Showing papers on "Finite element method published in 2006"

Finite element exterior calculus, homological techniques, and applications

Abstract: Finite element exterior calculus is an approach to the design and understand- ing of finite element discretizations for a wide variety of systems of partial differential equations. This approach brings to bear tools from differential geometry, algebraic topology, and homological algebra to develop discretiza- tions which are compatible with the geometric, topological, and algebraic structures which underlie well-posedness of the PDE problem being solved. In the finite element exterior calculus, many finite element spaces are re- vealed as spaces of piecewise polynomial differential forms. These connect to each other in discrete subcomplexes of elliptic differential complexes, and are also related to the continuous elliptic complex through projections which commute with the complex differential. Applications are made to the finite element discretization of a variety of problems, including the Hodge Lapla- cian, Maxwell's equations, the equations of elasticity, and elliptic eigenvalue problems, and also to preconditioners.

Variational formulation for the stationary fractional advection dispersion equation

Abstract: In this article a theoretical framework for the Galerkin finite element approximation to the steady state fractional advection dispersion equation is presented. Appropriate fractional derivative spaces are defined and shown to be equivalent to the usual fractional dimension Sobolev spaces Hs. Existence and uniqueness results are proven, and error estimates for the Galerkin approximation derived. Numerical results are included that confirm the theoretical estimates. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

A method for dynamic crack and shear band propagation with phantom nodes

Abstract: A new method for modelling of arbitrary dynamic crack and shear band propagation is presented. We show that by a rearrangement of the extended finite element basis and the nodal degrees of freedom, the discontinuity can be described by superposed elements and phantom nodes. Cracks are treated by adding phantom nodes and superposing elements on the original mesh. Shear bands are treated by adding phantom degrees of freedom. The proposed method simplifies the treatment of element-by-element crack and shear band propagation in explicit methods. A quadrature method for 4-node quadrilaterals is proposed based on a single quadrature point and hourglass control. The proposed method provides consistent history variables because it does not use a subdomain integration scheme for the discontinuous integrand. Numerical examples for dynamic crack and shear band propagation are provided to demonstrate the effectiveness and robustness of the proposed method. Copyright © 2006 John Wiley & Sons, Ltd.

ISOGEOMETRIC ANALYSIS: APPROXIMATION, STABILITY AND ERROR ESTIMATES FOR h-REFINED MESHES

Abstract: We begin the mathematical study of Isogeometric Analysis based on NURBS (non-uniform rational B-splines). Isogeometric Analysis is a generalization of classical Finite Element Analysis (FEA) which possesses improved properties. For example, NURBS are capable of more precise geometric representation of complex objects and, in particular, can exactly represent many commonly engineered shapes, such as cylinders, spheres and tori. Isogeometric Analysis also simplifies mesh refinement because the geometry is fixed at the coarsest level of refinement and is unchanged throughout the refinement process. This eliminates geometrical errors and the necessity of linking the refinement procedure to a CAD representation of the geometry, as in classical FEA. In this work we study approximation and stability properties in the context of h-refinement. We develop approximation estimates based on a new Bramble–Hilbert lemma in so-called "bent" Sobolev spaces appropriate for NURBS approximations and establish inverse estimates similar to those for finite elements. We apply the theoretical results to several cases of interest including elasticity, isotropic incompressible elasticity and Stokes flow, and advection-diffusion, and perform numerical tests which corroborate the mathematical results. We also perform numerical calculations that involve hypotheses outside our theory and these suggest that there are many other interesting mathematical properties of Isogeometric Analysis yet to be proved.

"Finite-Element" Displacement Fields Analysis from Digital Images: Application to Portevin-Le Châtelier Bands

Abstract: A new methodology is proposed to estimate displacement fields from pairs of images (reference and strained) that evaluates continuous displacement fields. This approach is specialized to a finite-element decomposition, therefore providing a natural interface with a numerical modeling of the mechanical behavior used for identification purposes. The method is illustrated with the analysis of Portevin–Le Châtelier bands in an aluminum alloy sample subjected to a tensile test. A significant progress with respect to classical digital image correlation techniques is observed in terms of spatial resolution and uncertainty.

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.

A new family of solvers for some classes of multidimensional partial differential equations encountered in kinetic theory modeling of complex fluids

Abstract: Kinetic theory models involving the Fokker-Planck equation can be accurately discretized using a mesh support (finite elements, finite differences, finite volumes, spectral techniques, etc.). However, these techniques involve a high number of approximation functions. In the finite element framework, widely used in complex flow simulations, each approximation function is related to a node that defines the associated degree of freedom. When the model involves high dimensional spaces (including physical and conformation spaces and time), standard discretization techniques fail due to an excessive computation time required to perform accurate numerical simulations. One appealing strategy that allows circumventing this limitation is based on the use of reduced approximation basis within an adaptive procedure making use of an efficient separation of variables. (c) 2006 Elsevier B.V. All rights reserved.

