Showing papers on "Smoothed finite element method 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.

The Finite Element Method for Fluid Dynamics

Abstract: The Finite Element Method for Fluid Dynamics offers a complete introduction the application of the finite element method to fluid mechanics. The book begins with a useful summary of all relevant partial differential equations before moving on to discuss convection stabilization procedures, steady and transient state equations, and numerical solution of fluid dynamic equations. The character-based split (CBS) scheme is introduced and discussed in detail, followed by thorough coverage of incompressible and compressible fluid dynamics, flow through porous media, shallow water flow, and the numerical treatment of long and short waves. Updated throughout, this new edition includes new chapters on: * Fluid-structure interaction, including discussion of one-dimensional and multidimensional problems. * Biofluid dynamics, covering flow throughout the human arterial system. Focusing on the core knowledge, mathematical and analytical tools needed for successful computational fluid dynamics (CFD), The Finite Element Method for Fluid Dynamics is the authoritative introduction of choice for graduate level students, researchers and professional engineers. * A proven keystone reference in the library of any engineer needing to understand and apply the finite element method to fluid mechanics. * Founded by an influential pioneer in the field and updated in this seventh edition by leading academics who worked closely with Olgierd C. Zienkiewicz. * Features new chapters on fluid-structure interaction and biofluid dynamics, including coverage of one-dimensional flow in flexible pipes and challenges in modeling systemic arterial circulation.

High-order extended finite element method for cracked domains

Abstract: The aim of the paper is to study the capabilities of the Extended Finite Element Method (XFEM) to achieve accurate computations in non smooth situations such as crack problems. Although the XFEM method ensures a weaker error than classical finite element methods, the rate of convergence is not improved when the mesh parameter h is going to zero because of the presence of a singularity. The difficulty can be overcome by modifying the enrichment of the finite element basis with the asymptotic crack tip displacement solutions as well as with the Heaviside function. Numerical simulations show that the modified XFEM method achieves an optimal rate of convergence (i.e. like in a standard finite element method for a smooth problem)

Adaptive finite element methods for differential equations.

