Meshfree methods

About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.

A compatible high-order meshless method for the Stokes equations with applications to suspension flows

Abstract: A stable numerical solution of the steady Stokes problem requires compatibility between the choice of velocity and pressure approximation that has traditionally proven problematic for meshless methods. In this work, we present a discretization that couples a staggered scheme for pressure approximation with a divergence-free velocity reconstruction to obtain an adaptive, high-order, finite difference-like discretization that can be efficiently solved with conventional algebraic multigrid techniques. We use analytic benchmarks to demonstrate equal-order convergence for both velocity and pressure when solving problems with curvilinear geometries. In order to study problems in dense suspensions, we couple the solution for the flow to the equations of motion for freely suspended particles in an implicit monolithic scheme. The combination of high-order accuracy with fully-implicit schemes allows the accurate resolution of stiff lubrication forces directly from the solution of the Stokes problem without the need to introduce sub-grid lubrication models.

Natural element approximations involving bubbles for treating mechanical models in incompressible media

Abstract: In this paper, a new approach is proposed to address issues associated with incompressibility in the context of the meshfree natural element method (NEM). The NEM possesses attractive features such as interpolant shape functions or auto-adaptive domain of influence, which alleviates some of the most common difficulties in meshless methods. Nevertheless, the shape functions can only reproduce linear polynomials, and in contrast to moving least squares methods, it is not easy to define interpolations with arbitrary approximation consistency. In order to treat mechanical models involving incompressible media in the framework of mixed formulations, the associated functional approximations must satisfy the well-known inf–sup, or LBB condition. In the proposed approach, additional degrees of freedom are associated with some topological entities of the underlying Delaunay tessellation, i.e. edges, triangles and tetrahedrons. The associated shape functions are computed from the product of the NEM shape functions related to the original nodes. Different combinations can be used to construct new families of NEM approximations. As these new approximations functions are not related to any node, as they vanish at the nodes, from now on we refer these shape functions as bubbles. The shape functions can be corrected enforcing different reproducing conditions, when they are used as weights in the moving least square (MLS) framework. In this manner, the effects of the obtained higher approximation consistency can be evaluated. In this work, we restrict our attention to the 2D case, and the following constructions will be considered: (a) bubble functions associated with the Delaunay triangles, called b1-NEM and (b) bubble functions associated with the Delaunay edges, called b2-NEM. We prove that all these approximation schemes allow direct enforcement of essential boundary conditions. The bubble-NEM schemes are then used to approximate the displacements in the linear elasticity mixed formulation, the pressure being approximated by the standard NEM. The numerical LBB test is passed for all the bubble-NEM approximations, and pressure oscillations are removed in the incompressible limit.

An Adaptive Radial Point Interpolation Meshless Method for Simulation of Electromagnetic and Optical Fields

Abstract: This paper is going to handle an adaptive meshless method by which a wide range of microwave problems in terms of frequency are numerically solved whose accuracy and computational time are acceptable with respect to some other numerical schemes. As the origin, in the process of imposing the conventional radial point interpolation method (CRPIM) to laser problems, a special function was found, which results in a well-behaved basis function for CRPIM. This basis function possesses two fundamental advantages in view of meshless methods. At first, and in contradiction with conventional basis functions, the shape parameters are deterministic, which results in a higher accuracy than conventional basis functions. Second, it will construct the shape functions without any need for the middle matrix inversion step. Also, the adaptive basis function inherits the fundamental properties of fields. Hence, the computational time is reduced, approximately by half, comparing with the conventional basis functions. To investigate the proposed adaptive method named quantum radial point interpolation method in different areas of interest, it has been employed to solve three classes of partial differential equations in computational electromagnetics, i.e., Schrodinger's equation in a quantum wave laser, Laplace and electromagnetic wave equations. The results are more accurate and faster than the CRPIM and the finite-element method.

Numerical solutions of two-dimensional flow fields by using the localized method of approximate particular solutions

Abstract: A combination of the localized method of approximate particular solutions (LMAPS), the implicit Euler method and the Newton’s method is adopted in this paper for transient solutions of two-dimensional velocity–vorticity formulation of the Navier–Stokes equations. The LMAPS, which is truly free from time-consuming mesh generation and numerical quadrature, and the implicit Euler method are, respectively, used for spatial and temporal discretizations of the velocity–vorticity formulation. Using the approximations of particular solutions in every local domain, the derivatives at nodes with respect to space coordinates via the LMAPS can be approximated by linear summations of nearby function values. After the discretizations for space and time derivatives, a system of nonlinear algebraic equations will be yielded at every time step and then the Newton’s method is used for efficiently analyzing these systems. Three numerical examples are provided to validate the accuracy and the simplicity of the proposed scheme and the numerical results are compared well with other numerical and analytical solutions. Besides, the numerical solutions, acquired by using different numbers of total nodes, different numbers of nodes in sub-domain, different shape parameters and different Reynolds numbers, are provided to show the merits of the proposed meshless scheme.

A Meshless Local Petrov-Galerkin (MLPG) Approach Based on the Regular Local Boundary Integral Equation for Linear Elasticity

Abstract: The moving local Petrov-Galerkin (MLPG) approach based on a regular local boundary integral equation (RLBIE) to solve problems in elasto-statics is developed. The present method is a truly meshless method, as absolutely no mesh connectivity is required for interpolating the solution variables and for integrating the weak form. Compared to the original MLPG method, the present method does not need the derivatives of the shape functions in constructing the stiffness matrix for those nodes with no displacement specified on their local boundaries. The numerical examples presented in the paper show that high rates of convergence with mesh refinement for the displacement and energy norms are achievable.