Meshless collocation method by Delta-shaped basis functions for default barrier model

In the present work, the nonlinear static analysis of functionally graded hyperelastic thin plate is investigated. The neo‐Hookean strain energy function and right Cauchy‐Green tensor are utilized to formulate the potential energy function. Also, the first‐order shear deformation theory of the plate is used to form the strain‐displacement relations. The strong form of the governing equations is derived using Euler‐Lagrange relations, and these relations are discretized using the meshless collocation method (MCM). The thin‐plate spline (TPS) radial basis function (RBF) is utilized to discretize the nonlinear governing equations, and the arc‐length continuation algorithm is used to solve the nonlinear algebraic system of equations. Square and circular plates with clamped and simply supported boundary conditions and various power‐law indexes are investigated using the MCM. The results are compared to those of the finite element method (FEM). The results show that the MCM is an accurate and efficient method in comparison to FEM for circular and square functionally graded (FG) hyperelastic thin plates.

In this paper, a pseudospectral meshless radial point interpolation (PSMRPI) technique is applied to the three-dimensional wave equation with variable coefficients subject to given appropriate initial and Dirichlet boundary conditions. The present method is a kind of combination of meshless methods and spectral collocation techniques. The point interpolation method along with the radial basis functions is used to construct the shape functions as the basis functions in the frame of the spectral collocation methods. These basis functions will have Kronecker delta function property, as well as unitary possession. In the proposed method, operational matrices of higher order derivatives are constructed and then applied. The merit of this innovative method is that, it does not require any kind of integration locally or globally over sub-domains, as it is essential in meshless methods based on Galerkin weak forms, such as element-free Galerkin and meshless local Petrov–Galerkin methods. Therefore, computational cost of PSMRPI method is low. Further, it is proved that the procedure is stable with respect to the time variable over some conditions on the 3D wave model, and the convergence of the technique is revealed. These latest claims are also shown in the numerical examples, which demonstrate that PSMRPI provides excellent rate of convergence.

In this paper, we consider the direct and inverse problem for time-fractional diffusion in a domain with an impenetrable subregion. Here we assume that on the boundary of the subregion the solution satisfies a generalized impedance boundary condition. This boundary condition is given by a second order spacial differential operator imposed on the boundary. A generalized impedance boundary condition can be used to model corrosion and delimitation. The well-posedness for the direct problem is established where the Laplace Transform is used to study the time dependent boundary value problem. The inverse impedance problem of determining the parameters from the Cauchy data is also studied provided the boundary of the subregion is known. The uniqueness of recovering the boundary parameters from the Neumann to Dirichlet mapping is proven.

Compact RBF meshless methods for photonic crystal modelling

Multiobjective optimization for node adaptation in the analysis of composite plates using a meshless collocation method

Theory of flooded porous electrodes I. Galvanostatic transients and generalised impedance

Well posedness of fluid-solid mixture models for biofilm spread

A study is presented using a mesh-free approach and a radial basis function generated finite difference (RBF-FD) method for numerically modelling 3-D gravity data. The gravity responses, that is, vertical gravity and gravity gradients, are obtained by solving the partial differential equation (PDE), that is, the Poisson’s equation for gravitational potential. The mesh-free approach discretizes PDEs using exclusively a cloud of unconnected nodes, instead of traditional tessellated meshes as used by mesh-based numerical methods such as finite difference, finite element and finite volume. Thus, the potentially computationally expensive and unstable creation and manipulation of 3-D meshes can be entirely avoided. A new type of finite-smoothness radial basis functions (RBFs), namely, the quintic-order polyharmonic spline (PHS) RBF, is proposed here in the RBF-FD frame for solving the gravity problem. Previous geophysical data modelling studies using RBF-FD have employed the infinitely smooth RBFs, such as the popular Gaussian (GA) RBFs. Here, both GA and PHS RBFs were tested with different numbers of nodes per mesh-free subdomain and with various shape parameter values (only GA RBFs have a shape parameter). The test results show that the PHS RBF-FD method is more computationally efficient than the GA RBF-FD counterpart. To achieve more efficiency, unstructured node distributions are proposed in discretizing the density models. For both quasi-uniform and unstructured node distributions, numerical results from the proposed PHS RBF-FD demonstrate that the computed vertical gravity and gravitational potential values agree well with analytical solutions with a reasonable number of degrees of freedom. A comparison study of modelling a complex density model with the PHS RBF-FD scheme and nodal finite-element method shows that the RBF-FD scheme generates sparse, asymmetric, linear systems of equations, supports unstructured nodal discretization and local refinement, and can have nonlinear h-convergence under refinement. Finally, the proposed RBF-FD method was applied to obtain the vertical gravity and gravity gradients over a real-world density model, where the benefits of mesh-free discretization are clearly illustrated.

