Radial Basis Function Meshless Method Research Articles

A LOCALIZED MESHLESS METHOD FOR TRANSIENT HEAT CONDUCTION WITH APPLICATIONS

The localized radial basis function (RBF) meshless approach is well suited for modeling transient heat conduction. The advantages of meshless methods, such as ease of discretization, are well known. However, there are still few examples of the method extended to three-dimensional (3D) transient heat conduction for geometries of practical engineering importance; in particular, with respect to graphics processing units (GPUs). In this study, we investigated the localized RBF meshless method in 3D transient heat conduction and explored its application for GPUs. Numerical examples are provided with an application to modeling chip cooling and friction welding.

Numerical assessment of meshless method for studying nanofluid natural convection in a corrugated wall square cavity

This study aims to investigate the impact of various factors, such as wall shape, Rayleigh number, and volume fraction of nanoparticles, on natural convection in a square cavity that is filled with a mixture of Al2O3 solid particles and liquid water. The research employs numerical simulations based on the radial basis function meshless method and the artificial compressibility technique. The results of the study showed that the temperature distribution in the cavity was mostly uniform, except in the vicinity of the hot wall, while the flow was primarily dominated by convection as the Rayleigh number increased. Furthermore, the heat transfer rate increased with the volume fraction of nanoparticles, indicating the significance of nanoparticles in improving the thermal performance of the system. Additionally, the study found that the average Nusselt number, which characterizes the heat transfer efficiency, was highest when the cavity had a wavy wall. For single and double wavy walls, there were respective enhancements of 32% and 6% compared to a regular wall. Additionally, the Nusselt number increased as the volume fraction of nanoparticles, indicating a significant influence of nanoparticle concentration and wall geometry on the fluid flow and heat transfer characteristics in the square cavity. Consequently, this study's outcomes provide crucial insights into designing and optimizing thermal management systems, particularly those utilizing nanofluids.

Laminar Rayleigh–Benard convection in a closed square field with meshless radial basis function method

Abstract Research on natural convection is exciting in some experimental and numerical cases, especially in rectangular cavities with relatively low heat dissipation and thermal control systems with low cost, reliability, and ease of use. The present study will use the meshless radial basis function method to solve the velocity formulation of the Navier–Stokes equations by varying some nominal Rayleigh numbers of 104, 105, and 106. The numerical accuracy is compared with the previous research. The advantages of the meshless method are that it does not require a structured mesh and does not require inter-nodal connectivity. The results show that the temperature pattern is identical to the previous research. The calculations have been done for three different Rayleigh numbers of 104, 105, and 106 for 151 × 151 nodes. The variations of the Ra number will affect the isothermal, velocity contours, and Nusselt number.

A radial basis function approximation method for conservative Allen–Cahn equations on surfaces

In this paper, we present a meshless radial basis function method to solve conservative Allen–Cahn equation on smooth compact surfaces embedded in R3, which can inherits the mass conservation property. The proposed method is established on the operator splitting scheme. We approximate the surface Laplace–Beltrami operator by an iterative radial basis function approximation method and discretize the diffusion equation in time by the Euler method. The reaction equation containing the nonlinear function is solved analytically. Moreover, to make the mass conservation, we employ a kernel-based quadrature formula to approximate the Lagrange multiplier. The salient feature of the meshless conservative scheme is that it is explicit and more efficient than narrow band methods since few scattered nodes on the surface are adopted in spatial approximation. Several numerical experiments are performed to illustrate the accuracy and the conservation property of the scheme on spheres and other surfaces.

