Local Radial Basis Function Collocation Research Articles

Three-dimensional analysis for functionally graded piezoelectric semiconductors

This article presents numerical analyses of functionally graded piezoelectric beam with piezoelectric semiconducting material properties using meshless method. Unlike piezoelectric dielectric materials, electron density and electric current are additionally considered in constitutive equations for piezoelectric semiconductors. Mutual coupling of elastic displacements, electric potentials, and electron density increases the complexity of the analyzed problem. For the solution of the set of partial differential equations with non-constant coefficients, the local radial basis function collocation method is proposed in this work. Approximating the spatial variations of all physical fields in the partial differential equations by the multiquadric radial basis function entails an ensuing system for time-dependent problems solved by the Houbolt finite-difference scheme as a time-stepping method. The presented local radial basis function collocation method is verified using the corresponding results obtained using the finite element method. The influence of material parameter gradation and initial electron density is then investigated, along with transient analyses.

Improved localized radial basis function collocation method for multi-dimensional convection-dominated problems

In this paper, the localized radial basis function collocation method (LRBFCM) is combined with the partial upwind scheme for solving convection-dominated fluid flow problems. The localization technique adopted in LRBFCM has shown to be effective in avoiding the well known ill-conditioning problem of traditional meshless collocation method with globally defined radial basis functions (RBFs). For convection–diffusion problems with dominated convection, stiffness in the form of boundary/interior layers and shock waves emerge as convection overwhelms diffusion. We show in this paper that these kinds of stiff problems can be well tackled by combining the LRBFCM with partial upwind scheme. For verification, several numerical examples are given to demonstrate that this scheme improves the LRBFCM in providing stable, accurate, and oscillation-free solutions to one- and two-dimensional Burgers׳ equations with shock waves and singular perturbation problems with turning points and boundary layers.

A multiphysics and multiscale model for low frequency electromagnetic direct-chill casting

Simulation and control of macrosegregation, deformation and grain size in low frequency electromagnetic (EM) direct-chill casting (LFEMC) is important for downstream processing. Respectively, a multiphysics and multiscale model is developed for solution of Lorentz force, temperature, velocity, concentration, deformation and grain structure of LFEMC processed aluminum alloys, with focus on axisymmetric billets. The mixture equations with lever rule, linearized phase diagram, and stationary thermoelastic solid phase are assumed, together with EM induction equation for the field imposed by the coil. Explicit diffuse approximate meshless solution procedure [1] is used for solving the EM field, and the explicit local radial basis function collocation method [2] is used for solving the coupled transport phenomena and thermomechanics fields. Pressure-velocity coupling is performed by the fractional step method [3]. The point automata method with modified KGT model is used to estimate the grain structure [4] in a post-processing mode. Thermal, mechanical, EM and grain structure outcomes of the model are demonstrated. A systematic study of the complicated influences of the process parameters can be investigated by the model, including intensity and frequency of the electromagnetic field. The meshless solution framework, with the implemented simplest physical models, will be further extended by including more sophisticated microsegregation and grain structure models, as well as a more realistic solid and solid-liquid phase rheology.

Band structure computation of in-plane elastic waves in 2D phononic crystals by a meshfree local RBF collocation method

In this paper, the band structures of in-plane elastic waves in two-dimensional (2D) phononic crystals are calculated by using a meshfree local radial basis functions (RBF) collocation method. In order to improve the stability of the local RBF collocation method, special techniques suggested in our previous work for anti-plane waves are further improved and extended for calculating the primary field quantities and their normal derivatives required by the treatment of the boundary conditions in the local RBF collocation method for computing the band structures of the in-plane elastic waves in 2D phononic crystals. The developed meshfree local RBF collocation method for the band structure calculations of in-plane elastic waves propagating in 2D phononic crystals is validated by using the corresponding numerical results obtained with the finite element method (FEM). The band structures of different material combinations or acoustic impedance ratios, different filling fractions, various lattice forms and scatterer shapes are computed numerically to show the accuracy and the efficiency of the meshfree local RBF collocation method for computing the band structures of in-plane elastic waves in 2D phononic crystals.

