Meshfree methods

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

Numerical Simulation of Capillary Flows Through Molecular Dynamics

Abstract: Capillary pore imbibition represents a very challenging field of research because of its crucial role in several physical phenomenona. This paper deals with capillary dynamics in connection with non destructive test technique used to evidence eventual defects present in mechanical parts whose perfect integrity is vital for the reliability of all the systems (car, planes, etc.) which they are part of. For the description of such phenomenona, essentially two different approaches are possible. The first approach consists in considering the fluid motion as a continuum to be described in terms of mathematical models governed by ordinary or partial differential equation. Subsequently, according to the prescribed initial and boundary conditions to these models, the solution can be worked out numerically through one of the available numerical techniques such as finite difference, finite elements, finite volumes, meshless methods, etc. Conversely, the second approach considers the fluid motion as an atomistic ensemble of discrete particles (atoms or molecules), whose macroscopic features are strongly determined by the inner interactions occurring among them. In this context, the two most important conceptual available tools are the Lattice Boltzmann theory, mainly used by physicists and the Molecular Dynamics, MD for the sequel, that jointly with its numerical treatment represents the topic of the present paper. Capillary imbibition represents a specific aspect of the wider area of wetting solid by liquids and its description in terms of MD has revealed to be a very promising approach for the description of the capillary flows alternative to continuous differential models. At the beginning, MD was used to simulate essentially the behavior of biological materials, but over the years MD has proved to be a powerful tool to model and simulate nanotechnology structures. In this paper large scale MD is used in order to model the intrusion of water into a defect of a few given solid materials. Finally numerical results have been obtained for cylindrical defects of Titanium or Aluminum both internally smooth or rough.

Algorithms for scientific computing

Abstract: There has long been interest in algorithms for simulating physical systems. We are concernedwith two areaswithin this field: fastmultipolemethods andmeshlessmethods. Since Greengard and Rokhlin’s seminal paper in 1987, considerable interest has arisen in fast multipole methods for finding the energy of particle systems in two and three dimensions, and more recently in many other applications where fast matrix-vector multiplication is called for. We develop a new fast multipole method that allows the calculation of the energy of a system of N particles in O(N) time, where the particles’ interactions are governed by the 2D Yukawa potential which takes the form of a modified Bessel function Kv. We then turn our attention to meshless methods. We formulate and test a new radial basis function finite differencemethod for solving an eigenvalue problemon a periodic domain. We then applymeshlessmethods to modelling photonic crystals. After an initial background study of the field, we detail the Maxwell equations, which govern the interaction of the light with the photonic crystal, and show how photonic band gaps may be given rise to. We present a novel meshless weak-strong form method with reduced computational cost compared to the existing meshless weak form method. Furthermore, we develop a new radial basis function finite differencemethod for photonic band gap calculations. Throughout the work we demonstrate the application of cutting-edge technologies such as cloud computing to the development and verification of algorithms for physical simulations.

Numerical methods for The Modelling Of Debonding In Composites

Abstract: This monograph starts with a discussion of various phenomena in laminated composite structures that can lead to failure: matrix cracking, delamination between plies, and debonding and subsequent pull-out between fibres and the matrix material. Next, the different scales are discussed at which the effect of these nonlinearities can be analysed and the ways to couple analyses at these different length scales. From these scales—the macro, meso and micro-levels — the meso-level is normally used for the analysis of delamination, which is the focus of this monograph. At this level, the plies are modelled as continua and interface elements between them conventionally serve as the framework to model delami-nation and debonding. After a brief discussion of the cohesive—zone concept and its importance for the analysis of delamination, a particular finite element model for the plies is elaborated: the solid—like shell. This is followed by a derivation of interface elements. In the second part of this monograph more recent methods to numerically model delamination are discussed: meshfree methods, methods that exploits the partition—of—unity property of finite element shape functions, and discontinuous Galerkin methods. These approaches offer advantages over the more traditional approach that uses interface elements, as will be discussed in detail. From these more modern discretisation concepts the partition-of-unity approach seems the most promising for modelling debonding in composite structures, one advantage being that it can rather straightforwardly be incorporated in solid-like shell elements, thus enabling large-scale analyses of layered composite structures that take into account the possibility of debonding.

Modeling of rock discontinuity with meshless method

Abstract: Various interface elements are successfully applied in traditional finite element method(FEM) to model discontinuities,in which Goodman element is the most representative one.The paper discusses the feasibility of introducing Goodman element to model geological discontinuities in meshless method in full length,and points out the problems existing in some current studies.The Goodman element is presented in the framework of FEM;the displacement mode of Goodman element is designed to be compatible with the finite element along the common boundary between finite element and Goodman element.But,in the meshless method,the displacement mode of Goodman element generally is not compatible with the meshless displacement mode which is based on a number of discrete nodes,the number,however,cannot be known beforehand.The key to solve this problem is that the stiffness matrix of the Goodman element must be computed through numerical integration,then the computed matrix but not the analytical stiffness matrix in the traditional FEM is added to the general matrix of the system when the contribution of the Goodman element to the general stiffness matrix is considered.So,discontinuities can be represented by Goodman element in analytical model in implicit or explicit mode.In the framework of the natural element method(NEM),the implementation scheme of the Goodman element,implicitly and explicitly respectively,is addressed in detail for illustration of the idea.The presented method is general and suitable for all existing meshless methods.

Intelligent instrumentation and numerical methods

Abstract: In this paper we discuss the choice of numerical methods that yield the least numerical error for a given number of discretized spatial nodes, and a proper eigenvalue procedure that avoids numerical integration in time with the corresponding saving in computer requirements. These procedures are suited for small computing systems to be used in field instrumentation. We check the effectiveness of these methods with the solution of a diffusion equation with recombination, time dependent and in one dimension in space. We compare the results obtained with a second order finite difference scheme with those obtained with a spectral method solution, showing the superiority of the last approach.