in the literature on least squares RBF approximation

[10]

The choice of basis function w is an important step in the

design of a truly meshfree method There are various

well-known popular smooth functions used in many papers in

the mathematics and engineering literature In this study we

use the multiquadric (MQ), w(r)=1+ (1r)2

, where

r ¼x 2 xj2 Many RBFs contain a positive shape

parameter 1 that is very important in the theory and practice

of meshfree methods This parameter influences not only

the accuracy of the solution but also its numerical stability

In particular, it is known that if the shape parameter

becomes large, the accuracy will be low (but the matrix A

will approach a diagonal – and thus very well-conditioned –

matrix) On the other hand, if the shape parameter becomes

small, that is, 1 0, corresponding to flat RBFs, then the

computation will be unstable due to an increasingly

ill-conditioned interpolation matrix Since it is also known that

the accuracy tends to increase with decreasing shape

parameter, practitioners therefore look for an ‘optimal’ value

of 1 that balances accuracy and stability For a much

more detailed discussion of this phenomenon we refer the

reader to[7]

Here we use the method of cross validation well-known in the statistics literature to estimate the shape parameter based

on the given data xi, u(xi), i ¼ 1, , N In this algorithm, the ‘optimal’ shape parameter is found by minimising the error for a fit to the data based on an approximation for which one of the centres was ‘left out’ Therefore it is called LOOCV As presented in[8, 9], the algorithm for the error estimator can be simplified to a single formula of cost vector components as

ek = ck

A−1 kk

(12)

where A−1kk is the diagonal element of A21 Since an equivalent way to write (10) is as

A(LG)T= (AL)T (13) the components of the cost matrix for the RBF-PS version of the LOOCV algorithm corresponding to (12) are given by

Ekl =((LG)T)kl

A−1 kk

(14)

More on the LOOCV algorithm can be found in[7, 8] The modification for the least squares method consists of interpreting (13) and (14) in the least squares sense, that is, using a pseudoinverse instead of an inverse

In order to test the least squares RBF-PS method with the LOOCV shape parameter strategy for a 3D problem, we select a benchmark electromagnetic problem because it has

an analytical solution and the variation of the electrostatic potential is strong in the upper corners[1, 3] This problem

is defined by the following Laplace equation as

∇2

V (x, y, z)= 0, x, y, z [ V = [0, 1]3

(15) and the boundary conditions are assumed as inFig 1 [3]

To solve the above 3D Laplace equation we use: (i) uniformly distributed collocation points as in Fig 2, where the dots, the circles and the X’s mark the interior collocation points, boundary collocation points and additional outside RBF centres, respectively (see [7] for more discussion of the choice of these points); (ii) the conditionally negative definite MQ RBF; and (iii) the LOOCV algorithm for choosing an ‘optimal’ shape parameter

Fig 1 Electrostatic cubical box as in [3]

In order to evaluate the accuracy and applicability of the

proposed methods, we use the root-mean square error

(RMS-error) of (16) and relative error (E) of (17) to

compare the errors of our RBF-PS meshfree solutions

against those obtained with FEM and FDM For this

comparison, we also used uniformly distributed

discretisation points for FDM, and for the FEM we used the

same points in a Delaunay mesh

RMS-error=



1 P

P i=1

(uh(xi)− u(xi))2



(16)

E=1

P



P i=1(uh(xi)− u(xi))2 P

i=1(u(xi)2

(17)

Here P is the total number of points at which the solution is

computed

The results are illustrated inFig 3 The error curves show

that the basic RBF-PS method coupled with the LOOCV

algorithm can be successfully applied to our 3D benchmark problem and that it yields more accurate solutions than the other standard numerical methods based on the same number of discretisation points As presented in the conclusion of [3], here we can again see that using a few hundred collocation points, one can obtain a solution that has the same level of accuracy as those of FEM and FDM methods

Fig 3shows the behaviour of the RMS-error and E for the 3D numerical solutions while increasing the total number of collocation points To illustrate the application of the least squares RBF-PS method to the 3D electromagnetic problem, Fig 4shows the distribution of points on a cross-section of the domain at y ¼ 0.5, where the dots, the stars, the circles and the X’s mark the collocation, auxiliary, boundary and outside boundary points, respectively In this case, we can see that the number and location of RBF auxiliary points are different from those of the collocation points as indicated above

To evaluate the accuracy of the least squares RBF-PS method, Fig 5 shows the convergence of the RMS-error and E norms of the 3D solution while increasing the total number M of RBFs and fixing the number of collocation points at 1331 We can observe that when the number of RBF centres is similar to those of collocation points, that is,

Fig 3 Error norms while increasing the total number of

collocation points of the RBF-PS method

Fig 4 Distribution of collocation, auxiliary, boundary and outside boundary points on cross-section at y ¼ 0.5

Fig 2 Distribution of 1331 uniform collocation points in [0, 1]3

along with extra boundary collocation points

Fig 5 Convergence of RMS-error and E norms while increasing the number of RBFs for the least squares RBF-PS method

M ¼ N ¼ 1331, the solution of the least squares RBF-PS will

be similar to the one of the RBF-PS method

In this paper, two new approaches for the numerical solution

of a 3D benchmark electromagnetic problem were proposed:

the RBF-PS and least squares RBF-PS meshfree methods In

particular, we presented to our knowledge for the first time the

formulation of a least squares RBF-PS method for a general

problem In addition, parameters such as points outside the

boundary, type of RBFs and LOOCV algorithm for

choosing ‘optimal’ shape parameters were used to increase

the accuracy of solutions The numerical results of the

proposed methods were compared with other standard

numerical methods, and it was shown that our proposed

methods are more accurate than other numerical methods

when using the same number of discretisation points

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To evaluate the accuracy of the least squares... convergence of the RMS-error and E norms of the 3D solution while increasing the total number M of RBFs and fixing the number of collocation points at 1331 We can observe that when the number of RBF