International
Workshop on MeshFree Methods 2003 Instituto
Superior
Técnico
July
2123, 2003

Abstracts
Contributed presentations
[x] Participants that will not be able to attend the workshop. 
Guidelines for extended abstracts
E. Arantes e Oliveira Dep. Eng. Civil e Arquitectura, Instituto Superior Técnico, Portugal arantes civil.ist.utl.pt 

Extended
Abstract (pdf)
Outline:

for Domain & BIE Discretizations Satya N. Atluri University of California at Irvine, USA satluri uci.edu 
Abstract
(pdf)
This talk will discuss how the Meshless Local Petrov Galerkin Method [Atluri and Zhu(1998), Atluri and Shen (2002 a,b)] is the basis of a variety of meshless methods, based on the use of a variety of trial functions and a variety of test fucntions, for both domainsolution approaches [Atluri and Shen(2003)], as well as boundary solution approaches using integral equations [Atluri, Han, and Shen (2003)]. Some new and recent results for 3dimensional solid mechanics problems involving strong singularities [Li, Shen, Han and Atluri (2003)], 3dimensional nonplanar fatigue crackgrowth in complexly loaded solids and structures [Han and Atluri (2003 a, b)], and in straingradient theories of material inelasticity [Tang, Shen and Atluri (2003)], will be presented and critically examined. The novel use of the MLPG method in generating O(N) algorithms for molecular dynamics, and for multiplelength&time scale simulations, is illustrated. References:  Atluri SN, Zhu T(1998): A new meshless local PetrovGalerkin (MLPG) approach in computational mechanics. Comput. Mech., 22: 117127  Atluri SN, Shen SP(2002): The mesh local PetrovGalerkin (MLPG) method: a simple & lesscostly alternative to the finite element and boundary element methods. CMES: Computer Modeling in Engineering and Sciences, 3: 1151  Atluri SN, Shen SP(2002): The meshless local PetrovGalerkin (MLPG) method. Tech. Science Press.  Atluri SN, Shen S (2003):"The Basis of Meshless Domain Discretization: The Meshless Local Petrov Galerkin (MLPG) Method" , UCI Report, February 9, 2003.  Han ZD, and Atluri SN (2003a): On Simple Formulations of WeaklySingular Traction & Displacement BIE, and Their Solutions through PetrovGalerkin Approaches, CMES: Computer Modeling in Engineering & Sciences, 4(1): 520.  Han ZD, and Atluri SN (2003b): Truly Meshless Local PatrovGalerkin (MLPG) solutions of traction & displacement BIEs. CMES: Computer Modeling in Engineering & Sciences, (accepted).  Li Q, Shen S, Han ZD, and Atluri SN(2003):Application of the Meshless Local PetrovGalerkin (MLPG) Method to Problems with Singularities, and Material Discontinuities, in 3D Elasticity,CMES: Computer Modeling in Engineering & Sciences, (accepted).  Tang Z, Shen S, Atluri SN (2003) Analysis of materials with strain gradient effects: A Meshless local PetrovGalerkin approach, with nodal displacements only. CMES: Computer Modeling in Engineering & Sciences 4(1): 177196. 
I. Boztosun ^{(*)} Department of Physics, Erciyes University, Turkey boztosun erciyes.edu.tr 
Extended
Abstract (pdf)
Short Abstract: A novel adaptive radial basis function scheme based on the radial basis function methods is presented for the numerical solution of the BlackScholes equation, which has been used extensively for the evaluation of European and American options. The accurate and efficient solution of this equation is very important and has remained as a long standing problem in financial engineering. We apply our novel approach to plain vanilla for the European case and compare the numerical solution of the BlackScholes equation with the results of different numerical methods. It is shown that the new approach achieves a major improvement on all the previous numerical calculations for the solution of the BlackScholes equation. Keywords: Meshless Methods, BlackScholes Equation, European Options, Thin Plate Radial Basis Functions, Multiquadrics, Partial Differential Equation (*) joint work with:  A. Charafi (Computational Mathematics Group, University of Portsmouth, UK)  D. Boztosun (Institute of Social Sciences, Faculty of Economic and Administrative Sciences, Gazi University, Ankara, Turkey) 
Martin Buhmann JustusLiebig University, Germany martin.buhmann math.unigiessen.de 
Extended
Abstract (pdf)
Short Abstract: There is a number of ways to generalise the known results for interpolation and approximation with radial basis functions on the infinite integer grid.This concerns in particular the approximation order results and the attempt to get away from the strict setting of infinite grids towards scattered data. Among many good reasons for this approach are the need for results on data without meshes in order to apply them to the numerical analysis of partial differential equations; moreover, radial basis functions originate from the interpolation to scattered data and therefore one wishes to obtain approximation orders estimates for this setting. In this talk we review some of the approaches to generalise theorems on integer grids to nongridded settings and compare their results. Both interpolation and (general) approximation  for instance the socalled quasiinterpolation  shall be considered, and approximation results in L^{p}norms deserve special mention. As is traditional, multiquadrics and inverse multiquadrics, as well as the thinplate splines, are the main source of examples, applications and used to exemplify the typical proof techniques. 
