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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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Proceedings ArticleDOI
14 Aug 2009
TL;DR: An adaptive meshfree method based on local thin-plate spline radial basis interpolation and collocation schemes was applied to solve partial differential equations (PDEs) with high gradient, demonstrating that a good improvement of approximate accuracy can be obtained after using adative node refinement in high gradients.
Abstract: An adaptive meshfree method based on local thin-plate spline radial basis interpolation and collocation schemes was applied to solve partial differential equations (PDEs) with high gradient. Special attention was paid to the detection of high gradient regions, and the adjustment coefficient is adopted to gain these nodes which need to be refined. The numerical results demonstrate that a good improvement of approximate accuracy can be obtained after using adative node refinement in high gradients.

1 citations

Book ChapterDOI
01 Jan 2019
TL;DR: Expansion of the goal-oriented error estimation procedures presented in the preceding chapter to the finite hyperelasticity problem within both Newtonian and Eshelbian mechanics are derived for compressible and (nearly) incompressible materials in this chapter.
Abstract: Coming full circle in this chapter, expansions of the goal-oriented error estimation procedures presented in the preceding chapter to the finite hyperelasticity problem within both Newtonian and Eshelbian mechanics are derived for compressible and (nearly) incompressible materials. These error estimation procedures represent the most challenging ones presented in this monograph from both theoretical and numerical points of view. As a consequence, attention is focused on the derivation of error approximations rather than upper- or lower-bound error estimates. In the nonlinear case, a natural norm, such as the energy norm does not exist. The estimation of the general error measures introduced in the preceding chapter, on the other hand, does not necessarily rely on norm-based error estimators and thus allows for the derivation of a more versatile approach in a posteriori error estimation that can be employed in this chapter. Throughout this chapter, we confine ourselves to Galerkin mesh-based methods although similar error estimation procedures can also be developed for Galerkin meshfree methods.

1 citations

Journal Article
TL;DR: Introductions are made on meshless methods and on the basic principles and methods necessary for the implementation of essential boundary conditions.
Abstract: Introductions are made on meshless methods and on the basic principles and methods necessary for the implementation of essential boundary conditions Various possible implementing ways and characteristics are discussedThe drawbacks of the present methods are pointed out,and some remarks of author's own are proposed

1 citations

Book ChapterDOI
01 Jan 2007
TL;DR: In this article, meshless discretisation methods are explored in the implementation of nonlocal continuum damage theories, where the main advantage of using a meshless implementation is that the higher-order continuity requirements imposed by gradient-type nonlocality can be accomodated straightforwardly.
Abstract: In this chapter, meshless discretisation methods are explored in the implementation of nonlocal continuum damage theories. Integral-type and gradient-type nonlocality are both considered. The main advantage of using a meshless implementation (compared to more established discretisation methods such as the finite element method) is that the higher-order continuity requirements imposed by gradient-type nonlocality can be accomodated straightforwardly Thus, meshless methods are particularly suited as an implementational framework to test and compare various nonlocal theories. Here, the element-free Galerkin (EFG) method is used. In particular, second-order and fourth order gradient damage models are compared to integral-type damage models whereby the integral nonlocal operator acts on the equivalent strain or on the displacements. No signficant differences in response are found, which implies that the inclusion of a fourth-order term in the gradient-type nonlocality is of lesser importance. Finally, the mathematical non-locality of EFG interpolation functions is tested to ascertain whether it provides a mechanical nonlocality to the description. It is shown that this is not the case. However, despite this lack of intrinsic mechanical nonlocality, the EFG method is an excellent tool for the numerical implementation of a nonlocal continuum theory.

1 citations

A. Bracci1
01 Jan 2010
TL;DR: The main goal of the present work is to provide an easy-to-use Mathematica package that implements the basic features of some existing meshless methods for the simulation of solid elasticity in both 2D and 3D cases.
Abstract: Meshless methods are emerging techniques for the numerical solution of problems in the fields of solid mechanics, fluid-dynamics and, in general, partial differential equations (PDE) with boundary conditions (see [1] for a comprehensive introduction and [2, 3, 4, 5] for some of the last developments). Briefly, a meshless method consists in discretizing the problem-specific equations using a set of scattered points (nodes) inside the integration domain, instead of using an element-based mesh discretization as in the finite-element method (FEM). One of the main differences between the FEM and a Meshless method is the definition of the shape functions. In the former, shape functions are element-based and depend only on the nodal values of the current element. In the latter, instead, since there are no elements, the shape functions at a given point depend on some basis functions and on a set of neighbor nodes of the current point itself. Once the shape functions are defined, the PDE problem is discretized in a similar fashion to the FEM approach thus involving both surface and volume integrals to be numerically solved. As a result the discretized PDE problem is reformulated as a linear system to be solved. For instance, in a problem of linear elasticity, the PDE problem becomes Kx = f where K is the stiffness matrix, x and f are the unknown nodal displacement and the force vector respectively. The research in the field of meshless methods is relatively new and then several problems still have to be solved. Currently there is not a widelyused commercial software that implements these methods and hence no extensive test campaigns can be run. The main goal of the present work is to provide an easy-to-use Mathematica package that implements the basic features of some existing meshless methods for the simulation of solid elasticity in both 2D and 3D cases. The package is implemented in an object-oriented fashion and provides the user with the following features:

1 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202355
2022112
2021102
202092
201996
201897