Showing papers on "Finite element method published in 1986"

Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms

Abstract: I. Mathematical Foundation of the Stokes Problem.- 1. Generalities on Some Elliptic Boundary Value Problems.- 1.1. Basic Concepts on Sobolev Spaces.- 1.2. Abstract Elliptic Theory.- 1.3. Example 1: Dirichlet's Problem for the Laplace Operator.- 1.4. Example 2: Neumann's Problem for the Laplace Operator.- 1.5. Example 3: Dirichlet's Problem for the Biharmonic Operator.- 2. Function Spaces for the Stokes Problem.- 2.1. Preliminary Results.- 2.2. Some Properties of Spaces Related to the Divergence Operator.- 2.3. Some Properties of Spaces Related to the Curl Operator.- 3. A Decomposition of Vector Fields.- 3.1. Decomposition of Two-Dimensional Vector Fields.- 3.2. Application to the Regularity of Functions of H(div ?) ? H(curl ?).- 3.3. Decomposition of Three-Dimensional Vector Fields.- 3.4. The Imbedding of H(div ?) ? H0 (curl ?) into H1(?)3.- 3.5. The Imbedding of H0(div ?) ? H (curl ?) into H1(?)3.- 4. Analysis of an Abstract Variational Problem.- 4.1. A General Result.- 4.2. A Saddle-Point Approach.- 4.3. Approximation by Regularization or Penalty.- 4.4. Iterative Methods of Gradient Type.- 5. The Stokes Equations.- 5.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 5.2. The Stream Function Formulation of the Dirichlet Problem in Two Dimensions.- 5.3. The Three-Dimensional Case.- Appendix A. Results of Standard Finite Element Approximation.- A.l. Triangular Finite Elements.- A.2. Quadrilateral Finite Elements.- A.3. Interpolation of Discontinuous Functions.- II. Numerical Solution of the Stokes Problem in the Primitive Variables.- 1. General Approximation.- 1.1. An Abstract Approximation Result.- 1.2. Decoupling the Computation of uh and ?h.- 1.3. Application to the Homogeneous Stokes Problem.- 1.4. Checking the inf-sup Condition.- 2. Simplicial Finite Element Methods Using Discontinuous Pressures.- 2.1. A First Order Approximation on Triangular Elements.- 2.2. Higher-Order Approximation on Triangular Elements.- 2.3. The Three-Dimensional case: First and Higher-Order Schemes.- 3. Quadrilateral Finite Element Methods Using Discontinuous Pressures.- 3.1. A quadrilateral Finite Element of Order One.- 3.2. Higher-Order Quadrilateral Elements.- 3.3. An Example of Checkerboard Instability: the Q1 - P0 Element.- 3.4. Error Estimates for the Q1 - P0 Element.- 4. Continuous Approximation of the Pressure.- 4.1. A First Order Method: the "Mini" Finite Element.- 4.2. The "Hood-Taylor" Finite Element Method.- 4.3. The "Glowinski-Pironneau" Finite Element Method.- 4.4. Implementation of the Glowinski-Pironneau Scheme.- III. Incompressible Mixed Finite Element Methods for Solving the Stokes Problem.- 1. Mixed Approximation of an Abstract Problem.- 1.1. A Mixed Variational Problem.- 1.2. Abstract Mixed Approximation.- 2. The "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions.- 2.1. A Mixed Formulation.- 2.2. Mixed Approximation and Application to Finite Elements of Degree l.- 2.3. The Technique of Mesh-Dependent Norms.- 3. Further Topics on the "Stream Function-Vorticity-Pressure" Scheme.- 3.1. Refinement of the Error Analysis.- 3.2. Super Convergence Using Quadrilateral Finite Elements of Degree l.- 4. A "Stream Function-Gradient of Velocity Tensor" Method in Two Dimensions.- 4.1. The Hellan-Herrmann-Johnson Formulation.- 4.2. Approximation with Triangular Finite Elements of Degree l.- 4.3. Additional Results for the Hellan-Herrmann-Johnson Scheme.- 4.4. Discontinuous Approximation of the Pressure.- 5. A "Vector Potential-Vorticity" Scheme in Three Dimensions.- 5.1. A Mixed Formulation of the Three-Dimensional Stokes Problem.- 5.2. Mixed Approximation in H(curl ?).- 5.3. A Family of Conforming Finite Elements in H(curl ?).- 5.4. Error Analysis for Finite Elements of Degree l.- 5.5. Discontinuous Approximation of the Pressure.- IV. Theory and Approximation of the Navier-Stokes Problem.- 1. A Class of Nonlinear Problems.- s Problem for the Laplace Operator.- 1.5. Example 3: Dirichlet's Problem for the Biharmonic Operator.- 2. Function Spaces for the Stokes Problem.- 2.1. Preliminary Results.- 2.2. Some Properties of Spaces Related to the Divergence Operator.- 2.3. Some Properties of Spaces Related to the Curl Operator.- 3. A Decomposition of Vector Fields.- 3.1. Decomposition of Two-Dimensional Vector Fields.- 3.2. Application to the Regularity of Functions of H(div ?) ? H(curl ?).