Showing papers on "Meshfree methods published in 1999"

Moving-least-squares-particle hydrodynamics-i. consistency and stability

Abstract: The Smooth-Particle-Hydrodynamics (SPH) method is derived in a novel manner by means of a Galerkin approximation applied to the Lagrangian equations of continuum mechanics as in the finite-element method. This derivation is modified to replace the SPH interpolant with the Moving-Least-Squares (MLS) interpolant of Lancaster and Saulkaskas, and define a new particle volume which ensures thermodynamic compatibility. A variable-rank modification of the MLS interpolants which retains their desirable summation properties is introduced to remove the singularities that occur when divergent flow reduces the number of neighbours of a particle to less than the minimum required. A surprise benefit of the Galerkin SPH derivation is a theoretical justification of a common ad hoc technique for variable-h SPH. The new MLSPH method is conservative if an anti-symmetric quadrature rule for the stiffness matrix elements can be supplied. In this paper, a simple one-point collocation rule is used to retain similarity with SPH, leading to a non-conservative method. Several examples document how MLSPH renders dramatic improvements due to the linear consistency of its gradients on three canonical difficulties of the SPH method: spurious boundary effects, erroneous rates of strain and rotation and tension instability. Two of these examples are non-linear Lagrangian patch tests with analytic solutions with which MLSPH agrees almost exactly. The examples also show that MLSPH is not absolutely stable if the problems are run to very long times. A linear stability analysis explains both why it is more stable than SPH and not yet absolutely stable and an argument is made that for realistic dynamic problems MLSPH is stable enough. The notion of coherent particles, for which the numerical stability is identical to the physical stability, is introduced. The new method is easily retrofitted into a generic SPH code and some observations on performance are made. Copyright © 1999 John Wiley & Sons, Ltd.

Numerical integration of the Galerkin weak form in meshfree methods

Abstract: The numerical integration of Galerkin weak forms for meshfree methods is investigated and some improvements are presented. The character of the shape functions in meshfree methods is reviewed and compared to those used in the Finite Element Method (FEM). Emphasis is placed on the relationship between the supports of the shape functions and the subdomains used to integrate the discrete equations. The construction of quadrature cells without regard to the local supports of the shape functions is shown to result in the possibility of considerable integration error. Numerical studies using the meshfree Element Free Galerkin (EFG) method illustrate the effect of these errors on solutions to elliptic problems. A construct for integration cells which reduces quadrature error is presented. The observations and conclusions apply to all Galerkin methods which use meshfree approximations.

A critical assessment of the truly Meshless Local Petrov-Galerkin (MLPG), and Local Boundary Integral Equation (LBIE) methods

Abstract: The essential features of the Meshless Local Petrov-Galerkin (MLPG) method, and of the Local Boundary Integral Equation (LBIE) method, are critically examined from the points of view of a non-element interpolation of the field variables, and of the meshless numerical integration of the weak form to generate the stiffness matrix. As truly meshless methods, the MLPG and the LBIE methods hold a great promise in computational mechanics, because these methods do not require a mesh, either to construct the shape functions, or to integrate the Petrov-Galerkin weak form. The characteristics of various meshless interpolations, such as the moving least square, Shepard function, and partition of unity, as candidates for trial and test functions are investigated, and the advantages and disadvantages are pointed out. Emphasis is placed on the characteristics of the global forms of the nodal trial and test functions, which are non-zero only over local sub-domains ΩtrJ and ΩteI, respectively. These nodal trial and test functions are centered at the nodes J and I (which are the centers of the domains ΩtrJ and ΩteI), respectively, and, in general, vanish at the boundaries ∂ΩtrJ and ∂ΩteI of ΩtrJ and ΩteI, respectively. The local domains ΩtrJ and ΩteI can be of arbitrary shapes, such as spheres, rectangular parallelopipeds, and ellipsoids, in 3-Dimensional geometries. The sizes of ΩtrJ and ΩteI can be arbitrary, different from each other, and different for each J, and I, in general. It is shown that the LBIE is but a special form of the MLPG, if the nodal test functions are specifically chosen so as to be the modified fundamental solutions to the differential equations in ΩteI, and to vanish at the boundary ∂ΩteI. The difficulty in the numerical integration of the weak form, to generate the stiffness matrix, is discussed, and a new integration method is proposed. In this new method, the Ith row in the stiffness matrix is generated by integrating over the fixed sub-domain ΩteI (which is the support for the test function centered at node I); or, alternatively the entry KIJ in the global stiffness matrix is generated by integrating over the intersections of the sub-domain ΩtrJ (which is the sub-domain, with node J as its center, and over which the trial function is non-zero), with ΩteI (which is the sub-domain centered at node I over which the test function is non-zero). The generality of the MLPG method is emphasized, and it is pointed that the MLPG can also be the basis of a Galerkin method that leads to a symmetric stiffness matrix. This paper also points out a new but elementary method, to satisfy the essential boundary conditions exactly, in the MLPG method, while using meshless interpolations of the MLS type. This paper presents a critical appraisal of the basic frameworks of the truly meshless MLPG/LBIE methods, and the numerical examples show that the MLPG approach gives good results. It now apears that the MLPG method may replace the well-known Galerkin finite element method (GFEM) as a general tool for numerical modeling, in the not too distant a future.

