Development of the Kansa method for solving seepage problems using a new algorithm for the shape parameter optimization

In this research, the Kansa or Multiquadric method (MQ) has been developed for solving the seepage problems in 2D and 3D arbitrary domains. This research is the first application of this method for seepage analysis in both confined and unconfined porous media. The domain decomposition approach has been employed for applying MQ method easily in inhomogeneous and irregular complex geometries and decreasing the computational costs. For determining the optimum shape parameter that affects strongly the accuracy of MQ and other RFB methods, a new scheme that decreases drastically the computational time is introduced. The efficiency of the proposed algorithm has been examined under various radial basis functions, variations of number of interpolating points and points distribution, through a numerical example with analytical solution. Eventually, three examples including different boundary conditions are presented. Comparing results of the examples with other numerical methods indicates that the present approach has high capability and accuracy in solving seepage problems.     

Laminar fluid flow and heat transfer in an internally corrugated tube by means of the method of fundamental solutions and radial basis functions

The paper shows application of the method of fundamental solutions in combination with the radial basis functions for analysis of fluid flow and heat transfer in an internally corrugated tube. Cross-section of such a tube is mathematically described by a cosine function and it can potentially represent a natural duct with internal corrugations, e.g. inside arteries. The boundary value problem is described by two partial differential equations (one for fluid flow problem and one for heat transfer problem) and appropriate boundary conditions. During solving this boundary value problem the average fluid velocity and average fluid temperature are calculated numerically. In the paper the Nusselt number and the product of friction factor and Reynolds number are presented for some selected geometrical parameters (the number and amplitude of corrugations). It is shown that for a given number of corrugations a minimal value of the product of friction factor and Reynolds number can be found. As it was expected the Nusselt number increases with increasing amplitude and number of corrugations.     

基于径向基函数网络的SFS算法研究

分析了现有从明暗恢复形状(SFS)的几种方法普遍存在对恢复的形状的连续性和光滑性的缺点,提出了一种基于径向基函数网络模型进行从明暗恢复形状的新算法。该算法先采用网络构造一个曲面方程,再利用反射函数作为约束条件,通过调整权因子和径向基函数中心和宽度对网络进行自学习,得到一个满意的曲面方程。理论和实验证明,该算法在恢复形状的准确性和曲面的光滑性,连续性上有较大改进。     

Particular solutions of products of Helmholtz-type equations using the Matern function

We derive closed-form particular solutions for Helmholtz-type partial differential equations. These are derived explicitly using the Matern basis functions. The derivation of such particular solutions is further extended to the cases of products of Helmholtz-type operators in two and three dimensions. The main idea of the paper is to link the derivation of the particular solutions to the known fundamental solutions of certain differential operators. The newly derived particular solutions are used, in the context of the method of particular solutions, to solve boundary value problems governed by a certain class of products of Helmholtz-type equations. The leave-one-out cross validation (LOOCV) algorithm is employed to select an appropriate shape parameter for the Matern basis functions. Three numerical examples are given to demonstrate the effectiveness of the proposed method.     

On the numerical solution of Fredholm integral equations utilizing the local radial basis function method

The current investigation describes a computational technique to solve one- and two-dimensional Fredholm integral equations of the second kind. The method estimates the solution using the discrete collocation method by combining locally supported radial basis functions (RBFs) constructed on a small set of nodes instead of all points over the analysed domain. In this work, we employ the Gauss–Legendre integration rule on the influence domains of shape functions to approximate the local integrals appearing in the method. In comparison with the globally supported RBFs for solving integral equations, the proposed method is stable and uses much less computer memory. The scheme does not require any cell structures, so it is meshless. We also obtain the error analysis of the proposed method and demonstrate that the convergence rate of the approach is high. Illustrative examples clearly show the reliability and efficiency of the new method.     

A numerical method for solution of the two-dimensional sine-Gordon equation using the radial basis functions

The nonlinear sine-Gordon equation arises in various problems in science and engineering. In this paper, we propose a numerical scheme to solve the two-dimensional damped/undamped sine-Gordon equation. The proposed scheme is based on using collocation points and approximating the solution employing the thin plate splines (TPS) radial basis function (RBF). The new scheme works in a similar fashion as finite difference methods. Numerical results are obtained for various cases involving line and ring solitons.     

