Integrated Radial Basis Function Research Articles

A numerical study of compact approximations based on flat integrated radial basis functions for second-order differential equations

In this paper, we propose a simple but effective preconditioning technique to improve the numerical stability of Integrated Radial Basis Function (IRBF) methods. The proposed preconditioner is simply the inverse of a well-conditioned matrix that is constructed using non-flat IRBFs. Much larger values of the free shape parameter of IRBFs can thus be employed and better accuracy for smooth solution problems can be achieved. Furthermore, to improve the accuracy of local IRBF methods, we propose a new stencil, namely Combined Compact IRBF (CCIRBF), in which (i) the starting point is the fourth-order derivative; and (ii) nodal values of first- and second-order derivatives at side nodes of the stencil are included in the computation of first- and second-order derivatives at the middle node in a natural way. The proposed stencil can be employed in uniform/nonuniform Cartesian grids. The preconditioning technique in combination with the CCIRBF scheme employed with large values of the shape parameter are tested with elliptic equations and then applied to simulate several fluid flow problems governed by Poisson, Burgers, convection–diffusion, and Navier–Stokes equations. Highly accurate and stable solutions are obtained. In some cases, the preconditioned schemes are shown to be several orders of magnitude more accurate than those without preconditioning.

A three-point coupled compact integrated RBF scheme for second-order differential problems

In this paper, we propose a three-point coupled compact integrated radial basis function (CCIRBF) approximation scheme for the discretisation of second-order differential problems in one and two dimensions. The CCIRBF employs integrated radial basis functions (IRBFs) to construct the approximations for its first and second derivatives over a three-point stencil in each direction. Nodal values of the first and second derivatives (i.e. extra information), incorporated into approximations by means of the constants of integration, are simultaneously employed to compute the first and second derivatives. The essence of the CCIRBF scheme is to couple the extra information of the nodal first and second derivative values via their identity equations. Owing to its coupling of the information of the nodal first and second derivatives, the CCIRBF scheme becomes more accurate, stable and efficient than the normal compact integrated radial basis function (CIRBF) schemes proposed by [Thai-Quang, N.; Mai-Duy, N.; Tran, C.-D.; Tran-Cong, T. (2012)]. The main features of the CCIRBF scheme include: three-point, high-order accuracy, stability, efficiency and inclusion of boundary values. Several analytic problems are considered to verify the present scheme and to compare its accuracy, stability and efficiency with those of the CIRBF, higher-order compact finite difference (HOC) and some other high-order schemes. Numerical results show that highly accurate and stable results are obtained with the proposed scheme. Additionally, the present scheme also takes less time to achieve target accuracy in comparison with the CIRBF and HOC schemes.

Parallel control-volume method based on compact local integrated RBFs for the solution of fluid flow problems

In this paper, a high performance computing method based on the Integrated Radial Basis Function (IRBF), Control Volume (CV) and Domain Decomposition technique for solving Partial Differential Equations is presented. The goal is to develop an efficient parallel algorithm based on the Compact Local IRBF method using the CV approach, especially for problems with non-rectangular domain. The results showed that the goal is achieved as the computational efficiency is quite significant. For the case of square lid driven cavity problem with Renoylds number 100, super-linear speed-up is also achieved. The parallel algorithm is implemented in the Matlab environment using Parallel Computing Toolbox based on Distributed Computing Engine.

A meshless numerical approach based on Integrated Radial Basis Functions and level set method for interfacial flows

This paper reports a new meshless Integrated Radial Basis Function Network (IRBFN) approach to the numerical simulation of interfacial flows in which the two-way interaction between a moving interface and the ambient viscous flow is fully investigated. When an interface between two immiscible fluids moves, not only its position and shape but also the flow variables (i.e. velocity field and pressure) change due to the presence of surface tension along the moving interface. The velocity field of the ambient flow, on the other hand, causes the interface to move and deform as a result of momentum transport between the two immiscible fluids on both sides of the interface. Numerical investigations of such a two-way interaction is reported in this paper where the level set method is used in combination with high-order projection schemes in the meshless framework of the IRBFN method. Numerical investigations on the meshless projection schemes are performed with typical benchmark incompressible viscous flow problems for verification purposes. The approach is then demonstrated with the numerical simulation of two bubbles moving, stretching and merging in an incompressible ambient fluid under the action of buoyancy force.

