In this work we utilize the boundary integral equation and the Dual Reciprocity Boundary Element Method (DRBEM) for the solution of the steady state convection-diffusion-reaction equations with variable convective coefficients in two-dimension. The DRBEM is a numerical method to transform the domain integrals into the boundary only integrals by using the fundamental solution of Helmholtz equation. Some examples are calculated to confirm the accuracy of the approach. The results obtained by the analytic solutions are in good agreement with ones provided by the DRBEM technique.

Aiming at complex configuration and large deformation, an efficient hybrid method for dynamic mesh generation is presented in this paper, which is based on Radial Basis Functions (RBFs) and Delaunay graph mapping. Based on the computational mesh, a set of very coarse grid named as background grid is generated firstly, and then the computational mesh can be located at the background grid by Delaunay graph mapping technique. After that, the RBFs method is applied to deform the background grid by choosing partial mesh points on the boundary as the control points. Finally, Delaunay graph mapping method is used to relocate the computational mesh by employing area or volume weight coefficients. By applying different dynamic mesh methods to a moving NACA0012 airfoil, it can be found that the RBFs-Delaunay graph mapping hybrid method is as accurate as RBFs and is as efficient as Delaunay graph mapping technique. Numerical results show that the dynamic meshes for all test cases including one two-dimensional (2D) and two three-dimensional (3D) problems with different complexities, can be generated in an accurate and efficient manner by using the present hybrid method.

The method of fundamentalsolutions (MFS)is known as aneffective boundary meshless method. However, the formulation of the MFS results in a dense and extremely ill-conditioned matrix. In this paper we investigate the MFS for solving large-scale problems for the nonhomogeneous modified Helmholtz equation. The key idea is to exploit the exponential decay of the fundamental solution of the modified Helmholtz equation, and consider a sparse or diagonal matrix instead of the original dense matrix. Hence, the homogeneous solution can be obtained efficiently and accurately. A standard two-step solution process which consists of evaluating the particular solution and the homogeneous solution is applied. Polyharmonic spline radial basis functions are employed to evaluate the particular solution. Five numerical examples in irregular domains and a large number of boundary collocation points are presented to show the simplicity and effectiveness of our approach for solving large-scale problems.

Aiming at a complex multi-block structured grid, an efficient dynamic mesh generation method is presented in this paper, which is based on radial basis functions (RBFs) and transfinite interpolation (TFI). When the object is moving, the multi-block structured grid would be changed. The fast mesh deformation is critical for numerical simulation. In this work, the dynamic mesh deformation is completed in two steps. At first, we select all block vertexes with known deformation as center points, and apply RBFs interpolation to get the grid deformation on block edges. Then, an arc-length-based TFI is employed to efficiently calculate the grid deformation on block faces and inside each block. The present approach can be well applied to both two-dimensional (2D) and three-dimensional (3D) problems. Numerical results show that the dynamic meshes for all test cases can be generated in an accurate and efficient manner.

This paper deals with the solvability and the convergence of a class of unsymmetric Meshless Local Petrov-Galerkin (MLPG) method with radial basis function (RBF) kernels generated trial spaces. Local weak-form testings are done with step-functions. It is proved that subject to sufficiently many appropriate testings, solvability of the unsymmetric RBF-MLPG resultant systems can be guaranteed. Moreover, an error analysis shows that this numerical approximation converges at the same rate as found in RBF interpolation. Numerical results (in double precision) give good agreement with the provided theory.

It is difficult to obtain the derivative values from most mesh dependent numerical procedures in general. This study proposes an efficient computational tool to accurately evaluate the multi-order derivatives by the radial basis functions and local differential quadrature (LRBF-DQ) algorithm. Most of the traditional derivative calculations can be only adopted to evaluate the differential values with the regular meshes. Moreover, the traditional numerical schemes are very restricted by the order of the shape functions. The present technique can be applied to both the structured and unstructured meshes as well as meshless numerical algorithms – such as RBFs and LDQ method. In addition, the proposed model can be also used to estimate multi-order or mixed partial derivative values because its test function using RBFs is a multi-order differentiable function. All of the evaluation of derivative results will be compared with the exact solutions and other numerical techniques. Consequently, this study provides an effective algorithm of post process to accurately calculate the multi-order derivative values for any numerical schemes.

2D shallow water equations with depth-averaged k−εmodel is considered. A meshless method based on multiquadric radial basis functions isdescribed. This methods is based on the collocation formulation and does not require thegeneration of a grid and any integral evaluation. The application of this method to a flowin horizontal channel, taken as an experimental device, is presented. The results ofcomputations are compared with experimental data and are found to be satisfactory

In this paper, we will discuss the meshless polyharmonic reconstruction of vector fieldsfrom scattered data, possibly, contaminated by noise. We give an explicit solution of theproblem. After some theoretical framework, we discuss some numerical aspect arising in theproblems related to the reconstruction of vector fields

A novel, automatic method for the detection, extraction and characterization of marine mammal tonal calls is presented. Signals are automatically detected from the spectrogram, isolated using region-based segmentation, extracted and finally characterized by means of a fixed number of radial basis function (RBF) coefficients. A total of sixteen RBF coefficients are sufficient to accurately capture the time–frequency information contained in the calls. These coefficients can be later used to classify signals based on their characteristics. New specific functions for contour extraction and cross-resolution have been developed. The performance of the method has been extensively tested using simulated signals and a set of recordings covering a significant range of situations that can be encountered at sea.