J. T. Chen [X] Dep. of Harbor and River Engineering, National Taiwan Ocean University, Taiwan jtchen mail.ntou.edu.tw 
Abstract:
In this paper, a new meshfree method for solving eigenproblems using the radial basis function (RBF) is proposed. By employing the imaginarypart fundamental solution as the RBF, the diagonal and offdiagonal coefficients of influence matrices are easily determined. True eigen solutions in conjunction with spurious eigen solution occur at the same time. To verify this finding, the circulant is adopted to analytically derive the true and spurious eigenequation in the discrete system of a circular domain. In addition, the spurious eigenequation is derived in the continuous system using degenerate kernels and Fourier series. In order to obtain the true and spurious eigenvalues and their corresponding boundary modes, the singular value decomposition (SVD) technique of updating terms and documents are utilized, respectively. Several examples, including 2D and 3D interior acoustics, membrane and plate eigenproblems, are demonstrated analytically and numerically to see the validity of the present method. References: [1] Chen JT, Chang MH, Chen KH, Chen IL, Boundary collocation method for acoustic eigenanalysis of threedimensional cavities using radial basis function, Computational Mechanics, 2002, Vol.29, pp.392408. [2] Chen JT, Chang MH, Chen KH and Lin SR, Boundary collocation method with meshless concept for acoustic eigenanalysis of twodimensional cavities using radial basis function, Journal of Sound and Vibration, 2002, Vol.257, No.4, pp.667711 [3] Chen JT, Chen IL, Chen KH, Yeh YT, Lee YT, A meshless method for free vibration analysis of arbitrarily shaped plates with clamped boundaries using radial basis function. Engineering Analysis with Boundary Elements 2002, Accepted. [4] Chen JT, Kuo SR, Chen KH, Cheng YC, Comments on "Vibration analysis of arbitrary shaped membranes using nondimensional dynamic influence function." Journal of Sound and Vibration 2000, 235, No.1, pp.156171. [5] Chen JT, Chang MH, Chung IL, Cheng YC, Comments on "Eigenmode analysis of arbitrarily shaped twodimensional cavities by the method of point matching." Journal of Acoustical Society of America, 2002, Vol.111, No.1, 3336. [6] Chen JT, Chen IL, Chen KH, Lee YT, Comment on "Free vibration analysis of arbitrarily shaped plates with clamped edges using wavetype function". Journal of Sound and Vibration 2002; Accepted. 
Wen Chen Simula Research Laboratory, Lysaker, Norway wenc simula.no 
Extended
Abstract (pdf)
Short Abstract: The boundary knot method (BKM) is a recent meshfree boundarytype radial basis function (RBF) collocation technique. Compared with the method of fundamental solution, the BKM uses the nonsingular general solution instead of the singular fundamental solution to evaluate the homogeneous solution, while as such the dual reciprocity method (DRM) is still employed to approximate the particular solution. However, it is noted that the nonsingular general solution of Laplace equation is a constant, the BKM can not thus directly applied to it. This paper is an extension of reference [1] (W.Chen, 2001), where a simple BKM scheme was presented for solving Laplace equations. The scheme opens the door to use the BKM for general linear and nonlinear problems. 
Gregory E. Fasshauer Department of Applied Mathematics, Illinois Institute of Technology, USA fass amadeus.math.iit.edu 
Extended
Abstract (pdf)
Short Abstract: If globally supported functions (such as Gaussians or multiquadrics) are used with the usual RBF approximation/ interpolation method then one usually faces the problem of having to solve large and dense linear systems. The advantage of this is that  at the cost of illconditioning  one obtains very accurate solutions. We suggest a new approximate approximation scheme (due to Maz'ya and Schmidt) which also yields high accuracy, but without having to solve any linear systems at all. In order to speed up the evaluation of such an approximant even more the method can be coupled with a fast evaluation algorithm or parallel computation. 
Csába Gáspár Department of Mathematics, Széchenyi István University, Hungary csgaspar nimrod.math.szif.hu 
Extended
Abstract (pdf)
Short Abstract: The method of radial basis functions (rbfs) has proved an efficient tool to solve scattered data interpolation problems. From computational point of view, however, it exhibits serious drawbacks. If the applied rbfs are globally supported, as it is often the case, the resulting linear system is full, not necessarily selfadjoint and often severely illconditioned, which causes numerical difficulties. These disadvantages can be reduced by using e.g. fast multipole evaluation techniques, domain decomposition and/or compactly supported radial basis functions. Our approach differs from these and is based on the use of the fundamental solutions of certain higher order partial differential operators as radial basis functions. However, these functions are not used in an explicit way: instead, the interpolation problem itself is reformulated as a solution of a higher order partial differential equation supplied with the interpolation conditions as special boundary conditions. To solve this new problem, robust, quadtree/octtreebased multilevel solvers are used which require much less computational cost that the traditional rbftechniques. The approach has proved suitable to handle not only scalar but vectorial interpolation problems as well. Based on this interpolation technique, both domain and boundarytype meshless methods for partial differential equations are constructed utilizing the idea of the particular solutions. First, using a domain interpolation, a particular solution of the pde is defined without taking into account the boundary conditions. Next, a homogeneous problem is to be solved with modified boundary conditions. Instead of using a BEM approach to this boundary problem as usual, a boundary interpolation is used with a carefully chosen rbf (and higher order partial differential operator, respectively). This makes it possible to completely avoid both domain and boundary meshes as well as large, full and illconditioned matrices. Theoretical results and numerical examples are also presented. 
Y. C. Hon ^{(*)} Department of Mathematics, City University of Hong Kong, China Benny.Hon cityu.edu.hk 
Abstract:
The
recent development of a meshless method by using radial basis functions
will be reported in this talk.
Applications to both solving partial differential equations and inverse problems have demonstrated the spectral convergence of the method for some particular radial basis functions like multiquadric. This talk will also discuss some of the recent proposed techniques for solving the illconditioning problem resulted from solving the full resultant matrix. 
for Nonlinear Transport Equations Armin Iske ^{(*)} Zentrum Mathematik, Technische Universität München, Germany iske ma.tum.de 
Extended
Abstract (pdf)
Short Abstract: A recent adaptive meshfree advection scheme for numerically solving nonlinear transport equations is discussed. The scheme, being a combination of an adaptive semiLagrangian particle method and local polyharmonic spline interpolation, is essentially a method of backward characteristics.The adaptivity of the resulting meshfree advection scheme relies on customized rules for the refinement and coarsening of scattered nodes. The design of these adaption rules is based on recent results concerning the approximation behaviour and the numerical stability of local scattered data interpolation by polyharmonic splines. This talk first discusses key features and computational aspects of the meshfree advection scheme, before its good performance is shown by using specific model problems arising from applications in multiscale fluid flow simulation. (*) Parts of the presented results are based on joint work with: Joern Behrens and Martin Kaeser 
Michael J. Johnson [X] Kuwait University, Kuwait johnson mcs.sci.kuniv.edu.kw 
Abstract:
A
radial basis function interpolant to some data f_{}_{X}
can usually be viewed as the function s, in some Hilbert space H,
which minimizes s_{H} subject to the interpolation
conditions s(x)
= f(x),
x
ÎX. For suitable choices
of the Hilbert space H, the minimal norm interpolant s
can be easily computed and this has made rbf interpolation a popular method
for interpolating scattered data.
