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

www.ietdl.org 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 functionpseudospectral meshfree methods to threedimensional electromagnetic problems P Vu1 G.E Fasshauer Department of Power Systems, Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam 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 Introduction 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 206 & The Institution of Engineering and Technology 2011 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 IET Sci Meas Technol., 2011, Vol 5, Iss 6, pp 206 –210 doi: 10.1049/iet-smt.2010.0125 www.ietdl.org 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 Section contains a brief introduction to the LOOCV algorithm In Section we apply the proposed methods and other standard numerical methods to a 3D benchmark electromagnetic problem Conclusions will be presented in Section 2.1 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 The approximate solution u h is 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 N Luh (x) = (1) The approximate RBF solution is formulated as a linear combination of RBFs as N uh (x) = cj w(||x − xj ||2 ) (2) j=1 where the basis functions w( † xj 2) 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 c=A u (4) The interpolation matrix A in (3) is usually symmetric and positive definite (or becomes positive definite when it is IET Sci Meas Technol., 2011, Vol 5, Iss 6, pp 206–210 doi: 10.1049/iet-smt.2010.0125 (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 = AL c (7) where uL ¼ [Lu h(x1), , Lu h(xNI), Bu h(xNI+1), , Bu h(xN)]T, the N × N matrix AL has 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 = AL A−1 u (8) Finally, the approximate solution of the boundary value problem (5) at the collocation points of the RBF-PS method is given by (3) where u ¼ [u h(x1), , u h(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 −1 cj Bw(||x − xj ||2 ) j=1 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 u h which interpolates (or more generally approximates) the function u on the set of given data, that is i = 1, , N (6a) N Buh (x) = RBF-pseudospectral method u (xi ) = u(xi ) cj Lw(||x − xj ||2 ) j=1 RBF-PS meshfree methods h (5) Bu = g u = L−1 G f g (9) in which LG = AL A−1 (10) is referred to as a differentiation matrix, and AL is as above As mentioned earlier, using the traditional RBF collocation method of [6] one first computes the coefficients c by solving 207 & The Institution of Engineering and Technology 2011 www.ietdl.org (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 LG is well-defined Note that this does not imply that LG is always guaranteed to be invertible (for more details see [7]) 2.2 ek = 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 LG u = f g (11) with LG as 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] 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 Choosing an ‘optimal’ shape parameter The choice of basis function w is an important step in the design of a truly meshfree method There are various wellknown popular smooth functions used in many papers in the mathematics and engineering literature In this study we use the multiquadric (MQ), w(r) = + (1r)2 , where r ¼ x xj Many RBFs contain a positive shape parameter 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, 0, corresponding to flat RBFs, then the computation will be unstable due to an increasingly illconditioned 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 that balances accuracy and stability For a much more detailed discussion of this phenomenon we refer the reader to [7] 208 & The Institution of Engineering and Technology 2011 ck A−1 kk (12) 21 where A−1 kk is the diagonal element of A 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 Numerical results 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 in Fig [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 Electrostatic cubical box as in [3] IET Sci Meas Technol., 2011, Vol 5, Iss 6, pp 206 –210 doi: 10.1049/iet-smt.2010.0125 www.ietdl.org Fig Distribution of 1331 uniform collocation points in [0, 1]3 along with extra boundary collocation points 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 = E= P P P i=1 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 shows 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 shows the distribution of points on a crosssection 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 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, P (uh (xi ) − u(xi ))2 (16) 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 in Fig The error curves show that the basic RBF-PS method coupled with the LOOCV Fig Distribution of collocation, auxiliary, boundary and outside boundary points on cross-section at y ¼ 0.5 Fig Error norms while increasing the total number of collocation points of the RBF-PS method IET Sci Meas Technol., 2011, Vol 5, Iss 6, pp 206–210 doi: 10.1049/iet-smt.2010.0125 Fig Convergence of RMS-error and E norms while increasing the number of RBFs for the least squares RBF-PS method 209 & The Institution of Engineering and Technology 2011 www.ietdl.org M ¼ N ¼ 1331, the solution of the least squares RBF-PS will be similar to the one of the RBF-PS method Conclusion 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 References Parreira, G.F., Silva, E.J., Fouseca, A.R., Mesquita, R.C.: ‘The elementfree galerkin method in three-dimensional electromagnetic problems’, IEEE Trans Magn., 2006, 42, (4), pp 711–714 210 & The Institution of Engineering and Technology 2011 Foncesa, A.R., Viana, S.A., Silva, E.J., Mesquita, R.C.: ‘Imposing boundary condition in the Meshless local Petrov-Galerkin method’, IET Sci Meas Technol., 2008, 2, (6), pp 387–394 Zhang, Y., Shao, K.R., Guo, Y., et al.: ‘An improved multiquadric collocation method for 3-D electromagnetic problems’, IEEE Trans Magn., 2007, 43, (4), pp 1509– 1512 Ferreira, A.J.M., Fasshauer, G.E., Batra, R.C., Rodrigues, J.D.: ‘Static deformations and vibration analysis of composite and sandwich plates using a Layerwise theory and RBF-PS discretizations with optimal shape parameter’, Compos Struct., 2008, 86, pp 328 –343 Ferreira, A.J.M., Fasshauer, G.E., Batra, R.C.: ‘Natural frequencies of thick plates made of orthotropic, monoclinic, and hexagonal materials by a Meshless method’, J Sound Vib., 2009, 319, pp 984 –992 Kansa, E.J.: ‘Multiquadrics – a scattered data approximation scheme with applications to computational fluid dynamics – Part ii: solutions to parabolic, hyperbolic and elliptic partial differential equations’, Comput Math Appl., 1990, 19, (8/9), pp 147–161 Faksshauer, G.E.: ‘Meshfree approximation methods with Matlab’ (World Scientific Publishing Co., 2007) Fasshauer, G.E., Zhang, J.G.: ‘On choosing “optimal” shape parameter for RBF approximation’, Numer Algorithms, 2007, 45, pp 346–368 Rippa, S.: ‘An algorithm for selecting a good value of the parameter c in radial basis function interpolation’, Adv Comput Math., 1999, 11, pp 193–210 10 Sivakumar, N., Ward, J.D.: ‘On the least squares fit by radial functions to multidimensional scattered data’, Numer Math., 1993, 65, pp 219–243 IET Sci Meas Technol., 2011, Vol 5, Iss 6, pp 206 –210 doi: 10.1049/iet-smt.2010.0125 ... 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. .. boundary collocation points 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... 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