Workshop on

Instituto Superior Técnico
Lisbon, Portugal

July 21-23, 2003

Main Page



Invited talks
E. R. Arantes e Oliveira
Satya N. Atluri,
Ismail Boztosun
Martin Buhmann
J. T. Chen, [x]
Wen Chen
Gregory Fasshauer
Csába Gáspár,
Y.C. Hon,
Armin Iske
Michael Johnson, [x]
Edward J. Kansa
Andreas Karageorghis
Eisuke Kita,
Gui-Rong Liu,
Donald E. Myers
Paul Partridge
Bozidar Sarler
Robert Schaback,
Chang Shu,
J. Teixeira de Freitas
Thanh Tran-Cong
Leopold Vrankar
Zongmin Wu, [x]
Der-Liang Young. [x]

Contributed presentations

P. Antunes
Michal Cialkowski
M. Ghorbany, [x]
Evgeny Glushkov,
Jan A. Kolodziej
I. Kovacevic,
L. Ling,
Ahmed Naji,
B. Pluymers
P. Ribeiro
A. L. Silvestre
Y. S. Smyrlis,
C. M. Tiago
Andrei I. Tolstykh
Y. Vidal Seguí.

[x] Participants that will not be able to attend the workshop.

Guidelines for extended abstracts

Invited Talks

From Infinitesimal Calculus To Avoiding Passages To The Limit
E. Arantes e Oliveira
Dep. Eng. Civil e Arquitectura, 
Instituto Superior Técnico, Portugal
Extended Abstract (pdf)
1- Introduction: the methodology of infinitesimal calculus
2- Solving differential equations
 2.1- Seeking for formal solutions 
 2.2- Finite difference methods and mesh free methods
 2.3- Finite element methods and block methods
3- Avoiding passages to the limit
 3.1- Suspension bridges
 3.2- Idealizing grids as plates
 3.3- Methods of fundamental solutions
4-  Conclusions

The Meshless Local Petrov-Galerkin (MLPG) Method
for Domain & BIE Discretizations
Satya N. Atluri
University of California at Irvine, USA
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 domain-solution 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 3-dimensional solid mechanics problems involving strong singularities [Li, Shen, Han and Atluri (2003)], 3-dimensional non-planar fatigue crack-growth in complexly loaded solids and structures [Han and Atluri (2003 a, b)],  and in strain-gradient 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 multiple-length&-time scale simulations, is illustrated.
- Atluri SN, Zhu T(1998): A new meshless local Petrov-Galerkin (MLPG) approach in computational mechanics. Comput. Mech., 22: 117-127
- Atluri SN, Shen SP(2002): The mesh local Petrov-Galerkin (MLPG) method: a simple & less-costly alternative to the finite element and boundary element methods. CMES: Computer Modeling in Engineering and Sciences, 3: 11-51
- Atluri SN, Shen SP(2002): The meshless local Petrov-Galerkin (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 Weakly-Singular Traction & Displacement BIE, and Their Solutions through Petrov-Galerkin Approaches, CMES: Computer Modeling in Engineering & Sciences, 4(1): 5-20.
- Han ZD, and Atluri SN (2003b): Truly Meshless Local Patrov-Galerkin (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 Petrov-Galerkin (MLPG) Method to Problems with Singularities, and Material Discontinuities, in 3-D 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 Petrov-Galerkin approach, with nodal displacements only. CMES: Computer Modeling in Engineering & Sciences 4(1): 177-196.

A Novel Adaptive Radial Basis Function Scheme for Options Pricing
I. Boztosun (*)
Department of Physics, Erciyes University, Turkey
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 Black-Scholes 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 Black-Scholes 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 Black-Scholes equation.
Keywords: Meshless Methods, Black-Scholes 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)

Interpolation with Radial Basis Function
Martin Buhmann
Justus-Liebig University, Germany
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 non-gridded settings and compare their results. Both interpolation and (general) approximation - for instance the so-called quasi-interpolation - shall be considered, and
approximation results in Lp-norms deserve special mention. As is traditional, multiquadrics and inverse multiquadrics, as well as the thin-plate splines, are the main source of examples, applications and used to exemplify the typical proof techniques.