Marangoni convection within thermosolute and absorptive aqueous LiBr solution

The Marangoni convection during steam absorption into an aqueous lithium bromide (LiBr) solution is investigated numerically. In order to obtain a better understanding of the physical mechanism of thermal convection in the solution, the influence of surface tension on the absorbing solution is explored. For this purpose, this paper proposes the radial basis function (RBF) meshless method for numerical simulation of thermosolutal convection and vapor absorption in aqueous LiBr solution problems. The spatial terms of the governing equation are discretized using the RBF meshless method. The fractional step method is used for solving the continuity equations and momentum equations. The coupled energy and species equations are solved iteratively. The semi-algebraic system is solved using an implicit backward Euler scheme to maintain stability. The no-slip boundary conditions are imposed on solid wall surface. The surface tension gradient and the energy balance due the phase change are imposed on the interface of vapor and solution interface. The thermosolutal convection problem is presented to examine the accuracy of proposed method. An analytical solution is also presented for validating the vapor absorption problem results for zero Marangoni number (Ma). The Marangoni number used has a range of 0 ≤ Ma ≤ 1000. The increase in Marangoni number for low Prandtl number values does not significantly affect heat and mass transfer. However, increasing Ma for LiBr solution has a significant effect. It is also found that the velocity of convection flow can be improved by adding a solid object to the interface.

Developing a leap-frog meshless methods with radial basis functions for modeling of electromagnetic concentrator

<abstract><p>The main goal of this paper is to develop a fast and effective meshless method by using radial basis function (RBF) for the time domain model equations of electromagnetic wave concentration device. This is mainly because the complex model equations involve different partial differential equations in different subdomains, which makes the meshless method very attractive and also very challenging. In order to simulate the propagation of electromagnetic waves in the electromagnetic concentrator, perfect matching layer technology was used to reduce an unbounded domain problem into a bounded domain problem. Borrowing the idea of the leap-frog finite-difference time-domain scheme, I develop the leap-frog RBF meshless method to solve the coupled complex modeling equations. The numerical results obtained by using a multiquadric RBF and Gaussian RBF demonstrate that our RBF method is very effective.</p></abstract>

Efficiency improvement of radial basis function meshless method in conjunction with bayesian theorem for electrical tomography of heterogeneous concrete

Electric potential measurement technique (tomography) has been presented as a non-destructive method to evaluate concrete's physical properties and durability. In this study, a radial basis function (RBF) meshless method combined with the Bayesian probabilistic theorem was proposed to solve Poisson's differential equation that simulates electric potential distribution for heterogeneous concrete with inclusion in 2D. Thus, prior information about shape coefficient (c) and electrical conductivity could contribute to problem-solving, leading to an accurate high-speed tomography. Two types of concrete mixes with significant differences in electrical conductivity were used to create heterogeneity in concrete. For this purpose, concrete samples with iron block inclusion in various locations were made. Probabilistic tomography was performed using optimized c and Bayes' method. Since c plays a significant role in RBF, different validation patterns were used to optimize it. Results showed that c is highly dependent on the validation pattern, injecting pair electrode, location of inclusions, and degree of uncertainty of concrete. Deterministic and probabilistic tomography was performed to detect the iron block inclusion. The experimental results showed that using various validation patterns, the proposed hybrid method of meshless and Bayesian theorem can eliminate the substantial physical uncertainty, and tomography can be performed with appropriate accuracy.

Research on nonlinear waves of blood flow in arterial vessels

In this paper, considering blood as an incompressible Newtonian fluid, we explore the propagation of blood flow. Based on the Navier–Stokes equations and the continuity equation, the vorticity equation is deduced. Then, using the multi-scale analysis and iterative perturbed method, we derive a new higher order nonlinear Schrödinger equation to describe the blood flow in the blood vessels. The traveling wave solution of the higher order equation is obtained with the help of the extended tanh-function method. By analyzing the higher-order term of the equation, we find that the width and velocity of the blood wave increase during the propagation process. To further study the numerical characteristic, the numerical solutions of the higher order nonlinear Schrödinger equation are obtained by the meshless radial basis function method. The absolute errors between the exact and numerical red solutions show that the meshless radial basis function method gives a more accurate solution. Finally, the influencing factors of stroke volume are discussed and analyzed which provide a strong theoretical basis for some blood diseases.