Plastic Hinge Integration Methods for Force-Based Beam¿Column Elements

Abstract: A new plastic hinge integration method overcomes the problems with nonobjective response caused by strain-softening behavior in force-based beam-column finite elements. The integration method uses the common concept of a plastic hinge length in a numerically consistent manner. The method, derived from the Gauss-Radau quadrature rule, integrates deformations over specified plastic hinge lengths at the ends of the beam-column element, and it has the desirable property that it reduces to the exact solution for linear problems. Numerical examples show the effect of plastic hinge integration on the response of force-based beam-column elements for both strain-hardening and strain-softening section behavior in the plastic hinge regions. The incorporation of a plastic hinge length in the element integration method ensures objective element and section response, which is important for strain-softening behavior in reinforced concrete structures. Plastic rotations are defined in a consistent manner and clearly related to deformations in the plastic hinges.

An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes – II. The three-dimensional isotropic case

Abstract: SUMMARY We present a new numerical method to solve the heterogeneous elastic wave equations formulated as a linear hyperbolic system using first-order derivatives with arbitrary high-order accuracy in space and time on 3-D unstructured tetrahedral meshes. The method combines the Discontinuous Galerkin (DG) Finite Element (FE) method with the ADER approach using Arbitrary high-order DERivatives for flux calculation. In the DG framework, in contrast to classical FE methods, the numerical solution is approximated by piecewise polynomials which allow for discontinuities at element interfaces. Therefore, the well-established theory of numerical fluxes across element interfaces obtained by the solution of Riemann-Problems can be applied as in the finite volume framework. To define a suitable flux over the element surfaces, we solve so-called Generalized Riemann-Problems (GRP) at the element interfaces. The GRP solution provides simultaneously a numerical flux function as well as a time-integration method. The main idea is a Taylor expansion in time in which all time-derivatives are replaced by space derivatives using the so-called Cauchy–Kovalewski or Lax–Wendroff procedure which makes extensive use of the governing PDE. The numerical solution can thus be advanced for one time step without intermediate stages as typical, for example, for classical Runge–Kutta time stepping schemes. Due to the ADER time-integration technique, the same approximation order in space and time is achieved automatically. Furthermore, the projection of the tetrahedral elements in physical space on to a canonical reference tetrahedron allows for an efficient implementation, as many computations of 3-D integrals can be carried out analytically beforehand. Based on a numerical convergence analysis, we demonstrate that the new schemes provide very high order accuracy even on unstructured tetrahedral meshes and computational cost and storage space for a desired accuracy can be reduced by higher-order schemes. Moreover, due to the choice of the basis functions for the piecewise polynomial approximation, the new ADER–DG method shows spectral convergence on tetrahedral meshes. An application of the new method to a well-acknowledged test case and comparisons with analytical and reference solutions, obtained by different well-established methods, confirm the performance of the proposed method. Therefore, the development of the highly accurate ADER–DG approach for tetrahedral meshes provides a numerical technique to approach 3-D wave propagation problems in complex geometry with unforeseen accuracy.

An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - I. The two-dimensional isotropic case with external source terms

Abstract: SUMMARY We present a new numerical approach to solve the elastic wave equation in heterogeneous media in the presence of externally given source terms with arbitrary high-order accuracy in space and time on unstructured triangular meshes. We combine a discontinuous Galerkin (DG) method with the ideas of the ADER time integration approach using Arbitrary high-order DERivatives. The time integration is performed via the so-called Cauchy-Kovalewski procedure using repeatedly the governing partial differential equation itself. In contrast to classical finite element methods we allow for discontinuities of the piecewise polynomial approximation of the solution at element interfaces. This way, we can use the well-established theory of fluxes across element interfaces based on the solution of Riemann problems as developed in the finite volume framework. In particular, we replace time derivatives in the Taylor expansion of the time integration procedure by space derivatives to obtain a numerical scheme of the same high order in space and time using only one single explicit step to evolve the solution from one time level to another. The method is specially suited for linear hyperbolic systems such as the heterogeneous elastic wave equations and allows an efficient implementation. We consider continuous sources in space and time and point sources characterized by a Delta distribution in space and some continuous source time function. Hereby, the method is able to deal with point sources at any position in the computational domain that does not necessarily need to coincide with a mesh point. Interpolation is automatically performed by evaluation of test functions at the source locations. The convergence analysis demonstrates that very high accuracy is retained even on strongly irregular meshes and by increasing the order of the ADER‐DG schemes computational time and storage space can be reduced remarkably. Applications of the proposed method to Lamb’s Problem, a problem of strong material heterogeneities and to an example of global seismic wave propagation finally confirm its accuracy, robustness and high flexibility.

