Showing papers on "Meshfree methods published in 1998"

A new Meshless Local Petrov-Galerkin (MLPG) approach in computational mechanics

Abstract: A local symmetric weak form (LSWF) for linear potential problems is developed, and a truly meshless method, based on the LSWF and the moving least squares approximation, is presented for solving potential problems with high accuracy. The essential boundary conditions in the present formulation are imposed by a penalty method. The present method does not need a “finite element mesh”, either for purposes of interpolation of the solution variables, or for the integration of the “energy”. All integrals can be easily evaluated over regularly shaped domains (in general, spheres in three-dimensional problems) and their boundaries. No post-smoothing technique is required for computing the derivatives of the unknown variable, since the original solution, using the moving least squares approximation, is already smooth enough. Several numerical examples are presented in the paper. In the example problems dealing with Laplace & Poisson's equations, high rates of convergence with mesh refinement for the Sobolev norms ||·||0 and ||·||1 have been found, and the values of the unknown variable and its derivatives are quite accurate. In essence, the present meshless method based on the LSWF is found to be a simple, efficient, and attractive method with a great potential in engineering applications.

EFG approximation with discontinuous derivatives

Abstract: A technique for incorporating discontinuities in derivatives into meshless methods is presented. The technique enriches the approximation by adding special shape functions that contain discontinuities in derivatives. The special shape functions have compact support which results in banded matrix equations. The technique is described in element-free Galerkin context, but is easily applicable to other meshless methods and projections. Numerical results for problems in one and two dimensions are reported. © 1998 John Wiley & Sons, Ltd.

Reducing the phase error for finite-difference methods without increasing the order

Abstract: The phase error in finite-difference (FD) methods is related to the spatial resolution and thus limits the maximum grid size for a desired accuracy. Greater accuracy is typically achieved by defining finer resolutions or implementing higher order methods. Both these techniques require more memory and longer computation times. In this paper, new modified methods are presented which are optimized to problems of electromagnetics. Simple methods are presented that reduce numerical phase error without additional processing time or memory requirements. Furthermore, these methods are applied to both the Helmholtz equation in the frequency domain and the finite-difference time-domain (FDTD) method. Both analytical and numerical results are presented to demonstrate the accuracy of these new methods.

Some meshless methods for electromagnetic field computations

Abstract: A presentation of some meshless methods of approximation is proposed These numerical methods are very attractive since they have good convergence rates for the unknown function and its derivatives This behaviour is observed on several numerical examples A meshless Galerkin method is developed Application to the post-treatment and the computation of electromagnetic fields is reviewed

Progress in smooth particle hydrodynamics

Abstract: Smooth Particle Hydrodynamics (SPH) is a meshless, Lagrangian numerical method for hydrodynamics calculations where calculational elements are fuzzy particles which move according to the hydrodynamic equations of motion. Each particle carries local values of density, temperature, pressure and other hydrodynamic parameters. A major advantage of SPH is that it is meshless, thus large deformation calculations can be easily done with no connectivity complications. Interface positions are known and there are no problems with advecting quantities through a mesh that typical Eulerian codes have. These underlying SPH features make fracture physics easy and natural and in fact, much of the applications work revolves around simulating fracture. Debris particles from impacts can be easily transported across large voids with SPH. While SPH has considerable promise, there are some problems inherent in the technique that have so far limited its usefulness. The most serious problem is the well known instability in tension leading to particle clumping and numerical fracture. Another problem is that the SPH interpolation is only correct when particles are uniformly spaced a half particle apart leading to incorrect strain rates, accelerations and other quantities for general particle distributions. SPH calculations are also sensitive to particle locations. The standard artificial viscosity treatment in SPH leads to spurious viscosity in shear flows. This paper will demonstrate solutions for these problems that they and others have been developing. The most promising is to replace the SPH interpolant with the moving least squares (MLS) interpolant invented by Lancaster and Salkauskas in 1981. SPH and MLS are closely related with MLS being essentially SPH with corrected particle volumes. When formulated correctly, JLS is conservative, stable in both compression and tension, does not have the SPH boundary problems and is not sensitive to particle placement. The other approach to solving SPH problems, pioneered by Randles and Libersky, is to use a different SPH equation and to renormalize the kernel gradient sums. Finally the authors present results using the SPH statistical fracture model (SPHSFM). It has been applied to a series of ball on plate impacts performed by Grady and Kipp. A description of the model and comparison with the experiments will be given.