A comparative analysis of local meshless formulation for multi-asset option models

A local meshless radial basis function collocation differential quadrature (LMRBFCDQ) is proposed for the numerical solution of a single and multi-asset option pricing PDE models arising in computational finance. Spatial discretization is performed by both local and a standard global meshless collocation procedures coupled with a set of different time integrators based on the forward Euler difference formula (FEDF), the fully Implicit method (FIM), the Crank–Nicolson method (CNM), the explicit Runge–Kutta method of order two (ERK2), the Crank–Nicolson Runge–Kutta method of order two (CNRK2), the fully Implicit Runge–Kutta method of order two (IRK2), the Runge–Kutta method of order four (RK4), the Embedded Runge–Kutta method (RK23). Operator splitting techniques like the ordinary operator splitting (OOS), the Lie–Trotter splitting, the additive splitting and the Strang splitting are also tested for time integration. The proposed hybrid schemes are the amalgamation of the meshless differential quadrature procedure and the finite difference approximations. Different types of radial basis functions (RBFs) i.e. the multiquadric (MQ), the inverse quadric (IQ) and the Gaussian (GA) are utilized for the spatial discretization of the PDE models. Numerical analysis of a range of computational finance related models are shown to demonstrate accuracy, efficiency and ease of implementation of the proposed meshless-finite difference procedure.

A meshfree local RBF collocation method for anti-plane transverse elastic wave propagation analysis in 2D phononic crystals

In this paper, a meshfree or meshless local radial basis function (RBF) collocation method is proposed to calculate the band structures of two-dimensional (2D) anti-plane transverse elastic waves in phononic crystals. Three new techniques are developed for calculating the normal derivative of the field quantity required by the treatment of the boundary conditions, which improve the stability of the local RBF collocation method significantly. The general form of the local RBF collocation method for a unit-cell with periodic boundary conditions is proposed, where the continuity conditions on the interface between the matrix and the scatterer are taken into account. The band structures or dispersion relations can be obtained by solving the eigenvalue problem and sweeping the boundary of the irreducible first Brillouin zone. The proposed local RBF collocation method is verified by using the corresponding results obtained with the finite element method. For different acoustic impedance ratios, various scatterer shapes, scatterer arrangements (lattice forms) and material properties, numerical examples are presented and discussed to show the performance and the efficiency of the developed local RBF collocation method compared to the FEM for computing the band structures of 2D phononic crystals.

Local radial basis function collocation method for linear thermoelasticity in two dimensions

Purpose The purpose of this paper is to upgrade our previous developments of Local Radial Basis Function Collocation Method (LRBFCM) for heat transfer, fluid flow and electromagnetic problems to thermoelastic problems and to study its numerical performance with the aim to build a multiphysics meshless computing environment based on LRBFCM.

Using a local radial basis function collocation method to approximate radiation boundary conditions

The local radial-basis-function collocation method (LRBFCM) based on the multiquadric type radial basis function is used to simulate the water wave propagation with oblique incidence and radiation. In this study, the mild-slope equation is applied and the radiation boundary condition is expanded to the third-order approximation. Two numerical cases are considered for verification, which include wave scattering around a vertical cylinder and shallow water wave passed over a submerged shoal. Numerical results based on the third-order radiation boundary condition are found to be in a better agreement than those of the first-order and second-order radiation boundary conditions especially for large incidence or radiation angles. Overall, the LRBFCM is a meshless numerical method and its applications to the high-order radiation boundary condition are much simpler compared with the traditional numerical methods, such as the finite difference method and the finite element method.

The radial basis function finite collocation approach for capturing sharp fronts in time dependent advection problems