The Immersed Interface Method: Numerical Solutions of PDEs Involving Interfaces and Irregular Domains

Abstract: Preface 1. Introduction 2. The IIM for one-dimensional elliptic interface problems 3. The IIM for two-dimensional elliptic interface problems 4. The IIM for three-dimensional elliptic interface problems 5. Removing source singularities for certain interface problems 6. Augmented strategies 7. The fourth-order IIM 8. The immersed finite element methods 9. The IIM for parabolic interface problems 10. The IIM for Stokes and Navier-Stokes equations 11. Some applications of the IIM Bibliography Index.

Stochastic finite element: a non intrusive approach by regression

Abstract: The stochastic finite element method allows to solve stochastic boundary value problems where material properties and loads are random. The method is based on the expansion of the mechanical response onto the so-called polynomial chaos. In this paper, a non intrusive method based on a least-squares minimization procedure is presented. This method is illustrated by the study of the settlement of a foundation. Different analysis are proposed: the computation of the statistical moments of the response, a reliability analysis and a parametric sensitivity analysis.

Stabilization of Low-order Mixed Finite Elements for the Stokes Equations

Abstract: We present a new family of stabilized methods for the Stokes problem. The focus of the paper is on the lowest order velocity-pressure pairs. While not LBB compliant, their simplicity and attractive computational properties make these pairs a popular choice in engineering practice. Our stabilization approach is motivated by terms that characterize the LBB "deficiency" of the unstable spaces. The stabilized methods are defined by using these terms to modify the saddle-point Lagrangian associated with the Stokes equations. The new stabilized methods offer a number of attractive computational properties. In contrast to other stabilization procedures, they are parameter free, do not require calculation of higher order derivatives or edge-based data structures, and always lead to symmetric linear systems. Furthermore, the new methods are unconditionally stable, achieve optimal accuracy with respect to solution regularity, and have simple and straightforward implementations. We present numerical results in two and three dimensions that showcase the excellent stability and accuracy of the new methods.

The influence of friction models on finite element simulations of machining

Abstract: In the analysis of orthogonal cutting process using finite element (FE) simulations, predictions are greatly influenced by two major factors; a) flow stress characteristics of work material at cutting regimes and b) friction characteristics mainly at the tool-chip interface. The uncertainty of work material flow stress upon FE simulations may be low when there is a constitutive model for work material that is obtained empirically from high-strain rate and temperature deformation tests. However, the difficulty arises when one needs to implement accurate friction models for cutting simulations using a particular FE formulation. In this study, an updated Lagrangian finite element formulation is used to simulate continuous chip formation process in orthogonal cutting of low carbon free-cutting steel. Experimentally measured stress distributions on the tool rake face are utilized in developing several different friction models. The effects of tool-chip interfacial friction models on the FE simulations are investigated. The comparison results depict that the friction modeling at the tool-chip interface has significant influence on the FE simulations of machining. Specifically, variable friction models that are developed from the experimentally measured normal and frictional stresses at the tool rake face resulted in most favorable predictions. Predictions presented in this work also justify that the FE simulation technique used for orthogonal cutting process can be an accurate and viable analysis as long as flow stress behavior of the work material is valid at the machining regimes and the friction characteristics at the tool-chip interface is modeled properly.

Lower bound limit analysis of cohesive‐frictional materials using second‐order cone programming

Abstract: The formulation of limit analysis by means of the finite element method leads to an optimization problem with a large number of variables and constraints. Here we present a method for obtaining strict lower bound solutions using second-order cone programming (SOCP), for which efficient primal-dual interior-point algorithms have recently been developed. Following a review of previous work, we provide a brief introduction to SOCP and describe how lower bound limit analysis can be formulated in this way. Some methods for exploiting the data structure of the problem are also described, including an efficient strategy for detecting and removing linearly dependent constraints at the assembly stage. The benefits of employing SOCP are then illustrated with numerical examples. Through the use of an effective algorithm/software, very large optimization problems with up to 700 000 variables are solved in minutes on a desktop machine. The numerical examples concern plane strain conditions and the Mohr–Coulomb criterion, however we show that SOCP can also be applied to any other problem of lower bound limit analysis involving a yield function with a conic quadratic form (notable examples being the Drucker–Prager criterion in 2D or 3D, and Nielsen's criterion for plates). Copyright © 2005 John Wiley & Sons, Ltd.

