Radial Basis Function Approximation Research Articles

Nonlinear bending analysis of functionally graded porous beams using the multiquadric radial basis functions and a Taylor series-based continuation procedure

In this paper, the nonlinear bending analysis of Functionally Graded Porous (FGP) beams is investigated using an efficient numerical algorithm associating a meshless collocation technique uses the Multiquadric Radial Basis Function (MQRBF) approximation method and a higher-order Taylor series-based continuation procedure. Material properties of the FGP beams are described by adopting a modified power-law function taking into account the effect of porosities. Based on the First Order Shear Deformation Theory (FSDT) of beams with the von-Kármán kinematic hypothesis, the strong form of nonlinear equations is established. For an efficient application of the proposed numerical approach, a quadratic matrix strong form of the problem is presented. The resulting nonlinear equations are solved numerically with the proposed algorithm which leaned on the following three steps: a higher-order Taylor series expansion to transform the quadratic nonlinear equations into a sequence of linear ones, a meshless collocation technique based on MQRBF approximation method to solve numerically the resulting linear equations and a continuation procedure to get the whole solution branch. To demonstrate the robustness of the developed algorithm, convergence and validation studies have been carried out. Furthermore, the effects of power-law index, porosity volume fraction, Young’s modulus ratio, loads and boundary conditions are investigated.

3D high-quality temperature-field reconstruction method in furnace based on acoustic tomography

Reconstructed 3D high-quality temperature distribution in furnace can provide important information to better control boiler operations and to optimize the combustion process. Acoustic travel-time tomography is considered to be a promising method that uses the dependence of sound speed on temperature to reconstruct the temperature field. In this paper, a new 3D temperature field reconstruction method based on radial basis function approximation with polynomial reproduction (RBF-PR) and truncated generalized singular value decomposition (TGSVD) is proposed to solve the inverse problem. To improve the reconstruction quality, we consider the refraction effect of sound waves in the non-uniform temperature field and estimate the sound wave propagation path model in the reconstruction method. The shooting method with improved Powell algorithms is used to solve the propagation path model and obtain the relationship between the acoustic travel-time over the propagation path and temperature distribution. Simulations are implemented to evaluate the effectiveness of the proposed reconstruction method. The results indicate that compared with other methods, the new reconstruction method that considers the refraction effect can reconstruct temperature distribution with higher accuracy and better anti-noise ability. Experimental results also demonstrate that the temperature distribution can be reconstructed effectively by the proposed method.

Ghost point method using RBFs and polynomial basis functions

In this paper, we propose a new Kansa method with fictitious centre approach in which the radial basis function (RBF) approximation is augmented by polynomial basis functions. The proposed approach considerably improves not only the accuracy but also the stability of the recently proposed ghost point method using radial basis functions. The difficulty of selecting a good RBF shape parameter is no longer an issue. Two numerical examples are presented to show the effectiveness and the improvement of the proposed method over previous methods.

Local RBF-based penalized least-squares approximation on the sphere with noisy scattered data

In this paper we derive local L2 error estimates for penalized least-squares approximation on the d-dimensional unit sphere Sd⊆Rd+1, given noisy, scattered, local data representing an underlying function from a Sobolev space of order s>d∕2 defined on a non-empty connected open set Ω⊆Sd with Lipschitz-continuous boundary. The quadratic regularization functional has two terms, one measuring the squared pointwise ℓ2-discrepancy from the local data, the other containing the squared native space norm of a radial basis function (RBF), multiplied by a regularization parameter. The RBF is chosen so that its native space is equivalent to the (global) Sobolev space of order s on Sd. While both the data and the approximated function are local, we minimize the quadratic functional over all functions in the native space of the RBF, and obtain as exact minimizer a (global) radial basis function approximation. By choosing the RBF to be a Wendland function the resulting linear system has a sparse matrix which is easily computed. We consider three different strategies for choosing the smoothing parameter, namely Morozov’s discrepancy principle and two a priori strategies, and derive L2(Ω) error estimates for each strategy. As auxiliary tools for proving the local L2 error estimates we develop both a local L2 sampling inequality and a suitable Sobolev extension theorem. The paper concludes with numerical experiments.