We propose a node-based local meshless method for advective transport problems that is capable of operating on centrally defined stencils and is suitable for shock-capturing purposes. High spatial convergence rates can be achieved; in excess of eighth-order in some cases. Strongly-varying smooth profiles may be captured at infinite Péclet number without instability, and for discontinuous profiles the solution exhibits neutrally stable oscillations that can be damped by introducing a small artificial diffusion parameter, allowing a good approximation to the shock-front to be maintained for long travel times without introducing spurious oscillations.The proposed method is based on local collocation with radial basis functions (RBFs) in a “finite collocation” configuration. In this approach the PDE governing and boundary equations are enforced directly within the local RBF collocation systems, rather than being reconstructed from fixed interpolating functions as is typical of finite difference, finite volume or finite element methods. In this way the interpolating basis functions naturally incorporate information from the governing PDE, including the strength and direction of the convective velocity field. By using these PDE-enhanced interpolating functions an “implicit upwinding” effect is achieved, whereby the flow of information naturally respects the specifics of the local convective field. This implicit upwinding effect allows high-convergence solutions to be obtained on centred stencils for advection problems.The method is formulated using a high-convergence implicit timestepping algorithm based on Richardson extrapolation. The spatial and temporal convergence of the proposed approach is demonstrated using smooth functions with large gradients. The capture of discontinuities is then investigated, showing how the addition of a dynamic stabilisation parameter can damp the neutrally stable oscillations with limited smearing of the shock front.

Modelling of stress fields during LFEM DC casting of aluminium billets by a meshless method

Direct Chill (DC) casting of aluminium alloys is a widely established technology for efficient production of aluminium billets and slabs. The procedure is being further improved by the application of Low Frequency Electromagnetic Field (LFEM) in the area of the mold. Novel LFEM DC processing technique affects many different phenomena which occur during solidification, one of them being the stresses and deformations present in the billet. These quantities can have a significant effect on the quality of the cast piece, since they impact porosity, hot-tearing and cold cracking. In this contribution a novel local radial basis function collocation method (LRBFCM) is successfully applied to the problem of stress field calculation during the stationary state of DC casting of aluminium alloys. The formulation of the method is presented in detail, followed by the presentation of the tackled physical problem. The model describes the deformations of linearly elastic, inhomogeneous isotropic solid with a given temperature field. The temperature profile is calculated using the in-house developed heat and mass transfer model. The effects of low frequency EM casting process parameters on the vertical, circumferential and radial stress and on the deformation of billet surface are presented. The application of the LFEM appears to decrease the amplitudes of the tensile stress occurring in the billet.

A meshless technique based on the local radial basis functions collocation method for solving parabolic–parabolic Patlak–Keller–Segel chemotaxis model

In this paper local radial basis functions (LRBFs) collocation method is proposed for solving the (Patlak-) Keller–Segel model. We use the Crank–Nicolson difference scheme for the time derivative to obtain a finite difference scheme with respect to the time variable for the Keller–Segel model. Then we use the local radial basis functions (LRBFs) collocation method to approximate the spatial derivative. We obtain the numerical results for the mentioned model. As we know, recently some approaches presented for preventing the blow up of the cell density. In the current paper we use the multiquadric (MQ) radial basis function. The aim of this paper is to show that the meshless methods based on the local RBFs collocation approach are also suitable for solving models that have the blow up of the cell density. Also, six test problems are given that show the acceptable accuracy and efficiency of the proposed schemes.

Local radial basis function collocation method for solving thermo-driven fluid-flow problems with free surface

This paper explores the application of the meshless Local Radial Basis Function Collocation Method (LRBFCM) for the solution of coupled heat transfer and fluid flow problems with a free surface. The method employs the representation of temperature, velocity and pressure fields on overlapping five-noded sub-domains through collocation by using Radial Basis Functions (RBFs). This simple representation is then used to compute the first and second derivatives of the fields from the respective derivatives of the RBFs. The energy and momentum equations are solved through explicit time integration scheme. For numerical efficiency, the Artificial Compressibility Method (ACM) with Characteristic Based Split (CBS) technique is firstly adopted to solve the pressure–velocity coupled equations. The performance of the method is assessed based on solving the classical two-dimensional De Vahl Davis steady natural convection benchmark problem with an upper free surface for Rayleigh number ranged from 103 to 105 and Prandtl number equals to 0.71.

