Showing papers on "Meshfree methods published in 2000"

Simulating surface tension with smoothed particle hydrodynamics

Abstract: A method for simulating two-phase flows including surface tension is presented. The approach is based upon smoothed particle hydrodynamics (SPH). The fully Lagrangian nature of SPH maintains sharp fluid–fluid interfaces without employing high-order advection schemes or explicit interface reconstruction. Several possible implementations of surface tension force are suggested and compared. The numerical stability of the method is investigated and optimal choices for numerical parameters are identified. Comparisons with a grid-based volume of fluid method for two-dimensional flows are excellent. The methods presented here apply to problems involving interfaces of arbitrary shape undergoing fragmentation and coalescence within a two-phase system and readily extend to three-dimensional problems. Boundary conditions at a solid surface, high viscosity and density ratios, and the simulation of free-surface flows are not addressed. Copyright © 2000 John Wiley & Sons, Ltd.

Correction and stabilization of smooth particle hydrodynamics methods with applications in metal forming simulations

Abstract: Smooth particle hydrodynamics (SPH) is a robust and conceptually simple method which suffers from unsatisfactory performance due to lack of consistency. The kernel function can be corrected to enforce the consistency conditions and improve the accuracy. For simplicity in this paper the SPH method with the corrected kernel is referred to as corrected smooth particle hydrodynamics (CSPH). The numerical solutions of CSPH can be further improved by introducing an integration correction which also enables the method to pass patch tests. It is also shown that the nodal integration of this corrected SPH method suffers from spurious singular modes. This spatial instability results from under integration of the weak form, and it is treated by a least-squares stabilization procedure which is discussed in detail in Section 4. The effects of the stabilization and improvement in the accuracy are illustrated via examples. Further, the application of CSPH method to metal-forming simulations is discussed by formulating the governing equation associated with the process. Finally, the numerical examples showing the effectiveness of the method in simulating metal-forming problems are presented. Copyright © 2000 John Wiley & Sons, Ltd.

The method of finite spheres

Abstract: The objective of this paper is to present some of our recent developments in meshless methods In particular, a technique is given – the method of finite spheres – that is truly meshless in nature in the sense that the nodes are placed and the numerical integration is performed without a mesh The method can be viewed as a special case of the general formulation known as the meshless local Petrov–Galerkin (MLPG) procedure Some of the novel features of the method of finite spheres are the numerical integration scheme and the way in which the Dirichlet boundary conditions are incorporated A new way of modeling doubly-connected domains is also presented Various example problems are solved to demonstrate the method

New concepts in meshless methods

Abstract: Meshless methods have been extensively popularized in literature in recent years, due to their flexibility in solving boundary value problems. Two kinds of truly meshless methods, the meshless local boundary integral equation (MLBIE) method and the meshless local Petrov–Galerkin (MLPG) approach, are presented and discussed. Both methods use the moving least-squares approximation to interpolate the solution variables, while the MLBIE method uses a local boundary integral equation formulation, and the MLPG employs a local symmetric weak form. The two methods are truly meshless ones as both of them do not need a ‘finite element or boundary 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. Numerical examples presented in the paper show that high rates of convergence with mesh refinement are achievable. 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. Copyright © 2000 John Wiley & Sons, Ltd.

Normalized SPH with stress points

Abstract: Smoothed particle hydrodynamics is extended to a normalized, staggered particle formulation with boundary conditions. A companion set of interpolation points is introduced that carry the stress, velocity gradient, and other derived field variables. The method is stable, linearly consistent, and has an explicit treatment of boundary conditions. Also, a new method for finding neighbours is introduced which selects a minimal and robust set and is insensitive to anisotropy in the particle arrangement. Test problems show that these improvements lead to increased accuracy and stability. Published in 2000 by John Wiley & Sons, Ltd.

Enrichment and coupling of the finite element and meshless methods

Abstract: A mixed hierarchical approximation based on finite elements and meshless methods is presented. Two cases are considered. The first one couples regions where finite elements or meshless methods are used to interpolate: continuity and consistency is preserved. The second one enriches a finite element mesh with particles. Thus, there is no need to remesh in adaptive refinement processes. In both cases the same formulation is used, convergence is studied and examples are shown. Copyright © 2000 John Wiley & Sons, Ltd.