- 3.3. Decomposition of Three-Dimensional Vector Fields.- 3.4. The Imbedding of H(div ?) ? H0 (curl ?) into H1(?)3.- 3.5. The Imbedding of H0(div ?) ? H (curl ?) into H1(?)3.- 4. Analysis of an Abstract Variational Problem.- 4.1. A General Result.- 4.2. A Saddle-Point Approach.- 4.3. Approximation by Regularization or Penalty.- 4.4. Iterative Methods of Gradient Type.- 5. The Stokes Equations.- 5.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 5.2. The Stream Function Formulation of the Dirichlet Problem in Two Dimensions.- 5.3. The Three-Dimensional Case.- Appendix A. Results of Standard Finite Element Approximation.- A.l. Triangular Finite Elements.- A.2. Quadrilateral Finite Elements.- A.3. Interpolation of Discontinuous Functions.- II. Numerical Solution of the Stokes Problem in the Primitive Variables.- 1. General Approximation.- 1.1. An Abstract Approximation Result.- 1.2. Decoupling the Computation of uh and ?h.- 1.3. Application to the Homogeneous Stokes Problem.- 1.4. Checking the inf-sup Condition.- 2. Simplicial Finite Element Methods Using Discontinuous Pressures.- 2.1. A First Order Approximation on Triangular Elements.- 2.2. Higher-Order Approximation on Triangular Elements.- 2.3. The Three-Dimensional case: First and Higher-Order Schemes.- 3. Quadrilateral Finite Element Methods Using Discontinuous Pressures.- 3.1. A quadrilateral Finite Element of Order One.- 3.2. Higher-Order Quadrilateral Elements.- 3.3. An Example of Checkerboard Instability: the Q1 - P0 Element.- 3.4. Error Estimates for the Q1 - P0 Element.- 4. Continuous Approximation of the Pressure.- 4.1. A First Order Method: the "Mini" Finite Element.- 4.2. The "Hood-Taylor" Finite Element Method.- 4.3. The "Glowinski-Pironneau" Finite Element Method.- 4.4. Implementation of the Glowinski-Pironneau Scheme.- III. Incompressible Mixed Finite Element Methods for Solving the Stokes Problem.- 1. Mixed Approximation of an Abstract Problem.- 1.1. A Mixed Variational Problem.- 1.2. Abstract Mixed Approximation.- 2. The "Stream Function-Vorticity-Pressure" Method for the Stokes Problem in Two Dimensions.- 2.1. A Mixed Formulation.- 2.2. Mixed Approximation and Application to Finite Elements of Degree l.- 2.3. The Technique of Mesh-Dependent Norms.- 3. Further Topics on the "Stream Function-Vorticity-Pressure" Scheme.- 3.1. Refinement of the Error Analysis.- 3.2. Super Convergence Using Quadrilateral Finite Elements of Degree l.- 4. A "Stream Function-Gradient of Velocity Tensor" Method in Two Dimensions.- 4.1. The Hellan-Herrmann-Johnson Formulation.- 4.2. Approximation with Triangular Finite Elements of Degree l.- 4.3. Additional Results for the Hellan-Herrmann-Johnson Scheme.- 4.4. Discontinuous Approximation of the Pressure.- 5. A "Vector Potential-Vorticity" Scheme in Three Dimensions.- 5.1. A Mixed Formulation of the Three-Dimensional Stokes Problem.- 5.2. Mixed Approximation in H(curl ?).- 5.3. A Family of Conforming Finite Elements in H(curl ?).- 5.4. Error Analysis for Finite Elements of Degree l.- 5.5. Discontinuous Approximation of the Pressure.- IV. Theory and Approximation of the Navier-Stokes Problem.- 1. A Class of Nonlinear Problems.- 2. Theory of the Steady-State Navier-Stokes Equations.- 2.1. The Dirichlet Problem in the Velocity-Pressure Formulation.- 2.2. The Stream Function Formulation of the Homogeneous Problem..- 3. Approximation of Branches of Nonsingular Solutions.- 3.1. An Abstract Framework.- 3.2. Approximation of Branches of Nonsingular Solutions.- 3.3. Application to a Class of Nonlinear Problems.- 3.4. Non-Differentiable Approximation of Branches of Nonsingular Solutions.- 4. Numerical Analysis of Centered Finite Element Schemes.- 4.1. Formulation in Primitive Variables: Methods Using Discontinuous Pressures.- 4.2. Formulation in Primitive Variables: the Case of Continuous Pressures.- 4.3. Mixed Incompressible Methods: the "Stream Function-Vorticity" Formulation.- 4.4. Remarks on the "Stream Function-Gradient of Velocity Tensor" Scheme.- 5. Numerical Analysis of Upwind Schemes.- 5.1. Upwinding in the Stream Function-Vorticity Scheme.- 5.2. Error Analysis of the Upwind Scheme.- 5.3. Approximating the Pressure with the Upwind Scheme.- 6. Numerical Algorithms.- 2.11. General Methods of Descent and Application to Gradient Methods.- 2.12. Least-Squares and Gradient Methods to Solve the Navier-Stokes Equations.- 2.13. Newton's Method and the Continuation Method.- References.- Index of Mathematical Symbols.

