Multiquadric Radial Basis Function Method Research Articles

Hybrid Haar wavelet and meshfree methods for hyperbolic double interface problems: Numerical implementations and comparative performance analysis

This paper introduces a variety of approaches for solving 2D and 3D hyperbolic double interface problems. The methods are based on the Haar wavelet method, multiquadric radial basis function method, and integrated multiquadric radial basis function method. Temporal derivatives are handled using the second central difference and the Houbolt method. Various numerical approaches based on these methods are developed, and their implementations are discussed in complete detail. The paper evaluates and compares the performances of these approaches using both linear and nonlinear 2D and 3D double interface hyperbolic problems. Error analysis, conducted using the L-infinity norm, and efficiency assessments measured through CPU times contribute to a comprehensive understanding of the applicability and comparative effectiveness of the proposed methods. This study provides valuable insights for researchers and practitioners dealing with the challenges posed by interface problems in general.

An Optimal Multiquadric Variable Shape Parameter for Boundary Value Problems Using Particle Swarm Optimization

The multiquadric radial basis function method has been widely used to solve partial differential equations-based problems regarding its flexibility and meshfree characteristics. The accuracy and stability of this method are derived and based on the use of a free-shape parameter that sensibly controls the comportment of the technique. Significant improvements have already been reported and show that variable shape parameters conduct the method to handle problems with striking results compared to global-based techniques. Nevertheless, choosing a suitable set of shape parameters is still an open topic because of the complexity of the method when the number of collocation points increases. The current work proposes a variant particle swarm optimization based on local displacement with attractors to determine the multi-quadratic function's ``best'' optimal variable shape parameter in solving boundary value problems. Based on an initially random set of variable shape parameters, the proposed algorithm first performs and evaluates the errors between the expected exact solution and the approximate solution thoroughly. In the first stage, the particle swarm algorithm search for an optimal set of shape parameters that minimize the error and the conditioning number of the radial basis system matrix. In the second stage, the obtained optimal set of shape parameters is applied to solve the considered problem. In this way, when the number of collocation points increases, the first stage based on particle swarm optimization stabilizes the strategy. It proposes an ``acceptable'' set of shape parameters for the given problem. The proposed method is applied to a set of well-known boundary value problems in one and two-dimensional spaces and compared to other techniques published in the literature. The results show that the proposed method achieves more accurate solutions than recently proposed in the literature.

Static analysis of graphene reinforced composite plate

In this paper static analysis of grapheme reinforced composite plate has been analyzed. A multiquadric radial basis function and meshless method has been analyzed toughly to determine the static axial stresses and in-plane stress over the domain of Graphene-reinforced (GRC) composite plate. Third order shear deformation theory (TSDT) is employed to solve such computational problem. Extended Halpin-Tsai homogenization techniques is used to evaluate the thermo mechanical properties of the composite media. An in-house mat lab code is generated to show the results by validating with some benchmark literatures.

An energy regularization of the MQ-RBF method for solving the Cauchy problems of diffusion-convection-reaction equations

The accuracy of the Kansa type multi-quadric radial basis function (MQ-RBF) method is heavily dependent on the distribution of source points. A proper choice of source points can enhance the stability and accuracy. In this paper we propose an energy regularization technique to choose the source points and the weighting factors preceding the MQ-RBFs in the numerical solution of the Cauchy problem for the steady-state diffusion-convection-reaction equation in an arbitrary plane domain. We derive an inequality, and the energy RBF (ERBF) method can preserve the energy when the inequality is satisfied. It is a criterion to pick up the source points and weighting factors. Through numerical tests under large noises, we find that the performance of the ERBF is good.

Optimal shape parameter in the MQ-RBF by minimizing an energy gap functional

By using the multiquadric radial basis function (MQ-RBF) method for solving the mixed boundary value problem as well as the Cauchy problem of the Laplace equation in an arbitrary plane domain, an energy gap functional (EGF) is proposed to choose the shape parameters. Upon minimizing the EGF we can pick up the optimal shape parameter and hence achieve the best accuracy of numerical solution. The performance of the minimizing EGF (MEGF) is assessed by numerical tests.