A direct forcing immersed boundary method employed with compact integrated RBF approximations for heat transfer and fluid flow problems

In this paper, we present a numerical scheme, based on the direct forcing immersed boundary (DFIB) approach and compact integrated radial basis function (CIRBF) approximations, for solving the Navier-Stokes equations in two dimensions. The problem domain of complicated shape is embedded in a Cartesian grid containing Eulerian nodes. Non-slip conditions on the inner boundaries, represented by Lagrangian nodes, are imposed by means of the DFIB method, in which a smoothed version of the discrete delta functions is utilised to transfer the physical quantities between two types of nodes. The velocities and pressure variables are approximated locally on Eulerian nodes using 3-node CIRBF stencils, where first- and second-order derivative values of the field variables are also included in the RBF approximations. The present DFIB-CIRBF scheme is verified through the solution of several test problems including Taylor-Green vortices, rotational flow, lid-driven cavity flow with multiple solid bodies, flow between rotating circular and fixed square cylinders, and natural convection in an eccentric annulus between two circular cylinders. Numerical results obtained using relatively coarse grids are in good agreement with available data in the literature.

Compact local IRBF and domain decomposition method for solving PDEs using a distributed termination detection based parallel algorithm

Compact Local Integrated Radial Basis Function (CLIRBF) methods based on Cartesian grids can be effective numerical methods for solving partial differential equations (PDEs) for fluid flow problems. The combination of the domain decomposition method and function approximation using CLIRBF methods yields an effective coarse-grained parallel processing approach. This approach has enabled not only each sub-domain in the original analysis domain to be discretised by a separate CLIRBF network but also compact local stencils to be independently treated. The present algorithm, namely parallel CLIRBF, achieves higher throughput in solving large scale problems by, firstly, parallel processing of sub-regions which constitute the original domain and, secondly, accelerating the convergence rate within each sub-region using groups of CLIRBF stencils in which function approximations are carried out by parallel processes. The procedure is illustrated with several numerical examples of PDEs and lid-driven cavity problem using Message Passing Interface supported by MATLAB.

RBF-based multiscale control volume method for second order elliptic problems with oscillatory coefficients

Many important engineering problems have multiple-scale solutions. Thermal conductivity of composite materials, flow in porous media, and turbulent transport in high Reynolds number flows are examples of this type. Direct numerical simulations for these problems typically require extremely large amounts of CPU time and computer memory, which may be too expensive or impossible on the present supercomputers. In this paper, we develop a high order computational method, based on multiscale basis function approach and integrated radial-basis-function (IRBF) approximant, for the solution of multiscale elliptic problems with reduced computational cost. Unlike other methods based on multiscale basis function approach, sets of basis and correction functions here are obtained through $C^2$-continuous IRBF element formulations. High accuracy and efficiency of this method are demonstrated by several one- and two-dimensional examples.

A global meshless collocation particular solution method (integrated Radial Basis Function) for two-dimensional Stokes flow problems

A global version of the Method of Approximate Particular Solutions (MAPS) is developed to solve two-dimensional Stokes flow problems in bounded domains. The velocity components and the pressure are approximated by a linear superposition of particular solutions of the non-homogeneous Stokes system of equations with a Multiquadric Radial Basis Function as forcing term. Although, the continuity equation is not explicitly imposed in the resulting formulation, the scheme is mass conservative since the particular solutions exactly satisfy the mass conservation equation. The present scheme is validated by comparing the obtained numerical result with the analytical solution of two boundary value problems constructed from the Stokeson exterior fundamental solution, i.e. regular everywhere except at infinity. For these two cases, convergence of the method and the influence of the value of the Multiquadric’s shape parameter on the numerical results are studied by computing the relative Root Mean Square (RMS) error for several homogeneous distributions of collocation points and values of the shape parameter. From this analysis is observed that the proposed MAPS results are stable and accurate for a wide range of shape parameter values. In addition, the lid-driven cavity and backward-facing step flow problems are solved and the obtained results compared with the solutions found with more conventional numerical schemes, showing good agreement between them.

