Showing papers on "Meshfree methods published in 2005"

An Introduction to Meshfree Methods and Their Programming

Abstract: This book aims to present meshfree methods in a friendly and straightforward manner, so that beginners can very easily understand, comprehend, program, implement, apply and extend these methods. It provides first the fundamentals of numerical analysis that are particularly important to meshfree methods. Typical meshfree methods, such as EFG, RPIM, MLPG, LRPIM, MWS and collocation methods are then introduced systematically detailing the formulation, numerical implementation and programming. Many well-tested computer source codes developed by the authors are attached with useful descriptions. The application of the codes can be readily performed using the examples with input and output files given in table form. These codes consist of most of the basic meshfree techniques, and can be easily extended to other variations of more complex procedures of meshfree methods. Readers can easily practice with the codes provided to effective learn and comprehend the basics of meshfree methods.

Physically Based Deformable Models in Computer Graphics.

Abstract: Physically based deformable models have been widely embraced by the Computer Graphics community. Many problems outlined in a previous survey by Gibson and Mirtich [ GM97] have been addressed, thereby making these models interesting and useful for both offline and real-time applications, such as motion pictures and video games. In this paper, we present the most significant contributions of the past decade, which produce such impressive and perceivably realistic animations and simulations: finite element/difference/volume methods, mass-spring systems, meshfree methods, coupled particle systems and reduced deformable models based on modal analysis. For completeness, we also make a connection to the simulation of other continua, such as fluids, gases and melting objects. Since time integration is inherent to all simulated phenomena, the general notion of time discretization is treated separately, while specifics are left to the respective models. Finally, we discuss areas of application, such as elastoplastic deformation and fracture, cloth and hair animation, virtual surgery simulation, interactive entertainment and fluid/smoke animation, and also suggest areas for future research.

Adaptivity for structured meshfree particle methods in 2D and 3D

Abstract: The implementation of h-adaptivity for meshfree particle methods within a structured framework is described In this framework, the initial particle arrangement is structured along with a background mesh, and outside boundaries and interior interfaces are described by implicit functions The advantage of meshfree approximations in this framework lies in the ease of implementing h-adaptivity and the simplicity of the data structures Particles can easily be added and removed without complications in the data structure, although there are some issues in the quadrature An a posteriori error estimation is used for the adaptive refinement An adaptive refinement strategy is applied to several linear elastic problems with high stress and strain gradients and singularities Several non-linear examples are also given Copyright © 2005 John Wiley & Sons, Ltd

A meshfree radial point interpolation method (RPIM) for three-dimensional solids

Abstract: A meshfree radial point interpolation method (RPIM) is developed for stress analysis of three-dimensional (3D) solids, based on the Galerkin weak form formulation using 3D meshfree shape functions constructed using radial basis functions (RBFs). As the RPIM shape functions have the Kronecker delta functions property, essential boundary conditions can be enforced as easily as in the finite element method (FEM). Numerical examples of 3D solids are presented to verify validity and accuracy of the present RPIM method, and intensive numerical study has been conducted to investigate the effects of some important parameters. It is demonstrated that the present meshfree RPIM is robust, stable, and reliable for stress analysis of 3D solids.

Meshfree methods and their comparisons

Abstract: In recent years, one of the hottest topics in computational mechanics is the meshfree or meshless method. Increasing number of researchers are devoting themselves to the research of the meshfree methods, and a group of meshfree methods have been proposed and used to solve the ordinary differential equations (ODEs) or the partial differential equations (PDE). In the meantime, meshfree methods are being applied to a growing number of practical engineering problems. In this paper, a detailed discussion will be provided on the development of meshfree methods. First, categories of meshfree methods are introduced. Second, the methods for constructing meshfree shape functions are discussed, and the interpolation qualities of them are also studied using the surface fitting. Third, several typical meshfree methods are introduced and compared with each others in terms of their accuracy, convergence and effectivity. Finally, the major technical issues in meshfree methods are discussed, and the future development of meshfree methods is addressed.

On the use of nonstandard finite difference methods

Abstract: Many real life problems are modelled by differential equations, for which analytical solutions are not always easy to find. One of the most difficult problems is how to solve these differential equations efficiently. Several researchers have tried to do this in various different ways (e.g. via Finite Element Methods, Standard Finite Difference Methods, Spline Approximation Methods, etc.). In recent years, to get reliable results with less effort, researchers have applied nonstandard finite difference methods (NSFDMs) and obtained competitive results to those obtained with other methods. In this survey article, the author tries to provide as much stimulating information as available regarding these NSFDMs to the researchers, which will be helpful for them as the research proceeds in this direction. While the author made the utmost efforts to include whatever he could, he would like to apologize if there are any omissions which are totally unintentional.