It is generally understood that if m_{1}, m_{2}, ... , m_{N} are continuous linear functionals on H and f ÎH, then the minimal norm interpolant to the data {m_{i}(f)}_{i=1,...,N }exists and can be computed provided the m_{i} are reasonably simple. One soon realizes that the linear functionals m_{i} can be chosen so that the resultant minimal norm interpolant approximates the solution of a given linear partial differential equation. In order to judge whether the approach is worth pursuing, the method has been implemented and tested on some standard PDEs like the wave equation and the heat equation. In this talk, I'll describe the implementation, primarily focusing on the construction of the interpolation matrix and the evaluation of the minimal norm interpolant. Additionally, some experiments will be presented which shed light on the accuracy and the numerical efficiency of the method. 
applied to gas dynamics Edward J. Kansa ^{(*)} EmbryRiddle Aeronautical University, U.S.A. kansa1 llnl.gov 
Extended
Abstract (pdf)
Short Abstract: A set of rotational and translation transformations are applied to the Euler gas dynamic equations. In such a transformed coordinate frame, the partial differential equations (PDEs) appear as a set of steady ordinary differential equations (ODEs) in the rotating, translating frame. By using appropriate linear combinations of the ODEs, we obtain a transformed set of ODEs that resemble the compatibility equations from the method of characteristics plus additional terms for the angular momentum or streamline bending. The new dependent variables are cast into radial basis functions that are volumetrically integrated over each piecewise continuous subregion. At discontinuities such as shocks or contact surfaces, these discontinuities are propagated by the RankineHugoniot jump conditions. For the case of weak shocks that are not important to track, they are captured and dampened away by the use of artificial viscosity. Knots over each continuous subregion may be added, deleted, or redistributed while constraining the appropriate volumetric dependent variables to be strictly conservative. Because volumetric integration is a smoothing operation, the numerical solutions converge faster compared with simple collocation. 
Andreas Karageorghis Department of Mathematics and Statistics, University of Cyprus, Cyprus andreask pythagoras.mas.ucy.ac.cy 
Extended
Abstract (pdf)
Short Abstract: The Method of Fundamental Solutions (MFS) is a boundarytype meshless method for the solution of certain elliptic boundary value problems. By exploiting the structure of the matrices appearing when this method is applied to certain threedimensional potential and biharmonic problems, we develop an efficient matrix decomposition algorithm for their solution. Numerical results are presented for various threedimensional regions. 
in Functionally Gradient Materials Eisuke Kita ^{(*)} Nagoya University, School of Informatics & Sciences, Japan kita info.human.nagoyau.ac.jp 
Extended
Abstract (pdf)
Short Abstract: This paper describes the application of Trefftz method to the steadystate heat conduction problem on the functionally gradient materials. Since the governing equation is expressed as the nonlinear Poisson equation, it is difficult to apply the ordinary Trefftz method to this problem. For overcoming this difficulty, we will present the combination scheme of the Trefftz method with the computing point analysis method. The inhomogeneous term of the Poisson equation is approximated by the polynomial of the Cartesian coordinates to determine the particular solution related to the inhomogeneous term. The solution of the problem is approximated with the linear combination of the particular solution and the Tcomplete functions of the Laplace equation. The unknown parameters are determined so that the approximate solution will satisfy the boundary conditions by means of the collocation method. Finally, the scheme is applied to some numerical examples. Keywords: Trefftz Method, Computing Point Analysis Method, SteadyState Heat Conduction, Functionally Gradient Materials (*) joint work with: Youichi Ikeda and Norio Kamyia 
G. R. Liu ^{(*)} Centre for Advanced Computations in Engineering Science, Dept. of Mechanical Engineering, National University of Singapore mpeliugr nus.edu.sg 
Extended
Abstract (pdf)
Short Abstract: In recent years, meshfree or meshless methods have been developed and used to solve partial differential equations (PDE). Mesh free methods can be largely categorized into two main categories: mesh free methods based on strong forms (e.g. collocation methods) and mesh free methods based on the weak forms (EFG, MLPG, PIM, etc.; see Mesh Free Methods, by G. R. Liu, CRC Press, 2002). The mesh free collocation method is simple to implement and computationally efficient. However, it is often found unstable and less accurate, especially for problems governed by partial differential equations with Neumann (derivative) boundary conditions, such as solid mechanics problems with stress (natural) boundary conditions. On the other hand, the mesh free methods based on the weak form exhibits very good stability and excellent accuracy. However, the numerical integration makes them computational expensive, and the background mesh (global or local) for integration is responsible for not being “truly” mesh free. In this paper, a new idea of combination of both the strong form and the local weak form is proposed to develop truly meshless method for 2D elastostatics. A novel truly meshfree method, the meshfree weakstrong (MWS) form method, is originated by Liu et al. (2002) based on a combined formulation of both the strong and local weak forms. This paper details the MWS method for solid and fluid mechanics problems. In the MWS method, the problem domain and its boundary is represented by a set of points or nodes. The strong form or collocation method is used for all the internal nodes and the nodes on the essential (Dirichlet) boundaries. The local weak form (PetrovGalerkin weak form) is used for nodes on the natural (Neumann) boundaries. There is no need for numerical integrations for all the internal nodes and the nodes on the essential boundaries. The local numerical integration is performed only for the nodes on the natural/Neumann boundaries. The natural/Neumann boundary conditions can then be easily imposed to produce stable and accurate solutions. The locally supported radial point interpolation method (RPIM) and moving least squares (MLS) approximation are used to construct the shape functions. The final system matrix will be sparse and banded for computational efficiency. Numerical examples of twodimensional solids and fluids are presented to demonstrate the efficiency, stability and accuracy of the proposed meshfree method. Keywords: Computational mechanics, Strong form, Weak form; Meshfree method, Meshless method, Collocation, Numerical analysis. (*) joint work with: Y. T. Gu (Dept. of Mechanical Engineering, National University of Singapore) 