Mesh-free method for eigenproblems
J. T. Chen 
Dep. of Harbor and River Engineering, 
National Taiwan Ocean University, Taiwan
In this paper, a new mesh-free method for solving eigenproblems using the radial basis function (RBF) is proposed. By employing the imaginary-part fundamental solution as the RBF, the diagonal and off-diagonal 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 2-D and 3-D interior acoustics, membrane and plate eigenproblems, are demonstrated analytically and numerically to see the validity of the present method.
[1] Chen JT, Chang MH, Chen KH, Chen IL, Boundary collocation method for acoustic eigenanalysis of three-dimensional cavities using radial basis function, Computational Mechanics, 2002, Vol.29, pp.392-408. 
[2] Chen JT, Chang MH, Chen KH and Lin SR, Boundary collocation method with meshless concept for acoustic eigenanalysis of two-dimensional cavities using radial basis function, Journal of Sound and Vibration, 2002, Vol.257, No.4, pp.667-711
[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 non-dimensional dynamic influence function." Journal of Sound and Vibration 2000, 235, No.1, pp.156-171.
[5] Chen JT, Chang MH, Chung IL, Cheng YC, Comments on "Eigenmode analysis of arbitrarily shaped two-dimensional cavities by the method of point matching." Journal of Acoustical Society of America, 2002, Vol.111, No.1, 33-36.
[6] Chen JT, Chen IL, Chen KH, Lee YT, Comment on "Free vibration analysis of arbitrarily shaped plates with clamped edges using wave-type function". Journal of Sound and Vibration 2002; Accepted.

Solving Poisson equations by the boundary knot method
Wen Chen
Simula Research Laboratory, Lysaker, Norway
Extended Abstract (pdf)
Short Abstract: The boundary knot method (BKM) is a recent meshfree boundary-type 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. 

Approximate Approximation as the Starting Point for Fast RBF Methods
Gregory E. Fasshauer
Department of Applied Mathematics,
Illinois Institute of Technology, USA
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 ill-conditioning - 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.

Fast meshless methods using rbf-like domain and boundary interpolation
Csába Gáspár
Department of Mathematics, Széchenyi István University, Hungary
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 self-adjoint and often severely ill-conditioned, 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/octtree-based multi-level solvers are used which require much less computational cost that the traditional rbf-techniques. The approach has proved suitable to handle not only scalar but vectorial interpolation problems as well. Based on this interpolation technique, both domain and boundary-type 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 ill-conditioned matrices. Theoretical results and numerical examples are also presented.

Meshless Computational Method By Using Radial Basis Functions
Y. C. Hon (*)
Department of Mathematics, City University of Hong Kong, China
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 ill-conditioning problem resulted from solving the full resultant matrix.

An Adaptive Meshfree Method of Backward Characteristics
for Nonlinear Transport Equations
Armin Iske (*)
Zentrum Mathematik, Technische Universität München, Germany
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 semi-Lagrangian 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

Minimal Norm Interpolation
Michael J. Johnson
Kuwait University, Kuwait
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 m1, m2, ... , mN are continuous linear functionals on H and f ÎH, then the minimal norm interpolant to the data {mi(f)}i=1,...,N exists and can be computed provided the mi are reasonably simple. One soon realizes that the linear functionals mi 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.

Volumetric radial basis functions methods 
applied to gas dynamics
Edward J. Kansa (*)
Embry-Riddle Aeronautical University, U.S.A.
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 Rankine-Hugoniot 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.