SPH simulations of transient viscoelastic flows at low Reynolds number

Abstract: A numerical study on the transient flow of a viscoelastic fluid is presented. The numerical framework is that of Smoothed Particle Hydrodynamics (SPH) already used by Ellero et al. in previous simulations of Non-Newtonian flows [J. Non-Newtonian Fluid. Mech. 105 (2002) 35–51]. In particular, the start-up flow between parallel plates is simulated for an Oldroyd-B and UCM fluid at low Reynolds number. Results for a Newtonian fluid are also shown for comparison. The numerical results are presented and compared with available theoretical solutions, showing a very good agreement. In particular, the simulations of an Oldroyd-B fluid have been found to be stable and accurate for a wide range of the Weissenberg number. In the case of a UCM fluid, the absence of a viscous term in the momentum equation makes its numerical modelling harder. Namely, the process is characterised by a travelling damped wave, which, if not accurately resolved, can lead to the rapid growth of small oscillations in time eventually causing divergence. On the other hand, if specifically dealing with transient flow problems, stabilising techniques such as BSD, EVSS or AVSS can not be used either; whenever used in conjunction with decoupled solution algorithms, they give an excessive oversmoothing in the results which deteriorates the final accuracy. In this work, we consider an exact SPH discretisation of the hyperbolic equation characterising the UCM model. SPH simulations are finally performed for different Weissenberg numbers showing very promising results. Finally, a discussion on the SPH treatment of the boundary conditions for general hydrodynamics problems is also outlined following the approach of ‘SPH boundary particles’ introduced by Morris for the simulations of low Reynolds number flows.

A meshless method based on least-squares approach for steady- and unsteady-state heat conduction problems

Abstract: The meshless method based on the least-squares approach, the meshless weighted leastsquares (MWLS) method, is extended to solve conduction heat transfer problems. The MWLS formulation is first established for steady-state problems and then extended to unsteady-state problems with time-stepping schemes. Theoretical analysis and numerical examples indicate that larger time steps can be used in the present method than in meshless methods based on the Galerkin approach. Numerical studies show that the proposed method is a truly meshless method with good accuracy, high convergence rate, and high efficiency.

Numerical comparison of two meshfree methods for acoustic wave scattering

Abstract: Density results using an infinite number of acoustic waves allow us to derive meshless methods for solving the homogeneous and the inhomogeneous Helmholtz equation. In this paper we consider the numerical simulation of acoustic scattering problems in a bounded domain using the plane waves method and the method of fundamental solutions. We establish a link between the two methods, namely the plane waves method may be seen as the asymptotic case of the method of fundamental solutions for distant source points. Several numerical tests comparing these methods are presented.

A meshfree weak-strong (MWS) form method for time dependent problems

Abstract: A meshfree weak-strong (MWS) form method, which is based on a combination of both the strong form and the local weak form, is formulated for time dependent problems. In the MWS method, the problem domain and its boundary are represented by a set of distributed field nodes. The strong form or the collocation method is used to discretize the time-dependent governing equations for all nodes whose local quadrature domains do not intersect with natural (derivative or Neumann) boundaries. Therefore, no numerical integration is required for these nodes. The local weak form, which needs the local numerical integration, is only used for nodes on or near the natural boundaries. The natural boundary conditions can then be easily imposed to produce stable and accurate solutions. The moving least squares (MLS) approximation is used to construct the meshfree shape functions in this study. Numerical examples of the free vibration and dynamic analyses of two-dimensional structures as well as a typical microelectromechanical system (MEMS) device are presented to demonstrate the effectivity, stability and accuracy of the present MWS formulation.

A meshless method for conjugate heat transfer problems

Abstract: Mesh reduction methods such as boundary element methods, method of fundamental solutions, and spectral methods all lead to fully populated matrices. This poses serious challenges for large-scale three-dimensional problems due to storage requirements and iterative solution of a large set of non-symmetric equations. Researchers have developed several approaches to address this issue including the class of fast-multipole techniques, use of wavelet transforms, and matrix decomposition. In this paper, we develop a domain decomposition, or the artificial sub-sectioning technique, along with a region-by-region iteration algorithm particularly tailored for parallel computation to address the coefficient matrix issue. The meshless method we employ is based on expansions using radial-basis functions (RBFs). An efficient physically based procedure provides an effective initial guess of the temperatures along the sub-domain interfaces. The iteration process converges very efficiently, offers substantial savings in memory, and features superior computational efficiency. The meshless iterative domain decomposition technique is ideally suited for parallel computation. We discuss its implementation under MPI standards on a small Windows XP PC cluster. Numerical results reveal the domain decomposition meshless methods produce accurate temperature predictions while requiring a much-reduced effort in problem preparation in comparison to other traditional numerical methods.