Donald E. Myers Department of Mathematics, University of Arizona, USA myers math.arizona.edu 
Extended
Abstract (pdf)
Short Abstract: The Radial Basis Function (RBF) interpolator is a linear combination of translates of basis functions, the basis functions being invariant with respect to rotations on the underlying space. This invariance is not necessary either in the derivation of the estimator nor for the existence of a unique solution for the coefficients in the linear combination. One of the easiest ways to see why directional dependence might be appropriate is to transform the interpolator into a weighted linear combination of the data values, in that case the only defines the interpolating function implicitly. A positive definite radial basis function is also a covariance function and in the data value form for the RBF, the weights are determined by the spatial correlation between the values at the data locations and the data locations vs the location to be interpolated. In that context then directional dependence is very plausible. In the case of basis functions which are only conditionally positive definite, they are only generalized covariances rather than “true” covariances. One of the ways in which a covariance can be directionally dependent is in the range, i.e., the distance at which the covariance is zero or nearly so. That is, the range depends on direction. This form of directional dependence is called a geometric anisotropy and can be incorporated by an affine transformation on a “radial” model. Thus all radial basis functions can be used to generate directionally dependent basis functions by the use of an affine transformation. 
Paul W. Partridge ^{(*)} Department of Civil and Environmental Engineering, University of Brasília, Brazil paulp unb.br 
Extended
Abstract (pdf)
Short Abstract: In the case of the thermoelastic problem, the Navier governing equation can be treated with the Method of Fundamental Solutions, (MFS) and the thermal forces treated using the Dual Reciprocity Method, (DRM) as follows: It has been shown that the effect of an increase in temperature of q^{0} for an elastic body can be represented by a pseudo body force of b_{k}= a(3l+2m)q_{,k} (k=1,2), where a is the coefficient of thermal expansion and l, m are Lamé constants and a pseudo surface traction p_{k}= a(3l+2m)qn_{k} where n is the outward normal to the boundary. The former is approximated using DRM considering a Polyharmonic Spline approximating function f = r^{2m}log(r) where m=1,2 etc, and r is the distance function as employed in the Boundary Element Method, (BEM). The Polyharmonic approximating functions are augmented with different numbers of polynomial terms. If the linear terms, 1, x, y are employed with a function f of order m=1, one obtains the Augmented Thin Plate Spline, or ATPS function. An example is considered on a rectangular geometry, of size 2m by 1m, with the coordinate origin at the baricenter, with displacement boundary conditions u_{1}=0 at x_{1}=+1 and u_{2} =0 at x_{2}=0 and with a cubic temperature field q = 50(y^{3}+y^{2}+y+1) applied. If cubic augmentation is applied to the DRM approximation function, for any order m the expected result s_{11}=Eaq where s is a stress and E is the Youngs Modulus, is obtained. In this case no internal points are necessary. If it is considered that the exact variation of the temperature field will not be known, then it is reasonable to employ the linear augmentation functions, 1, x, y in this case convergence to the exact solution can be obtained defining internal points. A solution with an error of less than 1% may be obtained with 60 internal points for m =1. For a fixed number of internal points, further convergence may be obtained increasing the order of the Polyharmonic Spline function, considering m =2, 3 etc. 28 boundary points are considered. In relation to the MFS employed to solve the homogeneous equation, here the circle of fictitious points as first introduced by Bogomolny is considered. If the cubic augmentation functions are employed in such a way that the expected solution is obtained for any order of m , then results are independent of the radius of the circle of points. If the linear augmentation is employed, in such a way that convergence is obtained using internal points, it is found that the results are practically unchanged over a range of values of the radius of the circle of fictitious points, and outside this range results are inaccurate. In the case considered above the values of the radius of the circle of fictitious points over which the results were practically unchanged were 250 to 2000m. It was further found that if cubic augmentation is employed, results can be obtained using the LU decomposition algorithm for obtaining the unknown MFS coefficients, however, if the linear augmentation is employed, Singular Value Decomposition (SVD) produces better results. (*) joint work with: G. C. de Medeiros 
Solution of Fluid Flow Problems Bozidar Sarler Laboratory for Multiphase Processes, Nova Gorica Polytechnic, Slovenia bozidar.sarler png.si 
Abstract:
This
paper describes the solution of a steadystate natural convection problem
by the Radial Basis Function Collocation Method (RBFCM). This meshfree
(polygonfree) numerical method is for coupled set of mass, momentum, and
energy equations in two dimensions structured by the Hardy's multiquadrics
with different shape parameter and different order of polynomial augmentation.
The solution is formulated in primitive variables and involves iterative
treatment of coupled pressure, velocity, pressure correction, velocity
correction, and energy equations. Numerical examples include convergence
studies with different collocation point density and arrangements for a
twodimensional differentially heated rectangular cavity problem filled
with different fluids (Newtonian, NonNewtonian, Darcy, ...) at different
Rayleigh numbers and different aspect ratios. A classical and more accurate
modified type of discretisation with double consideration of boundary nodes
are numerically implemented. The solution is assessed by comparison with
reference results of the onemesh unite volume method in terms of midplane
velocity components, midplane and insulated surface temperatures, stream
function minimum, and Nusselt number.