A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuscka-Brezzi condition: A stable Petrov-Galerkin formulation of

Abstract: A new Petrov-Galerkin formulation of the Stokes problem is proposed. The new formulation possesses better stability properties than the classical Galerkin/variational method. An error analysis is performed for the case in which both the velocity and pressure are approximated by C0 interpolations. Combinations of C0 interpolations which are unstable according to the Babuska-Brezzi condition (e.g., equal-order interpolations) are shown to be stable and convergent within the present framework. Calculations exhibiting the good behavior of the methodology are presented.

A new family of mixed finite elements in IR 3

Abstract: We introduce two families of mixed finite element on conforming inH(div) and one conforming inH(curl). These finite elements can be used to approximate the Stokes' system.

A new finite element formulation for computational fluid dynamics: II. Beyond SUPG

Abstract: A discontinuity-capturing term is added to the streamline-upwind/Petrov-Galerkin weighting function for the scalar advection-diffusion equation. The additional term enhances the ability of the method to produce smooth yet crisp approximations to internal and boundary layers.

A First Course in the Finite Element Method

Abstract: 1 INTRODUCTION Brief History Introduction to Matrix Notation Role of the Computer General Steps of the Finite Element Method Applications of the Finite Element Method Advantages of the Finite Element Method Computer Programs for the Finite Element Method 2 INTRODUCTION TO THE STIFFNESS (DISPLACEMENT) METHOD Definition of the Stiffness Matrix Derivation of the Stiffness Matrix for a Spring Element Example of a Spring Assemblage Assembling the Total Stiffness Matrix by Superposition (Direct Stiffness Method) Boundary Conditions Potential Energy Approach to Derive Spring Element Equations 3 DEVELOPMENT OF TRUSS EQUATIONS Derivation of the Stiffness Matrix for a Bar Element in Local Coordinates Selecting Approximation Functions for Displacements Transformation of Vectors in Two Dimensions Global Stiffness Matrix for Bar Arbitrarily Oriented in the Plane Computation of Stress for a Bar in the x-y Plane Solution of a Plane Truss Transformation Matrix and Stiffness Matrix for a Bar in Three-Dimensional Space Use of Symmetry in Structure Inclined, or Skewed, Supports Potential Energy Approach to Derive Bar Element Equations Comparison of Finite Element Solution to Exact Solution for Bar Galerkin's Residual Method and Its Use to Derive the One-Dimensional Bar Element Equations Other Residual Methods and Their Application to a One-Dimensional Bar Problem Flowchart for Solutions of Three-Dimensional Truss Problems Computer Program Assisted Step-by-Step Solution for Truss Problem 4 DEVELOPMENT OF BEAM EQUATIONS Beam Stiffness Example of Assemblage of Beam Stiffness Matrices Examples of Beam Analysis Using the Direct Stiffness Method Distribution Loading Comparison of the Finite Element Solution to the Exact Solution for a Beam Beam Element with Nodal Hinge Potential Energy Approach to Derive Beam Element Equations Galerkin's Method for Deriving Beam Element Equations 5 FRAME AND GRID EQUATIONS Two-Dimensional Arbitrarily Oriented Beam Element Rigid Plane Frame Examples Inclined or Skewed Supports - Frame Element Grid Equations Beam Element Arbitrarily Oriented in Space Concept of Substructure Analysis 6 DEVELOPMENT OF THE PLANE STRESS AND STRAIN STIFFNESS EQUATIONS Basic Concepts of Plane Stress and Plane Strain Derivation of the Constant-Strain Triangular Element Stiffness Matrix and Equations Treatment of Body and Surface Forces Explicit Expression for the Constant-Strain Triangle Stiffness Matrix