Radial basis function partition of unity method for modelling water flow in porous media

Water flow in variably-saturated porous media is modelled by using the highly nonlinear parabolic Richards’ equation. The nonlinearity is due to the hydraulic conductivity and moisture content variables. The latter were estimated by using experimental models, including Gardner, Burdine, Mualem and van Genuchten models. The aim of this work is to develop a new technique based on the radial basis function partition of unity method (RBFPUM) and Gardner model in order to solve Richards’ equation in one and two dimensions. We have used Gardner model to handle the nonlinearity issue and the RBFPUM is used to approximate the solution of the linearized Richards’ equation. Our proposed algorithm is based on testing many configurations of the partitions number and selecting the optimal shape parameter for each case. Then we pick up the optimal configuration (partitions number-shape parameter) that yields the best solution in terms of error and conditioning number. By following this procedure, an optimal solution is ensured for our given problem. As numerical tests, we consider the vertical infiltration of water in soils in order to validate our proposed method.     

An approach to the selection of optimal sensor locations in distributed parameter systems

This paper presents an approach to the selection of optimal sensor locations in distributed parameter systems, which distinguishes the purposes of state estimation from the purposes of parameter estimation. In the first case, the optimality criterion is based on a measure of independence between the sensor responses, while in the second case, it is based on a measure of independence between the parameter sensitivity functions. The procedure, which is general and can be applied to models with any degree of complexity, is illustrated with the optimal placement of temperature sensors in a catalytic fixed-bed reactor. Some numerical results for the on-line estimation of temperature and concentration profiles as well as for the estimation of unknown model parameters are discussed.     

Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: A survey

Since neural networks have universal approximation capabilities, therefore it is possible to postulate them as solutions for given differential equations that define unsupervised errors. In this paper, we present a wide survey and classification of different Multilayer Perceptron (MLP) and Radial Basis Function (RBF) neural network techniques, which are used for solving differential equations of various kinds. Our main purpose is to provide a synthesis of the published research works in this area and stimulate further research interest and effort in the identified topics. Here, we describe the crux of various research articles published by numerous researchers, mostly within the last 10 years to get a better knowledge about the present scenario.     

The solitary wave solution of the two-dimensional regularized long-wave equation in fluids and plasmas

This paper investigates the solitary wave solutions of the two-dimensional regularized long-wave equation which is arising in the investigation of the Rossby waves in rotating flows and the drift waves in plasmas. The main idea behind the numerical solution is to use a combination of boundary knot method and the analog equation method. The boundary knot method is a meshless boundary-type radial basis function collocation technique. In contrast with the method of fundamental solution, the boundary knot method uses the non-singular general solution instead of the singular fundamental solution to obtain the homogeneous solution. Similar to method of fundamental solution, the radial basis function is employed to approximate the particular solution via the dual reciprocity principle. In the current paper, we applied the idea of analog equation method. According to the analog equation method, the nonlinear governing operator is replaced by an equivalent nonhomogeneous linear one with known fundamental solution and under the same boundary conditions. Furthermore, in order to show the efficiency and accuracy of the proposed method, the present work is compared with finite difference scheme. The new method is analyzed for the local truncation error and the conservation properties. The results of several numerical experiments are given for both the single and double-soliton waves.     

Evaluation of the performance of backpropagation and radial basis function neural networks in predicting the drill flank wear

This study compares the performance of backpropagation neural network (BPNN) and radial basis function network (RBFN) in predicting the flank wear of high speed steel drill bits for drilling holes on mild steel and copper work pieces. The validation of the methodology is carried out following a series of experiments performed over a wide range of cutting conditions in which the effect of various process parameters, such as drill diameter, feed-rate, spindle speed, etc. on drill wear has been considered. Subsequently, the data, divided suitably into training and testing samples, have been used to effectively train both the backpropagation and radial basis function neural networks, and the individual performance of the two networks is then analyzed. It is observed that the performance of the RBFN fails to match that of the BPNN when the network complexity and the amount of data available are the constraining factors. However, when a simpler training procedure and reduced computational times are required, then RBFN is the preferred choice.     

On the optimal shape parameter for Gaussian radial basis function finite difference approximation of the Poisson equation

We investigate the influence of the shape parameter in the meshless Gaussian radial basis function finite difference (RBF-FD) method with irregular centres on the quality of the approximation of the Dirichlet problem for the Poisson equation with smooth solution. Numerical experiments show that the optimal shape parameter strongly depends on the problem, but insignificantly on the density of the centres. Therefore, we suggest a multilevel algorithm that effectively finds a near-optimal shape parameter, which helps to significantly reduce the error. Comparison to the finite element method and to the generalised finite differences obtained in the flat limits of the Gaussian RBF is provided.     

