Multiquadric Radial Basis Function Research Articles

Numerical analysis of fully non-linear sloshing waves in an arbitrary shape tank by meshless method

This paper is devoted to develop a truly meshless numerical procedure for the fully non-linear analysis of sloshing phenomenon in an arbitrary-shape tank. For this purpose, the potential theory is considered as a liquid sloshing model, and the Lagrangian form of kinematic and dynamic boundary conditions in a moving coordinate system fixed to the tank is used to accurately capture the free surface. The multi-quadric radial basis function (MQ-RBF) augmented with the polynomial terms is employed to determine RBF-FD weights on neighboring nodes surrounding the center point for the spatial derivatives. These weight coefficients are used to solve both boundary value problems for velocity and acceleration potential. Compared with mash-based methods, the present method has the advantage of being easy to construct and flexible in dealing with the moving boundary problems. The free surface elevations, hydrodynamic pressure, and first-order natural sloshing frequencies have been reported for various geometries tanks under horizontal excitations at resonance and non-resonance cases. Fairly satisfactory agreement is observed in the numerical results and exiting results, especially for free surface elevation and first-order natural sloshing frequency.

Nonlinear free vibration and flexural analysis of hyperelastic beam utilizing a meshless method based on radial basis function

This paper investigates nonlinear free and flexural analysis of hyperelastic beams. The constitutive relations of the hyperelastic beam were derived using the neo-Hookean strain energy function and Timoshenko beam theory. Also, the nonlinear governing equations and nonlinear natural boundary conditions were derived using Hamilton’s principle. The meshless collocation method based on the multiquadric radial basis function (MQ-RBF) was utilized to discretize the nonlinear governing equations. Also, the arc-length algorithm was used to solve the nonlinear system of equations. To validate the results of the meshless method, different boundary conditions (clamped–clamped, simply supported–simply supported, clamped–simply supported, and clamped-free) were examined, and the results obtained from the meshless method were compared with those of the finite element method in Abaqus finite element software. The results show that the maximum and minimum differences between meshless and finite element methods occur in clamped and free-boundary conditions, respectively. The results also show that the meshless method based on the MQ-RBF has good accuracy compared to the finite element method for bending and free vibration analysis of hyperelastic beams.

Development of Three-Dimensional LES Based Meshless Model of Continuous Casting of Steel

A large-eddy simulation (LES) based meshless model is developed for the three-dimensional (3D) problem of continuous casting (CC) of steel billet. The local collocation meshless method based on radial basis functions (RBF) is applied in 3D. The method applies scaled multiquadric (MQ) RBF with a shape parameter on seven nodded local sub-domains. The incompressible turbulent fluid flow is described using mass, energy, and momentum conservation equations and the LES turbulence model. The solidification system is solved with the mixture continuum model. The Boussinesq approximation for buoyancy and the Darcy approximation for porous media are used. Chorin’s fractional step method is used to couple velocity and pressure. The microscopic model is closed with the lever rule model. The LES model is compared to the two-equation Low Re k−ε turbulence Reynolds Averaged Navier–Stokes (RANS) model in terms of temperature, velocity and computational times. The LES model resolves transient character of vortices which RANS-type turbulence models are unable to tackle. The computational cost of LES models is considerably higher than in RANS. On the other hand, it results in a much lower computational cost than the direct numerical simulation (DNS). The paper demonstrates the ability of the method to solve realistic industrial 3D examples. Trivial adjustment of nodal densities, high accuracy, and low numerical diffusivity are the main advantages of this meshless method.

Inverse Multiquadric Function to Price Financial Options under the Fractional Black–Scholes Model

The inverse multiquadric radial basis function (RBF), which is one of the most important functions in the theory of RBFs, is employed on an adaptive mesh of points for pricing a fractional Black–Scholes partial differential equation (PDE) based on the modified RL derivative. To solve this problem, discretization along space is carried out on a non-uniform grid in order to focus on the hot area, at which the initial condition of the pricing model, i.e., the payoff, has discontinuity. The L1 scheme having the convergence order 2−α is used along the time fractional variable. Then, our proposed numerical method is built by matrices of differentiations to be as efficient as possible. Computational pieces of evidence are brought forward to uphold the theoretical discussions and show how the presented method is efficient in contrast to the exiting solvers.