Matrix Decomposition MFS Algorithms
Andreas Karageorghis
Department of Mathematics and Statistics, 
University of Cyprus, Cyprus
Extended Abstract (pdf)
Short Abstract: The Method of Fundamental Solutions (MFS) is a boundary-type 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 three-dimensional potential and biharmonic problems, we develop an efficient matrix decomposition algorithm for their solution. Numerical results are presented for various three-dimensional regions.

Trefftz Solution for Steady-State Heat Conduction Problem 
in Functionally Gradient Materials
Eisuke Kita (*)
Nagoya University, School of Informatics & Sciences, Japan
Extended Abstract (pdf)
Short Abstract: This paper describes the application of Trefftz method to the steady-state heat conduction problem on the functionally gradient materials. Since the governing equation is expressed as the non-linear 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 T-complete 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, Steady-State Heat Conduction, Functionally Gradient Materials
(*) joint work with:
Youichi Ikeda and Norio Kamyia

A Meshfree Weak-Strong-form (MWS) method 
G. R. Liu (*)
Centre for Advanced Computations in Engineering Science,
 Dept. of Mechanical Engineering, National University of Singapore
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 2-D elasto-statics.
A novel truly meshfree method, the meshfree weak-strong (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 (Petrov-Galerkin 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 two-dimensional 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)

Directional Dependence and Radial Basis Functions
Donald E. Myers
Department of Mathematics, University of Arizona, USA
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. 

The Method of Fundamental Solutions with Dual Reciprocity for Thermoelasticity
Paul W. Partridge (*)
Department of Civil and Environmental Engineering, 
University of Brasília, Brazil
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 q0 for an elastic body can be represented by a pseudo body force of bk= 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 pk= a(3l+2m)qnk  where n is the outward normal to the boundary. The former is approximated using DRM considering a Polyharmonic Spline approximating function f = r2mlog(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 u1=0  at x1=+1  and u2 =0 at x2=0 and with a cubic temperature field q = 50(y3+y2+y+1) applied. If cubic augmentation is applied to the DRM approximation function, for any order m the expected result s11=-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 

Radial Basis Function Collocation Method
Solution of Fluid Flow Problems
Bozidar Sarler
Laboratory for Multiphase Processes, 
Nova Gorica Polytechnic, Slovenia
Abstract: This paper describes the solution of a steady-state natural convection problem by the Radial Basis Function Collocation Method (RBFCM). This mesh-free (polygon-free) 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 two-dimensional differentially heated rectangular cavity problem filled with different fluids (Newtonian, Non-Newtonian, 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 one-mesh unite volume method in terms of mid-plane velocity components, mid-plane and insulated surface temperatures, stream function minimum, and Nusselt number.
Keywords: Natural convection, mesh-free methods, radial basis function collocation method, Hardy's multiquadrics, polynomial augmentation, primitive variables, fluid constitutive equations.

Special Techniques for Kernel-Based Reconstruction 
of Functions from Meshless Data
R. Schaback
Göttingen Universität, Germany
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.

Computation of Incompressible Navier-Stokes Equations by 
Local RBF-based Differential Quadrature Method 
C. Shu
Department of Mechanical Engineering,
National University of Singapore, Singapore
Extended Abstract (pdf)
Short Abstract: Local radial basis function-based differential quadrature (RBF-DQ) method was recently proposed by us. The method is a natural mesh-free 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” mesh-free 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 Navier-Stokes equations. Excellent numerical results are obtained on an irregular knot distribution.