Keywords: Natural convection, meshfree methods, radial basis function collocation method, Hardy's multiquadrics, polynomial augmentation, primitive variables, fluid constitutive equations. 
of Functions from Meshless Data R. Schaback Göttingen Universität, Germany schaback math.unigoettingen.de 
Extended
Abstract (pdf)
Short Abstract: Multivariate functions, e.g. solutions of partial differential equations, can be reconstructed from meshless data by techniques employing (conditionally) positive definite kernels. The latter may be radial basis functions, but radiality is not important for understanding the background, while it is of course useful for problems with many variables. This contribution looks at reconstruction techniques from a general point of view, including current applications to learning machines and the mathematical theory of learning. It will then focus on the systems that arise from symmetric reconstruction settings and address some of the techniques for solving them. A central issue is the question for good approximate solutions with only few nonzero coefficients, and this is closely related to "support vectors" of learning machines. This will lead to the investigation of greedy methods and the problem of optimal data locations. Both will be addressed in some detail, together with some illustrating toy examples. If time permits, some preliminary results concerning domain decomposition and preconditioning will also be presented. 
Local RBFbased Differential Quadrature Method C. Shu Department of Mechanical Engineering, National University of Singapore, Singapore mpeshuc nus.edu.sg 
Extended
Abstract (pdf)
Short Abstract: Local radial basis functionbased differential quadrature (RBFDQ) method was recently proposed by us. The method is a natural meshfree approach. Like the conventional differential quadrature (DQ) method, it discretizes any derivative at a knot by a weighted linear sum of functional values at its neighbouring knots, which may be distributed randomly. However, different from the conventional DQ method, the weighting coefficients in present method are determined by taking the radial basis functions (RBFs) instead of high order polynomials as the test functions. The method works in a similar fashion as conventional finite difference schemes but with “truly” meshfree property. In this presentation, we mainly concentrate on the multiquadric (MQ) radial basis functions since they have exponential convergence. The effects of shape parameter c on the accuracy of numerical solution of linear and nonlinear partial differential equations are studied, and how the value of optimal c varies with the number of local support knots is also numerically demonstrated. The proposed method is validated by its application to solve incompressible NavierStokes equations. Excellent numerical results are obtained on an irregular knot distribution. 
J. Teixeira de Freitas ^{(*)} Dep. Eng. Civil e Arquitectura, Instituto Superior Tecnico, Portugal freitas civil.ist.utl.pt 
Extended
Abstract (pdf)
Short Abstract: A stress model of the hybridmixed formulation is implemented on a compact radial basis and applied to the solution of elliptic problems. The formulation is based on a twofield domain approximation coupled with an independent boundary approximation. It is strictly meshless, as its implementation does not require the decomposition of the domain to define the approximation bases or to support the numerical integration of the coefficients of the solving system. The performance of the formulation is illustrated on a twodimensional linear elastostatic problem. (*) joint work with: P.M. Pimenta and S.P.B. Proença 
Thanh TranCong ^{(*)} Faculty of Engineering and Surveying University of Southern Queensland, Australia trancong usq.edu.au 
Extended
Abstract (pdf)
Short Abstract: In our previous work, Indirect Radial Basis Function Network (IRBFN) method has proved to be a highly accurate tool for approximating multivariate functions and solving elliptic PDEs. A recent development of the method for solving timedependent PDEs will be presented in this paper. The proposed method can be used with various time integration schemes of which those based on semidiscrete scheme are preferred when solving timedependent nonlinear PDEs. At least for the problems considered in this work, the accuracy of the method is not very sensitive to the network parameters. In particular, the method can tolerate a rather wide range of values of the shape parameter while yielding highly accurate results. Example problems, including those governed by parabolic PDEs, hyperbolic PDEs and advectiondiffusion equations are solved by the proposed method, and the results compare favourably in terms of accuracy with those from other numerical methods such as finite difference, finite element, boundary element and direct RBFN methods. (*) joint work with: Lan MaiCao 
Migration Using Radial Basis Functions Leopold Vrankar ^{(*)} Slovenian Nuclear Safety Administration, Ljubljana, Slovenia Leopold.Vrankar gov.si 
Extended
Abstract (pdf)
Short Abstract: Many problems in science and engineering are reduced to a set of partial differential equations (PDEs) through the process of mathematical modelling. Although the model equation based on established physical laws may be constructed, analytical tools are frequently inadequate for the purpose of obtaining their closed form solution. The numerical solution of PDEs has been usually obtained by either finite difference methods (FDM), finite element methods (FEM), or finite volume methods(FVM). These methods require a mesh to support the localised approximations. Kansa introduced the concept of solving PDEs using radial basic functions (RBFs) for hyperbolic, parabolic and elliptic PDEs. As for most interpolation methods, the errors in RBFs approximations tend to be much larger near boundaries. Due to this fact, it makes sense to impose more information right there. Fedoseyev, Friedman and Kansa formulated a method that collocates both the boundary condition and the PDE at the boundary. This paper presents two applications of the RBFs. The first one is intended for the determination of torsion and stress functions in mechanical analysis of torsion, and the second one for the modelling of the movement of radionuclides through geosphere at disposing of radioactive waste. (*) joint work with: Goran Turk (Faculty of Civil and Geodesic Engineering, University of Ljubljana, Slovenia) Franc Runovc (Faculty of Natural Sciences and Engineering, University of Ljubljana, Slovenia) 
solving time dependent propagation equation Zongmin Wu [X] Department of Mathematics, Fudan University Shanghai, China zmwu fudan.edu.cn 
Abstract: If we want display a function on monitor, we should have more sampling data near the point where the function vary more quickly and is more osculate. From the finite elements method, we should have more fine element near the singularity for example such as the boundary elements method. It is no problem for the static function because we often know where the singularity will happen. However, we don’t know, where will the osculation or even shock wave happen for a function which is a solution of propagation partial differential equation. Therefor we can not presetting the fine elements and can only set a uniformly fine knots or elements for such problem.A simple consideration is to move the knots or the mesh of the finite elements according to the varying function. We see that, it is very difficulty to succeed the consideration for finite difference method, because the knots will going to scattered and we can not construct the finite difference to approximate the differential or derivatives of the function. It is difficulty to succeed the consideration for finite elements methods too, because the moving knots will often destroy the topology of the moving mesh. The meshless method for solving partial differential equation do not require a mesh or a structure of the knots, thus supply a possibility of the moving knots to simulate our problem. We require only to keep the knots no overlapping. This paper will test the approach for the Burgers equation. 
