Showing papers on "Finite element method published in 2001"

An optimal control approach to a posteriori error estimation in finite element methods

Abstract: This article surveys a general approach to error control and adaptive mesh design in Galerkin finite element methods that is based on duality principles as used in optimal control. Most of the existing work on a posteriori error analysis deals with error estimation in global norms like the ‘energy norm’ or the L2 norm, involving usually unknown ‘stability constants’. However, in most applications, the error in a global norm does not provide useful bounds for the errors in the quantities of real physical interest. Further, their sensitivity to local error sources is not properly represented by global stability constants. These deficiencies are overcome by employing duality techniques, as is common in a priori error analysis of finite element methods, and replacing the global stability constants by computationally obtained local sensitivity factors. Combining this with Galerkin orthogonality, a posteriori estimates can be derived directly for the error in the target quantity. In these estimates local residuals of the computed solution are multiplied by weights which measure the dependence of the error on the local residuals. Those, in turn, can be controlled by locally refining or coarsening the computational mesh. The weights are obtained by approximately solving a linear adjoint problem. The resulting a posteriori error estimates provide the basis of a feedback process for successively constructing economical meshes and corresponding error bounds tailored to the particular goal of the computation. This approach, called the ‘dual-weighted-residual method’, is introduced initially within an abstract functional analytic setting, and is then developed in detail for several model situations featuring the characteristic properties of elliptic, parabolic and hyperbolic problems. After having discussed the basic properties of duality-based adaptivity, we demonstrate the potential of this approach by presenting a selection of results obtained for practical test cases. These include problems from viscous fluid flow, chemically reactive flow, elasto-plasticity, radiative transfer, and optimal control. Throughout the paper, open theoretical and practical problems are stated together with references to the relevant literature.

Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues

Abstract: The finite element analysis of delamination in laminated composites is addressed using interface elements and an interface damage law. The principles of linear elastic fracture mechanics are indirectly used by equating, in the case of single-mode delamination, the area underneath the traction/relative displacement curve to the critical energy release rate of the mode under examination. For mixed-mode delamination an interaction model is used which can fulfil various fracture criteria proposed in the literature. It is then shown that the model can be recast in the framework of a more general damage mechanics theory. Numerical results are presented for the analyses of a double cantilever beam specimen and for a problem involving multiple delamination for which comparisons are made with experimental results. Issues related with the numerical solution of the non-linear problem of the delamination are discussed, such as the influence of the interface strength on the convergence properties and the final results, the optimal choice of the iterative matrix in the predictor and the number of integration points in the interface elements. Copyright © 2001 John Wiley & Sons, Ltd.

Arbitrary discontinuities in finite elements

Abstract: A technique for modelling arbitrary discontinuities in finite elements is presented. Both discontinuities in the function and its derivatives are considered. Methods for intersecting and branching discontinuities are given. In all cases, the discontinuous approximation is constructed in terms of a signed distance functions, so level sets can be used to update the position of the discontinuities. A standard displacement Galerkin method is used for developing the discrete equations. Examples of the following applications are given: crack growth, a journal bearing, a non-bonded circular inclusion and a jointed rock mass. Copyright © 2001 John Wiley & Sons, Ltd.

A new method for modelling cohesive cracks using finite elements

Abstract: A model which allows the introduction of displacements jumps to conventional finite elements is developed. The path of the discontinuity is completely independent of the mesh structure. Unlike so-called ‘embedded discontinuity’ models, which are based on incompatible strain modes, there is no restriction on the type of underlying solid finite element that can be used and displacement jumps are continuous across element boundaries. Using finite element shape functions as partitions of unity, the displacement jump across a crack is represented by extra degrees of freedom at existing nodes. To model fracture in quasi-brittle heterogeneous materials, a cohesive crack model is used. Numerical simulations illustrate the ability of the method to objectively simulate fracture with unstructured meshes. Copyright © 2001 John Wiley & Sons, Ltd.

The finite element method and its reliability

Abstract: Preface 1. Introduction 2. Mathematical formulation of the model problem 3. The finite element method 4. Local behaviour in the finite element method 5. A-posteriori estimation of the error 6. Guaranteed a-posteriori error estimation, and a-posteriori estimation of the pollution error Appendix Index

A point interpolation method for two-dimensional solids

Abstract: A point interpolation method (PIM) is presented for stress analysis for two-dimensional solids. In the PIM, the problem domain is represented by properly scattered points. A technique is proposed to construct polynomial interpolants with delta function property based only on a group of arbitrarily distributed points. The PIM equations are then derived using variational principles. In the PIM, the essential boundary conditions can be implemented with ease as in the conventional finite element methods. The present PIM has been coded in FORTRAN. The validity and efficiency of the present PIM formulation are demonstrated through example problems. It is found that the present PIM is very easy to implement, and very flexible for obtained displacements and stresses of desired accuracy in solids. As the elements are not used for meshing the problem domain, the present PIM opens new avenues to develop adaptive analysis codes for stress analysis in solids and structures. Copyright © 2001 John Wiley & Sons, Ltd.