Finite Element Solution of a Plane Stress Problem Rectangular Plane Element (Bilinear Rectangle, Q4) 7 PRACTICAL CONSIDERATIONS IN MODELING: INTERPRETING RESULTS AND EXAMPELS OF PLANE STRESS/STRAIN ANALYSIS Finite Element Modeling Equilibrium and Compatibility of Finite Element Results Convergence of Solution Interpretation of Stresses Static Condensation Flowchart for the Solution of Plane Stress-Strain Problems Computer Program Assisted Step-by-Step Solution, Other Models, and Results for Plane Stress-Strain Problems 8 DEVELOPMENT OF THE LINEAR-STRAIN TRAINGLE EQUATIONS Derivation of the Linear-Strain Triangular Element Stiffness Matrix and Equations Example of LST Stiffness Determination Comparison of Elements 9 AXISYMMETRIC ELEMENTS Derivation of the Stiffness Matrix Solution of an Axisymmetric Pressure Vessel Applications of Axisymmetric Elements 10 ISOPARAMETRIC FORMULATION Isoparametric Formulation of the Bar Element Stiffness Matrix Isoparametric Formulation of the Okabe Quadrilateral Element Stiffness Matrix Newton-Cotes and Gaussian Quadrature Evaluation of the Stiffness Matrix and Stress Matrix by Gaussian Quadrature Higher-Order Shape Functions 11 THREE-DIMENSIONAL STRESS ANALYSIS Three-Dimensional Stress and Strain Tetrahedral Element Isoparametric Formulation 12 PLATE BENDING ELEMENT Basic Concepts of Plate Bending Derivation of a Plate Bending Element Stiffness Matrix and Equations Some Plate Element Numerical Comparisons Computer Solutions for Plate Bending Problems 13 HEAT TRANSFER AND MASS TRANSPORT Derivation of the Basic Differential Equation Heat Transfer with Convection Typical Units Thermal Conductivities K and Heat-Transfer Coefficients, h One-Dimensional Finite Element Formulation Using a Variational Method Two-Dimensional Finite Element Formulation Line or Point Sources Three-Dimensional Heat Transfer by the Finite Element Method One-Dimensional Heat Transfer with Mass Transport Finite Element Formulation of Heat Transfer with Mass Transport by Galerkin's Method Flowchart and Examples of a Heat-Transfer Program 14 FLUID FLOW IN POROUS MEDIA AND THROUGH HYDRAULIC NETWORKS AND ELECTRICAL NETWORKS AND ELECTROSTATICS Derivation of the Basic Differential Equations One-Dimensional Finite Element Formulation Two-Dimensional Finite Element Formulation Flowchart and Example of a Fluid-Flow Program Electrical Networks Electrostatics 15 THERMAL STRESS Formulation of the Thermal Stress Problem and Examples 16 STRUCTURAL DYNAMICS AND TIME-DEPENDENT HEAT TRANSFER Dynamics of a Spring-Mass System Direct Derivation of the Bar Element Equations Numerical Integration in Time Natural Frequencies of a One-Dimensional Bar Time-Dependent One-Dimensional Bar Analysis Beam Element Mass Matrices and Natural Frequencies Truss, Plane Frame, Plane Stress, Plane Strain, Axisymmetric, and Solid Element Mass Matrices Time-Dependent Heat-Transfer Computer Program Example Solutions for Structural Dynamics APPENDIX A - MATRIX ALGEBRA Definition of a Matrix Matrix Operations Cofactor of Adjoint Method to Determine the Inverse of a Matrix Inverse of a Matrix by Row Reduction Properties of Stiffness Matrices APPENDIX B - METHODS FOR SOLUTION OF SIMULTANEOUS LINEAR EQUATIONS Introduction General Form of the Equations Uniqueness, Nonuniqueness, and Nonexistence of Solution Methods for Solving Linear Algebraic Equations Banded-Symmetric Matrices, Bandwidth, Skyline, and Wavefront Methods APPENDIX C - EQUATIONS FOR ELASTICITY THEORY Introduction Differential Equations of Equilibrium Strain/Displacement and Compatibility Equations Stress-Strain Relationships APPENDIX D - EQUIVALENT NODAL FORCES APPENDIX E - PRINCIPLE OF VIRTUAL WORK APPENDIX F - PROPERTIES OF STRUCTURAL STEEL AND ALUMINUM SHAPES ANSWERS TO SELECTED PROBLEMS INDEX

