scispace - formally typeset
Search or ask a question

Showing papers on "Finite element method published in 2006"


Journal ArticleDOI
TL;DR: Finite element exterior calculus as mentioned in this paper is an approach to the design and understand- ing of finite element discretizations for a wide variety of systems of partial differential equations, which 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.
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.

1,044 citations



Journal ArticleDOI
TL;DR: In this paper, the concept of k-refinement is explored and shown to produce more accurate and robust results than corresponding finite elements, including rods, thin beams, membranes, and thin plates.

1,008 citations


Journal ArticleDOI
TL;DR: In this paper, a theoretical framework for the Galerkin finite element approximation to the steady state fractional advection dispersion equation is presented, and appropriate fractional derivative spaces are defined and shown to be equivalent to the usual fractional dimension Sobolev spaces Hs.
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

707 citations


Journal ArticleDOI
TL;DR: In this article, a new method for modeling arbitrary dynamic crack and shear band propagation is presented, where cracks are treated by adding phantom nodes and superposing elements on the original mesh.
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.

686 citations


Journal ArticleDOI
TL;DR: In this paper, the authors studied the stability properties of Isogeometric Analysis in the context of mesh refinement and developed approximation estimates based on a new Bramble-Hilbert lemma in so-called "bent" Sobolev spaces.
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.

681 citations


Journal ArticleDOI
TL;DR: Outflow boundary conditions are derived for any downstream domain where an explicit relationship of pressure as a function of flow rate or velocities can be obtained at the coupling interface.

652 citations


Journal ArticleDOI
TL;DR: In this paper, three-dimensional geometrical models for concrete are generated taking the random structure of aggregates at the mesoscopic level into consideration, where the aggregate particles are generated from a certain aggregate size distribution and then placed into the concrete specimen in such a way that there is no intersection between the particles.

594 citations


Journal ArticleDOI
TL;DR: This work states thatKinetic theory models involving the Fokker-Planck equation can be accurately discretized using a mesh support using a reduced approximation basis within an adaptive procedure making use of an efficient separation of variables.
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.

546 citations


Proceedings ArticleDOI
TL;DR: In this article, a semi-analytical finite element (SAFE) method for modeling wave propagation in waveguides of arbitrary cross-section is proposed, and the dispersive solutions are obtained in terms of phase velocity, group velocity, energy velocity, attenuation and cross-sectional mode shapes.

534 citations


Journal ArticleDOI
TL;DR: Based on the ABAQUS software, uncoupled thermal-mechanical three-dimensional and two-dimensional (2-D) finite element models are developed in this article to evaluate the transient temperature and the residual stress fields during welding.

Journal ArticleDOI
TL;DR: In this paper, the authors developed a new method to simulate blood flow in 3D deformable models of arteries, which couples the equations of the deformation of the vessel wall at the variational level as a boundary condition for the fluid domain.

Journal ArticleDOI
TL;DR: A new plastic hinge integration method overcomes the problems with nonobjective response caused by strain-softening behavior in force-based beam-column finite elements by using the common concept of a plastic hinge length in a numerically consistent manner.
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.

Journal ArticleDOI
TL;DR: 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.
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.

Journal ArticleDOI
TL;DR: A discontinuous Galerkin (DG) method combined with the ideas of the ADER time integration 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 is presented.
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.

Book
01 Jul 2006
TL;DR: The IIM for Stokes and Navier-Stokes equations and some applications of the IIM Bibliography Index are reviewed.
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.

Journal ArticleDOI
TL;DR: In this paper, the discrete cohesive zone model (DCZM) is implemented using the finite element (FE) method to simulate fracture initiation and subsequent growth when material non-linear effects are significant.

Journal ArticleDOI
TL;DR: In this article, a non-intrusive method based on a least-squares minimization procedure is presented to solve stochastic boundary value problems where material properties and loads are random.
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.

Journal ArticleDOI
TL;DR: A novel numerical algorithm for simulating interfacial dynamics of non-Newtonian fluids using an efficient adaptive meshing scheme governed by the phase-field variable that easily accommodates complex flow geometry and makes it possible to simulate large-scale two-phase systems of complex fluids.

Journal ArticleDOI
TL;DR: In this paper, a new family of stabilized methods for the Stokes problem is proposed, which are parameter free, do not require calculation of higher order derivatives or edge-based data structures, and always lead to symmetric linear systems.
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.

