Topic
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: A one‐parameter family of approximation schemes that bridges continuously two important limits: Delaunay triangulation and maximum‐entropy (max‐ent) statistical inference are presented.
Abstract: We present a one-parameter family of approximation schemes, which we refer to as local maximum-entropy approximation schemes, that bridges continuously two important limits: Delaunay triangulation and maximum-entropy (max-ent) statistical inference. Local max-ent approximation schemes represent a compromise—in the sense of Pareto optimality—between the competing objectives of unbiased statistical inference from the nodal data and the definition of local shape functions of least width. Local max-ent approximation schemes are entirely defined by the node set and the domain of analysis, and the shape functions are positive, interpolate affine functions exactly, and have a weak Kronecker-delta property at the boundary. Local max-ent approximation may be regarded as a regularization, or thermalization, of Delaunay triangulation which effectively resolves the degenerate cases resulting from the lack or uniqueness of the triangulation. Local max-ent approximation schemes can be taken as a convenient basis for the numerical solution of PDEs in the style of meshfree Galerkin methods. In test cases characterized by smooth solutions we find that the accuracy of local max-ent approximation schemes is vastly superior to that of finite elements.
368 citations
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01 Oct 2003TL;DR: This work presents a new computational paradigm, the meshfree particle method, where the object representation and the numerical calculation are purely based on the nodal points and do not require the meshing of the analysis domain, and can naturally handle large deformation and domain discontinuity issues.
Abstract: Many of the computer vision algorithms have been posed in various forms of differential equations, derived from minimization of specific energy functionals, and the finite element representation and computation have become the de facto numerical strategies for solving these problems. However, for cases where domain mappings between numerical iterations or image frames involve large geometrical shape changes, such as deformable models for object segmentation and nonrigid motion tracking, these strategies may exhibit considerable loss of accuracy when the mesh elements become extremely skewed or compressed. We present a new computational paradigm, the meshfree particle method, where the object representation and the numerical calculation are purely based on the nodal points and do not require the meshing of the analysis domain. This meshfree strategy can naturally handle large deformation and domain discontinuity issues and achieve desired numerical accuracy through adaptive node and polynomial shape function refinement. We discuss in detail the element-free Galerkin method, including the shape function construction using the moving least square approximation and the Galerkin weak form formulation, and we demonstrate its applications to deformable model based segmentation and mechanically motivated left ventricular motion analysis.
367 citations
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TL;DR: In this paper, a method for treating fluid-structure interaction of fracturing structures under impulsive loads is described, which does not require any modifications when the structure fails and allows fluid to flow through openings between crack surfaces.
Abstract: A method for treating fluid-structure interaction of fracturing structures under impulsive loads is described. The coupling method is simple and does not require any modifications when the structure fails and allows fluid to flow through openings between crack surfaces. Both the fluid and the structure are treated by meshfree methods. For the structure, a Kirchhoff-Love shell theory is adopted and the cracks are treated by introducing either discrete (cracking particle method) or continuous (partition of unity-based method) discontinuities into the approximation. Coupling is realized by a master-slave scheme where the structure is slave to the fluid. The method is aimed at problems with high-pressure and low-velocity fluids, and is illustrated by the simulation of three problems involving fracturing cylindrical shells coupled with fluids.
362 citations
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TL;DR: In this article, the authors describe the foundation and properties of the so-called dynamic boundary particles (DBPs), which share the same equations of continuity and state as the moving particles placed inside the domain, although their positions and velocities remain unaltered in time.
Abstract: Smoothed Particle Hydrodynamics is a purely Lagrangian method that can be applied to a wide variety of fields. The foundation and properties of the so called dynamic boundary particles (DBPs) are described in this paper. These boundary particles share the same equations of continuity and state as the moving particles placed inside the domain, although their positions and velocities remain unaltered in time or are externally prescribed. Theoretical and numerical calculations were carried out to study the collision between a moving particle and a boundary particle. The boundaries were observed to behave in an elastic manner in absence of viscosity. They allow the fluid particles to approach till a critical distance depending on the energy of the incident particle. In addition, a dam break confined in a box was used to check the validity of the approach. The good agreement between experiments and numerical results shows the reliability of DBPs. Keyword: Meshfree methods, SPH, smoothed particle hydrodynamics, boundary conditions
351 citations
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TL;DR: A survey of meshless methods can be found in this article, where the authors provide a unified mathematical theory with proofs, briefly address implementational aspects, present illustrative numerical examples, and provide a list of references.
Abstract: In the past few years meshless methods for numerically solving partial differential equations have come into the focus of interest, especially in the engineering community. This class of methods was essentially stimulated by difficulties related to mesh generation. Mesh generation is delicate in many situations, for instance, when the domain has complicated geometry; when the mesh changes with time, as in crack propagation, and remeshing is required at each time step; when a Lagrangian formulation is employed, especially with nonlinear PDEs. In addition, the need for flexibility in the selection of approximating functions ( e.g. , the flexibility to use non-polynomial approximating functions), has played a significant role in the development of meshless methods. There are many recent papers, and two books, on meshless methods; most of them are of an engineering character, without any mathematical analysis. In this paper we address meshless methods and the closely related generalized finite element methods for solving linear elliptic equations, using variational principles. We give a unified mathematical theory with proofs, briefly address implementational aspects, present illustrative numerical examples, and provide a list of references to the current literature. The aim of the paper is to provide a survey of a part of this new field, with emphasis on mathematics. We present proofs of essential theorems because we feel these proofs are essential for the understanding of the mathematical aspects of meshless methods, which has approximation theory as a major ingredient. As always, any new field is stimulated by and related to older ideas. This will be visible in our paper.
350 citations