An accurate, stable and efficient domain-type meshless method for the solution of MHD flow problems

Abstract

a b s t r a c tThe aim of the present paper is the development of an efficient numerical algorithm for the solution of magnetohydrodynamics flow problems for regular and irregular geometries subject to Dirichlet, Neumann and Robin boundary conditions. Toward this, the meshless point collocation method (MPCM) is used for MHD flow problems in channels with fully insulating or partially insulating and partially conducting walls, having rectangular, circular, elliptical or even arbitrary cross sections. MPC is a truly meshless and computationally efficient method. The maximum principle for the discrete harmonic operator in the meshfree point collocation method has been proven very recently, and the convergence proof for the numerical solution of the Poisson problem with Dirichlet boundary conditions have been attained also. Additionally, in the present work convergence is attained for Neumann and Robin boundary conditions, accordingly. The shape functions are constructed using the Moving Least Squares (MLS) approximation. The refinement procedure with meshless methods is obtained with an easily handled and fully automated manner. We present results for Hartmann number up to 10 5 . The numerical evidences of the proposed meshless method demonstrate the accuracy of the solutions after comparing with the exact solution and the conventional FEM and BEM, for the Dirichlet, Neumann and Robin boundary conditions of interior problems with simple or complex boundaries.

Influence of geometrical characteristics of sidewall-type multiple aneurysms on hemodynamic factors

“…29 Recently, another attractive alternative is the meshless method, which discretizes the incompressible flow governing equations by making use of point clouds. 30,31 Without relying on stencils, elements, or control volumes, meshless methods also do not require mesh generation. This not only simplifies the costly and time consuming process but also avoids various topological and connectivity problems.…”

Transient thermal conduction with variable conductivity using the Meshless Local Petrov–Galerkin method

“…Several meshless methods have been reported thus far, such as the Diffuse Element Method (DEM) [6], the Element Free Galerkin (EFG) method [7][8][9], and the Reproducing Kernel Particle (RKP) method [10]. Meshless methods have undergone extensive development recently, largely due to their inherent potential of handling engineering problems with complex and irregular geometries more efficiently than standard computational techniques [11]. Meshless numerical techniques rely on local type interpolation on either regularly or irregularly distributed spatial points, usually called nodes.…”

Localized meshless point collocation method for time-dependent magnetohydrodynamics flow through pipes under a variety of wall conductivity conditions

“…Thus, meshless methods emerged as a potential alternative for solutions in computational mechanics, and a variety of such approaches have appeared. Several meshfree methods have been proposed; a review of the relative literature is presented in [24,25]. It should be noted that the majority of these methods are not really meshless, since they need to use a background mesh for the numerical integration.…”

“…To the authors' knowledge a very limited number of research works using meshless strongform collocation methods for solving MHD flow problems exists. Namely, the authors in [25] used the meshless point collocation method, to solve steady state MHD problems for rectangular, circular, elliptical and irregular cross section for high Hartmann numbers up to M = 100,000. element free-Galerkin (EFG) method was used in [26] to solve the steady-state MHD flow. The formulation was applied to the study of two-dimensional magnetohydrodynamic flow problems for moderate Hartmann numbers (M = 500), and as the authors claimed, the computed results confirm the accuracy and correctness of the proposed formulation.…”

“…Elimination of this difficulty requires the introduction of additional unknown parameters, such as Lagrange multipliers, however, it can lead to poor conditioning of the matrix equations [26]. In the present article, we use an interpolatory formulation for MLS approximants that allows the direct introduction of boundary conditions, reduces the processing time and improves the condition numbers, as in [25,26,29]. The formulation is applied to the study of two-dimensional magnetohydrodynamic flow problems, and the computed results confirm the accuracy and correctness of the proposed formulation.…”