Reproducing kernel hierarchical partition of unity, Part II—applications

Abstract: In this part of the work, the meshless hierarchical partition of unity proposed in [1], referred here as Part I, is used as a multiple scale basis in numerical computations to solve practical problems. The applications discussed in the present work fall into two categories: (1) a wavelet adaptivity refinement procedure; and (2) a so-called wavelet Petrov–Galerkin procedure. In the applications of wavelet adaptivity, the hierarchical reproducing kernels are used as a multiple scale basis to compute the numerical solutions of the Helmholtz equation, a model equation of wave propagation problems, and to simulate shear band formation in an elasto-viscoplastic material, a problem dictated by the presence of the high gradient deformation. In both numerical experiments, numerical solutions with high resolution are obtained by inserting the wavelet-like basis into the primary interpolation function basis, a process that may be viewed as a spectral p-type refinement. By using the interpolant that has synchronized convergence property as a weighting function, a wavelet Petrov–Galerkin procedure is proposed to stabilize computations of some pathological problems in numerical computations, such as advection–diffusion problems and Stokes' flow problem; it offers an alternative procedure in stablized methods and also provides some insight, or new interpretation of the method. Detailed analysis has been carried out on the stability and convergence of the wavelet Petrov–Galerkin method. Copyright © 1999 John Wiley & Sons, Ltd.

A reproducing kernel particle method for meshless analysis of microelectromechanical systems

Abstract: Many existing computer-aided design systems for microelectromechanical systems require the generation of a three-dimensional mesh for computational analysis of the microdevice. Mesh generation requirements for microdevices are very complicated because of the presence of mixed-energy domains. Point methods or meshless methods do not require the generation of a mesh, and computational analysis can be performed by sprinkling points covering the domain of the microdevice. A corrected smooth particle hydrodynamics approach also referred to as the reproducing kernel particle method is developed here for microelectromechanical applications. A correction function that establishes the consistency and the stability of the meshless method is derived. A simple approach combining the constraint elimination and the Lagrange multiplier technique is developed for imposition of boundary conditions. Numerical results are shown for static and dynamic analysis of microswitches and electromechanical pressure sensors. The accuracy of the meshless method is established by comparing the numerical results obtained with meshless methods with previously reported experimental and numerical data.

Boundary and interface conditions meshless methods [for EM field analysis]

Abstract: When using meshless methods for electromagnetic field analysis, imposing boundary conditions and interface conditions are important sources of difficulties. This paper provides several answers to the issues: plain truncature of the balls; Lagrange multipliers technique; substitution method; and jump function. All those approaches are developed and compared.

Application of finite element methods to the simulation of semiconductor devices

Abstract: In this paper a survey is presented of the use of finite element methods for the simulation of the behaviour of semiconductor devices. Both ordinary and mixed finite element methods are considered. We indicate how the various mathematical models of semiconductor device behaviour can be obtained from the Boltzmann transport equation and the appropriate closing relations. The drift-diffusion and hydrodynamic models are discussed in more detail. Some mathematical properties of the resulting nonlinear systems of partial differential equations are identified, and general considerations regarding their numerical approximations are discussed. Ordinary finite element methods of standard and non-standard type are introduced by means of one-dimensional illustrative examples. Both types of finite element method are then extended to two-dimensional problems and some practical issues regarding the corresponding discrete linear systems are discussed. The possibility of using special non-uniform fitted meshes is noted. Mixed finite element methods of standard and non-standard type are described for both one- and two-dimensional problems. The coefficient matrices of the linear systems corresponding to some methods of non-standard type are monotone. Ordinary and mixed finite element methods of both types are applied to the equations of the stationary drift-diffusion model in two dimensions. Some promising directions for future research are described.

Convergence of meshless methods for conservation laws applications to Euler equations

Abstract: This paper is devoted to analyse new meshless methods. They generalize classical weighted particle methods for conservation laws. We prove that they can be both conservative and consistent. We obtain convergence of the methods in scalar case with the only requirement that the ratio of the smoothing length (or size of the cut-off) to the characteristic size of the mesh be bounded. Applications for Euler equations are proposed.

Multigrid Approach to Adaptive Analysis of B. V. Problems by The Meshless GFDM

Abstract: Meshless methods are the subject of increasing interest nowadays. As such considered is here the adaptive finite difference method generalized for arbitrary irregular grids (GFDM). This is an integrated approach including original concepts of a’posteriori error analysis, solution smoothing mesh generation and modification as well as special emphasis posed on multigrid solution approach. Although the GFDM itself has been a well established method for years [15], its fully adaptive formulation has only recently been proposed and outlined [17,18,20]. The method is designed as a general and powerful tool of analysis of large and very large discrete boundary-value problems. A fully adaptive GFDM multigrid solution approach is outlined here. Presented are new concepts of prolongation, restriction and solution procedure.

Numerical simulation of 2D semiconductor devices using high-order finite difference methods

Abstract: Numerical tests of a new accurate compact finite difference scheme for solving the 2D drift-diffusion system are presented EMAIL:: pietra@dragon.ian.pv.cnr.it