Solving Parabolic and Hyperbolic Equations with Variable Coefficients Using Space-Time Localized Radial Basis Function Collocation Method

In this paper, we investigate the numerical approximation solution of parabolic and hyperbolic equations with variable coefficients and different boundary conditions using the space-time localized collocation method based on the radial basis function. The method is based on transforming the original d -dimensional problem in space into d + 1 -dimensional one in the space-time domain by combining the d -dimensional vector space variable and 1 -dimensional time variable in one d + 1 -dimensional variable vector. The advantages of such formulation are (i) time discretization as implicit, explicit, θ -method, method-of-line approach, and others are not applied; (ii) the time stability analysis is not discussed; and (iii) recomputation of the resulting matrix at each time level as done for other methods for solving partial differential equations (PDEs) with variable coefficients is avoided and the matrix is computed once. Two different formulations of the d -dimensional problem as a d + 1 -dimensional space-time one are discussed based on the type of PDEs considered. The localized radial basis function meshless method is applied to seek for the numerical solution. Different examples in two and three-dimensional space are solved to show the accuracy of such method. Different types of boundary conditions, Neumann and Dirichlet, are also considered for parabolic and hyperbolic equations to show the sensibility of the method in respect to boundary conditions. A comparison to the fourth-order Runge-Kutta method is also investigated.

Meshless numerical model based on radial basis function (RBF) method to simulate the Rayleigh–Taylor instability (RTI)

The Rayleigh-Taylor instability (RTI) is the instability at the interface between two fluids when a heavier fluid is placed on top of lighter fluid in a gravitational field. In the present work, the RTI was studied numerically by using a meshless radial basis function (RBF) method. The present manuscript describes the development of the meshless RBF method to solve the RTI problem in an incompressible viscous two-phase immiscible fluid. This method can address the difficulty of the classical base method which often requires much computing time for the generation of the computational mesh. Moreover, the meshless RBF method does not require connectivity information among the nodes. Consequently, the present manuscript provides a new numerical procedure in the solution of the RTI problem by the combination of meshless RBF and Cahn-Hilliard equations. In the present numerical study, the RBF method was combined with the domain decomposition method (DDM) to solve the large scale problem. The problem was governed by the Navier-Stokes and Cahn-Hilliard equations in a primitive variable formulation. The Cahn-Hilliard equations were used to capture the interface between two fluids systems. The RBF method was used for spatial discretization and the Euler implicit method was implemented for time discretization. The fractional step scheme was used to solve the pressure velocity coupling. Here, the effects of Atwood numbers as representing the density ratio on the RTI were investigated. As a result, it was found that the position of the rising bubble and falling spike during RTI conforms well to the results from the previous works.

A localized meshless approach using radial basis functions for conjugate heat transfer problems in a heat exchanger

An understanding of conjugate heat transfer is important for enhancing the thermal performance of heat transfer devices. This study discusses the numerical solutions of the conjugate natural convective heat transfer problem using a local radial basis function meshless method. The temperature difference is assumed to be small enough to ensure the validity of the Boussinesq approximation. An artificial compressibility method is used to couple pressure to the continuity equation. The meshless method requires only one set of nodes, which are distributed in the spatial domain. The semi-algebraic equations are integrated using the second-order Runge–Kutta method. Three benchmark problems of conjugate natural convection with one vertical wall, conjugate natural convection with centrally vertical partition, and natural convection with a conducting solid in the middle of the square cavity are analyzed. The numerical results are compared with three benchmark solutions and experimental results; it was observed that the experimental and predicted results showed good agreement with each other. This numerical study is to contribute to understanding the knowledge of heat and fluid flow mechanisms for conjugate heat transfer in a heat exchanger acquired through experiments.

On the Approximation of a Nonlinear Biological Population Model Using Localized Radial Basis Function Method

A localized radial basis function meshless method is applied to approximate a nonlinear biological population model with highly satisfactory results. The method approximates the derivatives at every point corresponding to their local support domain. The method is well suited for arbitrary domains. Compared to the finite element and element free Galerkin methods, no integration tool is required. Four examples are demonstrated to check the efficiency and accuracy of the method. The results are compared with an exact solution and other methods available in literature.