One and two dimensional elliptic and Maxwell problems

Abstract: 1D PROBLEMS 1D Model Elliptic Problem A Two-Point Boundary Value Problem Algebraic Structure of the Variational Formulation Equivalence with a Minimization Problem Sobolev Space H1(0, l) Well Posedness of the Variational BVP Examples from Mechanics and Physics The Case with "Pure Neumann" BCs Exercises Galerkin Method Finite Dimensional Approximation of the VBVP Elementary Convergence Analysis Comments Exercises 1D hp Finite Element Method 1D hp Discretization Assembling Element Matrices into Global Matrices Computing the Element Matrices Accounting for the Dirichlet BC Summary Assignment 1: A Dry Run Exercises 1D hp Code Setting up the 1D hp Code Fundamentals Graphics Element Routine Assignment 2: Writing Your Own Processor Exercises Mesh Refinements in 1D The h-Extension Operator. Constrained Approximation Coefficients Projection-Based Interpolation in 1D Supporting Mesh Refinements Data-Structure-Supporting Routines Programming Bells and Whistles Interpolation Error Estimates Convergence Assignment 3: Studying Convergence Definition of a Finite Element Exercises Automatic hp Adaptivity in 1D The hp Algorithm Supporting the Optimal Mesh Selection Exponential Convergence. Comparing with h Adaptivity Discussion of the hp Algorithm Algebraic Complexity and Reliability of the Algorithm Exercises Wave Propagation Problems Convergence Analysis for Noncoercive Problems Wave Propagation Problems Asymptotic Optimality of the Galerkin Method Dispersion Error Analysis Exercises 2D ELLIPTIC PROBLEMS 2D Elliptic Boundary-Value Problem Classical Formulation Variational (Weak) Formulation Algebraic Structure of the Variational Formulation Equivalence with a Minimization Problem Examples from Mechanics and Physics Exercises Sobolev Spaces Sobolev Space H1(O) Sobolev Spaces of an Arbitrary Order Density and Embedding Theorems Trace Theorem Well Posedness of the Variational BVP Exercises 2D hp Finite Element Method on Regular Meshes Quadrilateral Master Element Triangular Master Element Parametric Element Finite Element Space. Construction of Basis Functions Calculation of Element Matrices Modified Element. Imposing Dirichlet Boundary Conditions Postprocessing- Local Access to Element d.o.f Projection-Based Interpolation Exercises 2D hp Code Getting Started Data Structure in FORTRAN 90 Fundamentals The Element Routine Modified Element. Imposing Dirichlet Boundary Conditions Assignment 4: Assembly of Global Matrices The Case with "Pure Neumann" Boundary Conditions Geometric Modeling and Mesh Generation Manifold Representation Construction of Compatible Parametrizations Implicit Parametrization of a Rectangle Input File Preparation Initial Mesh Generation The hp Finite Element Method on h-Refined Meshes Introduction. The h Refinements 1-Irregular Mesh Refinement Algorithm Data Structure in Fortran 90 (Continued) Constrained Approximation for C0 Discretizations Reconstructing Element Nodal Connectivities Determining Neighbors for Midedge Nodes Additional Comments Automatic hp Adaptivity in 2D The Main Idea The 2D hp Algorithm Example: L-Shape Domain Problem Example: 2D "Shock" Problem Additional Remarks Examples of Applications A "Battery Problem" Linear Elasticity An Axisymmetric Maxwell Problem Exercises Exterior Boundary-Value Problems Variational Formulation. Infinite Element Discretization Selection of IE Radial Shape Functions Implementation Calculation of Echo Area Numerical Experiments Comments Exercises 2D MAXWELL PROBLEMS 2D Maxwell Equations Introduction to Maxwell's Equation Variational Formulation Exercises Edge Elements and the de Rham Diagram Exact Sequences Projection-Based Interpolation De Rham Diagram Shape Functions Exercises 2D Maxwell Code Directories. Data Structure The Element Routine Constrained Approximation. Modified Element Setting up a Maxwell Problem Exercises hp Adaptivity for Maxwell Equations Projection-Based Interpolation Revisited The hp Mesh Optimization Algorithm Example: The Screen Problem Exterior Maxwell Boundary-Value Problems Variational Formulation Infinite Element Discretization in 3D Infinite Element Discretization in 2D Stability Implementation Numerical Experiments Exercises A Quick Summary and Outlook Appendix Bibliography Index