Abstract: Gegenstand des Buches ist die Dual Weighted Residual method (DWR), ein sehr effizientes numerisches Verfahren zur Behandlung einer großen Klasse von variationell formulierten Differentialgleichungen. Das numerische Verfahren ist adaptiv, d.h. es konstruiert eigenständig eine Folge von Approximationen für eine gegebene Fragestellung. Typische Fragestellungen sind die Bestimmung gewichteter Mittelwerte der Lösung oder ihrer Ableitungen, die Bestimmung von Randintegralen über Lösungskomponenten (relevant z.B. für die Berechnung von strömungsmechanischen Kenngrößen) oder die Bestimmung von Spannungsintensitätsfaktoren (z.B. in der Bruchmechanik). Das Verfahren basiert auf Projektionsmethoden wie z.B. der Finiten Elemente Methode (FEM). Dort wird die Approximationsgüte durch die Wahl der Gitter gesteuert. Der Kern jeder adaptiven FEM ist deshalb die Art, wie die Gitter gewählt werden. Typischerweise geschieht dies in einer adaptiven Schleife, in der in mehreren Durchgängen schrittweise das Gitter verbessert wird, bis eine gewünschte Genauigkeit erreicht ist. Bei der DWR wird in jedem Schleifendurchgang ein lineares Hilfsproblem—das sog. duale Problem, welches von der vorliegenden Fragestellung abhängt—(näherungsweise) gelöst. Weiterhin wird eine Approximation der Differentialgleichung bestimmt. Aus diesen nun vorliegenden Daten wird dann herausdestilliert, wo das Gitter verfeinert werden sollte bzw. vergröbert werden kann, um eine genauere Lösung zu erhalten. Ziel eines adaptiven Algorithmus ist, das gewünschte Ergebnis möglichst effizient zu bestimmen, d.h. mit möglichst geringem Bedarf an Resourcen (Rechenzeit, Speicherbedarf etc.). Mit zahlreichen Beispielen belegt das Buch, daß die DWR dieses Ziel erreicht. Es sei hier besonders hervorgehoben, daß eine Kosten-Nutzen-Betrachtung für die DWR besonders bei nichtlinearen Problemen günstig ausfällt, da die Kosten für die Lösung des linearen Hilfsproblems vergleichbar mit denen eines Newtonschrittes sind und somit nur einen kleinen Teil der Gesamtkosten ausmachen. Das Buch gibt einen sehr guten Überblick über die Technik und die Möglichkeiten der DWR. In einleitenden Kapiteln wird die DWR an gewöhnlichen Differentialgleichungen und dann an einfachen linearen, elliptischen partiellen Differentialgleichungen sehr klar und verständlich vorgeführt. Anschließend wird die DWR in einem abstrakten funktionalanalytischen Rahmen vorgestellt. Der Rest des Buches illustriert auf eindrucksvolle Weise die Leistungsfähigkeit und Breite der Anwendungsfähigkeit des Konzeptes an Hand von Fallbeispielen: Es werden Eigenwertprobleme, Optimierungsaufgaben mit Zwangsbedingungen, die durch eine partielle Differentialgleichung gegeben sind, Strukturmechanikprobleme (lineare Elastizität, Plastizität), Strömungsmechanik (hydrodynamische Stabilitätsanalyse, Berechnung von Strömungskennwerten) behandelt. Auch zeitabhängige Probleme wie die Lösung der Wellengleichung werden mit der DWR erfolgreich bearbeitet. Insgesamt wird klar ersichtlich, daß die DWR eine sehr flexible und vielseitig anwendbare Technik ist. Die ausgewählten numerischen Beispiele, die vor allem aus umfangreichen numerischen Untersuchungen der Gruppe von Rolf Rannacher aus den letzten 10 Jahren ausgewählt wurden, sind sehr illustrativ. Die Erläuterungen zu den Beispielen sind auch deshalb interessant, weil eine Menge zusätzlicher Informationen über die numerische Behandlung des vorliegenden Problems quasi nebenbei einfließen. Das Buch entstand aus einer fortgeschrittenen Spezialvorlesung, die an der ETH Zürich gehalten wurde. Einen Lehrbuchcharakter erhält das Buch dadurch, daß Übungsaufgaben (mit detailierten Lösungen im Anhang) jedes Kapitel abschließen. Die Aufgaben enthal-

A Multiscale Finite Element Method for Numerical Homogenization

Abstract: This paper is concerned with a multiscale finite element method for numerically solving second-order scalar elliptic boundary value problems with highly oscillating coefficients. In the spirit of previous other works, our method is based on the coupling of a coarse global mesh and a fine local mesh, the latter being used for computing independently an adapted finite element basis for the coarse mesh. The main idea is the introduction of a composition rule, or change of variables, for the construction of this finite element basis. In particular, this allows for a simple treatment of high-order finite element methods. We provide optimal error estimates in the case of periodically oscillating coefficients. We illustrate our method in various examples.

Finite element formulation for a digital image correlation method

Abstract: A finite element formulation for a digital image correlation method is presented that will determine directly the complete, two-dimensional displacement field during the image correlation process on digital images. The entire interested image area is discretized into finite elements that are involved in the common image correlation process by use of our algorithms. This image correlation method with finite element formulation has an advantage over subset-based image correlation methods because it satisfies the requirements of displacement continuity and derivative continuity among elements on images. Numerical studies and a real experiment are used to verify the proposed formulation. Results have shown that the image correlation with the finite element formulation is computationally efficient, accurate, and robust.

