Meshfree methods

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

The Generalized Interpolation Material Point Method

Abstract: The Material Point Method (MPM) discrete solution procedure for computational solid mechanics is generalized using a variational form and a Petrov- Galerkin discretization scheme, resulting in a family of methods named the Generalized Interpolation Material Point (GIMP) methods. The generalization permits iden- tification with aspects of other point or node based dis- crete solution techniques which do not use a body-fixed grid, i.e. the "meshless methods". Similarities are noted and some practical advantages relative to some of these methods are identified. Examples are used to demon- strate and explain numerical artifact noise which can be expected in MPM calculations. This noise results in non- physical local variations at the material points, where constitutive response is evaluated. It is shown to destroy the explicit solution in one case, and seriously degrade it in another. History dependent, inelastic constitutive laws can be expected to evolve erroneously and report inac- curate stress states because of noisy input. The noise is due to the lack of smoothness of the interpolation func- tions, and occurs due to material points crossing compu- tational grid boundaries. The next degree of smoothness available in the GIMP methods is shown to be capable of eliminating cell crossing noise. keyword: MPM, PIC, meshless methods, Petrov- Galerkin discretization.

Numerical methods for singularly perturbed differential equations : convection-diffusion and flow problems

Abstract: I. Ordinary Differential Equations: The analytical behaviour of solutions - numerical methods for second-order boundary value problems - numerical methods for higher-order problems II. Parabolic Initial-Boundary Value Problems in One Space Dimension: Analytical behaviour of solutions - finite difference methods - finite element methods - adaptive methods III. Elliptic Boundary Value Problems: Analytical behaviour of solutions - finite difference methods - finite element methods IV. Incompressible Navier-Stokes Equations: Existence and uniqueness results - an upwind finite element method - stabilized higher order methods - adaptive error control Appendix: Robust Solvers for Linear Systems

H‐p clouds—an h‐p meshless method

Abstract: A new methodology to build discrete models of boundary-value problems is presented. The h-pcloud method is applicable to arbitrary domains and employs only a scattered set of nodes to build approximate solutions to BVPs. This new method uses radial basis functions of varying size of supports and with polynomialreproducing properties of arbitrary order. The approximating properties of the h-p cloud functions are investigated in this article and a several theorems concerning these properties are presented. Moving least squares interpolants are used to build a partition of unity on the domain of interest. These functions are then used to construct, at a very low cost, trial and test functions for Galerkin approximations. The method exhibits a very high rate of convergence and has a greater -exibility than traditional h-p finite element methods. Several numerical experiments in I-D and 2-D are also presented. @ 1996 John Wiley & Sons, Inc. In most large-scale numerical simulations of physical phenomena, a large percentage of the overall computational effort is expended on technical details connected with meshing. These details include, in particular, grid generation, mesh adaptation to domain geometry, element or cell connectivity, grid motion and separation to model fracture, fragmentation, free surfaces, etc. Moreover, in most computer-aided design work, the generation of an appropriate mesh constitutes, by far, the costliest portion of the computer-aided analysis of products and processes. These are among the reasons that interest in so-called meshless methods has grown rapidly in recent years. Most meshless methods require a scattered set of nodal points in the domain of interest. In these methods, there may be no fixed connectivities between the nodes, unlike the finite element or finite difference methods. This feature has significant implications in modeling some physical phenomena that are characterized by a continuous change in the geometry of the domain under analysis. The analysis of problems such as crack propagation, penetration, and large deformations, can, in principle, be greatly simplified by the use of meshless methods. A growing crack, for example, can be modeled by simply extending the free surfaces that correspond to the crack [ 11. The analysis of large deformation problems by, e.g., finite element methods, may require the continuous remeshing of the domain to avoid the breakdown of the calculation due to

The Meshless Local Petrov-Galerkin (MLPG) Method: A Simple \& Less-costly Alternative to the Finite Element and Boundary Element Methods

Abstract: A comparison study of the efficiency and ac- curacy of a variety of meshless trial and test functions is presented in this paper, based on the general concept of the meshless local Petrov-Galerkin (MLPG) method. 5 types of trial functions, and 6 types of test functions are explored. Different test functions result in different MLPG methods, and six such MLPG methods are pre- sented in this paper. In all these six MLPG methods, absolutely no meshes are needed either for the interpo- lation of the trial and test functions, or for the integration of the weak-form; while other meshless methods require background cells. Because complicated shape functions for the trial function are inevitable at the present stage, in order to develop a fast and robust meshless method, we explore ways to avoid the use of a domain integral in the weak-form, by choosing an appropriate test function. The MLPG5 method (wherein the local, nodal-based test function, over a local sub-domain Ω s (or Ω te) centered at a node, is the Heaviside step function) avoids the need for both a domain integral in the attendant symmetric weak-form as well as a singular integral. Convergence studies in the numerical examples show that all of the MLPG methods possess excellent rates of convergence, for both the unknown variables and their derivatives. An analysis of computational costs shows that the MLPG5 method is less expensive, both in computational costs as well as definitely in human-labor costs, than the FEM, or BEM. Thus, due to its speed, accuracy and robustness, the MLPG5 method may be expected to replace the FEM, in the near future.

Meshfree Particle Methods

Abstract: Meshfree Particle Methods is a comprehensive and systematic exposition of particle methods, meshfree Galerkin and partitition of unity methods, molecular dynamics methods, and multiscale methods Most theories, computational formulations, and simulation results presented are recent developments in meshfree methods They were either just published recently or even have not been published yet, many of them resulting from the authors own research The presentation of the technical content is heuristic and explanatory with a balance between mathematical rigor and engineering practice It can be used as a graduate textbook or a comprehensive source for researchers, providing the state-of-the art on Meshfree Particle Methods