Hybrid Compact-WENO Finite Difference Scheme with Radial Basis Function Based Shock Detection Method for Hyperbolic Conservation Laws

A hybrid scheme, based on the high order nonlinear characteristicwise weighted essentially nonoscillatory (WENO) conservative finite difference scheme and the spectral-like linear compact finite difference scheme, has been developed for capturing shocks and strong gradients accurately and resolving fine scale structures efficiently for hyperbolic conservation laws. The key issue in any hybrid scheme is the design of an accurate, robust, and efficient high order shock detection algorithm which is capable of determining the smoothness of the solution at any given grid point and time. An improved iterative adaptive multiquadric radial basis function (IAMQ-RBF-Fast) method [W. S. Don, B. S. Wang, and Z. Gao, J. Sci. Comput, 75 (2018), pp. 1016--1039], which employed the $O(N^2)$ recursive Levinson--Durbin method and the Sherman--Morrison--Woodbury method for solving the perturbed Toeplitz matrix system, has been successfully developed as an efficient and accurate edge detector of the piecewise smooth functions. In this study, the method, together with Tukey's boxplot method and the domain segmentation technique, is extended to serve as a novel shock detection algorithm for solving the Euler equations. The applicability and performance of the RBF edge detection method as the shock detector in the hybrid scheme in terms of accuracy, robustness, efficiency, resolution, and other implementation issues are given. Several one- and two-dimensional benchmark problems in shocked flow demonstrate that the proposed hybrid scheme can reach a speedup of the CPU times by a factor up to 2--3 compared with the pure fifth order WENO-Z scheme.

Fast Iterative Adaptive Multi-quadric Radial Basis Function Method for Edges Detection of Piecewise Functions—I: Uniform Mesh

In Jung et al. (Appl Numer Math 61:77–91, 2011), an iterative adaptive multi-quadric radial basis function (IAMQ-RBF) method has been developed for edges detection of the piecewise analytical functions. For a uniformly spaced mesh, the perturbed Toeplitz matrices, which are modified by those columns where the shape parameters are reset to zero due to the appearance of edges at the corresponding locations, are created. Its inverse must be recomputed at each iterative step, which incurs a heavy $$O(n^3)$$ computational cost. To overcome this issue of efficiency, we develop a fast direct solver (IAMQ-RBF-Fast) to reformulate the perturbed Toeplitz system into two Toeplitz systems and a small linear system via the Sherman–Morrison–Woodbury formula. The $$O(n^2)$$ Levinson–Durbin recursive algorithm that employed Yule–Walker algorithm is used to find the inverse of the Toeplitz matrix fast. Several classical benchmark examples show that the IAMQ-RBF-Fast based edges detection method can be at least three times faster than the original IAMQ-RBF based one. And it can capture an edge with fewer grid points than the multi-resolution analysis (Harten in J Comput Phys 49:357–393, 1983) and just as good as if not better than the L1PA method (Denker and Gelb in SIAM J Sci Comput 39(2):A559–A592, 2017). Preliminary results in the density solution of the 1D Mach 3 extended shock–density wave interaction problem solved by the hybrid compact-WENO finite difference scheme with the IAMQ-RBF-Fast based shocks detection method demonstrating an excellent performance in term of speed and accuracy, are also shown.

Solving Singularly Perturbed Problems Using Multi-quadric/Inverse Multi-quadric Radial Basis Function Method

This work focuses on the implementation of a multi-quadric/inverse multi-quadric (MQ/IMQ) radial basis function method on Singularly Perturbed Problems (SPP). Elliptic equation and convection–diffusion differential equation are solved using MQ/IMQ RBF methods and results are compared with analytical results. Numerical results are computed using stationary approximation and non-stationary approximation for SPP problems. Condition number of the system matrix in both cases is tuned with shape parameters. Maximum error is reduced for small shape parameter although it is depending on the perturbation parameter. Accuracy and CPU time are computed with increasing the number of distinct centres with a given shape parameter also. Better accuracy is obtained in the case of elliptic SPP as compared to convective–diffusion SPP for a very small perturbation parameter.