Computational study on constant and sinusoidal heating of skin tissue using radial basis functions

This paper concerns the study of heat distribution in skin tissue with constant and sinusoidal heat flux at the skin surface. Three types of heat transfer models, i.e., Pennes bioheat model, single-phase lag model, and dual-phase lag model based on Fourier and non-Fourier heat conduction, are taken into consideration. Also, Fourier, as well as non-Fourier boundary conditions, are considered at both the end of skin tissue for the single and dual-phase lag model of heat conduction. The resulting models are solved using finite difference and radial basis function (RBF) approximations for the temporal and spatial variables, respectively. The effects of these three models and other parameters involved in models on heat transfer in skin tissue are studied. The effects of Fourier and non-Fourier boundary conditions on heat transfer for single and dual-phase lag models are also studied.

Multiquadric radial basis function approximation method and asymptotic numerical method for nonlinear and linear static analysis of single-walled carbon nanotubes

In this paper, a numerical algorithm combining the meshless collocation technique based on the globally supported Multiquadric Radial Basis Function (MQRBF) approximation method and the Asymptotic Numerical Method (ANM) is developed to investigate the nonlinear and linear static behaviors of Single-Walled Carbone Nanotubes (SWCNTs) based on a Nonlocal Continuum Beam Model (NCBM). The NCBM which accounts for the effects of small scale, shear deformation and geometric nonlinearity of von-Kármán type is constructed by implementing the Nonlocal Elasticity Theory of Eringen (NET) into the First-Order Shear Deformation Elastic Beam Theory (FOSDEBT) and used to model the SWCNTs as a Continuous Elastic Nanobeam (CEN). Based on the Total Lagrangian Formulation (TLF) the nonlocal nonlinear governing equations of SWCNT are elaborated in a quadratic matrix strong form. These equations are then solved numerically by applying the proposed algorithm. This algorithm permits to transform the quadratic nonlocal nonlinear governing equations into a sequence of linear ones shared the same tangent stiffness matrix which are then solved numerically by the MQRBF approximation method, to get the whole branch of solution a continuation procedure is required. In order to highlight the reliability and efficiency of the present numerical algorithm, comparative studies are conducted.

Intelligent Approach to Robust Design Optimization of a Rotor System due to Its Support Stiffness Uncertainty

In this paper, an intelligent robust design approach combined with different techniques such as polynomial chaos expansion (PCE), radial basis function (RBF) neural network, and evolutionary algorithms is presented with a focus on the optimization of the dynamic response of a rotor system considering support stiffness uncertainty. In the proposed method, the PCE method instead of the traditional Monte Carlo uncertainty analysis is applied to analyze the uncertain propagation of system performance. The RBF network is introduced to establish the approximate models of the objective and constraint functions. Taking the low-pressure rotor of a gas turbine with support stiffness uncertainty as an example, the optimization model is established with the mean and variance of unbalanced response of the rotor system at different operating speeds as the objective function, and the maximum unbalance response is less than the upper limit as the constraint function. The polynomial chaos expansion is generated to facilitate a rapid analysis of robustness in the presence of support stiffness uncertainties that is defined in terms of tolerance with good accuracy. The optimal Hypercubus are used as experimental plans for building RBF approximation models of the objective and constraint functions. Finally, the robust solutions are obtained with the multiobject optimization algorithm NSGA-II. Monte Caro simulation analysis demonstrates that the qualified rate of maximum vibration responses of the low-pressure rotor system can be increased from 83.6% to over 99%. This approach to robust design optimization is shown to lead to designs that significantly decrease vibration responses of the rotor system and improved system performance with reduced sensitivity to support stiffness uncertainty.

Local-adaptive and outlier-tolerant image alignment using RBF approximation

Image alignment is a crucial step to generate a high quality panorama. The state-of-the-art approaches use local-adaptive transformations to deal with multi-view parallax, but still suffer from unreliable feature correspondences and high computational cost. In this paper, we propose a local-adaptive and outlier-tolerant image alignment method using RBF (radial basis function) approximation. To eliminate the visible artifacts, the input images are warped according to a constructed projection error function, whose parameters are estimated by solving a linear system. The outliers are efficiently removed by screening out the abnormal weights of RBFs, such that better alignment quality can be achieved compared to the existing approaches. Moreover, a weight assignment strategy is introduced to further address the overfitting issues caused by extrapolation, and hence the global projectivity can be well preserved. The proposed method is computationally efficient, whose performance is verified by comparative experiments on several challenging cases.

A numerical method based on boundary integral equations and radial basis functions for plane anisotropic thermoelastostatic equations with general variable coefficients

A boundary integral method with radial basis function approximation is proposed for numerically solving an important class of boundary value problems governed by a system of thermoelastostatic equations with variable coefficients. The equations describe the thermoelastic behaviors of nonhomogeneous anisotropic materials with properties that vary smoothly from point to point in space. No restriction is imposed on the spatial variations of the thermoelastic coefficients as long as all the requirements of the laws of physics are satisfied. To check the validity and accuracy of the proposed numerical method, some specific test problems with known solutions are solved.

