Showing papers on "Meshfree methods published in 1996"

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

Admissible approximations for essential boundary conditions in the reproducing kernel particle method

Abstract: In the reproducing kernel particle method (RKPM), and meshless methods in general, enforcement of essential boundary conditions is awkward as the approximations do not satisfy the Kronecker delta condition and are not admissible in the Galerkin formulation as they fail to vanish at essential boundaries Typically, Lagrange multipliers, modified variational principles, or a coupling procedure with finite elements have been used to circumvent these shortcomings

Alternatives to Finite Difference Methods in Numerical Relativity

Abstract: Finite difference techniques have played a dominant role in numerical relativity. This situation will likely prevail; for instance, the current grand challenge effort in the United States to simulate, by the end of the century, black-hole collisions is entirely based on finite difference codes. Furthermore, the power of finite difference techniques has recently been enhanced with the implementation of adaptive mesh refinements. In spite of the finite difference success, there have been a significant number of numerical studies in gravitation in which finite difference methods are either not used, or applied in combination with other techniques. These lectures review four alternative approaches to numerical relativity from those of finite difference. The first lecture addresses solutions to the initial-data problem in general relativity using multiquadrics and finite elements methods. The second lecture reviews particle-mesh and smoothed particle hydrodynamics methods used in conjunction with finite differences to solve the Einstein-hydro field equations.