Meshless method for the numerical solution of the Fokker–Planck equation

Abstract In this paper numerical meshless method for solving Fokker–Planck equation is considered. This meshless method is based on multiquadric radial basis function and collocation method to approximate the solution. Here we apply θ -weighted finite difference method. The stability analysis of the method is dealt with, using a linearized stability method. Numerical examples illustrate the validity and applicability of the method.

An adaptive multiquadric radial basis function method for expensive black-box mixed-integer nonlinear constrained optimization

Many real-world optimization problems comprise objective functions that are based on the output of one or more simulation models. As these underlying processes can be time and computation intensive, the objective function is deemed expensive to evaluate. While methods to alleviate this cost in the optimization procedure have been explored previously, less attention has been given to the treatment of expensive constraints. This article presents a methodology for treating expensive simulation-based nonlinear constraints alongside an expensive simulation-based objective function using adaptive radial basis function techniques. Specifically, a multiquadric radial basis function approximation scheme is developed, together with a robust training method, to model not only the costly objective function, but also each expensive simulation-based constraint defined in the problem. The article presents the methodology developed for expensive nonlinear constrained optimization problems comprising both continuous and integer variables. Results from various test cases, both analytical and simulation-based, are presented.

Multiquadric Radial Basis Function Method for Boundary Value and Free Vibration Problems

Radial basis function method is a generalized version of the multiquadric radial basis function method (MQRBF). It is considered an important tool for interpolating multidimensional scattered data. Its ability to handle arbitrarily scattered data, to easily generalize to several space dimensions, and to provide spectral accuracy have made it popular in various applications. In this paper development of the multi quadric radial basis function (MQRBF) has been discussed. Shape parameter plays important role, when MQRBF is applied to solve boundary value problems and it significantly affects the stability and accuracy. Value of the shape parameter decreases as number of data points increases. The effect of shape parameter is explained in details. Approaches for solving boundary value problems using multiquadric radial basis function have been mentioned.

Nonlinear flexural analysis of functionally graded plates under different loadings using RBF based meshless method

The nonlinear flexural response of functionally graded plates under different transverse loading is investigated using multiquadric radial basis function (MQRBF) method. The mathematical formulation of the actual physical problem of the plate subjected to mechanical loading is presented utilizing Levinson shear deformation theory and von Karman nonlinear kinematics. These non-linear governing differential equations of equilibrium are linearized using quadratic extrapolation technique. The effect of the volume fraction exponent on the nonlinear flexural response of simply supported and clamped plates under uniformly distributed, sinusoidal, line and point loads is studied.

The Influence of Fiber Treatment on the Mechanical Behavior of Jute-Coir Reinforced Epoxy Resin Hybrid Composite Plate

A novel surface modification of coir fiber was done by oxidation followed by furfuryl alcohol treatment. The surface modified coir fiber showed better interfacial adhesion with epoxidized phenolic novolac (EPN) thermoset resin compared to unmodified coir fiber reinforced composites. Experimentally, tensile and flexural properties are evaluated at a different percentage of the coir fibers. Stiffness and damping behavior of the composite plates are evaluated theoretically, using a meshless multiquadric radial basis function method. Numerical results are compared with those obtained by other analytical methods. The aim of the study is to observe the behavior of a chemically treated and untreated coir fiber-reinforced composite plate at a different percentage of the coir fibers.