희소벡터 복원 기법 구축을 위한 방사형 기저 함수망 설계 및 방사형 기저 함수의 비교 연구

Three radial basis functions (RBFs): Gaussian RBF, multiquadric RBF, and inverse multiquadric RBF, are compared for the radial basis function network (RBFN) as sparse reconstruction of a non-intrusive reduced order model (ROM) based on proper orthogonal decomposition (POD). Steady-state temperature distributions of an industrial-scale gas boiler by computational fluid dynamic (CFD) simulations are applied as training dataset for the ROM. The optimal number of training samples and truncated eigenmodes are selected by adaptive sampling and projection error, respectively. Sparse data of 16 sensors located in the inner wall of the boiler was applied as an input for the RBFNs of this study for the consideration of realistic application. Parameter study was performed for the shape factor on each RBF and reconstruction error are analyzed as the performance of each RBFN. The RBFN with Gaussian RBF showed the best predictability upon the optimal shape factor. Gaussian RBF is recommended for RBFN as sparse reconstruction under the premise of prior search for optimal shape factor.

Numerical Investigation of the Fredholm Integral Equations with Oscillatory Kernels Based on Compactly Supported Radial Basis Functions

The integral equations with oscillatory kernels are of great concern in applied sciences and computational engineering, particularly for large-scale data points and high frequencies. Therefore, the interest of this work is to develop an accurate, efficient, and stable algorithm for the computation of the Fredholm integral equations (FIEs) with the oscillatory kernel. The oscillatory part of the FIEs is evaluated by the Levin quadrature coupled with a compactly supported radial basis function (CS-RBF). The algorithm exhibits sparse and well-conditioned matrix even for large-scale data points, as compared to its counterpart, multi-quadric radial basis function (MQ-RBF) coupled with the Levin quadrature. Usually, the RBFs behave with spherical symmetry about the centers, known as radial. The comparison of convergence and stability analysis of both types of RBFs are performed and numerically verified. The proposed algorithm is tested with benchmark problems and compared with the counterpart methods in the literature. It is concluded that the algorithm in this work is accurate, robust, and stable than the existing methods in the literature based on MQ-RBF and the Chebyshev interpolation matrix.

Radial basis collocation method for free vibration analysis of elastically supported porous bi-directional FGM plate under various types of porosity distribution

ABSTRACT In this paper, the radial basis collocation method (RBCM) for the free vibration response of elastically supported porous bi-directional functionally graded (BDPFG) plate is investigated in a framework of higher-order shear deformation theory (HSDT). The material properties of BDPFG plates are assumed to be varying continuously in the length and thickness direction. The material properties are evaluated based on Voigt’s micro-mechanical model considering power law distribution method with arbitrary power index. Governing differential equations (GDEs) of the porous BFGM plate area achieved utilising Hamilton’s principle and multiquadric radial basis function (MQRBF) is used for discretizing the GDEs. To incorporate porosity in the BDPFG plate, five types of porosity distribution models are carried out and analysed. The present results are validated with the available published results in the literature to certify the efficacy and accuracy. The influence of grading index in the length and thickness direction, porosity distribution, porosity index, span to thickness ratio, and elastic foundation over natural frequency of bi-directional BDPFG plate is investigated which is the main contribution.

Initial Buckling and Free Vibration Analysis of Elastically Supported Laminated Plates Using the Meshless Technique

In this paper, a five-variable displacement field is developed within the framework of the higher-order shear deformation theory (HSDT) for the examination of the free vibrations and the buckling loads of simply supported laminated plate resting on an elastic foundation. The theory is applicable to frame-up with thick and thin plates. Hamilton’s principle is employed to obtain the governing differential equations (GDEs), which are discretized via multiquadric radial basis function (MQRBF). The present study highlights the influence of the elastic foundation and different types of transverse loading, as well as the effects of the span to thickness, aspect ratio, orthotropy ratio, and number of layers, on the results. The paper targets inclining reliable benchmarks for the most recent advancements of the topic to be utilized as references in future validation tests.