Energy release rate along a three-dimensional crack front in a thermally stressed body

Abstract: Based on a line-integral expression for the energy release rate in terms of crack tip fields, which is valid for general material response, a (area/volume) domain integral expression for the energetic force in a thermally stressed body is derived. The general three-dimensional finite domain integral expression and the two-dimensional and axisymmetric specializations for the energy release rate are given. The domain expression is naturally compatible with the finite element formulation of the field equations. As such it is ideally suited for efficient and accurate calculation of the pointwise values of the energy release rate along a three-dimensional crack front. The finite element implementation of the domain integral corresponds to the virtual crack extension technique. Procedures for calculating the energy release rate using the numerically determined field solutions are discussed. For illustrative purposes several numerical examples are presented.

The construction of preconditioners for elliptic problems by substructuring. I

Abstract: In earlier parts of this series of papers, we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems. The resulting algorithms are well suited for implementation on computers with parallel architecture. In this paper, we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners. The iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns. These preconditioners involve an incomplete Chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work. Theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods.

Random field finite elements

Abstract: The probabilistic finite element method (PFEM) is formulated for linear and non-linear continua with inhomogeneous random fields. Analogous to the discretization of the displacement field in finite element methods, the random field is also discretized. The formulation is simplified by transforming the correlated variables to a set of uncorrelated variables through an eigenvalue orthogonalization. Furthermore, it is shown that a reduced set of the uncorrelated variables is sufficient for the second-moment analysis. Based on the linear formulation of the PFEM, the method is then extended to transient analysis in non-linear continua. The accuracy and efficiency of the method is demonstrated by application to a one-dimensional, elastic/plastic wave propagation problem and a two-dimensional plane-stress beam bending problem. The moments calculated compare favourably with those obtained by Monte Carlo simulation. Also, the procedure is amenable to implementation in deterministic FEM based computer programs.