Estimating the temperature evolution of foodstuffs during freezing with a 3D meshless numerical method

Freezing processes are characterised by sharp changes in specific heat capacity and thermal conductivity for temperatures close to the freezing point. This leads to strong nonlinearities in the governing PDE that may be difficult to resolve using traditional numerical methods. In this work we present a meshless numerical method, based on a local Hermite radial basis function collocation approach in finite differencing mode, to allow the solution of freezing problems. By introducing a Kirchhoff transformation and solving the governing equations in Kirchhoff space, the strength of nonlinearity is reduced while preserving the structure of the heat equation. In combination with the high-resolution meshless numerical method, this allows efficient and stable numerical solutions to be obtained for freezing problems using 3D unstructured datasets. We demonstrate the proposed numerical formulation for the freezing of foodstuffs. By approximating the shape of the thermal conductivity and heat capacity curves in a piecewise linear fashion the temperature-dependent material curves may be described using eight independent parameters. We consider the optimisation of these parameters to match experimental data for the freezing of a hemispherical sample of mashed potato by using a simple manual procedure that requires a minimal number of simulations to be performed. Working in this way, a good approximation is obtained to the temperature profile throughout the sample without introducing instability into the numerical results.

Extrapolated local radial basis function collocation method for shallow water problems

This paper provides a novel meshfree numerical method – the extrapolated local-radial-basis-function collocation method (ELRBFCM) – to solve the well-known shallow-water equations (or de Saint-Venant equations). To promote the accuracy and stability of the local-radial-basis-function collocation method (LRBFCM), the extrapolation method is employed in the ELRBFCM. The numerical experiments include the simulation of two linear and three non-linear problems. For the linear problems, the results of the extrapolation method not only have good agreement with analytical solutions, but also show better efficiency than the Euler and RK4 scheme. It is convinced that by the extrapolation process, the approximating order and stability of time integration is automatically controlled. For the non-linear problems, the solutions by the proposed method capture the physical reaction that also obtained in the benchmark by a complex high-order finite-volume method. As a result, it is concluded that even for very long period, the meshfree ELRBFCM with high-order temporal approximation can provide robust, high accurate and great efficient numerical simulation for the shallow-water problems.

An efficient accurate Local Method of Approximate Particular Solutions for solving convection–diffusion problems

An efficient accurate Local Method of Approximate Particular Solutions (LMAPS) using Multiquadric Radial Basis Functions (RBFs) for solving convection–diffusion problems is proposed. It consists in adding auxiliary points to the local interpolation stencil at which the governing PDE is enforced, known as PDE points, besides imposing the boundary condition at the stencil in contact with the problem boundary. Two convection–diffusion problems are considered as test problems and solved with two previous local direct RBF collocation schemes (with and without PDE points) and two LMAPS (with and without PDE points), as well as the Global MAPS, in order to compare accuracy, convergence order and their behaviour in terms of the shape parameter. If PDE points are added, the result accuracy is improved as well as the convergence rate when using both local direct and MAPS formulations.

A meshless local RBF collocation method using integral operators for linear elasticity

This work presents a meshless numerical method for the solution of the linear elasticity equation, based on local collocation with radial basis functions and global reconstruction using an integral representational formula. Starting from a set of nodes distributed over the domain interior and boundaries, a small local stencil is formed around each interior node by connecting it to its neighbours. Over each local stencil a radial basis function (RBF) collocation is defined, which enforces the PDE governing and boundary operators. In this way the solution domain is covered by a series of small overlapping local RBF collocation systems. At each interior node the displacement field is approximated from the surrounding RBF collocation system and expressed in terms of its corresponding integral representational formula (Betti׳s formula) defined over a local contour. In this way the approach can be considered to be an improved local Boundary Element method (LBEM) where besides using a direct interpolation of the displacement field, the PDE and boundary conditions are also imposed within the local interpolation. The resulting sparse global assembly is then solved to obtain the displacement field at each interior node.Two benchmark problems with known analytical solutions are considered to test the performance of the method: a thin plate with a circular hole under traction, and a cylindrical annulus under internal and external pressure. Using these two numerical examples the accuracy of the proposed method is investigated, considering different configurations for the local integration contour, different quadrature schemes, the effect of basis function flatness, and spatial convergence rates. To allow a consistent analysis of these features regular nodal distributions are used for most investigations, however the capability of the method to operate on irregular datasets is also demonstrated.