CO₂ emission is considerably dependent on energy consumption and on share of energy sources as well as on the extent of economic activities. Consequently, these factors must be considered for CO₂ emission prediction for seven middle eastern countries including Iran, Kuwait, United Arab Emirates, Turkey, Saudi Arabia, Iraq and Qatar. In order to propose a predictive model, a Multilayer Perceptron Artificial Neural Network (MLP ANN) is applied. Three transfer functions including logsig, tansig and radial basis functions are utilized in the hidden layer of the network. Moreover, various numbers of neurons are applied in the structure of the models. It is revealed that using MLP ANN makes it possible to accurately predict CO₂ emission of these countries. In addition, it is concluded that using logsig transfer function leads to the highest accuracy with minimum value of mean squared error (MSE) which is followed by the networks with radial basis and tansig transfer functions. The R-squared of the networks with logsig, radial basis and tansig transfer functions are 0.9998, 0.9997 and 0.9996, respectively. Finally, comparison of the proposed model with a similar study, considered five countries in the same region, reveals higher accuracy in term of MSE.

On the use of radial basis functions in a local weighted meshless method

This paper proposes a method of gravity inversion based on an adaptive mesh-free approach by using a modified radial basis function (RBF). We parametrize the density distribution by using a mesh-free approach. Scattered points are introduced in most mesh-free methods to discretize the given equations. The subsurface space is generally discretized into regular grid cells, while mesh-free methods can avoid the expensive mesh generation and manipulation required in traditional approaches. To deal with the problem of unstructured nodal discretization, we use a mesh-free discretization strategy to establish a mapping of subsurface grid cells to a cloud of discrete points. The nodes are adaptively refined during the inversion process to better recover abnormal bodies. In addition, the hybrid basis function and the modified radial basis function are used to improve the accuracy and stability of the solution. We verify the effectiveness of the proposed method by using several synthetic and real tests.

Angular basis functions formulation for 2D potential flows with non-smooth boundaries

A two-dimensional adaptive nodes technique in irregular regions applied to meshless-type methods

Localized radial basis function scheme for multidimensional transient generalized newtonian fluid dynamics and heat transfer

ABSTRACTA local radial basis function meshless (LRBFM) method is developed to solve coupled radiative and conductive heat transfer problems in multidimensional participating media, in which compact support radial basis functions (RBFs) augmented on a polynomial basis are employed to construct the trial function, and the radiative transfer equation (RTE) and energy conservation equation are discretized directly at nodes by the collocation method. LRBFM belongs to a class of truly meshless methods which require no mesh or grid, and can be readily implemented in a set of uniform or irregular node distributions with no node connectivity. Performances of the LRBFM is compared to numerical results reported in the literature via a variety of coupled radiative and conductive heat transfer problems in 1D and 2D geometries. It is demonstrated that the local radial basis function meshless method provides high accuracy and great efficiency to solve coupled radiative and conductive heat transfer problems in multidimensional participating media with uniform and irregular node distribution, especially for coupled heat transfer problems in irregular geometry with Cartesian coordinates. In addition, it is extremely simple to implement.