An analysis of the discontinuous Galerkin method for a scalar hyperbolic equation

Abstract: We prove Lp stability and error estimates for the discontinuous Galerkin method when applied to a scalar linear hyperbolic equation on a convex polygonal plane domain. Using finite element analysis techniques, we obtain L2 estimates that are valid on an arbitrary locally regular triangulation of the domain and for an arbitrary degree of polynomials. Lp estimates for p * 2 are restricted to either a uniform or piecewise uniform triangulation and to polynomials of not higher than first degree. The latter estimates are proved by combining finite difference and finite element analysis techniques.

A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems

Abstract: A finite element method based on the SUPG concept is presented for multidimensional advective-diffusive systems. Error estimates for the linear case are established which are valid over the full range of advective-diffusive phenomena.

A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier—Stokes equations and the second law of thermodynamics

Abstract: Results of Harten and Tadmor are generalized to the compressible Navier-Stokes equations including heat conduction effects. A symmetric form of the equations is derived in terms of entropy variables. It is shown that finite element methods based upon this form automatically satisfy the second law of thermodynamics and that stability of the discrete solution is thereby guaranteed ab initio.

The $h-p$ version of the finite element method with quasiuniform meshes

Abstract: : The classical error estimates for the h-version of the finite element method are extended for the h-p version. The estimates are expressed as explicit functions of h and p are shown to be optimal. The estimates are given for the case where the solution u (H sub k and the case when u has singularities at the corners of the domain. (Author)

Computational heat transfer

Abstract: Preface 1.Introduction Part 1:Mathematical Background 2. Governing Equations 3. Finite Differences 4.Finite Elements Part 2: Simulation of Transport Processes 5.Numerical Methods for Conduction Heat Transfer 6. Numerical Methods for Convection Heat Trasnfer 7.Numerical Methods for Radiation Heat Transfer Part 3: Combined Modes and Process Applications Appendixes

Probabilistic finite elements for nonlinear structural dynamics

Abstract: A finite element method applicable to truss structures for the determination of the probabilistic distribution of the dynamic response has been developed. Several solutions have been obtained for the mean and variance of displacements and stresses of a truss structure; nonlinearities due to material and geometrical effects have also been included. In addition, to test this method, Monte Carlo simulations have been used and a new method with implicit and/or explicit time integration and Hermite-Gauss quadrature has also been developed and used. All these methodologies have been implemented into a pilot computer code with two-dimensional bar elements.

Dynamics of viscoelastic structures: a time-domain finite element formulation

Abstract: Mathematical models of elastic structures have become very sophisticated: given the crucial material properties (mass density and the several elastic moduli), computer-based techniques can be used to construct exotic finite element models. By contrast, the modeling of damping is usually very primitive, often consisting of no more than mere guesses at “modal damping factors.” The aim of this paper is to raise the modeling of viscoelastic structures to a level consistent with the modeling of elastic structures. Appropriate material properties are identified which permit the standard finite element formulations used for undamped structures to be extended to viscoelastic structures. Through the use of “dissipation” coordinates, the canonical “M , K ” form of the undamped motion equations is expanded to encompass viscoelastic damping. With this formulation finite element analysis can be used to model viscoelastic damping accurately.