A point interpolation method for two-dimensional solids

Abstract: A Point Interpolation Method (PIM) is presented for stress analysis for two-dimensional solids. In the PIM, the problem domain is represented by properly scattered points. A technique is proposed to construct polynomial interpolants with delta function property based only on a group of arbitrarily distributed points. The PIM equations are then derived using variational principles. In the PIM, the essential boundary conditions can be implemented with ease as in the conventional Finite Element Methods. The present PIM has been coded in FORTRAN. The validity and efficiency of the present PIM formulation are demonstrated through example problems. It is found that the present PIM is very easy to implement, and very flexible for obtained displacements and stresses of desired accuracy in solids. As the elements are not used for meshing the problem domain, the present PIM opens new avenue to develop adaptive analysis codes for stress analysis in solids and structures.

Cross-sectional mapping of residual stresses by measuring the surface contour after a cut

Abstract: A powerful new method for residual stress measurement is presented. A part is cut in two, and the contour, or profile, of the resulting new surface is measured to determine the displacements caused by release of the residual stresses. Analytically, for example using a finite element model, the opposite of the measured contour is applied to the surface as a displacement boundary condition. By Bueckner's superposition principle, this calculation gives the original residual stresses normal to the plane of the cut. This contour method is more powerful than other relaxation methods because it can determine an arbitrary cross-sectional area map of residual stress, yet more simple because the stresses can be determined directly from the data without a tedious inversion technique. The new method is verified with a numerical simulation, then experimentally validated on a steel beam with a known residual stress profile.

Higher-Order Numerical Methods for Transient Wave Equations

Abstract: Solving efficiently the wave equations involved in modeling acoustic, elastic or electromagnetic wave propagation remains a challenge both for research and industry. To attack the problems coming from the propagative character of the solution, the author constructs higher-order numerical methods to reduce the size of the meshes, and consequently the time and space stepping, dramatically improving storage and computing times. This book surveys higher-order finite difference methods and develops various mass-lumped finite (also called spectral) element methods for the transient wave equations, and presents the most efficient methods, respecting both accuracy and stability for each sort of problem. A central role is played by the notion of the dispersion relation for analyzing the methods. The last chapter is devoted to unbounded domains which are modeled using perfectly matched layer (PML) techniques. Numerical examples are given.

Theoretical Numerical Analysis: A Functional Analysis Framework

Abstract: Preface 1 Linear Spaces 2 Linear Operators on Normed Spaces 3 Approximation Theory 4 Nonlinear Equations and Their Solution by Iteration 5 Finite Difference Method 6 Sobolev Spaces 7 Variational Formulations of Elliptic Boundary Value Problems 8 The Galerkin Method and Its Variants 9 Finite Element Analysis 10 Elliptic Variational Inequalities and Their Numerical Approximations 11 Numerical Solution of Fredholm Integral Equations of the Second Kind 12 Boundary Integral Equations References Index.

Dynamic real-time deformations using space & time adaptive sampling

Abstract: This paper presents a robust, adaptive method for animating dynamic visco-elastic deformable objects that provides a guaranteed frame rate. Our approach uses a novel automatic space and time adaptive level of detail technique, in combination with a large-displacement (Green) strain tensor formulation. The body is partitioned in a non-nested multiresolution hierarchy of tetrahedral meshes. The local resolution is determined by a quality condition that indicates where and when the resolution is too coarse. As the object moves and deforms, the sampling is refined to concentrate the computational load into the regions that deform the most. Our model consists of a continuous differential equation that is solved using a local explicit finite element method. We demonstrate that our adaptive Green strain tensor formulation suppresses unwanted artifacts in the dynamic behavior, compared to adaptive mass-spring and other adaptive approaches. In particular, damped elastic vibration modes are shown to be nearly unchanged for several levels of refinement. Results are presented in the context of a virtual reality system. The user interacts in real-time with the dynamic object through the control of a rigid tool, attached to a haptic device driven with forces derived from the method.

