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Meshfree methods
About: Meshfree methods is a research topic. Over the lifetime, 2216 publications have been published within this topic receiving 69596 citations.
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TL;DR: Numerical results demonstrate that this extended meshfree approach significantly improves the solution accuracy with commensurately less computational effort compared to the conventional meshfree formulation.
22 citations
TL;DR: This work proposes a numerical algorithm to reduce the ill-conditioning in both the method of fundamental solutions and the plane waves method, which allows to obtain new basis functions that span exactly the same space as the original meshless method, but are much better conditioned.
Abstract: Some meshless methods have been applied to the numerical solution of boundary value problems involving the Helmholtz equation. In this work, we focus on the method of fundamental solutions and the plane waves method. It is well known that these methods can be highly accurate assuming smoothness of the domains and the boundary data. However, the matrices involved are often ill-conditioned and the effect of this ill-conditioning may drastically reduce the accuracy. In this work, we propose a numerical algorithm to reduce the ill-conditioning in both methods. The idea is to perform a suitable change of basis. This allows to obtain new basis functions that span exactly the same space as the original meshless method, but are much better conditioned. In the case of circular domains, this technique allows to obtain errors close to machine precision, with condition numbers of order O(1), independently of the number of basis functions in the expansion.
22 citations
TL;DR: The unconditional stability of the method has been proved by avoiding the algebra developments related to the usually adopted von Neumann analysis and an alternating direction implicit leapfrog scheme for time evolution is proposed.
Abstract: Numerical solution of partial differential equations governing time domain simulations in computational electromagnetics, is usually based on grid methods in space and on explicit schemes in time. A predefined grid in the problem domain and a stability step size restriction need. Recently, the authors have reformulated the meshless framework based on smoothed particle hydrodynamics, in order to be applied for time domain electromagnetic simulation. Despite the good spatial properties, the numerical explicit time integration introduces, also in a meshless context, a severe constraint. In this paper, at first, the stability condition is addressed in a general way by allowing the time step increment get away from the minimum points spacing. Then, an alternating direction implicit leapfrog scheme for time evolution is proposed. The unconditional stability of the method is analytically provided and numerically validated. The stability of the method has been proved by avoiding the algebra developments related to the usually adopted von Neumann analysis. Three case studies are investigated by achieving a satisfactory agreement by comparing both numerical and analytical results.
22 citations
TL;DR: In this paper, the Navier-Cauchy equations were solved with strong and weak form meshless numerical methods and the results were compared to the solution obtained from ABAQUS, which is based on finite element method (FEM).
22 citations
TL;DR: In this article, three unit cell models for the IPCs with the simple cubic (SC), face-centered cubic (FCC), and body-centred cubic (BCC) microstructures are developed using the mesh-free radial point interpolation method.
Abstract: Interpenetrating phase composites (IPCs) have recently been fabricated using three-dimensional (3D) printing methods. In a two-phase IPC, the two phases are topologically interconnected and mutually reinforced in the three dimensions. As a result, such IPCs exhibit higher stiffness, strength, and toughness than particle- or fiber-reinforced composites. In the current study, three unit cell models for the IPCs with the simple cubic (SC), face-centered cubic (FCC), and body-centered cubic (BCC) microstructures are developed using the meshfree radial point interpolation method. Radial basis functions with polynomial reproduction are applied to construct shape functions, and the Galerkin method is employed to formulate discretized equations. These unit cell-based meshfree models are used to evaluate effective elastic properties of 3D printable IPCs. The simulation results are compared with those based on the finite element (FE) method and various analytical bounding techniques in micromechanics, inclu...
22 citations