Laminar natural convection in a 2-D pentagonal house with sloping roof heated by constant heat flux

Natural convection in a pentagonal house with a protruding cooler for summer is numerically investigated. The sloping roof is subjected to constant heat flux, and the cooler maintains at a uniform cold temperature. The remaining walls of the house are insulated. Local multi-quadratic radial basis function (LMQRBF) method is applied in solution of Navier–Stokes equations in stream function-vorticity form. The working fluid in the house is air with Pr = 0.71. The effects of Rayleigh number (103 ≤ Ra ≤ 106), heat flux of the roof (0.1 ≤ q ≤ 2.0), and the cooler’s vertical position (0.65 ≤ H/L ≤ 0.85) are explored. Streamlines, isotherms, velocity profiles, and the distribution of Nusselt number are displayed. Numerical results indicate that for the top and right wall of the cooler, the heat transfer is stronger than that for the bottom wall. It reveals that all the governing parameters efficiently impact the generation and evolution of the secondary and tertiary vortex in the gable roof. It is also observed that the variation of average Nusselt number on the right wall with Rayleigh number is under the combined influence of the heat flux and the cooler’s vertical position.

A novel adaptive mesh free approach for even- parity neutron transport equation

In this paper, a novel spatially adaptive Mesh free approach using local nodes and radial basis functions, based on weak-form formulation is presented to solve the even-parity neutron transport equation. A residual based error indicator is proposed and used in an adaptive scheme for error estimation. The local balance of neutrons is used as an indicator to determine the appropriate regions of problem for node refinement. In this regard our previous computer program, MFreent, is upgraded for solving neutron transport equation using K+ variational principle in an adaptive scheme for two-dimensional X-Y geometry. The multi-quadrics radial basis function is used to construct the radial point interpolation shape functions for spatial approximation of even-parity angular flux. Also, the angular part of flux is implemented via spherical harmonics expansion. Some numerical test cases are investigated to evaluate the performance of the locally adaptive node-refinement approach for both fixed source and eigenvalue problems solution. The obtained results confirm the solution accuracy and efficiency of the proposed method.

Coupled Kansa and hybrid optimization methodological approach for Kolmogorov–Feller equations

Methodological approaches coupling a meshfree method with multiquadric radial basis function and a hybrid optimization method are elaborated to solve nonlinear stochastic differential equations with a Poisson white noise. The associated probability density function is governed by the Kolmogorov Feller equation named also Generalized Fokker–Planck equation. Actually, the shape parameter of RBFs affects the numerical solution strongly. Choosing arbitrary this parameter may lead to inaccurate solutions and thus parameter should be chosen. A hybrid optimization method is elaborated to predict an efficient shape parameter based on a well adapted residual error. The resulting coupled procedure is applied to the considered stationary and instationary generalized Fokker-Planck equations using an implicit time scheme with a regularized initial condition. The multiquadric radial basis function with optimized parameter is also coupled with the Monte Carlo method. The effectiveness and accuracy of the elaborated methodological approaches are demonstrated and the probability density functions are computed for several nonlinear stochastic differential equations under a Poisson white noise.

Meshless procedure for highly oscillatory kernel based one-dimensional Volterra integral equations

A meshless procedure with a multi-quadric radial basis function is proposed for the numerical solution of oscillatory Volterra integral equations. Integral operator of the Volterra integral equation is transformed into a differential operator by Liven’s formulation. Consequently, the numerical solution of the differential equation is obtained by a meshless procedure on global and local support domains. Theoretical error bound is derived in the inverse power of the frequency parameter. The proposed technique is also applied to the perturbed form of the Volterra integral equation of the second kind. Perturbation technique is used to overcome challenges in the numerical solution of the Volterra integral equations. Accuracy and efficiency of the proposed procedure are validated through some numerical benchmark problems.