Classical and Computational Solid Mechanics

Abstract: Tensor analysis stress tensor analysis of strain conservation laws elastic and plastic behaviour of materials linearized theory of elasticity solutions of problems in linearized theory of elasticity by potentials two-dimensional problems in linearized theory of elasticity variational calculus, energy theorems, Saint-Venant's principle Hamilton's principle, wave propagation, applications of generalized co-ordinates elasticity and thermodynamics irreversible thermodynamics and viscoelasticity thermoelasticity viscoelasticity large deformation incremental approach to solving some nonlinear problems finite element methods mixed and hybrid formulations finite element methods for plates and shells finite element modelling of nonlinear elasticity, viscoelasticity, plasticity, viscoplasticity and creep.

Discontinuous hp -Finite Element Methods for Advection-Diffusion-Reaction Problems

Abstract: We consider the hp-version of the discontinuous Galerkin finite element method (DGFEM) for second-order partial differential equations with nonnegative characteristic form. This class of equations includes second-order elliptic and parabolic equations, advection-reaction equations, as well as problems of mixed hyperbolic-elliptic-parabolic type. Our main concern is the error analysis of the method in the absence of streamline-diffusion stabilization. In the hyperbolic case, an hp-optimal error bound is derived; here, we consider only advection-reaction problems which satisfy a certain (standard) positivity condition. In the self-adjoint elliptic case, an error bound that is h-optimal and p-suboptimal by $\frac{1}{2}$ a power of p is obtained. These estimates are then combined to deduce an error bound in the general case. For elementwise analytic solutions the method exhibits exponential rates of convergence under p-refinement. The theoretical results are illustrated by numerical experiments.

High order finite difference and finite volume WENO schemes and discontinuous Galerkin methods for CFD

Abstract: In recent years high order numerical methods have been widely used in computational fluid dynamics (CFD), to effectively resolve complex flow features using meshes which are reasonable for today''s computers. In this paper we review and compare three types of high order methods being used in CFD, namely the weighted essentially non-oscillatory (WENO) finite difference methods, the WENO finite volume methods, and the discontinuous Galerkin (DG) finite element methods. We summarize the main features of these methods, from a practical user''s point of view, indicate their applicability and relative strength, and show a few selected numerical examples to demonstrate their performance on illustrative model CFD problems.

Computational Mechanics of the Heart

Abstract: Finite elasticity theory combined with finite element analysis provides the framework for analysing ventricular mechanics during the filling phase of the cardiac cycle, when cardiac cells are not actively contracting. The orthotropic properties of the passive tissue are described here by a “pole–zero” constitutive law, whose parameters are derived in part from a model of the underlying distributions of collagen fibres. These distributions are based on our observations of the fibrous-sheet laminar architecture of myocardial tissue. We illustrate the use of high order (cubic Hermite) basis functions in solving the Galerkin finite element stress equilibrium equations based on this orthotropic constitutive law and for incorporating the observed regional distributions of fibre and sheet orientations. Pressure–volume relations and 3D principal strains predicted by the model are compared with experimental observations. A model of active tissue properties, based on isolated muscle experiments, is also introduced in order to predict transmural distributions of 3D principal strains at the end of the contraction phase of the cardiac cycle. We end by offering a critique of the current model of ventricular mechanics and propose new challenges for future modellers.

Three Dimensional Absolute Nodal Coordinate Formulation for Beam Elements: Theory

Abstract: The description of a beam element by only the displacement of its centerline leads to some difficulties in the representation of the torsion and shear effects. For instance such a representation does not capture the rotation of the beam as a rigid body about its own axis. This problem was circumvented in the literature by using a local coordinate system in the incremental finite element method or by using the multibody floating frame of reference formulation. The use of such a local element coordinate system leads to a highly nonlinear expression for the inertia forces as the result of the large element rotation. In this investigation, an absolute nodal coordinate formulation is presented for the large rotation and deformation analysis of three dimensional beam elements. This formulation leads to a constant mass matrix, and as a result, the vectors of the centrifugal and Coriolis forces are identically equal to zero. The formulation presented in this paper takes into account the effect of rotary inertia, torsion and shear, and ensures continuity of the slopes as well as the rotation of the beam cross section at the nodal points. Using the proposed formulation curved beams can be systematically modeled.