Meshfree Simulations of Ductile Crack Propagations

Abstract: In this work, a meshfree method is used to simulate ductile crack growth and propagation under finite deformation and large scale yielding conditions. A so-called parametric visibility condition and its related particle splitting procedure have been developed to automatically adapt the evolving strong continuity or fracture configuration due to an arbitrary crack growth in ductile materials. It is shown that the proposed meshfree crack adaption and re-interpolation procedure is versatile in numerical simulations, and it can capture some essential features of ductile fracture and ductile crack surface morphology, such as the rough zig-zag pattern of crack surface and the ductile crack front damage zone, which have been difficult to capture in previous numerical simulations.

On Approximation in Meshless Methods

Abstract: We analyze the approximation properties of some meshless methods Three types of functions systems are discussed: systems of functions that reproduce polynomials, a class of radial basis functions, and functions that are adapted to a differential operator Additionally, we survey techniques for the enforcement of essential boundary conditions in meshless methods

Meshfree particle simulation of micro channel flows with surface tension

Abstract: This paper presents a study of micro channel flows using a meshfree particle approach. The approach is based on smoothed particle hydrodynamics (SPH) and its variant, adaptive smoothed particle hydrodynamics (ASPH). The incompressible flow in the micro channels is modeled as an artificially compressible flow. The surface tension is incorporated into the equations of motion. The classic Poiseuille flow and a practical micro channel flow problem of flip-chip underfill encapsulation process are investigated. It is found that the adaptive kernel can well match the computational geometry with long channels and can greatly save computational time. The simulation results are in close agreement with the analytical solutions.

On solving large strain hyperelastic problems with the natural element method

Abstract: In this paper, an extension of the natural element method (NEM) is presented to solve finite deformation problems. Since NEM is a meshless method, its implementation does not require an explicit connectivity definition. Consequently, it is quite adequate to simulate large strain problems with important mesh distortions, reducing the need for remeshing and projection of results (extremely important in three-dimensional problems). NEM has important advantages over other meshless methods, such as the interpolant character of its shape functions and the ability of exactly reproducing essential boundary conditions along convex boundaries. The α-NEM extension generalizes this behaviour to non-convex boundaries. A total Lagrangian formulation has been employed to solve different problems with large strains, considering hyperelastic behaviour. Several examples are presented in two and three dimensions, comparing the results with the ones of the finite element method. NEM performs better showing its important capabilities in this kind of applications. Copyright © 2004 John Wiley & Sons, Ltd.

Analysis and Numerics for Conservation Laws

Abstract: S.Andreae, J.Ballmann, S.Muller: Wave Processes at Interfaces.- J.Heiermann, M.Auweter-Kurtz, Ch.Sleziona: Numerics for Magnetoplasmadynamic Propulsion.- H.Babovsky: Hexagonal Kinetic Models and the Numerical Simulation of Kinetic Boundary Layers.- R.Deiterding, G.Bader: High-resolution Simulation of Detonations with Detailed Chemistry.- A.S.Bormann: Numerical Linear Stability Analysis for Compressible Fluids.- M.Schussler, J.H.M.J.Bruls, A.Vogler, P. Vollmoller: Simulation of Solar Radiative Magneto-Convection.- W.Dahmen, S.Muller, A.Voss: Riemann Problem for the Euler Equation with Non-Convex Equation of State Including Phase Transitions.- A.Dedner, D.Kroner, C.Rohde, M.Wesenberg: Radiation Magnetohydrodynamics: Analysis for Model Problems and Efficient 3d-Simulations for the Full System.- W.Dreyer, M.Herrmann, M.Kunik, S.Qamar: Kinetic Schemes for Selected Initial and Boundary Value Problems.- F.Volker, R.Vilsmeier, D.Hanel: A Local Level-Set Method under Involvement of Topological Aspects.- T.Hillen, K.P.Hadeler: Hyperbolic Systems and Transport Equations in Mathematical Biology.- J.Harterich, St. Liebscher: Travelling Waves in Systems of Hyperbolic Balance Laws.- R.Hartmann: The Role of the Jacobian in the Adaptive Discontinuous Galerkin Method for the Compressible Euler Equations.- Ch.Helling, R.Klein, E.Sedlmayr: The Multi-Scale Dust Formation in Substellar Atmospheres.- D.Hietel, M.Junk, J. Kuhnert, S.Tiwari: Meshless Methods for Conservation Laws.- W.Hillbrandt, M.Reinecke, W.Schmidt, F.K.Ropke, C.Travaglio, J.C.Niemeyer: Simulations of Turbulent Thermonuclear Burining in Type 1a Supernovae.- Y.L.Lee, R.Schneider, C.-D.Munz, F.Kemm: Hyperbolic GLM Scheme for Elliptic Constraints in Computational Electromagnetics and MHD.- H.Schmidt, R.Klein: Flexible Flame Structure Modelling in a Flame Front Tracking Scheme.- T.Kroger, S.Noelle: Riemann-Solver Free Schemes.- H.Liu: Relaxation Dynamics, Scaling Limits and Convergence of Relaxation Schemes.-S.Noelle, W.Rosenbaum, M.Rumpf: Multidimensional Adaptive Staggered Grids.- W.-A.Yong, W.Jager: On Hyperbolic Relaxation Problems.- Appendix: Color Plates.