The Finite Element Method: Basic Concepts and Applications

Abstract: Preface 1.Introduction 2.The Method of Weighted Residuals and Galerkin Approximations 3.The Finite Element Method in One Dimension 4.The Two-Dimensional Triangular Element 5.The Two-Dimensional Quadrilateral ELement 6.Isoparametric Two-dimensional Elements 7.The Three-Dimensional Element 8.Additional Applications References Appendices Index

Finite Difference, Finite Element and Finite Volume Methods for Partial Differential Equations

Abstract: There are three important steps in the computational modelling of any physical process: (i) problem definition, (ii) mathematical model, and (iii) computer simulation

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.

Finite element methods and their applications

Abstract: Elementary Finite Elements.- Nonconforming Finite Elements.- Mixed Finite Elements.- Discontinuous Finite Elements.- Characteristic Finite Elements.- Adaptive Finite Elements.- Solid Mechanics.- Fluid Mechanics.- Fluid Flow in Porous Media.- Semiconductor Modeling.

A Mixed Finite Element Method for Elasticity in Three Dimensions

Abstract: We describe a stable mixed finite element method for linear elasticity in three dimensions.

Parallel Octree-Based Finite Element Method for Large-Scale Earthquake Ground Motion Simulation

Abstract: We present a parallel octree-based finite element method for large-scale earthquake ground motion simulation in realistic basins. The octree representation combines the low memory per node and good cache performance of finite difference methods with the spatial adaptivity to local seismic wavelengths characteristic of unstructured finite element methods. Several tests are provided to verify the numerical performance of the method against Green’s function solutions for homogeneous and piecewise homogeneous media, both with and without anelastic attenuation. A comparison is also provided against a finite difference code and an unstructured tetrahedral finite element code for a simulation of the 1994 Northridge Earthquake. The numerical tests all show very good agreement with analytical solutions and other codes. Finally, performance evaluation indicates excellent single-processor performance and parallel scalability over a range of 1 to 2048 processors for Northridge simulations with up to 300 million degrees of freedom. keyword: Earthquake ground motion modeling, octree, parallel computing, finite element method, elastic wave propagation

A Moving Grid Finite Element Method for the Simulation of Pattern Generation by Turing Models on Growing Domains

Abstract: Numerical techniques for moving meshes are many and varied. In this paper we present a novel application of a moving grid finite element method applied to biological problems related to pattern formation where the mesh movement is prescribed through a specific definition to mimic the growth that is observed in nature. Through the use of a moving grid finite element technique, we present numerical computational results illustrating how period doubling behaviour occurs as the domain doubles in size.

A remark on the application of the Newton-Raphson method in non-linear finite element analysis

Abstract: Usually the notion “Newton-Raphson method” is used in the context of non-linear finite element analysis based on quasi-static problems in solid mechanics. It is pointed out that this is only true in the case of non-linear elasticity. In the case of constitutive equations of evolutionary-type, like in viscoelasticity, viscoplasticity or elastoplasticity, the “Multilevel-Newton algorithm” is usually applied yielding the notions of global and local level (iteration), as well as the consistent tangent operator. In this paper, we investigate the effects of a consistent application of the classical Newton-Raphson method in connection with the finite element method, and compare it with the classical Multilevel-Newton algorithm. Furthermore, an improved version of the Multilevel-Newton method is applied.

Basics and some applications of the mortar element method

Abstract: We describe the main characteristics of the mortar element method, together with the usual arguments for its numerical analysis in an abstract framework. We illustrate this presentation by focussing on mortar spectral element and mortar finite element methods. We give three examples of recent applications, concerning the treatment of non homogeneous media, eddy currents in moving conductors, and finite element mesh adaptivity. (© 2014 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A multilevel finite element method in space‐time for the Navier‐Stokes problem

Abstract: A multilevel finite element method in space-time for the two-dimensional nonstationary Navier-Stokes problem is considered. The method is a multi-scale method in which the fully nonlinear Navier-Stokes problem is only solved on a single coarsest space-time mesh; subsequent approximations are generated on a succession of refined space-time meshes by solving a linearized Navier-Stokes problem about the solution on the previous level. The a priori estimates and error analysis are also presented for the J-level finite element method. We demonstrate theoretically that for an appropriate choice of space and time mesh widths: hj ∼ h, kj ∼ k, j = 2, …, J, the J-level finite element method in space-time provides the same accuracy as the one-level method in space-time in which the fully nonlinear Navier-Stokes problem is solved on a final finest space-time mesh. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

Models of near-field spectroscopic studies: comparison between Finite-Element and Finite-Difference methods.