Electromagnetic Wave Scattering Problems D. L. Young ^{(*)} [X] Department of Civil Engineering & Hydrotech Research Institute National Taiwan University, Taiwan dlyoung hy.ntu.edu.tw 
Extended
Abstract (pdf)
Short Abstract: In this paper we attempt to construct the electromagnetic wave scattering field by a given incident wave. For twodimensional problems the normal incident plane wave scattering by conducting cylinder with infinite dimension in the zdirection would be discussed. For threedimensional problem we focus on scattering wave by a conducting sphere. Both cases have analytic solutions and other numerical results so that a comparison could be made to assess the potential of the present study. The method of fundamental solutions (MFS) for the vector Helmholtz equations in the frequency domain together with the singular value decomposition (SVD) are employed to simulate the electromagnetic wave problems. Both the 2D and 3D homogeneous wave scattering are compared with the analytical as well as other numerical methods, such as finite elements (FE) or boundary elements (BE) schemes. The MFS has shown very efficient and accurate results as comparing with the analytical and other numerical solutions. The MFS will provide a very promising and powerful tool according to present study as far as computational electromagnetics is concerned. (*) joint work with: J. W. Ruan 
in a bounded domain using the MFS Pedro Antunes ^{(*)} Department of Mathematics, Instituto Superior Técnico, Portugal l45935 isabelle.math.ist.utl.pt 
Extended
Abstract (pdf)
Short Abstract: In this work we present a numerical algorithm for thedetermination of the eigenvalues and eigenfunctions associated to the Dirichlet problem for the Laplacian, in a bounded domain. The determination of higher eigenfrequencies is a well known numerical problem that has been addressed with other numerical methods. Here we propose to use the method of fundamental solutions. Since the MFS produces highly ill conditioned matrices, a particular technique was derived to overcome the difficulty of determining accurately those eigenfrequencies. Extensive numerical simulations will be presented. [poster presentation] (*) joint work with: C. J. S. Alves 
with the use of Inverse Operations Michal Cialkowski ^{(*)} Heat Engineering Chair, Poznan University of Technology, Poland Michal.Cialkowski put.poznan.pl 
Extended
Abstract (pdf)
(*) joint work with: A. Frackowiak 
PetrovGalerkin Viscous Methods Mehrzad Ghorbany ^{(*)} [X] Department of Mathematics, Sistan & Baluchestan University, Zahedan, Iran ghorbany hamoon.usb.ac.ir 
Extended
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Short Abstract: Moving Meshless Methods (MMM) are new generation of numerical methods for unsteady or time dependent differential equations, that have shock, boundary layer, high gradient region, high oscillatory region,... . These methods link rrefinement and Moving Finite Element method (MFE) by Keith Miller to Meshless methods such as, DEM, EFGM, SPH, RKPH, PUM, hp Clouds, ... . In MMM, mesh coordinates are unknown and are found by equations. This implies: exertion of indirect or pseudo equidistribution of nodes. Weak form and system will be found by Galerkin or PetrovGalerkin method. In proceeding time steps, nodes move smoothly into the high gradient region and concentrates there, for better approximation. We appended a penalty of relative velocity of nodes to energy functional. Gas dynamic problems need a pseudoviscous region for preventing high velocity, colliding and collapsing of nodes and controls their motion. Numerical solution of Heat equation and Burger equation, demonstrate the potential of the MMM. Among Meshless methods we only use of EFGM to introduce Moving Element Free PetrovGalerkin viscous Method (MEFPGVM) by C^{2} cubic Hermite base function on nodes as test or weight functions. (*) joint work with: Ali Reza Soheili 
and singularly perturbed problems Evgeny Glushkov ^{(*)} Kuban State University, Krasnodar, Russia evg math.kubsu.ru 
Abstract:
The
use of axiallysymmetric deltalike functions as a basis and local Green's
functions as test ones in the Galerkin and PetrovGalerkin schemes is considered.
