Nam Mai-Duy

Numerical solution of differential equations using multiquadric radial basis function networks

This paper presents mesh-free procedures for solving linear differential equations (ODEs and elliptic PDEs) based on multiquadric (MQ) radial basis function networks (RBFNs). Based on our study of approximation of function and its derivatives using RBFNs that was reported in an earlier paper (Mai-Duy, N. & Tran-Cong, T. (1999). Approximation of function and its derivatives using radial basis function networks. Neural networks, submitted), new RBFN approximation procedures are developed in this paper for solving DEs, which can also be classified into two types: a direct (DRBFN) and an indirect (IRBFN) RBFN procedure. In the present procedures, the width of the RBFs is the only adjustable parameter according to a(i) = betad(i), where d(i) is the distance from the ith centre to the nearest centre. The IRBFN method is more accurate than the DRBFN one and experience so far shows that beta can be chosen in the range 7 < or = beta 10 for the former. Different combinations of RBF centres and collocation points (uniformly and randomly distributed) are tested on both regularly and irregularly shaped domains. The results for a 1D Poissons equation show that the DRBFN and the IRBFN procedures achieve a norm of error of at least O(1.0 x 10(-4)) and O(1.0 x 10(-8)), respectively, with a centre density of 50. Similarly, the results for a 2D Poissons equation show that the DRBFN and the IRBFN procedures achieve a norm of error of at least O(1.0 x 10(-3)) and O(1.0 x10(-6)) respectively, with a centre density of 12 X 12.

Approximation of function and its derivatives using radial basis function networks

Abstract This paper presents a numerical approach, based on radial basis function networks (RBFNs), for the approximation of a function and its derivatives (scattered data interpolation). The approach proposed here is called the indirect radial basis function network (IRBFN) approximation which is compared with the usual direct approach. In the direct method (DRBFN) the closed form RBFN approximating function is first obtained from a set of training points and the derivative functions are then calculated directly by differentiating such closed form RBFN. In the indirect method (IRBFN) the formulation of the problem starts with the decomposition of the derivative of the function into RBFs. The derivative expression is then integrated to yield an expression for the original function, which is then solved via the general linear least squares principle, given an appropriate set of discrete data points. The IRBFN method allows the filtering of noise arisen from the interpolation of the original function from a discrete set of data points and produces a greatly improved approximation of its derivatives. In both cases the input data consists of a set of unstructured discrete data points (function values), which eliminates the need for a discretisation of the domain into a number of finite elements. The results obtained are compared with those obtained by the feed forward neural network approach where appropriate and the “finite element” methods. In all examples considered, the IRBFN approach yields a superior accuracy. For example, all partial derivatives up to second order of the function of three variables y = x 1 2 + x 1 x 2 −2 x 2 2 − x 2 x 3 + x 3 2 are approximated with at least an order of magnitude better in the L 2 -norm in comparison with the usual DRBFN approach.

Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson's equations

This paper presents the combination of new mesh-free radial basis function network (RBFN) methods and domain decomposition (DD) technique for approximating functions and solving Poissons equations. The RBFN method allows numerical approximation of functions and solution of partial differential equations (PDEs) without the need for a traditional ‘finite element’-type (FE) mesh while the combined RBFN–DD approach facilitates coarse-grained parallelisation of large problems. Effect of RBFN parameters on the quality of approximation of function and its derivatives is investigated and compared with the case of single domain. In solving Poissons equations, an iterative procedure is employed to update unknown boundary conditions at interfaces. At each iteration, the interface boundary conditions are first estimated by using boundary integral equations (BIEs) and subdomain problems are then solved by using the RBFN method. Volume integrals in standard integral equation representation (IE), which usually require volume discretisation, are completely eliminated in order to preserve the mesh-free nature of RBFN methods. The numerical examples show that RBFN methods in conjunction with DD technique achieve not only a reduction of memory requirement but also a high accuracy of the solution.

A collocation method based on one‐dimensional RBF interpolation scheme for solving PDEs

Purpose – To present a new collocation method for numerically solving partial differential equations (PDEs) in rectangular domains.Design/methodology/approach – The proposed method is based on a Cartesian grid and a 1D integrated‐radial‐basis‐function scheme. The employment of integration to construct the RBF approximations representing the field variables facilitates a fast convergence rate, while the use of a 1D interpolation scheme leads to considerable economy in forming the system matrix and improvement in the condition number of RBF matrices over a 2D interpolation scheme.Findings – The proposed method is verified by considering several test problems governed by second‐ and fourth‐order PDEs; very accurate solutions are achieved using relatively coarse grids.Research limitations/implications – Only 1D and 2D formulations are presented, but we believe that extension to 3D problems can be carried out straightforwardly. Further, development is needed for the case of non‐rectangular domains.Originality/...