A Particle-Partition of Unity Method for the Solution of Elliptic, Parabolic, and Hyperbolic PDEs

Abstract: In this paper, we present a meshless discretization technique for instationary convection-diffusion problems. It is based on operator splitting, the method of characteristics, and a generalized partition of unity method. We focus on the discretization process and its quality. The method may be used as an h-version or a p-version. Even for general particle distributions, the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid. We discuss the implementational aspects of the proposed method. Furthermore, we present the results of numerical examples, where we considered instationary convection-diffusion, instationary diffusion, linear advection, and elliptic problems.

Numerical simulations of large deformation of thin shell structures using meshfree methods

Abstract: In this paper, meshfree simulations of large deformation of thin shell structures is presented. It has been shown that the window function based meshfree interpolants can be used to construct highly smoothed (high order “manifold”) shape functions for three-dimensional (3-D) meshfree discretization/interpolation, which can be used to simulate large deformation of thin shell structures while avoiding ill-conditioning as well as stiffening in numerical computations. The main advantage of such 3-D meshfree continuum approach is its simplicity in both formulation and implementation as compared to shell theory approach, or degenerated continuum approach. Moreover, it is believed that the accuracy of the computation may increase because of using 3-D exact formulation. Possible mechanism to relieve shear/volumetric locking due to the meshfree interpolation is discussed. Several examples have been computed by using a meshfree, explicit, total Lagrangian formulation. Towards to developing a self-contact algorithm, a novel meshfree contact algorithm is proposed in the end.

Dispersion analysis and element‐free Galerkin solutions of second‐ and fourth‐order gradient‐enhanced damage models

Abstract: Gradient-dependent damage formulations incorporate higher-order derivatives of state variables in the constitutive equations. Different formulations have been derived for this gradient enhancement, comparison of which is difficult in a finite element context due to higher-order continuity requirements for certain formulations. On the other hand, the higher-order continuity requirements are met naturally by element-free Galerkin (EFG) shape functions. Thus, the EFG method provides a suitable tool for the assessment of gradient enhanced continuum models. Dispersion analyses have been carried out to compare different gradient enhanced models with the non-local damage model. The formulation of the additional boundary conditions is addressed. Numerical examples show the objectivity with respect to the discretization and the differences between various gradient formulations with second- and fourth-order derivatives. It is shown that with the same underlying internal length scale, very different results can be obtained. Copyright © 2000 John Wiley & Sons, Ltd.

Imposing essential boundary conditions in the natural element method by means of density-scaled?-shapes

Abstract: A new method is proposed for the ‘exact’ imposition of essential boundary conditions in the context of the natural element method (NEM). This is a new technique in the field of Computational Mechanics and can be considered as a meshless method. Unlike most of these methods, the NE shape functions are strictly interpolant and the essential boundary conditions can be imposed by directly substituting the corresponding terms in the system of equations. However, these shape functions are not strictly linear over non-convex boundaries and the approach does not make the test functions vanish over the whole essential boundary region. A modification of the initial NEM version is considered based on α-shapes and α-complexes, which are widely used in the field of scientific visualization. Using α-shapes in the context of the NEM allows the construction of models entirely in terms of nodes and also ensures the linear precision of the interpolant over convex and non-convex boundaries. Results on some benchmark problems are presented after a theoretical description of the method. Copyright © 2000 John Wiley & Sons, Ltd.

The Finite Volume, Finite Difference, and Finite Elements Methods as Numerical Methods for Physical Field Problems

Abstract: Publisher Summary This chapter presents a set of conceptual tools for the formulation of physical field problems in discrete terms. These tools allow the representation of the geometry and of the fields in discrete terms and those of chain and cochain. Moreover, they allow bridging the gap between the continuous and the discrete concept of field by means of the idea of limit systems. Analysis of the structure of physical field theories is based on these tools. This analysis unveils the importance of thinking of physical quantities as associated with space–time oriented geometric objects. Moreover, this analysis exposes the distinction of topological laws from constitutive relations showing their different behavior from the point of view of their discretizability. A privileged discrete operator—the coboundary operator—exists for the representation of topological laws. The chapter discusses a reference discretization strategy that complies with these concepts. It is based on the idea of topological time stepping for time-dependent equations that operates on global quantities and derives from the application of the coboundary operator in space–time.