The use of radial functions as a trail subspace for solution boundary integral equations arising in elastodynamic diffraction and contact problems allowed us to reduce considerably the computing costs thanks to replacing multifold singular integrals by onedimensional ones. The selected shapeform of the basis functions assures good convergence in integral metrics so that the method yields good results when studying wave fields and energy scattered by a crack or radiated by a vibrating indentor. In singularly perturbed problems numerical solution meets generally with essential difficulties caused by sharp boundary and interior layers. Their presence leads to numerical instability and large error pollution spreading out over the whole domain as the perturbation parameter tends to its limiting value. The solution of such problems requires either local mesh refinement in the layers or basis functions accounting for the singular behaviour. As the latter the local Green's functions (fundamental solutions of the adjoint problem) can be used. By the example of a twodimensional convectiondiffusion Dirichlet problem a semianalytical approach to calculate the local Green's functions by means of the Fourier transform technique is proposed. They are used as projectors (test functions) in the PetrovGalerkin scheme. Besides very accurate approximation achieved even with a coarse mesh the distinctive feature of the method is fast and stable iterative solution of the large sparse algebraic systems arising here after discretization. (*) joint work with: Natalia Glushkova 
regular polygonal crosssection Jan A. Kolodziej ^{(*)} Institute of Applied Mechanics, Poznan University of Technology, Poland jan.kolodziej put.poznan.pl 
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Short Abstract: A special purpose Trefftz functions for solution of torsion problem of simply connected, two connected, and composite bars possessing regular polygon on cross section contour are proposed. Seven cases of bars are considered: 1) regular polygonal bars, 2) regular polygonal bars with circular centred holes, 3) cylindrical bars with regular polygonal centred holes, 4) regular polygonal bars with regular polygonal centred holes, 5) regular polygonal bars with circular centred reinforced rod, 6) cylindrical bars with regular polygonal reinforced rod, 7) regular polygonal bars with regular polygonal reinforced rod. Proposed Trefftz functions fulfil not only governing equation but also boundary conditions on part of boundary. The boundary collocation method in the least squares sense for solving appropriate boundary value problems for the stress function is used. By means of analytical integration of the stress functions, for the seven considered cases the analytical formulae for nondimensional stiffness of bars are obtained. (*) joint work with: Agnieszka Wachowska 
direct chill casting of aluminium alloys Igor Kovacevic ^{(*)} Laboratory for Multiphase Processes, Nova Gorica Polytechnic, Slovenia 
Abstract:
To appear
(*) joint work with: Bozidar Sarler 
Collocation Methods Leevan Ling Simon Fraser University Canada lling math.sfu.ca 
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Short Abstract: Preconditioning techniques are aimed to reduce the computational time and cost by recasting the problem in a different but better context. We present a simple preconditioning scheme that is based upon constructing leastsquares approximate cardinal basis functions (ACBFs) from linear combinations of the RBFPDE matrix elements. The ACBFs transforms a badly conditioned linear system into one that is very well conditioned, allowing us to solve the expansion coefficients iteratively so we can reconstruct the unknown solution everywhere on the domain. Our preconditioner requires O(mN^{2}) flops to set up, and O(mN^{2}) storage locations where m is a user define parameter of order O(10). For the 2D MQRBF with the shape parameter c ~ 1/sqrt(N), the number iterations required for convergence is of order O(10) for large values of $N$, making this a very attractive approach computationally. This scheme is numerically shown to have better performance than the methods. Our method is able to handle problems that are more ill conditioned, and allows one to solve a given problem with fewer data points. By using fewer data points, the computational time also reduces. In other words, a larger problem can now be handle with the same computational power. 
yieldstress fluid under imposed torques Ahmed Naji ^{(*)} Laboratoire de Mécanique de Milieux Hétérogènes, F.S.T de Tanger, Maroc najifstt2003 hotmail.com 
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Short Abstract: A Lagrangian description of the Couette flow between two coaxial cylinders, of a viscoplastic fluid (i.e. exhibiting a yield stress) under imposed torque is presented. Beyond a value of the shear stress, the viscosity variation is approximated by a layering of two fluid regions with different viscosities such that m_{2} /m_{1} <<1. So the rheological behaviour is described by the model of biviscosity which approaches the Bingham model. In this work meshless radial basis function method is used to build an approximation of the PDEs governing the Couette flow. The used technique is based on the application of globally multiquadrics radial basis function to compute the velocity field and the free surface separating the two phases. (*) joint work with: M. ErRiani : Laboratoire de Mécanique de Milieux Hétérogènes, F.S.T de Tanger, Maroc. C. Nouar, O. SeroGuillaume : LEMTA – CNRS, France 
steadystate structuralacoustic radiation analysis B. Pluymers ^{(*)} Dep. Mechanical Engineering, Katholieke Universiteit Leuven, Belgium bert.pluymers mech.kuleuven.ac.be 
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Short Abstract: The use of element based prediction techniques such as the finite element (FE) method, the infinite element (IE) method and the boundary element (BE) method, is generally accepted for steadystate dynamic analysis of coupled structuralacoustic radiation problems. The FE based methods truncate the unbounded radiation domain by introducing an artificial boundary surface. At this boundary surface appropriate impedance boundary conditions are applied so that no acoustic reflections occur. The IE method models explicitly the domain, exterior to the truncation surface, by coupling infinite elements to the bounded FE domain. Since model sizes increase with frequency, the use of these methods is restricted to lowfrequency applications. BE methods discretize only the boundaries of the considered problem and base their solutions on a boundary integral formulation that inherently satisfies the Sommerfeld radiation condition. In this way no truncation surfaces must be introduced. Drawbacks of these methods are the fully populated, frequency dependent and not always symmetric system matrices which lead to computational demanding calculations and restrict the use of the BE methods to lowfrequency applications. Recently a new wave based prediction technique (WBT), which is based on the Trefftz approach, has proven to be successful for low and midfrequency applications in bounded domains. Instead of using simple, approximating shape functions to describe the dynamic variables, exact solutions of the governing differential equations are used. No fine discretization of the domains is necessary so the model size is much smaller than with the element based methods. This allows to handle also midfrequency applications. This paper discusses how the WBT can be extended for radiation problems in unbounded domains. The technique is illustrated in a twodimensional analysis of the sound radiation of a bassreflex loudspeaker and its performance is compared with the conventional element based techniques. (*) joint work with: W. Desmet, D. Vandepitte and P. Sas 
elastic problems in nonsimply connected domains P. M. C. Ribeiro Escola Superior de Tecnologia, Universidade do Algarve, Portugal pribeiro ualg.pt 
Abstract:
A
mathematical motivation for the method of fundamental solutions is given
by the straightforward discretization of the single layer potential, leading
to a first kind integral equation. This motivation does not explain convergence
of the MFS for non analytic boundary data, and density results are needed.
Sequences of single layer potentials or the span of fundamental solutions
with point sources located outside the domain provide a further justification
for the approximation. In the case of nonsimply connected domains it is
known that the point sources must be located in all the components of the
non connected exterior domain. We show several numerical simulations concerning
the effect of the choice of point sources in the convergence of the MFS
for elastic problems in nonsimply connected domains.