Some procedures for function approximation based on the use of sample data and their application in heuristic methods for solving practical problems

Let f(x) be a member of a set of functions over a probability space. Samples of f(x) are 2-tuples (xⁱ,f(xⁱ) where xⁱ is a sample of the random variable X and f(xⁱ) is a sample of f(x) at x = xⁱ. Some procedures and analysis are presented for the approximation of such functions by systems of orthonormal functions. The approximations are based on the data samples. The analysis includes the case of error in the measurement of f(xⁱ). The properties of the expected square error in the approximation are examined for a number of different estimators for the coefficients in the expansion and these well-behaved and easily analyzed estimators are compared to those obtained using the method of least squares. The effectiveness of different sets of basis functions, those involved in the Karhunen-Loeve expansion and others, can be compared and an approach is suggested to adaptive basis selection in order to select that basis which is most efficient in approximating the particular function under examination. The connection between results and applications are discussed in the introduction and conclusion.     

Radial basis functions method for numerical solution of the modified equal width equation

The numerical solution of the modified equal width equation is investigated by using meshless method based on collocation with the well-known radial basis functions. Single solitary wave motion, two solitary waves interaction and three solitary waves interaction are studied. Results of the meshless methods with different radial basis functions are presented.     

求一类非线性偏微分方程精确解的简化试探函数法

利用试探函数法,将一个难于求解的非线性偏微分方程化为一个易于求解的代数方程,然后用待定系数法确定相应的常数,简洁地求得了一类非线性偏微分方程的精确解．将此方法应用到Burgers方程、KdV方程和KdV—Burgers方程,所得结果与已有结果完全吻合．本方法可望进一步推广用于求解其它非线性偏微分方程．     

Optimal design of radial basis function neural networks for fuzzy-rule extraction in high dimensional data

The design of an optimal radial basis function neural network (RBFNF) is not a straightforward procedure. In this paper we take advantage of the functional equivalence between RBFN and fuzzy inference systems to propose a novel efficient approach to RBFN design for fuzzy rule extraction. The method is based on advanced fuzzy clustering techniques. Solutions to practical problems are proposed. By combining these different solutions, a general methodology is derived. The efficiency of our method is demonstrated on challenging synthetic and real world data sets.     

Numerical method for the simultaneous determination of unknown parameters in a parabolic equation

Numerical procedures for the solution of an inverse problem of simultaneously determining unknown parameters in a linear parabolic equation are considered. The approach proposed is to approximate unknown functions using Chebyshev polynomials, which are determined consecutively from the solutions of the minimization problems based on overspecified data. Finally, the results of a numerical experiment are displayed.     

Iterative computational approach to the solution of the Hamilton-Jacobi-Bellman-Isaacs equation in nonlinear optimal control

In this paper, iterative or successive approximation methods for the Hamilton-Jacobi-Bellman-Isaacs equations (HJBIEs) arising in both deterministic and stochastic optimal control for affine nonlinear systems are developed. Convergence of the methods are established under fairly mild assumptions, and examples are solved to demonstrate the effectiveness of the methods. However, the results presented in the paper are preliminary, and do not yet imply in anyway that the solutions computed will be stabilizing. More improvements and experimentation will be required before a satisfactory algorithm is developed.     

Controller design for distributed parameter systems with time delays in the boundary feedbacks via the backstepping method

For a distributed parameter system with an input delay in the boundary, a feedback control law is presented by means of the backstepping method. The square integrability of input signal is verified based on the target system. Then, the boundedness and invertibility of the corresponding backstepping transformation are proved under the regularity of system and the admissibility of feedback operator. Thus, the resulting closed-loop system is shown to be exponentially stable. Finally, as an application, a numerical simulation of a one-dimensional Schrödinger equation with a delay input is carried out, and the simulation results demonstrate the effectiveness of the suggested control law.     

A new algorithm for importance analysis of the inputs with distribution parameter uncertainty

Importance analysis is aimed at finding the contributions by the inputs to the uncertainty in a model output. For structural systems involving inputs with distribution parameter uncertainty, the contributions by the inputs to the output uncertainty are governed by both the variability and parameter uncertainty in their probability distributions. A natural and consistent way to arrive at importance analysis results in such cases would be a three-loop nested Monte Carlo (MC) sampling strategy, in which the parameters are sampled in the outer loop and the inputs are sampled in the inner nested double-loop. However, the computational effort of this procedure is often prohibitive for engineering problem. This paper, therefore, proposes a newly efficient algorithm for importance analysis of the inputs in the presence of parameter uncertainty. By introducing a ‘surrogate sampling probability density function (SS-PDF)’ and incorporating the single-loop MC theory into the computation, the proposed algorithm can reduce the original three-loop nested MC computation into a single-loop one in terms of model evaluation, which requires substantially less computational effort. Methods for choosing proper SS-PDF are also discussed in the paper. The efficiency and robustness of the proposed algorithm have been demonstrated by results of several examples.