A high-order compact local integrated-RBF scheme for steady-state incompressible viscous flows in the primitive variables

This study is concerned with the development of integrated radialbasis- function (IRBF) method for the simulation of two-dimensional steady-state incompressible viscous flows governed by the pressure-velocity formulation on Cartesian grids. Instead of using low-order polynomial interpolants, a high-order compact local IRBF scheme is employed to represent the convection and diffusion terms. Furthermore, an effective boundary treatment for the pressure variable, where Neumann boundary conditions are transformed into Dirichlet ones, is proposed. This transformation is based on global 1D-IRBF approximators using values of the pressure at interior nodes along a grid line and first-order derivative values of the pressure at the two extreme nodes of that grid line. The performance of the proposed scheme is investigated numerically through the solution of several linear (analytic tests including Stokes flows) and non-linear (recirculating cavity flow driven by combined shear & body forces and lid-driven cavity flow) problems. Unlike the global 1D-IRBF scheme, the proposed method leads to a sparse system matrix. Numerical results indicate that (i) the present solutions are more accurate and converge faster with grid refinement in comparison with standard finite-difference results; and (ii) the proposed boundary treatment for the pressure is more effective than conventional direct application of the Neumann boundary condition.

Local moving least square-one-dimensional integrated radial basis function networks technique for incompressible viscous flows

This paper presents a local moving least square-one-dimensional integrated radial basis function networks method for solving incompressible viscous flow problems using stream function-vorticity formulation. In this method, the partition of unity method is employed as a framework to incorporate the moving least square and one-dimensional integrated radial basis function networks techniques. The major advantages of the proposed method include the following: (i) a banded sparse system matrix which helps reduce the computational cost; (ii) the Kronecker-i property of the constructed shape function which helps impose the essential boundary condition in an exact manner; and (iii) high accuracy and fast convergence rate owing to the use of integration instead of conventional differentiation to construct the local radial basis function approximations. Several examples including two-dimensional (2D) Poisson problems, lid-driven cavity flow and flow past a circular cylinder are considered, and the present results are compared with the exact solutions and numerical results from other methods in the literature to demonstrate the attractiveness of the proposed method.

Local moving least square - one-dimensional IRBFN technique: part 11 - unsteady incompressible viscous flows

In this study, local moving least square - one dimensional integrated radial basis function network (LMLS-1D-IRBFN) method is presented and demonstrated with the solution of time-dependent problems such as Burgers' equation, unsteady flow past a square cylinder in a horizontal channel and unsteady flow past a circular cylinder. The present method makes use of the partition of unity concept to combine the moving least square (MLS) and one-dimensional integrated radial basis function network (1D-IRBFN) techniques in a new approach. This approach offers the same order of accuracy as its global counterpart, the 1D-IRBFN method, while the system matrix is more sparse than that of the 1D-IRBFN, which helps reduce the computational cost significantly. For fluid flow problems, the diffusion terms are discretised by using LMLS-1D-IRBFN method, while the convection terms are explicitly calculated by using 1D-IRBFN method. The present numerical procedure is combined with a domain decomposition technique to handle largescale problems. The numerical results obtained are in good agreement with other published results in the literature.

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.

Computation of Laminated Composite Plates using Integrated Radial Basis Function Networks

This paper reports a meshless method, which is based on radial-basis-function networks (RBFNs), for the static analysis of moderately-thick laminated composite plates using the first-order shear deformation theory. Integrated RBFNs are employed to represent the field variables, and the governing equations are discretized by means of point collocation. The use of integration rather than conventional differentiation to construct the RBF approximations significantly stabilizes the solution and enhances the quality of approximation. The proposed method is verified through the solution of rectangular and non-rectangular composite plates. Numerical results obtained show that the method achieves a very high degree of accuracy and a fast convergence rate.

Integrated multiquadric radial basis function approximation methods

Promising numerical results using once and twice integrated radial basis functions have been recently presented. In this work we investigate the integrated radial basis function (IRBF) concept in greater detail, connect to the existing RBF theory, and make conjectures about the properties of IRBF approximation methods. The IRBF methods are used to solve PDEs.

An Auto-Tuning Algorithm for the IRBF Network of Brushless DC Motor

The integrated radial basis function (IRBF) network has been reported as an efficient algorithm to study the performance of brushless dc motors. However, such an algorithm cannot be implemented readily since it is difficult to auto-tune or even to find the undetermined coefficients in the integrated RBF network. In this paper, a novel auto-tuning algorithm that can effectively guarantee the automatic implementation of the integrated RBF network of a brushless dc motor is reported.