Meshfree method for solving the elliptic Monge-Ampère equation with Dirichlet boundary

Combination of meshless local weak and strong (MLWS) forms to solve the two dimensional hyperbolic telegraph equation

In this article, recently proposed spectral meshless radial point interpolation (SMRPI) method is applied to the two-dimensional diffusion equation with a mixed group of Dirichlet's and Neumann's and non-classical boundary conditions. The present method is based on meshless methods and benefits from spectral collocation ideas. The point interpolation method with the help of radial basis functions is proposed to construct shape functions which have Kronecker delta function property. Evaluation of high-order derivatives is possible by constructing and using operational matrices. The computational cost of the method is modest due to using strong form equation and collocation approach. A comparison study of the efficiency and accuracy of the present method and other meshless methods is given by applying on mentioned diffusion equation. Stability and convergence of this meshless approach are discussed and theoretically proven. Convergence studies in the numerical examples show that SMRPI method possesses excellent rates of convergence.

The Navier-Stokes equation, applied to the calculation of wind velocity without accounting for the turbulent motion of the atmosphere, is considered in this work. The main flow characteristics were computed with the use of the Lagrange discrete vortex method for finding the solution of the Poisson equation under the Dirichlet and Neumann boundary conditions. To do this, two mesh-free methods: the element-free Galerkin (EFG) and the Finite Pointset (FP) methods, as well as the modification of the latter, have been analyzed. It is shown that the computation speed of the EFG method is higher than of the FP-method. It is determined that a serious disadvantage of the FP-method is its low rate convergence, while the computational complexity of each iteration is reasonable. The use of the modified FP-method has shown its computational speed to be comparable with that of the EFG method, although the advantage of the FP-method is not obvious when the size of the problem increases.

Implementation of boundary conditions in BIEs-based meshless methods: A dual boundary node method

Explicit empirical formula evaluating original intensity factors of singular boundary method for potential and Helmholtz problems

This paper considers some multiscale radial basis function collocation methods for solving the two-dimensional Monge–Ampère equation with Dirichlet boundary. We discuss and study the performance of the three kinds of multiscale methods. The first method is the cascadic meshfree method, which was proposed by Liu and He (2013). The second method is the stationary multilevel method, which was proposed by Floater and Iske (1996), and is used to solve the fully nonlinear partial differential equation in the paper for the first time. The third is the hierarchical radial basis function method, which is constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. Compared with the first two methods, the hierarchical radial basis function method can not only solve the present problem on a single level with higher accuracy and lower computational cost but also produce highly sparse nonlinear discrete system. These observations are obtained by taking the direct approach of numerical experimentation.

This paper presents the thermal solution of cylindrical composite systems using meshless element free Galerkin (EFG) method. The EFG method utilizes the moving least square approximants, which are constructed by using a weight function, a basis function and a set of non-constant coefficients to approximate the unknown function of temperature. Dirichlet (essential) boundary conditions have been enforced using Lagrange multiplier and penalty methods. Existing rational weight function has been modified and used in the present analysis. MATLAB codes have been developed to obtain the numerical solution. The EFG results have been obtained using cubicspline, quarticspline, Gaussian, quadratic, hyperbolic, exponential, rational and cosine weight functions for a model problem. The results obtained using different EFG weight functions are also compared with those obtained by finite element method. The effect of scaling and penalty parameters has also been studied in detail.

In recent years, there has been a great deal of interest in developing meshless methods for computational fluid dynamics (CFD) applications. In this paper, a meshless finite difference method is developed for solving conjugate heat transfer problems in complex geometries. Traditional finite difference methods (FDMs) have been restricted to an orthogonal or a body-fitted distribution of points. However, the Taylor series upon which the FDM is based is valid at any location in the neighborhood of the point about which the expansion is carried out. Exploiting this fact, and starting with an unstructured distribution of mesh points, derivatives are evaluated using a weighted least squares procedure. The system of equations that results from this discretization can be represented by a sparse matrix. This system is solved with an algebraic multigrid (AMG) solver. The implementation of Neumann, Dirichlet and mixed boundary conditions within this framework is described, as well as the handling of conjugate heat transfer. The method is verified through application to classical heat conduction problems with known analytical solutions. It is then applied to the solution of conjugate heat transfer problems in complex geometries, and the solutions so obtained are compared with more conventional unstructured finite volume methods. Metrics for accuracy are provided and future extensions are discussed.