Meshfree modeling and analysis of physical fields in heterogeneous media

Abstract: Continuous and discrete variations in material properties lead to substantial difficulties for most mesh-based methods for modeling and analysis of physical fields. The meshfree method described in this paper relies on distance fields to boundaries and to material features in order to represent variations of material properties as well as to satisfy prescribed boundary conditions. The method is theoretically complete in the sense that all distributions of physical properties and all physical fields are represented by generalized Taylor series expansions in terms of powers of distance fields. We explain how such Taylor series can be used to construct solution structures – spaces of functions satisfying the prescribed boundary conditions exactly and containing the necessary degrees of freedom to satisfy additional constraints. Fully implemented numerical examples illustrate the effectiveness of the proposed approach.

Meshless Methods and Partition of Unity Finite Elements

Abstract: This paper encompasses the main conclusions obtained in the mini-symposium New and Advanced Numerical Strategies in Forming Processes Simulation, held during the 6th International ESAFORM Conference on Material Forming (Salerno 2003), particularly those aspects dealing with meshless and partition of unity methods applied to the simulation of forming processes

Meshless Galerkin least-squares method

Abstract: Collocation method and Galerkin method have been dominant in the existing meshless methods. Galerkin-based meshless methods are computational intensive, whereas collocation-based meshless methods suffer from instability. A new efficient meshless method, meshless Galerkin lest-squares method (MGLS), is proposed in this paper to combine the advantages of Galerkin method and collocation method. The problem domain is divided into two subdomains, the interior domain and boundary domain. Galerkin method is applied in the boundary domain, whereas the least-squares method is applied in the interior domain.The proposed scheme elliminates the posibilities of spurious solutions as that in the least-square method if an incorrect boundary conditions are used. To investigate the accuracy and efficiency of the proposed method, a cantilevered beam and an infinite plate with a central circular hole are analyzed in detail and numerical results are compared with those obtained by Galerkin-based meshless method (GBMM), collocation-based meshless method (CBMM) and meshless weighted least squares method (MWLS). Numerical studies show that the accuracy of the proposed MGLS is much higher than that of CBMM and is close to, even better than, that of GBMM, while the computational cost is much less than that of GBMM.

Meshless Methods for Conservation Laws

Abstract: In this article, two meshfree methods for the numerical solution of conservation laws are considered. The Finite Volume Particle Method (FVPM) generalizes the Finite Volume approach and the Finite Pointset Method (FPM) is a Finite Difference scheme which can work on unstructured and moving point clouds. Details of the derivation and numerical examples are presented for the case of incompressible, viscous, two-phase flow. In the case of FVPM, our main focus lies on the derivation of stability estimates.

Stochastic crack growth simulation in reinforced concrete structures by means of coupled finite element and meshless methods

Abstract: The complex failure process of concrete structures can not be described in detail by standard engineering design formulas. The numerical analysis of crack development in concrete is essential for several problems. In the last decades a large number of research groups have dealt with this topic and several models and algorithms were developed. However, most of these methods show some difficulties and are limited to special cases. The goal of this study was to develop an automatic algorithm for the efficient simulation of multiple cracking in plain and reinforced concrete structures of medium size. For this purpose meshless methods were used to describe the growth of crack surfaces. Two meshless interpolation schemes were improved for a simple application. The cracking process of concrete has been modeled using a stable criterion for crack growth in combination with an improved cohesive crack model which can represent the failure process under combined crack opening and crack sliding very well. This crack growth algorithm was extended in order to represent the fluctuations of the concrete properties by enlarging the single-parameter random field concept for multiple correlated material parameters.