A 3-D Dispersive Time-Domain Meshless Formulation for Frequency-Dependent Materials

This communication presents a novel transient meshless formulation for analyzing wave propagation through linear dispersive materials. Many real materials are linear dispersive; thus, their constitutive parameters are not constant and change with frequency. However, the conventional numerical techniques are not proper tools for numerical analysis of dispersive materials, considering that they are formulated based on the assuming constant constitutive parameters for the materials. Therefore, until now, a number of numerical techniques have been proposed for modeling frequency behavior of dispersive materials. On the other hand, meshless methods are new and powerful numerical techniques which their capability in simulating the problem domain without using connection information among nodes makes them efficient techniques for modeling problems with complex geometries. In this communication, we have demonstrated that by incorporating some approaches into the meshless methods, we can turn them into suitable tools for modeling dispersive media. In order to obtain a meshless analysis of a dispersive medium, we have considered the scalar radial basis function meshless method, and for taking the frequency behavior of the medium into account, the auxiliary differential equations method has been used. Hence, the proposed dispersive meshless method not only models the frequency behavior of the dispersive materials but also provides more flexibility in simulating problems with complex geometries. In addition, the efficiency and accuracy of our proposed method are investigated by two numerical examples.

Numerical Simulation of Dam Break Flows Using a Radial Basis Function Meshless Method with Artificial Viscosity

A simple radial basis function (RBF) meshless method is used to solve the two-dimensional shallow water equations (SWEs) for simulation of dam break flows over irregular, frictional topography involving wetting and drying. At first, we construct the RBF interpolation corresponding to space derivative operators. Next, we obtain numerical schemes to solve the SWEs, by using the gradient of the interpolant to approximate the spatial derivative of the differential equation and a third-order explicit Runge–Kutta scheme to approximate the temporal derivative of the differential equation. For the problems involving shock or discontinuity solutions, we use an artificial viscosity for shock capturing. Then, we apply our scheme for several theoretical two-dimensional numerical experiments involving dam break flows over nonuniform beds and moving wet-dry fronts over irregular bed topography. Promising results are obtained.

On the optimal shape parameters of distinct versions of RBF meshless methods for the bending analysis of plates

This study intends to computationally analyze single-layer plates. Reissner–Mindlin plate deformation theory was combined with the radial basis function (RBF) meshless method. Considering the first order shear deformation theory, the RBF meshless method formulation for isotropic plates analysis is presented. According to the motion equations in terms of displacement for isotropic homogeneous laminates, non-zero laminate stiffness is determined. Transverse displacement results are evaluated for various thickness values where the plate is simply supported in all edges. In this work, it is also shown that Wendland's RBF formulation with optimal values of the shape parameter formulation using several mesh densities produces deflections of single-layer laminate plates agreeing well with analytical solutions. The obtained RBF solution proves that the supporting numerical methodology is efficient and robust.

An Optimized RBF Analysis of an Isotropic Mindlin Plate in Bending

This study focuses on the numerical analysis of an isotropic plate under a uniform distributed load. The problem is solved considering the Reissner-Mindlin plate deformation theory combined with a Radial Basis Function (RBF) meshless method. Thus, in this work, the RBF meshless method formulation for the analysis of thick isotropic plates assuming a first order shear deformation theory is shown with detail. Additionally, the essential boundary conditions, material properties and geometrical characteristics of the model are fully presented. In this work, the nonzero laminate stiffnesses are determined through equations of motion in terms of displacements for homogeneous laminates.The transverse displacement results are evaluated for various thickness values where the plate is simply supported in all edges. Additionally, the shape parameter governing the RBF formulation is optimized using several mesh densities. The obtained numerical results are compared to the exact analytical solution available in the literature. Accurate RBF solutions are obtained, supporting the efficiency of the numerical methodology.