Physics-based Modeling of Brittle Fracture: Cohesive Formulations and the Application of Meshfree Methods

Abstract: Simulation of generalized fracture and fragmentation remains an ongoing challenge in computational fracture mechanics There are difficulties associated not only with the formulation of physically-based models of material failure, but also with the numerical methods required to treat geometries that change in time The issue of fracture criteria is addressed in this work through a cohesive view of material, meaning that a finite material strength and work to fracture are included in the material description In this study, we present both surface and bulk cohesive formulations for modeling brittle fracture, detailing the derivation of the formulations, fitting relations, and providing a critical assessment of their capabilities in numerical simulations of fracture Due to their inherent adaptivity and robustness under severe deformation, meshfree methods are especially well-suited to modeling fracture behavior We describe the application of meshfree methods to both bulk and surface approaches to cohesive modeling We present numerical examples highlighting the capabilities and shortcomings of the methods in order to identify which approaches are best-suited to modeling different types of fracture phenomena

Multi-scale methods

Abstract: In this paper four multiple scale methods are proposed. The meshless hierarchical partition of unity is used as a multiple scale basis. The multiple scale analysis with the introduction of a dilation parameter to perform multiresolution analysis is discussed. The multiple field based on a 1-D gradient plasticity theory with material length scale is also proposed to remove the mesh dependency difficulty in softening/localization problems. A non-local (smoothing) particle integration procedure with its multiple scale analysis are then developed. These techniques are described in the context of the reproducing kernel particle method. Results are presented for elastic-plastic one-dimensional problems and 2-D large deformation strain localization problems to illustrate the effectiveness of these methods. Copyright © 2000 John Wiley & Sons, Ltd.

Explicit and implicit meshless methods for linear advection–diffusion-type partial differential equations

Abstract: Simple, mesh/grid free, explicit and implicit numerical schemes for the solution of linear advection-diffusion problems is developed and validated herein. Unlike the mesh or grid-based methods, these schemes use well distributed quasi-random points and approximate the solution using global radial basis functions. The schemes can be seen as generalized finite differences with random points instead of a regular grid system. This allows the computation of problems with complex-shaped boundaries in higher dimensions with no need for complex mesh/grid structure and with no extra implementation difficulties.

Implementation of essential boundary conditions in a meshless method

Abstract: Accurate imposition of essential boundary conditions is a main drawback in the use of the element-free Galerkin (EFG) method. A way to solve the problem, is to use a constrained variational principle with a penalty function. This new treatment for essential boundary conditions is simple and logical and works very well in all numerical examples for 2-D potential problems that are presented here, considering an approximation close to an interpolation. It is shown that the present constrained variational formulation together with the EFG method and appropriated weighting function exhibit very high accuracy and stability, for regular and irregular grids of nodes. Copyright © 2000 John Wiley & Sons, Ltd.

Some recent improvements in meshfree methods for incompressible finite elasticity boundary value problems with contact

Abstract: Two major difficulties are encountered in the meshfree solution of incompressible boundary value problems. The first is due to the employment of higher-order quadrature rules that leads to an over-constrained discrete system in incompressible problems. The second is associated with the treatment of essential boundary conditions and contact conditions owing to the loss of Kronecker delta properties in the meshfree shape functions. This paper discusses some recent enhancements in meshfree methods for incompressible boundary value problems, carries out numerical convergence analysis, and compares accuracy and efficiency improvement of these methods. Presented methods are a pressure projection method to remedy the over-constrained discrete system, and a mixed transformation method and a boundary singular kernel method for imposition of essential boundary conditions and contact constraints. Several linear and nonlinear problems were analyzed to demonstrate the effectiveness of the new approaches.

An efficient linear‐precision partition of unity basis for unstructured meshless methods

Abstract: We describe an approach to construct approximation basis functions for meshless methods, which is based on the concept of a partition of unity. The approach has the following properties: (i) the grid consists of scattered nodes, (ii) the basis reproduces exactly complete linear polynomials, (iii) only the values of the approximated function at the nodes are used as unknowns, (iv) the construction of the basis is only slightly more expensive than the Shepard constant-precision method, and finally, (v) the method is applicable in any number of spatial dimensions.