Iterative adaptive RBF methods for detection of edges in two-dimensional functions

Radial basis functions have gained popularity for many applications including numerical solution of partial differential equations, image processing, and machine learning. For these applications it is useful to have an algorithm which detects edges or sharp gradients and is based on the underlying basis functions. In our previous research, we proposed an iterative adaptive multiquadric radial basis function method for the detection of local jump discontinuities in one-dimensional problems. The iterative edge detection method is based on the observation that the absolute values of the expansion coefficients of multiquadric radial basis function approximation grow exponentially in the presence of a local jump discontinuity with fixed shape parameters but grow only linearly with vanishing shape parameters. The different growth rate allows us to accurately detect edges in the radial basis function approximation. In this work, we extend the one-dimensional iterative edge detection method to two-dimensional problems. We consider two approaches: the dimension-by-dimension technique and the global extension approach. In both cases, we use a rescaling method to avoid ill-conditioning of the interpolation matrix. The global extension approach is less efficient than the dimension-by-dimension approach, but is applicable to truly scattered two-dimensional points, whereas the dimension-by-dimension approach requires tensor product grids. Numerical examples using both approaches demonstrate that the two-dimensional iterative adaptive radial basis function method yields accurate results.

Static and dynamic mechanical analysis of chemically modified randomly distributed short banana fiber reinforced high-density Polyethylene/Poly (Ε-caprolactonc) Composites

Randomly distributed short banana fiber reinforced HDPE/PCL (high-density polyethylene / poly (∈-caprolactone)) composites have been fabricated to determine the mechanical behavior at static and dynamic loading. To enhance the mechanical properties of matrix, poly caprolactone has been blended with high-density polyethylene. Three samples of banana fibers were treated with sodium hydroxide, sebacoyl chloride, and toluene diisocyanate solutions separately. It has been observed that banana fibers treated with sodium hydroxide give better mechanical properties compared to the other two solutions. The role of fiber/matrix interactions in chemically treated banana fibers reinforced composites was investigated to predict the stiffness and damping properties and their different mechanical behavior is compared with untreated banana fibers. In order to study the static and dynamic response of HDPE, HDPE/PCL blend, treated and untreated banana fiber reinforced composite plate, a multiquadric radial basis function (MQRBF) method was developed. MQRBF is applied for spatial discretization and a Newmark implicit scheme is used for temporal discretization. The discretization of the differential equations generates a larger number of algebraic equations than the unknown coefficients. To overcome this ill conditioning, the multiple linear regression analysis, which is based on the least square error norm, is employed to obtain the coefficients. Simple supported and clamped boundary conditions are considered. Numerical results are compared with those obtained by other analytical methods.

Mechanical Behavior of Polyethylene Fibers Reinforced Resol/VAC-EHA

Polyethylene fibers reinforced Resol/VAC-EHA composites have been prepared in the laboratory at different volume fraction of fibers for the evaluation of mechanical properties. Resol solution was blended with vinyl acetate-2-ethylhexyl acrylate (VAc-EHA) resin in an aqueous medium with varying volume fraction of polyethylene fibers. It has also been observed that reinforcement of the polyethylene fiber in Resol/VAC-EHA matrix shows improvement in ductility, tensile strength and Young's modulus. But damping property decreases as soon as percentage of the polyethylene fiber increases. In order to study the static and dynamic response of polyethylene fiber reinforced Resol/VAC-EHA composite plate at uniformly distributed load and fluctuating uniformly distributed load, a multi-quadric radial basis function (MQRBF) method is developed. MQRBF is applied for spatial discretization and a Newmark implicit scheme is used for temporal discretization. The discretization of the differential equations generates a greater number of algebraic equations than the unknown coefficients. To overcome this ill-conditioning, the multiple linear regression analysis, based on the least square error norm, is employed to obtain the coefficients. Simple supported and clamped boundary conditions are considered. Numerical results are compared with those obtained by other analytical methods.