[poster presentation] 
method of fundamental solutions applied to fluid flow A. L. Silvestre CEMAT Centro de Matemática e Aplicações, Instituto Superior Técnico, Portugal ana.silvestre math.ist.utl.pt 
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Short Abstract: The method of fundamental solutions has been used to solve fluid flow problems by reduction to Laplace problems. In this work we propose to use the fundamental solution of the Stokes system, using the socalled Stokeslets, to solve problems with conservative and nonconservative forces (homogeneous and nonhomogeneous Stokes system). In the nonhomogeneous case, we will also consider fundamental solutions of eigenvalue equations associated to the Stokes operator. We establish new density results in terms of fundamental solutions for the functional spaces used in the Stokes equations, by extending some density results recently obtained by Alves and Chen. Such density results are used to choose suitable basis functions for the method of fundamental solutions to approach the solution of the boundary value problem for the Stokes equations. We also show the convergence of this MFS based on the density results. Numerical simulations will be presented. 
Yiorgos S. Smyrlis ^{(*)} Department of Mathematics and Statistics, University of Cyprus, Cyprus smyrlis pythagoras.mas.ucy.ac.cy 
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Short Abstract: The Method of Fundamental Solutions (MFS) is a boundarytype meshless method for the solution of certain elliptic boundary value problems. In this work, we investigate the properties of the matrices that arise when the rotated MFS is applied to the Dirichlet problem for Laplace's equation in a disk. In particular, we study the behaviour of the eigenvalues of these matrices and the cases in which they vanish. Based on this, we propose a modified efficient numerical algorithm for the solution of the problem which is applicable even in the cases when the MFS matrix might be singular. We prove the convergence of the method for analytic boundary data and perform a stability analysis of the method with respect to the distance of the singularities from the origin and the number of degrees of freedom. (*) joint work with: A. Karageorghis 
applications to C1 structural problems. Carlos M. Tiago ^{(*)} ICIST  Dep. Eng. Civil e Arquitectura, Instituto Superior Técnico, Portugal ccarlos.tiago civil.ist.utl.pt 
Abstract:
Continuity
of the generalized displacement and stress fields are preserved by
meshless methods, such as the EFG (Element Free Galerkin), as long as an
appropriate basis together with an appropriate weight function
are used. Nevertheless, the possibility of using approximation functions
built not only on information from the unknown functions
at the nodal points but also of its derivatives, may provide
more efficient procedures in the numerical solution of boundary value problems
governed by fourth order differential equations. The GMLS (Generalized
Moving Least Squares Method) is, as its name suggests, a generalization
of the moving least squares concept (used in the EFG method) which takes
into account, to build the approximation, both the approximation values
and the corresponding derivatives.
In this work, application of the GMLS concept to thin beams and plates is carried out. Implementation aspects, namely the integration of the week form and the choice of the weight function, are discussed. Comparisons with the standard MLS (Moving Least Squares) are presented. (*) joint work with: Vítor M. A. Leitão 
Andrei I. Tolstykh ^{(*)} Computing Center of Russian Academy of Sciences, Russia tol ccas.ru 
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Short Abstract: In the context of PDE's, Radial Basis Functions (RBF) are usually associated with either boundary elements or collocation strategies. In the present talk, another way of using RBF is attempted. It essence is constructing approximate formulas for derivatives discretizations based on RBF interpolants with local supports similar to stencils in finite difference methods. Defining for each node a stencil as a set of its neighbor nodes, one can obtain an approximation to governing equations at the node (rather then satisfying them at the node as in the case of collocation approach) and the resulting global system with a sparse matrix. Results of numerical experiments with the Poisson and biharmonic equations showing good hconvergence properties of the technique as well as increasing solutions accuracy with increasing numbers of nodes in stencils are presented. As further illustrations, examples of solving linear and nonlinear solid mechanics problems are displayed, the emphasize being placed on benchmark calculations. (*) joint work with: Dmitrii A. Shirobokov 
using a meshfree planewaves method Svilen S. Valtchev ^{(*)} CEMAT Centro de Matemática e Aplicações, Instituto Superior Técnico, Portugal ssv math.ist.utl.pt 
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Short Abstract: Density results using an infinite number of plane acoustic waves allow to derive meshless methods to solve the homogeneous or the nonhomogeneous Helmholtz equation. In this work we consider the numerical simulation of acoustic source problems in a bounded domain using this method. We present several tests comparing with the method of fundamental solutions and a recent extension to nonhomogeneous problems. (*) joint work with: C. J. S. Alves 
method for incompressible fluid flow. Yolanda Vidal Seguí ^{(*)} Department of Applied Mathematics, Universitat Politècnica de Catalunya, Spain yolanda.vidal upc.es 
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Short Abstract: Accurate and efficient modelling of incompressible flows is an important issue in finite elements. The continuity equation for an incompressible fluid takes the peculiar form. It consists of a constraint on the velocity field which must be divergence free. Then the pressure has to be considered as a variable not related to any constitutive equation. Its presence in the momentum equation has the purpose of introducing an additional degree of freedom needed to satisfy the incompressibility constraint. The role of the pressure variables is thus to adjust itself instantaneously in order to satisfy the condition of divergencefree velocity. That is, the pressure is acting as a Lagrangian multiplier of the incompressibility constraint and thus there is a coupling between the velocity and the pressure unknowns. Incompressibility in meshfree methods is still an open topic. Even recently it was claimed that meshless methods do not exhibit volumetric locking. In a recent paper by Huerta A. and FernándezMéndez S. this issue is clarified determining the influence of the dilation parameter on the locking behavior of EFG near the incompressible limit. The major conclusion is that an increase of the dilation parameter attenuates, but never supresses the volumetric locking. Until now the remedies proposed in the literature are extensions of the methods developed for finite elements. Here a novel approach is explored: The PseudoDivergenceFree EFG method (PDF EFG). It consists in using interpolation functions that verify approximately the divergencefree constraint. This method is based on diffuse derivatives which converge to the derivatives of the exact solution when the radius of the support goes to zero (for a fixed dilation parameter).Here convergence of the approximation in incompressible flows is studied. In particular, it is shown that the PDF EFG method passes the numerical infsup test. And two wellknown examples of Stokes flow are used to compare different mixed formulations. (*) joint work with: Antonio Huerta 