This paper focuses on the comparative study of global and local mesh- less methods based on collocation with radial basis functions for solving two di- mensional initial boundary value diffusion-reaction problem with Dirichlet and Neumann boundary conditions. A similar study was performed for the boundary value problem with Laplace equation by Lee, Liu, and Fan (2003). In both global and local methods discussed, the time discretization is performed in explicit and implicit way and the multiquadric radial basis functions (RBFs) are used to inter- polate diffusion-reaction variable and its spatial derivatives. Five and nine nodded sub-domains are used in the local support of the local method. Uniform and non- uniform space discretization is used. Accuracy of global and local approaches is assessed as a function of the time and space discretizations, and value of the shape parameter. One can observe the convergence with denser nodes and with smaller time-steps in both methods. The global method is prone to instability due to ill- conditioning of the collocation matrix with the increase of the number of the nodes in cases N t 3000. On the other hand, the global method is more stable with re- spect to the time-step length. Numerical tests with and without noise are conducted based on the methodology proposed in Younga, Fana, Hua, and Atluri (2009). The results show larger stability of the local versions of the method in comparison with the global ones. The accuracy of the local method is comparable with the accuracy of the global method. The local method is more efficient because we solve only a small system of equations for each node in explicit case and a sparse system of equations in implicit case. Hence the local method represents a preferable choice to its global counter part.

The meshless local Pretrov-Galerkin method (MLPG) with the test function in view of the Heaviside step function is introduced to solve the system of coupled nonlinear reaction-diffusion equations in two-dimensional spaces subjected to Dirichlet and Neumann boundary conditions on a square domain. Two-field velocities are approximated by moving Kriging (MK) interpolation method for constructing nodal shape function which holds the Kronecker delta property, thereby enhancing the arrangement nodal shape construction accuracy, while the Crank-Nicolson method is chosen for temporal discretization. The nonlinear terms are treated iteratively within each time step. The developed formulation is verified in two numerical examples with investigating the convergence and the accuracy of numerical results. The numerical experiments revealing the solutions by the developed formulation are stable and more precise.

In this paper, an extended version of the method of minimizing an energy gap functional for determining the optimal source points in the method of fundamental solutions (MFS) is applied to the 3D Laplace operator subject to the Dirichlet and Neumann boundary conditions. As we know, the MFS is a more popular meshless method for solving boundary or initial-boundary value problems due to its simplicity and high accuracy. However, the accuracy of the MFS depends strongly on the distribution of the source points. Finally, some of the numerical experiments are carried out to express the simplicity and effectiveness of the presented method.

Domain and boundary type meshless methods based on the Direct Multi-Elliptic Interpolation Method are presented. The approach is equivalent to a special RBFmethod but completely avoids the solution of large, full and ill-conditioned systems, thus, the computational cost is significantly reduced. The method is illustrated through the example of the usual Poisson problem. Both Dirichlet and Neumann boundary conditions are investigated. The domain version of the method results in particular solutions, while the boundary version can be applied to solve homogeneous problems. Along Neumann boundaries, either off-boundary points can be introducedor a boundaryreconstructiontechniquebased on boundaryinterpolation can be applied. Some further possible applications are also outlined.

In this work, a hybrid localized meshless method is developed for solving transient groundwater flow in two dimensions by combining the Crank–Nicolson scheme and the generalized finite difference method (GFDM). As the first step, the temporal discretization of the transient groundwater flow equation is based on the Crank–Nicolson scheme. A boundary value problem in space with the Dirichlet or mixed boundary condition is then formed at each time node, which is simulated by introducing the GFDM. The proposed algorithm is truly meshless and easy to program. Four linear or nonlinear numerical examples, including ones with complicated geometry domains, are provided to verify the performance of the developed approach, and the results illustrate the good accuracy and convergency of the method.

This paper considered the spectral meshless radial point interpolation (SMRPI) method to unravel for the nonlinear p-Laplacian equation with mixed Dirichlet and Neumann boundary conditions. Through this assessment, which includes meshless methods and collocation techniques based on radial point interpolation, we construct the shape functions, with the Kronecker delta function property, as basis functions in the framework of spectral collocation methods. Studies in this regard require one to evaluate the high-order derivatives without any kind of integration locally over the small quadrature domains. Finally, some examples are given to illustrate the low computing costs and high enough accuracy and efficiency of this method to solve a p-Laplacian equation and it would be of great help to fulfill the implementation related to the element-free Galerkin (EFG) method. Both the SMRPI and the EFG methods have been compared by similar numerical examples to show their application in strongly nonlinear problems.

In this work, the problem of increasing the convergence order of the integral meshless method already proposed by the same authors is addressed. Solutions are determined through equations directly written in discrete form over a tributary region represented by the circle with center in the generic node and radius given by the average of the distances between the node itself and its neighbors, thus allowing a considerable ease in writing the discrete form of the governing equations. The proposed approach, besides avoiding global mesh generation, adopts interpolating polynomials, which exactly reproduce nodal values of field variables, and eliminates some problems typically encountered when posing Dirichlet and Neumann boundary conditions with the Finite Element Method. Several numerical schemes adopting extended or compact computational cells are proposed and tested for the Laplace equation, in line with the previous papers. Results show that, when using interpolating polynomials that satisfy also the differential operator in some nodes, compact computational cells characterized by the fifth-order of convergence may be constructed.