Hybrid-Mixed Meshless Formulation
J. Teixeira de Freitas (*)
Dep. Eng. Civil e Arquitectura, 
Instituto Superior Tecnico, Portugal
Extended Abstract (pdf)
Short Abstract: A stress model of the hybrid-mixed formulation is implemented on a compact radial basis and applied to the solution of elliptic problems. The formulation is based on a two-field 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 two-dimensional linear elastostatic problem.
(*) joint work with:
P.M. Pimenta  and S.P.B. Proença

Solving time-dependent PDEs with a meshless IRBFN-based method
Thanh Tran-Cong (*)
Faculty of Engineering and Surveying
University of Southern Queensland, Australia 
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 time-dependent PDEs will be presented in this paper. The proposed method can be used with various time integration schemes of which those based on semi-discrete scheme are preferred when solving time-dependent 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 advection-diffusion 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 Mai-Cao

Calculation of the Stress Function and Modelling of the Radionuclide
Migration Using Radial Basis Functions
Leopold Vrankar (*)
Slovenian Nuclear Safety Administration, 
Ljubljana, Slovenia
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)

Dynamically Knots Setting in meshless method for 
solving time dependent propagation equation
Zongmin Wu 
Department of Mathematics, Fudan University
Shanghai, China
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.

Method of Fundamental Solutions for Modeling 
Electromagnetic Wave Scattering Problems
D. L. Young (*)
Department of Civil Engineering & Hydrotech Research Institute
National Taiwan University, Taiwan
Extended Abstract (pdf)
Short Abstract: In this paper we attempt to construct the electromagnetic wave scattering field by a given incident wave. For two-dimensional problems the normal incident plane wave scattering by conducting cylinder with infinite dimension in the z-direction would be discussed. For three-dimensional 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

Contributed Presentations

Numerical determination of the resonance frequencies 
in a bounded domain using the MFS
Pedro Antunes (*)
Department of Mathematics, 
Instituto Superior Técnico, Portugal
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

Explicit Estimation of an Integer at a Domain in the Reciprocity Principle
with the use of Inverse Operations 
Michal Cialkowski (*)
Heat Engineering Chair, 
Poznan University of Technology, Poland
Extended Abstract (pdf)
(*) joint work with:
A. Frackowiak

Moving Meshless Methods (I): Moving Element Free
Petrov-Galerkin Viscous Methods
Mehrzad Ghorbany (*)
Department of Mathematics, 
Sistan & Baluchestan University, Zahedan, Iran
Extended Abstract (pdf)
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 r-refinement and Moving Finite Element method (MFE) by Keith Miller to Meshless methods such as, DEM, EFGM, SPH, RKPH, PUM, h-p Clouds, ... . In MMM, mesh coordinates are unknown and are found by equations. This implies: exertion of indirect or pseudo equi-distribution of nodes. Weak form and system will be found by Galerkin or Petrov-Galerkin 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 pseudo-viscous 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 Petrov-Galerkin viscous Method (MEFPGVM) by C2 cubic Hermite base function on nodes as test or weight functions.
(*) joint work with:
Ali Reza Soheili

Radial and local Green's functions in diffraction 
and singularly perturbed problems
Evgeny Glushkov (*)
Kuban State University, 
Krasnodar, Russia
Abstract: The use of axially-symmetric delta-like functions as a basis and local Green's functions as test ones in the Galerkin and Petrov-Galerkin 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 one-dimensional ones. The selected shape-form 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 two-dimensional convection-diffusion Dirichlet problem a semi-analytical 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 Petrov-Galerkin 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

Special purpose Trefftz functions for the torsion of bars with 
regular polygonal cross-section
Jan A. Kolodziej (*)
Institute of Applied Mechanics, 
Poznan University of Technology, Poland
Extended Abstract (pdf)
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 non-dimensional stiffness of bars are obtained.
(*) joint work with:
Agnieszka Wachowska

A radial basis function collocation solver for temperature field in 
direct chill casting of aluminium alloys
Igor Kovacevic (*)
Laboratory for Multiphase Processes, 
Nova Gorica Polytechnic, Slovenia
Abstract: To appear
(*) joint work with:
Bozidar Sarler

A Least Squares Preconditioner for Radial Basis Functions 
Collocation Methods
Leevan Ling
Simon Fraser University
Extended Abstract (pdf)
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 least-squares approximate cardinal basis functions (ACBFs) from linear combinations of the RBF-PDE 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(mN2) flops to set up, and O(mN2) storage locations where m is a user define parameter of order O(10).
For the 2D MQ-RBF 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.