Recovering a moving boundary from Cauchy data in an inverse problem which arises in modeling brain tumor treatment: the (quasi)linearization idea combined with radial basis functions (RBFs) approximation

In this paper, the dynamical modeling and behavior analysis of the inverse boundary Stefan problem which promising the understanding of modeling brain tumor treatment, are studied. To umerical simulate these models and to overcome their difficulties such as, non-linearity, free boundary property and having a non-rectangular domain, we propose the use of a strongly meshless technique based on radial basis functions in conjunction with a (quasi)linearization algorithm. Numerical examples are given to show the good accuracy and stability of the presented method.

Two-Grid Iterative Method for a Class of Fredholm Functional Integral Equations based on the Radial Basis Function Interpolation

In this paper, we discuss a two-grid iterative method for solving a class of Fredholm functional integral equations based on the radial basis function interpolation. Firstly, the existence and uniqueness of the solution are proved by Banach fixed point theorem. Secondly, the algorithm and convergence analysis of the radial basis function approximation method is given on the coarse grid. Thirdly, the fine grid iterative solution and convergence results are obtained. Finally, the validity and reliability of the theoretical analysis are verified by two numerical experiments.

Numerical Solution of MHD Incompressible Convection Flow in Channels

The purpose of this paper is to study numerically the influence of the magnetic field, buoyancy force and viscous dissipation on the convective flow and temperature of the fluid in a square cavity, lid-driven cavity, and lid-driven cavity with an obstacle at the center. The continuity, momentum and energy equations are coupled including buoyancy and magnetic forces, and energy equation contains Joule heating and viscous dissipation. The equations are solved in terms of stream function, vorticity and temperature by using polynomial radial basis function (RBF) approximation for the inhomogeneity and particular solution. The numerical solutions are obtained for several values of Grashof number (Gr), Hartmann number (M) for fixed Prandtl number Pr = 0:71 and fixed Reynolds number Re = 100 with or without viscous dissipation. It is observed that in the absence of obstacle, viscous dissipation changes the symmetry of the isotherms, and the dominance of buoyancy force increases with an increase in Gr, whereas decreases when the intensity of magnetic field increases. The obstacle in the lid-driven cavity causes a secondary flow on its left part. The effect of moving lid is weakened on the flow and isotherms especially for large Gr when the cavity contains obstacle.

Numerical and experimental study on temperature field reconstruction based on acoustic tomography

Obtaining a high-quality measurement of the temperature distribution is crucial to optimally adjust the combustion in a boiler. Acoustic tomography (AT) can be used to measure the temperature distribution from multipath acoustic time of flight (TOF). The reconstructed model and TOF measurement are crucial for practical applications of AT measurement. In this paper, a new temperature field reconstruction model based on radial basis function approximation with polynomial reproduction is presented for solving the AT inverse problem. In our reconstructed model, we also consider the refraction effect of sound wave paths in the non-uniform temperature field. A truncated singular value decomposition reconstruction method is used to improve the reconstruction quality and the anti-noise ability. In addition, a generalized cross-correlation with second correlation is applied to estimate the TOF to effectively suppress interference from noise. Numerical simulations and experimental study are implemented to evaluate the effectiveness of the proposed reconstruction method. The numerical results indicate that our model considering the refraction effect can reconstruct temperature distributions with higher accuracy and better anti-noise ability compared with other existing methods. The experimental results are compared against thermocouple measurements, which show that when considering the refraction effect, the proposed method has superior reconstruction performance than methods without the refraction effect.

Preconditioning RBF collocation method by adapted finite difference matrix

In this paper, we shall consider an adapted finite difference preconditioning technique for radial basis function (RBF) method. This technique, for the system of equations that arises from solving Helmholtz equation with Dirichlet boundary conditions by RBF collocation method, is considered. We prove that the eigenvalues of preconditioned matrices are bounded for one and two-dimensional cases. We also show that RBF interpolating matrix can be well-conditioned by using appropriate number of polynomial bases in augmented term of RBF approximation. Numerical experiments show the efficiency and robustness of the preconditioning procedure, particularly when there are non-zero boundary conditions.

