In this paper, an ADAPTIVE MESHLESS method of line is applied to distributethe nodes in the spatial domain. In many cases in MESHLESS METHODS, it isalso necessary for the chosen nodes to have certain smoothness properties. The set of nodes is also required to satisfy certain constraints. In this paper, one of these constraints is investigated. The aim of this manuscript is theimplementation of an algorithm for selection of the nodes satisfying a givenconstraint, in the MESHLESS method of line. This algorithm is applied to someillustrative examples to show the e ciency of the algorithm and its ability toincrease the accuracy.

We introduce a RBFs mesheless method of lines that decomposes theinterior and boundary centers to obtain the numerical solution of the timedependent PDEs. Then, the method is applied with an ADAPTIVE algorithmto obtain the numerical solution of one dimensional problems. We show thatin the problems in which the solutions contain region with rapid variation, the ADAPTIVE RBFs METHODS are successful so that the PDE solution can beapproximated well with a small number of basis functions. The method isdescribed in detail, and computational experiments are performed for one-dimensional Burgers' equations.

An overview on some MESHLESS METHODS, and a MESHLESS procedure for solving linear convection-diffusion equation in steady state and nonlinear burgear equation in transient are presented in this paper. The approach termed generically the "Element Free Galerkin method" is based on a weighted moving least square (MLS) interpolation of point data for evaluating the approximation integral. EFG MESHLESS method uses a background mesh to introducing the Guossian points for solution of the integral equation. In this paper we develop EFG MESHLESS procedure for linear and nonlinear convection-diffusion equation problems in steady and transient forms. A one dimensional example by different values of peclet number is solved using MATLAB, V6.5 programming and the results are compared with the exact solution. The method is tested for shock wave propagation in example no two.

The importance of focusing on the research of viable models to predict the behaviour of structures which may possess in some cases complex geometries is an issue that is growing in different scientific areas, ranging from the civil and mechanical engineering to the architecture or biomedical devices fields. In these cases, the research effort to find an efficient approach to fit laser scanning point clouds, to the desired surface, has been increasing, leading to the possibility of modelling as-built/as-is structures and components’ features. However, combining the task of surface reconstruction and the implementation of a structural analysis model is not a trivial task. Although there are works focusing those different phases in separate, there is still an effective need to find approaches able to interconnect them in an efficient way. Therefore, achieving a representative geometric model able to be subsequently submitted to a structural analysis in a similar based platform is a fundamental step to establish an effective expeditious processing workflow. With the present work, one presents an integrated methodology based on the use of MESHLESS approaches, to reconstruct shells described by points’ clouds, and to subsequently predict their static behaviour. These METHODS are highly appropriate ondealing with unstructured points clouds, as they do not need to have any specific spatial or geometric requirement when implemented, depending only on the distance between the points. Details on the formulation, and a set of illustrative examples focusing the reconstruction of cylindrical and double-curvature shells, and its further analysis, are presented.

Distance-based clustering METHODS categorize samples by optimizing a global criterion, finding ellipsoid clusters with roughly equal sizes. In contrast, density-based clustering techniques form clusters with arbitrary shapes and sizes by optimizing a local criterion. Most of these METHODS have several hyper-parameters, and their performance is highly dependent on the hyper-parameter setup. Recently, a Gaussian Density Distance (GDD) approach was proposed to optimize local criteria in terms of distance and density properties of samples. GDD can find clusters with different shapes and sizes without any free parameters. However, it may fail to discover the appropriate clusters due to the interfering of clustered samples in estimating the density and distance properties of remaining unclustered samples. Here, we introduce ADAPTIVE GDD (AGDD), which eliminates the inappropriate effect of clustered samples by ADAPTIVEly updating the parameters during clustering. It is stable and can identify clusters with various shapes, sizes, and densities without adding extra parameters. The distance metrics calculating the dissimilarity between samples can affect the clustering performance. The effect of different distance measurements is also analyzed on the method. The experimental results conducted on several well-known datasets show the effectiveness of the proposed AGDD method compared to the other well-known clustering METHODS.

The computational centers in the multiquadric radial basis functions MESHLESS method have high adaptability considering the lack of geometric and physical connection between the centers. In this research, a new ADAPTIVE algorithm is proposed based on the gradients of the physical variables of the problem with the aim of creating an optimal distribution. The resulted ADAPTIVE distribution generated by this algorithm improves significantly the accuracy and speed of the multiquadric method compared to the uniform distribution in steady and unsteady problems. In this approach, firstly, the domains with low and high physical variations are identified in a known time step, then the number of computational centers decreases and increases in these areas, respectively. Thus, the centers will be distributed more compact where needed and will be eliminated where not. Facing another important challenge of the multiquadric method, i.e. determining the optimal shape parameter, a simple and efficient method is introduced in such a way that there is no need to optimize the shape parameter at each time step and the computational costs are controlled. Finally, the effectiveness of the proposed method is shown by solving examples of diffusion, convection and convection-diffusion equations. The results are compared to their uniform distributions by measuring their efficiency and to the exact solution by evaluating the accuracy.

Interpolation and approximation are the most important parts of partial differential equation solution procedures, which significantly affect the cost and the accuracy of the results. This paper is aimed to exhaustively investigate the interpolation algorithms and trace their chronologically developments. The interpolation METHODS are classified based on their mathematical representation, and then surveyed separately. An abridgement of calculation steps of METHODS are presented and for details, the reader is referred by the main references. The usage records in applied science and engineering are included and their numerical dominance, stability and convergence rate are discussed.