Multiquadrics method for Couette flow of a 
yield-stress fluid under imposed torques
Ahmed Naji (*)
Laboratoire de Mécanique de Milieux Hétérogènes, 
F.S.T de Tanger, Maroc
Extended Abstract (pdf)
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 m2 /m1 <<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. Er-Riani : Laboratoire de Mécanique de Milieux Hétérogènes, F.S.T de Tanger, Maroc.
C. Nouar, O. Sero-Guillaume : LEMTA – CNRS, France

On the use of a Wave Based prediction technique for
steady-state structural-acoustic radiation analysis
B. Pluymers (*)
Dep. Mechanical Engineering, 
Katholieke Universiteit Leuven, Belgium
Extended Abstract (pdf)
Short Abstract: The use of element based prediction techniques such as the finite element (FE) method, the infi-nite element (IE) method and the boundary element (BE) method, is generally accepted for steady-state dynamic analysis of coupled structural-acoustic 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, ex-terior 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 low-frequency 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 low-frequency applications.
Recently a new wave based prediction technique (WBT), which is based on the Trefftz approach, has proven to be successful for low- and mid-frequency 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 mid-frequency applications.
This paper discusses how the WBT can be extended for radiation problems in unbounded domains. The technique is illustrated in a two-dimensional analysis of the sound radiation of a bass-reflex loudspeaker and its performance is compared with the conventional element based techniques.
(*) joint work with:
W. Desmet, D. Vandepitte and P. Sas

The method of fundamental solutions applied to 
elastic problems in non-simply connected domains
P. M. C. Ribeiro
Escola Superior de Tecnologia, 
Universidade do Algarve, Portugal
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 non-simply 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 non-simply connected domains.
[poster presentation]

Density results using Stokeslets and the
method of fundamental solutions applied to fluid flow 
A. L. Silvestre
CEMAT- Centro de Matemática e Aplicações, 
Instituto Superior Técnico, Portugal
Extended Abstract (pdf)
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 so-called Stokeslets, to solve problems with conservative and non-conservative 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.

Numerical Analysis of the MFS for Certain Harmonic Problems 
Yiorgos S. Smyrlis (*)
Department of Mathematics and Statistics, 
University of Cyprus, Cyprus
Extended Abstract (pdf)
Short Abstract: The Method of Fundamental Solutions (MFS) is a boundary-type 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

GMLS approximations in the EFG method: 
applications to C1 structural problems. 
Carlos M. Tiago (*)
ICIST - Dep. Eng. Civil e Arquitectura,
Instituto Superior Técnico, Portugal
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

Using radial basis functions in a "finite difference mode"
Andrei I. Tolstykh (*)
Computing Center of Russian Academy of Sciences, Russia
Extended Abstract (pdf)
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 h-convergence 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 non-linear solid mechanics problems are displayed, the emphasize being placed on benchmark calculations.
(*) joint work with:
Dmitrii A. Shirobokov

Numerical simulation of acoustic wave scattering 
using a meshfree plane-waves method
Svilen S. Valtchev (*)
CEMAT- Centro de Matemática e Aplicações, 
Instituto Superior Técnico, Portugal
Extended Abstract (pdf)
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

Pseudo-divergence-free element free Galerkin
method for incompressible fluid flow.
Yolanda Vidal Seguí (*)
Department of Applied Mathematics, 
Universitat Politècnica de Catalunya, Spain
Extended Abstract (pdf)
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 divergence-free 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 mesh-free 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 Fernández-Méndez 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 Pseudo-Divergence-Free EFG method (PDF EFG). It consists in using interpolation functions that verify approximately the divergence-free 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 inf-sup test. And two well-known examples of Stokes flow are used to compare different mixed formulations.
(*) joint work with:
Antonio Huerta