Numerical investigations on the compressibility of a DPD fluid

The compressibility of a dissipative particle dynamics (DPD) fluid is studied numerically through several newly developed test models, where both the density and the divergence of the velocity field are considered. In the case of zero conservative force, the DPD fluid turns out to be compressible. Effects of the compressibility are observed to be reduced as the particle mass is chosen to be smaller and the system temperature to be higher. In the case of non-zero conservative force, the condition of constant density and divergence-free of velocity can be approximately achieved at large values of the repulsion parameter (i.e., weakly compressible flow). Furthermore, the speed of sound and local Mach number are computed and found to be in good agreement with the theoretical estimation.

Compact local integrated-RBF approximations for second-order elliptic differential problems

This paper presents a new compact approximation method for the discretisation of second-order elliptic equations in one and two dimensions. The problem domain, which can be rectangular or non-rectangular, is represented by a Cartesian grid. On stencils, which are three nodal points for one-dimensional problems and nine nodal points for two-dimensional problems, the approximations for the field variable and its derivatives are constructed using integrated radial basis functions (IRBFs). Several pieces of information about the governing differential equation on the stencil are incorporated into the IRBF approximations by means of the constants of integration. Numerical examples indicate that the proposed technique yields a very high rate of convergence with grid refinement.

Investigation of particles size effects in Dissipative Particle Dynamics (DPD) modelling of colloidal suspensions

Abstract In the Dissipative Particle Dynamics (DPD) simulation of suspension, the fluid (solvent) and colloidal particles are replaced by a set of DPD particles and therefore their relative sizes (as measured by their exclusion zones) can affect the maximal packing fraction of the colloidal particles. In this study, we investigate roles of the conservative, dissipative and random forces in this relative size ratio (colloidal/solvent). We propose a mechanism of adjusting the DPD parameters to properly model the solvent phase (the solvent here is supposed to have the same isothermal compressibility to that of water).

A compact five-point stencil based on integrated RBFs for 2D second-order differential problems

In this paper, a compact 5-point stencil for the discretisation of second-order partial differential equations (PDEs) in two space dimensions is proposed. We employ integrated radial basis functions in one dimension (1D-IRBFs) to construct the approximations for the dependent variable and its derivatives over the three nodes in each direction of the stencil. Certain nodal values of the second-order derivatives are incorporated into the approximations with the help of the integration constants. In the case of elliptic PDEs, one algebraic equation is formed at each interior node, and the obtained final system, of which each row has 5 non-zero entries, is solved iteratively using a Picard scheme. In the case of parabolic PDEs discretised with a Crank-Nicolson procedure, a set of three simultaneous algebraic equations is established at each interior node and the three equations are then combined to form two tridiagonal equations through the implicit elimination approach. Linear and non-linear test problems, including lid-driven cavity flow and natural convection between the outer square and the inner cylinder, are considered to verify the proposed stencil.

An Effective Integrated-RBFN Cartesian-Grid Discretization for the Stream Function–Vorticity–Temperature Formulation in Nonrectangular Domains

This article presents a new numerical collocation procedure, based on Cartesian grids and one-dimensional integrated radial-basis-function networks (1D-IRBFNs), for the simulation of natural convection defined in two-dimensional, multiply connected domains and governed by the stream function–vorticity–temperature formulation. Special emphasis is placed on the handling of vorticity values at boundary points that do not coincide with grid nodes. A suitable formula for computing vorticity boundary conditions, which is based on the approximations with respect to one coordinate direction only, is proposed. Normal derivative boundary conditions for the stream function are forced to be satisfied identically. Several test problems, including natural convection in the annulus between square and circular cylinders, are considered to investigate the accuracy of the proposed technique.

A Galerkin approach incorporating integrated radial basis function networks for the solution of 2D biharmonic equations

This paper is concerned with the use of integrated radial basis function networks (IRBFNs) for the discretisation of Galerkin approximations for Dirichlet biharmonic problems in two dimensions. The field variable is approximated by global high-order IRBFNs on uniform grids without suffering from Runges phenomenon. Double boundary conditions, which can be of complicated shapes, are both satisfied identically. The proposed technique is verified through the solution of linear and nonlinear problems, including a benchmark buoyancy-driven flow in a square slot. Good accuracy and fast convergence are obtained.