Simulation of laminar backward facing step flow under magnetic field with explicit local radial basis function collocation method

The purpose of the present paper is to extend the use of a novel meshless Local Radial Basis Function Collocation Method (LRBFCM) for solving the two-dimensional, steady, laminar flow over a backward facing step under the influence of the Lorentz force. The incompressible Navier–Stokes equations are under the influence of predetermined static magnetic field numerically solved on a non-uniform node arrangement. In the numerical procedure, local collocation and Multiquadric Radial Basis Functions (MQ RBF) are used on five-nodded subdomains. The coupling between the pressure and the velocity is made by using Fractional Step Method (FSM). The considered problem is calculated for Reynolds numbers (Re) ranging from 300 to 800, Hartman numbers (Ha) ranging from 0 to 100, and for low magnetic Rem number. The numerical results demonstrate excellent agreement with previously published data, obtained with the classical numerical methods, such as Finite Volume Method (FVM) and Finite Element Method (FEM). Simplicity of the numerical implementation, accuracy and the absence of the polygonalisation are the main advantages of the LRBFCM.

Solving biharmonic equation using the localized method of approximate particular solutions

Some localized numerical methods, such as finite element and finite difference methods (FDMs), have encountered difficulties when solving fourth or higher order differential equations. Localized methods, which use radial basis functions, are considered the generalized FDMs and, thus, inherit the similar difficulties when solving higher order differential equations. In this paper, we deal with the use of the localized method of approximate particular solutions (LMAPS), a recently developed localized radial basis function collocation method, in solving two-dimensional biharmonic equation in a bounded region. The technique is based on decoupling the biharmonic problem into two Poisson equations, and then the LMAPS is applied to each Poisson's problem to compute numerical solutions. Furthermore, the influence of the shape parameter and different radial basis functions on the numerical solution is discussed. The effectiveness of the proposed method is demonstrated by solving three examples in both regular and irregular domains.

Simulation of macrosegregation with mesosegregates in binary metallic casts by a meshless method

Simulation of macrosegregation with mesosegregates as a consequence of solidification of a binary Sn–10%Pb alloy in a 2-dimensional rectangular cast is tackled in the present paper. Coupled volume averaged governing equations for mass, energy, momentum and species transfer are considered by incorporating Lever solidification rule and incompressible Newtonian fluid with Darcy limit in the mushy zone. Solid phase is assumed stationary. Double diffusive effects in the melt are modeled by the thermal and solutal Boussinesq hypothesis. The physical model is solved by the meshless Local Radial Basis Function Collocation Method (LRBFCM) by using 5-noded influence domains, multiquadrics radial basis functions and explicit time stepping. Pressure–velocity coupling is based on local pressure correction. Adaptive upwinding has to be used for stabilization of the convective terms. The numerical simulations reveal instabilities during solidification process that introduce anomalies in the final segregation map that scale with the typical cast as well as sub-cast dimensions. The main advantages of choosing the represented meshless approach for solving the problem are in its simplicity and similar coding in 2D and 3D, as well as straightforward applicability in non-uniform node arrangements. The locality of the proposed numerical approach is also convenient for parallel execution. It is demonstrated that LRBFCM can be advantageously used in casting simulations where the chemical segregation exhibits industrially relevant multi-scale patterns.

Applications of the Local RBF Collocation Method and the Fictitious Time Integration Method for Burgers’ Equations

In this paper, a meshfree numerical scheme, which is a combination of the local RBF (radial basis function) collocation method (LRBFCM) and the fictitious time integration method (FTIM), is proposed to efficiently and accurately analyze the two-dimensional Burgers’ equations. The Burgers’ equations, which can be regarded as the simplified form of the Navier-Stokes equations, are a system of time-and space-dependent nonlinear partial differential equations, so it is very difficult and important to be accurately solved. The LRBFCM, one of newly-developed meshfree methods, is adopted for spatial discretization and, at the meantime, the implicit Euler method is used for temporal discretization. Then, the system of nonlinear algebraic equations at every time step will be efficiently resolved by the FTIM, which can avoid the calculation of the Jacobian matrix. The numerical results and comparisons of one numerical example will be provided to validate the accuracy and the efficiency of the proposed meshfree numerical scheme.