Local RBF meshless scheme for coupled radiative and conductive heat transfer

ABSTRACTA local radial basis function meshless (LRBFM) method is developed to solve coupled radiative and conductive heat transfer problems in multidimensional participating media, in which compact support radial basis functions (RBFs) augmented on a polynomial basis are employed to construct the trial function, and the radiative transfer equation (RTE) and energy conservation equation are discretized directly at nodes by the collocation method. LRBFM belongs to a class of truly meshless methods which require no mesh or grid, and can be readily implemented in a set of uniform or irregular node distributions with no node connectivity. Performances of the LRBFM is compared to numerical results reported in the literature via a variety of coupled radiative and conductive heat transfer problems in 1D and 2D geometries. It is demonstrated that the local radial basis function meshless method provides high accuracy and great efficiency to solve coupled radiative and conductive heat transfer problems in multidimensional participating media with uniform and irregular node distribution, especially for coupled heat transfer problems in irregular geometry with Cartesian coordinates. In addition, it is extremely simple to implement.

A meshless radial basis function method for 2D steady-state heat conduction problems in anisotropic and inhomogeneous media

The paper presents a new meshless numerical method for solving 2D steady-state heat conduction problems in anisotropic and inhomogeneous media. The coefficients of the governing PDEs are spatially dependent functions including the main operator part. The boundary conditions of a most general form for the temperature and the heat flux are considered. The key idea of the method is the use of the basis functions which satisfy the homogeneous boundary conditions of the problem. Each basis function used in the algorithm is a sum of a RBF and a special correcting function which is chosen to satisfy the homogeneous BC of the problem. The conical radial basis functions, the Duchon splines and the multiquadric RBFs are used in approximation of the PDE. This allows us to seek an approximate solution in the form which satisfies the boundary conditions of the initial problem with any choice of the free parameters. As a result we separate the approximation of the boundary conditions and the approximation of the PDE inside the solution domain. The numerical experiments are carried out for accuracy and convergence investigations. The comparison of the numerical results obtained in the paper with the exact solutions and with the data obtained with the use of other numerical techniques is performed. The numerical examples demonstrate that the present method is accurate, convergent, stable, and computationally efficient in solving this kind of problems.

New implementation of local RBF meshless scheme for radiative heat transfer in participating media

The local radial basis function meshless method (LRBFM) is an effective numerical scheme for solving radiative heat transfer in participating media, in which radial basis functions (RBF) augmented with polynomial basis are employed to construct the trial functions, and radiative transfer equations are discretized directly at nodes by collocation method. The LRBFM belongs to a class of truly meshless methods which do not need any mesh, and can be implemented on a set of uniform or irregular nodes without nodes’ connectivity. The main drawback of this method is that like most of the other numerical methods, the solution to RTE by LRBFM also suffers much from nonphysical oscillations. In the present work a new implementation of local RBF meshless scheme (nLRBFM) is proposed, based on the RBF correction interpolation of outflow-boundary intensity which can easily eliminate numerical oscillations of results without any extra stabilization technique. Classical cases are studied to illustrate the numerical performance of this new implementation. From comparisons it is shown that the new implementation of local RBF meshless scheme (nLRBFM) is of high effectiveness, high accuracy and excellent stability, to solve radiative heat transfer in homogeneous participating media.

Local multiquadric RBF meshless scheme for radiative heat transfer in strongly inhomogeneous media

A local radial basis function meshless method (LRBFM) is developed to solve radiative heat transfer in participating media, in which multiquadric (MQ) radial basis functions (RBF) augmented with polynomial basis are employed to construct the trial functions, and the radiative transfer equation (RTE) is discretized directly at nodes by collocation method. The LRBFM belongs to a class of truly meshless methods which do not need any mesh, and can be implemented on a set of uniform or irregular nodes without nodes׳ connectivity. To improve numerical stability of LRBFM for the solution to radiative heat transfer in strongly inhomogeneous media, an upwind support domain scheme is introduced. The upwind scheme is implemented by moving the support domain of local radial basis function interpolation approximation to the opposite direction of each streamline, which can fully capture the information from upstream and improve the accuracy and stability of LRBFM. Performances of the LRBFM and upwind LRBFM (LRBFM_U) are compared with analytical solutions and other numerical results reported earlier in the literatures via a variety of problems in 1-D and 2-D geometries with strongly inhomogeneous media. It is demonstrated that the local radial basis function meshless method with upwind support domain scheme (LRBFM_U) provides high accuracy and great stability to solve radiative heat transfer in strongly inhomogeneous media.