Journal ArticleDOI
Tuğrul Özel1
TL;DR: In this article, an updated Lagrangian finite element formulation is used to simulate continuous chip formation process in orthogonal cutting of low carbon free-cutting steel, and the effects of tool-chip interfacial friction models on the finite element simulations are investigated.
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.

Book
01 Jan 2006
TL;DR: 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 theVariational BVP Examples from Mechanics and Physics
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

Journal ArticleDOI
TL;DR: In this article, a bilinear cohesive zone model (CZM) is employed in conjunction with a viscoelastic bulk (background) material to investigate fracture behavior of asphalt concrete.

Journal ArticleDOI
TL;DR: In this article, a simple micro-mechanical model for the homogenised limit analysis of in-plane loaded masonry is proposed, assuming brickwork under plane stress condition and adopting a polynomial expansion for the 2D stress field.

Journal ArticleDOI
TL;DR: The symmetric interior penalty discontinuous Galerkin finite element method is presented for the numerical discretization of the second‐order wave equation and error bounds are derived in the energy norm and the L^2$‐norm for the semidiscrete formulation.
Abstract: The symmetric interior penalty discontinuous Galerkin finite element method is presented for the numerical discretization of the second‐order wave equation. The resulting stiffness matrix is symmetric positive definite, and the mass matrix is essentially diagonal; hence, the method is inherently parallel and leads to fully explicit time integration when coupled with an explicit time‐ stepping scheme. Optimal a priori error bounds are derived in the energy norm and the $L^2$‐norm for the semidiscrete formulation. In particular, the error in the energy norm is shown to converge with the optimal order ${\cal O}(h^{\min\{s,\ell\}})$ with respect to the mesh size h, the polynomial degree $\ell$, and the regularity exponent s of the continuous solution. Under additional regularity assumptions, the $L^2$‐error is shown to converge with the optimal order ${\cal O}(h^{\min\{s,\ell\}})$. Numerical results confirm the expected convergence rates and illustrate the versatility of the method.

Journal ArticleDOI
TL;DR: A method was developed for calibrating the two-composite structure of the annulus fibrosus, the ground substance and collagen fibers to fulfil the required range of motion obtained from in vitro results within an accuracy of 99%.

Journal ArticleDOI
TL;DR: In this article, the authors proposed a Single Walled Carbon Nanotube (SWCNT) finite element (FE) model, based on the use of non-linear and torsional spring elements, to evaluate its mechanical properties.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the modeling of the interaction of fluid flow with flexibly supported rigid bodies, governed by the incompressible Navier-Stokes equations and modelled by employing stabilised low order velocity-pressure finite elements.

Journal ArticleDOI
TL;DR: This paper generalizes high order finite volume WENO schemes and Runge-Kutta discontinuous Galerkin (RKDG) finite element methods to the same class of hyperbolic systems to maintain a well-balanced property.

Journal ArticleDOI
TL;DR: In this article, the authors proposed an efficient numerical algorithm for computing deformations of very soft tissues (such as the brain, liver, kidney etc.), with applications to real-time surgical simulation, based on the finite element method using the total Lagrangian formulation.
Abstract: We propose an efficient numerical algorithm for computing deformations of ‘very’ soft tissues (such as the brain, liver, kidney etc.), with applications to real-time surgical simulation. The algorithm is based on the finite element method using the total Lagrangian formulation, where stresses and strains are measured with respect to the original configuration. This choice allows for pre-computing of most spatial derivatives before the commencement of the time-stepping procedure. We used explicit time integration that eliminated the need for iterative equation solving during the time-stepping procedure. The algorithm is capable of handling both geometric and material non-linearities. The total Lagrangian explicit dynamics (TLED) algorithm using eight-noded hexahedral under-integrated elements requires approximately 35% fewer floating-point operations per element, per time step than the updated Lagrangian explicit algorithm using the same elements. Stability analysis of the algorithm suggests that due to much lower stiffness of very soft tissues than that of typical engineering materials, integration time steps a few orders of magnitude larger than what is typically used in engineering simulations are possible. Numerical examples confirm the accuracy and efficiency of the proposed TLED algorithm. Copyright © 2006 John Wiley & Sons, Ltd.