Abstract: We compare the numerical results obtained by the Finite Element Method (FEM) and the Finite Difference Time Domain Method (FDTD) for near-field spectroscopic studies and intensity map computations. We evaluate their respective efficiencies and we show that an accurate description of the dispersion and of the geometry of the material must be included for a realistic modeling. In particular for the nano-objects, we show that a grid size around Δρa ≈ 4πa/λ (expressed in λ units) as well as a Drude-Lorentz’ model of dispersion for FDTD should be used in order to describe more accurately the confinement of the light around the nanostructures (i.e. the high gradients of the electromagnetic field) and to assure the convergence to the physical solution.

Time‐dependent algorithms for viscoelastic flow: Finite element/volume schemes

Abstract: Hybrid finite volume/element methods are investigated within the context of transient viscoelastic flows. A finite volume algorithm is proposed for the hyperbolic constitutive equation, of Oldroyd-form, whereas the continuity/momentum balance is accommodated through a Taylor-Galerkin finite element method. Various finite volume combinations are considered to derive accurate and stable implementations. Consistency of formulation is key, embracing fluctuation distribution and median-dual-cell constructs, within a cell-vertex discretisation on triangles. In addition, we investigate the effect of treating the time-term in a finite element fashion, using mass-matrix iteration instead of the standard finite volume mass-lumping approach. We devise an accurate transient scheme that captures the analytical solution at short and long time, both in core flow and near shear boundaries. In this respect, some difficulties are highlighted. A new method emerges, with the Low Diffusion B (LDB, with or without mass-matrix iteration) as the optimal choice. We progress to a complex flow application and demonstrate some provocative features due to the influence of true transient boundary conditions on evolutionary flow-structure in a 4:1 start-up rounded-corner contraction problem. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

Finite element method for radiative heat transfer in absorbing and anisotropic scattering media

Abstract: This article proposes a finite element method (FEM) for simulation of radiative heat transfer in absorbing, emitting and anisotropic scattering media. This simulation is developed on the base of discrete ordinates method and theories of finite element. The finite element formulations of triangle isoparametric element and detailed steps of numerical calculation are given. Unstructured triangular element grids are employed in spatial discretization, azimuthal discretization strategy is used in the angular discretization. Two efficient iterative solvers are employed to solve the sparse equations of FEM. Some typical two-dimensional problems are used to verify the method. Pure absorbing cases and anisotropic scattering cases with four different scattering phase functions are investigated and analyzed. A simple irregular geometry case is also used to verify the method. The results of present research have a good agreement with the Monte-Carlo method or published reference.

Finite volume multiscale finite element method for solving the groundwater flow problems in heterogeneous porous media

Abstract: [1] In this paper we present a numerical method, the finite volume multiscale finite element method (FVMSFEM), for solving the groundwater flow problems in heterogeneous porous media spanning over many scales This method is based on an efficient coupling between the finite volume discretization and the multiscale finite element base functions It can efficiently capture the large-scale structure of the solution on a coarse grid without resolving all the small-scale features and is locally conservative The underlying idea is to estimate the macroscopic fluxes across the finite control volume interface segments, which bring the local small-scale information of the medium property to the large scales, by employing the multiscale finite element base functions We describe the strategy for constructing such a method on the basis of the transient flow problems in porous media Numerical experiments are carried out for groundwater flow in porous media with a random lognormal conductivity field to demonstrate the efficiency and accuracy of the developed method

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