Application of meshless methods to the analysis and design of groundingsystems

Abstract: Analysis and design of grounding systems of electrical installations involves computing the potential distribution in the earth and the equivalent resistance of the system. Several numerical formulations based on the Boundary Element Method have recently been derived for grounding grids embedded in uniform soils and in stratified soils, which feasibility has been demonstrated with its application to large earthing systems in a two-layer soil. In cases of the analysis of grounding systems buried in more stratified soils or heterogeneous, the application of Boundary Element approaches can require a considerable computational effort. On the other hand, the specific geometry of earthing systems in practice (a grid of interconnected buried conductors) precludes the use of standard numerical techniques (such as finite elements or finite differences), since discretization of the domain (the earth) is required and the obtention of sufficiently accurate results should imply unacceptable computing efforts. For these reasons, we have turned our attention to investigate the applicability of numerical formulations based on meshless methods for the grounding analysis. In this paper, a meshless technique based on the Moving Least Square method with a point collocation approach is proposed.

A point collocation method for meshless analysis of microelectronic and microelectromechanical devices

Abstract: The present approach to modeling and design of microelectronic and microelectromechanical devices (MEMS) (hereafter simply referred to as a microdevice) involves the generation of a geometric model for the complicated two or three-dimensional microdevice, the generation of a mesh for the geometric model, a mesh based numerical analysis, and postprocessing steps such as visualization. The time consuming steps in such an approach are the generation of a mesh and mesh-based numerical analysis. Meshless methods, which do not require the generation of a mesh, are very attractive for numerical solution of partial differential equations. In this paper we introduce a new meshless technique, referred to as a point collocation method, for numerical solution of partial differential equations. The effectiveness of the point collocation method is demonstrated by solving the one and two-dimensional Poisson equation, the solution of which is required in the analysis of both electronic and microelectromechanical devices.

An Application of Meshless Methods in Modeling of Damaged Rock Masses

Abstract: A two-dimensional code coupling Element-Free Galerkin Method (EFGM) and Finite Elements Method (FEM) as described by Belytschko et al. (1995) is used to simulate the behavior of elasto-damaged materials. The EFGM is used in some regions of interest and the FEM in the other parts. Two damage models are implemented and tested. The first is a continuum damage model as described by Homand et al. (1998). The second is a micromechanical damage model based on the works of Kachanov (1982, 1994) and those of Nemat-Nasser & Obata (1988). Some simulations of laboratory tests using the micromechanical model are performed to check its applicability. Then the damage around a circular gallery was studied using the two models.

Numerical solution: Some finite difference methods

Abstract: As noted in Section 12.5, the use of a theoretical model of a materials synthesis or fabrication process requires that an equation of the form be $$\frac{{\partial f}}{{\partial t}} = - v \bullet abla f + {c_{3}}{ abla ^{2}}f + {c_{4}}{s_{\psi}}$$ (14.1.1) solved at all relevant point within the material, for all times of interest. This chapter, which deals with the solution of En. (14.1.1) using finite difference techniques, is the first of two chapters (14 and 15) that describe important and useful methods for determining the behavior of f--and thus that of the material itself--as a function of position and time. The finite difference methods are perhaps the simplest of the approximation methods to use. And, when using the finite difference methods, it is also possibly easiest to visualize the connections between the approximate form of the equation and the “original” equation that it approximates. The finite difference methods, moreover, are relatively easy to implement when the shape of the material (to which the model equations pertain) is geometrically simple, and when the equation that accounts for a “conserved quantity” (ψ) is mathematically well-behaved.1 Finally, the finite difference methods are used to convert the “original” equation [of the form of En. (14.1.1)] in such a way that it is amenable to numerical solution.

On the application of a meshless method to shear band problems

Abstract: Themultiple-scale Reproducing Kernel Particle Method (RKPM), one of the meshless methods, is applied to the analysis of a strain localization problem. The LagrangianRKPM formulation and explicit time integration are employed for the simulation of shear band formation in a viscoplastic material. The mul-tiresolution study is also performed using the multiple-scaleRKPM and then, an efficient high-scale component from the viewpoint of meshless method is proposed for mesh-free adaptive procedures that will be accomplished in the future.