A finite element pressure gradient stabilization¶for the Stokes equations based on local projections

Abstract: We present a variant of the classical weighted least-squares stabilization for the Stokes equations. Compared to the original formulation, the new method has improved accuracy properties, especially near boundaries. Furthermore, no modification of the right-hand side is needed, and implementation is simplified, especially for generalizations to more complicated equations. The approach is based on local projections of the residual terms which are used in order to achieve consistency of the method, avoiding local evaluation of the strong form of the differential operator. We prove stability and give a priori and a posteriori error estimates. We show convergence of an iterative method which uses a simplified stabilized discretization as preconditioner. Numerical experiments indicate that the approach presented is at least as accurate as the original method, but offers some algorithmic advantages. The ideas presented here also apply to the Navier–Stokes equations. This is the topic of forthcoming work.

A Priori Error Estimates for Finite Element Methods Based on Discontinuous Approximation Spaces for Elliptic Problems

Abstract: We analyze three discontinuous Galerkin approximations for solving elliptic problems in two or three dimensions. In each one, the basic bilinear form is nonsymmetric: the first one has a penalty term on edges, the second has one constraint per edge, and the third is totally unconstrained. For each of them we prove hp error estimates in the H1 norm, optimal with respect to h, the mesh size, and nearly optimal with respect to p, the degree of polynomial approximation. We establish these results for general elements in two and three dimensions. For the unconstrained method, we establish a new approximation result valid on simplicial elements. L2 estimates are also derived for the three methods.

Discretization Methods and Iterative Solvers Based on Domain Decomposition

Abstract: Discretization Techniques Based on Domain Decomposition.- 1.1 Introduction to Mortar Finite Element Methods.- 1.2 Mortar Methods with Alternative Lagrange Multiplier Spaces.- 1.2.1 An Approximation Property.- 1.2.2 The Consistency Error.- 1.2.3 Discrete Inf-sup Conditions.- 1.2.4 Examples of Lagrange Multiplier Spaces.- 1.2.4.1 The First Order Case in 2D.- 1.2.4.2 The First Order Case in 3D.- 1.2.4.3 The Second Order Case in 2D.- 1.3 Discretization Techniques Based on the Product Space.- 1.3.1 A Dirichlet-Neumann Formulation.- 1.3.2 Variational Formulations.- 1.3.3 Algebraic Formulations.- 1.4 Examples for Special Mortar Finite Element Discretizations.- 1.4.1 The Coupling of Primal and Dual Finite Elements.- 1.4.2 An Equivalent Nonconforming Formulation.- 1.4.3 Crouzeix-Raviart Finite Elements.- 1.5 Numerical Results.- 1.5.1 Influence of the Lagrange Multiplier Spaces.- 1.5.2 A Non-optimal Mortar Method.- 1.5.3 Influence of the Choice of the Mortar Side.- 1.5.4 Influence of the Jump of the Coefficients.- Iterative Solvers Based on Domain Decomposition.- 2.1 Abstract Schwarz Theory.- 2.1.1 Additive Schwarz Methods.- 2.1.2Multiplicative Schwarz Methods.- 2.1.3 Multigrid Methods.- 2.2 Vector Field Discretizations.- 2.2.1 Raviart-Thomas Finite Elements.- 2.2.2 An Iterative Substructuring Method.- 2.2.2.1 An Interpolation Operator onto VH.- 2.2.2.2 An Extension Operator onto VF.- 2.2.2.3 Quasi-optimal Bounds.- 2.2.3 A Hierarchical Basis Method.- 2.2.3.1 Horizontal Decomposition.- 2.2.3.2 Vertical Decomposition.- 2.2.4 Numerical Results.- 2.2.4.1 The 2D Case.- 2.2.4.2 The 3D Case.- 2.3 A Multigrid Method for the Mortar Product Space Formulation.- 2.3.1 Bilinear Forms.- 2.3.2 An Approximation Property.- 2.3.3 Smoothing and Stability Properties.- 2.3.4 Implementation of the Smoothing Step.- 2.3.5 Numerical Results in 2D and 3D.- 2.3.6 Extensions to Linear Elasticity.- 2.3.6.1 Uniform Ellipticity.- 2.3.6.2 Numerical Results.- 2.3.6.3 A Weaker Interface Condition.- 2.4 A Dirichlet-Neumann Type Method.- 2.4.1 The Algorithm.- 2.4.2 Numerical Results.- 2.5 A Multigrid Method for the Mortar Saddle Point Formulation.- 2.5.1 An Approximation Property.- 2.5.2 Smoothing and Stability Properties.- 2.5.2.1 A Block Diagonal Smoother.- 2.5.2.2 An Indefinite Smoother.- 2.5.3 Numerical Results.- List of Figures.- List of Tables.- Notations.