Mechanical Behavior of Unidirectional Glass Fibers Reinforced Resol/VAC‐EHA Composites at Different Volume Fraction of Fibers

Glass fibers reinforced Resol/VAC‐EHA Composites were fabricated at different volume fraction of fibers to determine the mechanical behavior. Resol solution was blended with vinyl acetate‐2‐ethylhexyl acrylate (VAc‐EHA) resin in an aqueous medium with varying volume fraction of glass fibers. The role of fiber/matrix interactions in glass fibers reinforced composites were investigated to predict the stiffness and damping properties and their different mechanical behavior was compared with pure Resol and Resol/VAC‐EHA blend. In order to study the static and dynamic response of Resol, Resol/VAC‐EHA blend, and glass fibers reinforced composites, a multiquadric radial basis function (MQRBF) method was developed. MQRBF was applied for spatial discretization and a Newmark implicit scheme was used for temporal discretization. The discretization of the differential equations generates a larger number of algebraic equations than the unknown coefficients. To overcome this ill conditioning, the multiple linear regression analysis, which is based on the least square error norm, was employed to obtain the coefficients. Simple supported and clamped boundary conditions were considered. Numerical results were compared with those obtained by other analytical methods.

An iterative adaptive multiquadric radial basis function method for the detection of local jump discontinuities

In [J.-H. Jung, Appl. Numer. Math. 57 (2007) 213–229], an adaptive multiquadric radial basis function method has been proposed for the reconstruction of discontinuous functions. Utilizing the vanishing shape parameters near the local jump discontinuity, the adaptive method considerably reduces the Gibbs oscillations and enhances convergence. In this paper, a new jump discontinuity detection method is developed based on the adaptive method. The global maximum of the expansion coefficients, λ i , exists at the strongest jump discontinuity and its magnitude increases exponentially with the number of the center points, N. The adaptive method, however, dynamically reduces its magnitude to O ( N ) once applied. The global maximum of λ i then exists at the next strongest jump discontinuity and its magnitude is exponentially large. In this way, the local jump discontinues are successively detected with the adaptive method applied iteratively. Numerical examples are provided using the piecewise analytic functions and the numerical solution of the shock interaction equations. Numerical results verify that the proposed method is efficient and accurate in finding local jump discontinuities.

Some Experimental and Theoretical Investigations on Fire Retardant Coir/Epoxy Micro-Composites

This article reports some experimental and theoretical investigations on fire retardant coir/epoxy micro-composites. The coir fiber is treated with saturated bromine water for increasing the electrical properties and then mixed with stannous chloride solution for improving the fire retardant properties. Only 5% (approximately) of fire retardant filler reduces the smoke density by 25% and the LOI value increases to 24%. The mechanical properties of the coir/epoxy micro-composites are not affected much after incorporation of filler. Flexural strength and flexural modulus of the NET increases tremendously compared to NEU and NE. Multiquadric radial basis function (MQRBF) method is applied for static and dynamic analysis of coir/epoxy micro-composite plate under uniformly distributed load. Damping behavior and natural frequencies are observed in NE, NEU, and NET. MQRBF is applied for spatial discretization and Newmark implicit scheme is used for temporal discretization. The discretization of the differential equations generates greater number of algebraic equations than the unknown coefficients. The multiple linear regression analysis, which is based on the least square error norm, is employed to obtain the coefficients. Simple supported and clamped boundary conditions are considered for verification of present results and they are compared with other existing analytical methods.

An Improved Multiquadric Collocation Method for 3-D Electromagnetic Problems

The multiquadric radial basis function method (MQ RBF or, simply, MQ) developed recently is a truly meshless collocation method with global basis functions. It was introduced for solving many 1- and 2-D partial differential equations (PDEs), including linear and nonlinear problems. However, few works are found for electromagnetic PDEs, especially for 3-D problems. This paper presents an improved MQ collocation method for 3-D electromagnetic problems. Numerical results show a considerable improvement in accuracy over the traditional MQ collocation method, although both methods are direct collocation method with exponential convergence