Multiple-scale meshfree adaptivity for the simulation of adiabatic shear band formation

Abstract: The Reproducing Kernel Particle Method (RKPM), one of the major meshfree methods, is applied to the analysis of shear localization problem. The total Lagrangian formulation of RKPM and explicit time integration are employed in order to simulate the adiabatic shear band formation in a thermo-viscoplastic material. The multiresolution study based on multiple-scale RKPM is then accomplished and an efficient measure of the high-scale component is accordingly proposed for meshfree adaptive procedures. A time-marching strategy is also suggested and successfully implemented for the process of transient adaptive refinement. Numerical examples are finally presented to demonstrate the performance of meshfree adaptivity and to ensure the feasibility of the high-scale detector for multiple-scale refinements.

A Meshless Method for the Numerical Solution of the 2- and 3-D Semiconductor Poisson Equation

Abstract: This paper describes the application of the meshless Finite Point (FP) method to the solution of the nonlinear semiconductor Poisson equation. The FP method is a true meshless method which uses a weighted least-squares fit and point collocation. The nonlinearity of the semiconductor Poisson equation is treated by Newton-Raphson iteration, and sparse matrices are employed to store the shape function and coefficient matrices. Using examples in twoand threedimensions (2and 3-D) for a prototypical n-channel MOSFET, the FP method demonstrates promise both as a means of mesh enhancement and for treating problems where arbitrary point placement is advantageous, such as for the simulation of carrier wave packet and dopant cloud effects in the ensemble Monte Carlo method. The validity of the solutions and the capability of the method to treat arbitrary boundary conditions is shown by comparison with finite difference results. keyword: finite point methods, meshless methods, mesh generation, monte carlo methods.

Modeling groundwater flow by the element-free Galerkin (EFG) method

Abstract: The element-free Galerkin (EFG) method is one of meshless methods, which is a very powerful, efficient and accurate method of modeling problems of fluid or solid mechanics with complex boundary shapes and large changes in boundary conditions. This paper reports the theory and the first-known application of the EFG method to groundwater flow modeling. The EFG method constructs shape functions based on moving least square (MLS) approximations, which do not require any element but only a set of nodes. Thus, the EFG method eliminates time-consuming mesh generation procedure with irregular shaped boundaries. The coupled EFG-FEM technique was used to treat Dirichlet boundary conditions. A computer code EFGGW was developed for the problems of steady-state and transient groundwater flow in homogeneous or heterogeneous aquifers. Solutions by the EFG method were similar in accuracy to that by the FEM. The main advantages of the method are the convenience of node generation and the enforced implementation of boundary conditions.

Finite increment gradient stabilization of point integrated meshless methods for elliptic equations

Abstract: This paper describes a new technique to stabilize meshless methods used in conjunction with point-based integration. The method proposed is based on the finite increment calculus (FIC) concepts for convection-dominated problems. In this paper a finite increment gradient operator is defined in such a way that second-order derivatives are included. This operator is then used in the context of a variational formulation of an elliptic problem in order to define a stabilized numerical procedure. For simplicity, the Poisson equation will be used in this paper to illustrate the method, although more general elliptic problems can be equally treated. An eigenvalue analysis will be carried out in order to demonstrate that no mechanisms are present in the resulting equations. Finally, a simple example will illustrate the technique. Copyright © 2000 John Wiley & Sons, Ltd.

Computational Mechanics: Techniques and Developments

Abstract: The papers in this volume include the following topics: Analysis of Mechanisms Meshless Methods Boundary Elements Finite Element & Boundary Element Methods Differential Quadrature Method Dynamics Problems Stochastic Methods of Analysis Analysis of Fracture and Cracking Contact and Contact Detection Fluid-Structure Interaction and Computational Fluid Mechanics.

Efficient interfacing of fluid and structure for aeroelastic instability predictions

Abstract: An efficient procedure for multidisciplinary computation of fluid and structure interaction problems of aerospace vehicles is presented. It features the use of Meshless Methods, Finite Elements, Rayleigh–Ritz, and Kernel Functions enabling the aeroelastic requirements to be included in design without demanding prohibitive computational efforts. Improvements made are in terms of choosing meshless fluid–structure interface for displacement and aerodynamic load transfer, including nine modes in Rayleigh–Ritz, introducing Gradient Adaptive Transfinite Element in Finite Element model, and taking into account deformations of structure in aerodynamic load calculations. Numerical results are presented for flutter and divergence type aeroelastic responses of lifting surfaces constructed of advanced composites and supper alloys at subsonic, supersonic, and hypersonic flight velocities. Copyright © 2000 John Wiley & Sons, Ltd.