Efficient radial basis functions approaches for solving a class of fractional optimal control problems

The method of radial basis functions (RBFs) is a method of scattered data interpolation that has many applications in different fields. Based on the RBFs approach, two simple and accurate methods are presented in this paper to favor the solution of fractional optimal control problems (FOCPs) which are nominated as indirect and direct methods. In the first one, the considered FOCP is converted into a system of fractional differential equations (FDEs) that is called two-point boundary value problem (TPBVP) of fractional order. Then, both the context of minimization the total error and a joint application of Volterra integral equation will be used to rewrite this problems as as an unconstrained optimization problem that can be solved by using RBFs approximation. In direct one, we rewrite the FOCP as a classical static optimization problem that can be easily solved using known formulas for computing fractional derivatives of RBFs. Numerical example are given to prove the accuracy, effectiveness and feasibility of these methods.

A boundary element and radial basis function approximation method for a second order elliptic partial differential equation with general variable coefficients

A numerical method based on boundary integral equation and radial basis function approximation is presented for solving boundary value problems governed by a second‐order elliptic partial differential equation with variable coefficients. The equation arises in the analysis of steady‐state anisotropic heat or mass diffusion in nonhomogeneous media with properties that vary according to general smoothly varying functions of space. The method requires only the boundary of the solution domain to be discretized into elements. To check the validity and accuracy of the numerical solution, some specific problems with known solutions are solved.

A RBFWENO finite difference scheme for Hamilton–Jacobi equations

The aim of this paper is to study the numerical application of radial basis functions (RBFs) approximation in the reconstruction process of well known ENO/WENO schemes. The resulted schemes are employed for approximating the viscosity solution of Hamilton–Jacobi (H–J) equations. The accuracy in the smooth area is enhanced by locally optimizing the shape parameter according to the results. It is revealed that the proposed schemes in this research prepare more accurate reconstructions and sharper solution near singularities by comparing the RBFENO/RBFWENO schemes and the classical ENO/WENO schemes for some benchmark examples. Looking at the several numerical examples in 1D, 2D and 3D illustrate that the proposed schemes in this paper perform better than the traditional ENO/WENO schemes for solving H–J equations.

A control volume scheme using compact integrated radial basis function stencils for solving the Richards equation

A new control volume approach is developed based on compact integrated radial basis function (CIRBF) stencils for solution of the highly nonlinear Richards equation describing transient water flow in variably saturated soils. Unlike the conventional control volume method, which is regarded as second-order accurate, the proposed approach has high-order accuracy owing to the use of a compact integrated radial basis function approximation that enables improved flux predictions. The method is used to solve the Richards equation for transient flow in 1D homogeneous and heterogeneous soil profiles. Numerical results for different boundary conditions, initial conditions and soil types are shown to be in good agreement with Warrick’s semi-analytical solution and simulations using the HYDRUS-1D software package. Results obtained with the proposed method were far less dependent upon the grid spacing than the HYDRUS-1D finite element solutions.

Numerical solution of non-Newtonian fluid flow and heat transfer problems in ducts with sharp corners by the modified method of fundamental solutions and radial basis function collocation

In this paper, non-Newtonian fluid flow and heat transfer in an internally finned duct is considered. The problems are governed by nonlinear equations with linear boundary conditions. In this paper, the solution is obtained with two Picard iteration processes, separately for the fluid flow and heat transfer problems. Furthermore, in both problems the solutions consist of the general and particular solutions. The method of fundamental solutions is employed in order to obtain the general solution while radial basis function approximations are used for obtaining the general solution by two possible methods. The results obtained using both approaches give similar results.

3D Temperature Distribution Reconstruction in Furnace Based on Acoustic Tomography

3D temperature distribution measurement in a furnace based on acoustic tomography (AT) calculates temperature field through multipath acoustic time-of-flight (TOF) data. In this paper, a new 3D temperature field reconstruction model based on radial basis function approximation with polynomial reproduction (RBF-PR) is proposed for solving the AT inverse problem. In addition, the modified reconstruction method that integrates the advantages of the TSVD and Tikhonov regularization methods is presented to reduce the sensitivity of noise on perturbations with the ill-posed problems and improve the reconstruction quality (RQ). Numerical simulations are implemented to evaluate the effectiveness of the proposed reconstruction method using different 3D temperature distribution models, which include the one-peak symmetry distribution, one-peak asymmetry distribution, and two-peak symmetry distribution. To study the antinoise ability of our method, noises are added to the value of TOF. 3D display of reconstructed temperature fields and reconstruction errors is given. The results indicate that our model can reconstruct the temperature distribution with higher accuracy and better antinoise ability compared with the truncated generalized singular value decomposition (TGSVD). Besides that, the proposed method can determine the hot spot position with higher precision, and the temperature error of the hot spot is lower than the other compared methods.