RBF approximation of three dimensional PDEs using tensor Krylov subspace methods

In this paper, we propose different algorithms for the solution of a tensor linear discrete ill-posed problem arising in the application of the meshless method for solving PDEs in three-dimensional space using multiquadric radial basis functions. It is well known that the truncated singular value decomposition (TSVD) is the most common effective solver for ill-conditioned systems, but unfortunately the operation count for solving a linear system with the TSVD is computationally expensive for large-scale matrices. In the present work, we propose algorithms based on the use of the well known Einstein product for two tensors to define the tensor global Arnoldi and the tensor Gloub Kahan bidiagonalization algorithms. Using the so-called Tikhonov regularization technique, we will be able to provide computable approximate regularized solutions in a few iterations. The tensor formulation will allow us to develop an RBF approximation for high dimensional PDEs based on Hierarchical tensors and Adaptive Cross Approximation, which in turn reduces significantly the storage and computational costs, while the accuracy of the method is preserved. The performance of the proposed methods is illustrated with a variety of benchmark examples and large-scale industrial applications with high degrees of freedom.

Geometrically nonlinear analysis for flexure response of FGM plate under patch load

Present work explores quadratic extrapolation (QE) technique for large deformation flexure study of FGM plate with five variables higher-order shear deformation theory (HSDT). Rule of mixtures is utilized in achieving effective material properties across the thickness. Governing differential equations (GDEs) and boundary conditions have been acquired using the principle of virtual work. Quadratic extrapolation (QE) and Newton-Raphson (NR) linearization techniques are considered and compared with respect to computational time. Von-Karman type nonlinearity is used in the present work. Multiquadric radial basis function (RBF) based meshfree technique discretizes GDEs in displacement form. FGM plates with different boundary conditions are investigated under different patch loadings. Present results are validated with some previously published results after a convergence study. Some new results are produced for the flexural response of the FGM plate under patch loads. Effect of types of loads, grading index, patch load location, and boundary conditions are presented.

A meshless two-stage scheme for the fifth-order dispersive models in the science of waves on water

In this study, we investigate the numerical solution of the higher odd-order PDEs, called the Kawahara and Kawahara-KdV equations, which are main models to study the dynamics of water waves. We combine the two-stage fourth order exponential Rosenbrock integrator with a meshfree scheme based on multiquadric-radial basis function (MQ-RBF) in time and space discretization, respectively. We demonstrate the scheme based on this combination for two test examples of the Kawahara-type dispersive equations with their solitary wave solutions. The obtained results are compared with the previous findings and are figured out graphically. It is shown that the proposed algorithm provides compatible and higher accurate results in order to efficiently illustrate the behavior of solitary waves.

Numerical solution of fractional Dullin–Gottwald–Holm equation for solitary shallow water waves

AbstractIn this paper, a numerical scheme to solve the time‐fractional Dullin–Gottwald–Holm equation has been proposed. The proposed scheme is based on approximating the solution using multiquadric radial basis function. The fractional derivatives are described in the Caputo sense. The obtained numerical results are compared with exact solution to confirm the accuracy and effectiveness of the presented scheme.

Optimal trajectory generation method to find a smooth robot joint trajectory based on multiquadric radial basis functions

Optimal trajectory generation method to find a smooth robot joint trajectory based on multiquadric radial basis functions

Meshfree Galerkin Method for a Rotating Non-Uniform Rayleigh Beam with Refinement of Radial Basis Functions

The rotating Rayleigh beam problem is solved with meshfree method where the radial basis functions are explored. Numbers of basis functions have been used for meshfree methods which also include radial basis function. In this paper, the Gaussian radial basis function and multiquadrics radial basis functions are combined to get the new basis function which provides accuracy for higher natural frequencies. The radial basis functions satisfy the Kronecker delta property and it is easy to apply the essential boundary conditions. The Galerkin method is used for weak formulation. The matrices have been derived. The results are obtained for Gaussian radial basis function and new basis function. The results show more accurate values for fourth and fifth natural frequency with new basis function where only six nodes are used within the subdomain of trial and test function. The results are also obtained with conventional finite element method where forty, two node elements are considered. Also, the results are obtained for rotating Euler-Bernoulli beam to observe the difference in results with rotating Rayleigh beam.

Solving calculus of variation problems via multiquadric radial basis function method with optimal shape parameter

Abstract In this paper we present callocation method for solving the problems in the calculus of variation (CV) using multiquadratic radial basis functions (MQRBFs). In this method we use the Gauss quadrature rule for approximating the integral in CV problems. The effects of the shape parameter of MQRBFs on the convergence of the method have been discussed. Illustrative examples are included to evaluate the capability of the proposed method.

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>