Computer modeling of heat flow in welds

Abstract: This paper summarizes progress in the development of methods, models, and software for analyzing or simulating the flow of heat in welds as realistically and accurately as possible. First the fundamental equations for heat transfer are presented and then a formulation for a nonlinear transient finite element analysis (FEA) to solve them is described. Next the magnetohydrodynamics of the arc and the fluid mechanics of the weld pool are approximated by a flux or power density distribution selected to predict the temperature field as accurately as possible. To assess the accuracy of a model, the computed and experimentally determined fusion zone boundaries are compared. For arc welds, accurate results are obtained with a power density distribution in which surfaces of constant power density are ellipsoids and on radial lines the power density obeys a Gaussian distribution. Three dimensional, in-plane and cross-sectional kinematic models for heat flow are defined. Guidelines for spatial and time discretization are discussed. The FEA computed and experimentally measured temperature field,T(x, y, z, t), for several welding situations is used to demonstrate the effect of temperature dependent thermal properties, radiation, convection, and the distribution of energy in the arc.

A new finite element formulation for computational fluid dynamics: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems

Abstract: A discontinuity-capturing operator is developed for the ‘streamline’ formulation of advective-diffusive systems extending previous work on the scalar advection-diffusion equation. The operator provides a mechanism for exerting control over strong gradients in the discrete solution which appear, for example, in boundary and interior layers.

Finite volume solution of the two-dimensional Euler equations on a regular triangular mesh

Abstract: The two-dimensional Euler equations have been solved on a triangular grid by a multigrid scheme using the finite volume approach. By careful construction of the dissipative terms, the scheme is designed to be secondorder accurate in space, provided the grid is smooth, except in the vicinity of shocks, where it behaves as firstorder accurate. In its present form, the accuracy and convergence rate of the triangle code are comparable to that of the quadrilateral mesh code of Jameson.

Euler calculations for multielement airfoils using Cartesian grids

Abstract: A finite volume formulation for the Euler equations using Cartesian grids is presented and used to study complex two-dimensional configurations. The formulation extends methods developed for the potential equation to the Euler equations. Results using this approach for single element airfoils are shown to be competitive with and as accurate as other methods that employ mapped grids. Further, it is demonstrated that this method provides a simple and accurate procedure for solving flow problems involving multielement airfoils.

On the multi-level splitting of finite element spaces

Abstract: In this paper we analyze the condition number of the stiffness matrices arising in the discretization of selfadjoint and positive definite plane elliptic boundary value problems of second order by finite element methods when using hierarchical bases of the finite element spaces instead of the usual nodal bases. We show that the condition number of such a stiffness matrix behaves like O((log κ)2) where κ is the condition number of the stiffness matrix with respect to a nodal basis. In the case of a triangulation with uniform mesh sizeh this means that the stiffness matrix with respect to a hierarchical basis of the finite element space has a condition number behaving like $$O\left( {\left( {\log \frac{1}{h}} \right)^2 } \right)$$ instead of $$O\left( {\left( {\frac{1}{h}} \right)^2 } \right)$$ for a nodal basis. The proofs of our theorems do not need any regularity properties of neither the continuous problem nor its discretization. Especially we do not need the quasiuniformity of the employed triangulations. As the representation of a finite element function with respect to a hierarchical basis can be converted very easily and quickly to its representation with respect to a nodal basis, our results mean that the method of conjugate gradients needs onlyO(log n) steps andO(n log n) computer operations to reduce the energy norm of the error by a given factor if one uses hierarchical bases or related preconditioning procedures. Heren denotes the dimension of the finite element space and of the discrete linear problem to be solved.

Discontinuity-capturing finite element formulations for nonlinear convection-diffusion-reaction equations

Abstract: Formulations which complement the streamline-upwind/Petrov-Galerkin procedure are presented. These formulations minimize the oscillations about sharp internal and boundary layers in convection-dominated and reaction-dominated flows. The proposed methods are tested on various single- and multi-component transport problems.

Theh,p andh-p versions of the finite element method in 1 dimension

Abstract: This paper is the first one in the series of three which are addressing in detail the properties of the three basic versions of the finite element method in the one dimensional setting The main emphasis is placed on the analysis when the (exact) solution has singularity of xα-type. The first part analyzes thep-version, the second theh-version and generalh-p version and the final third part addresses the problems of the adaptiveh-p version.

Efficient implementation of quadrilaterals with high coarse-mesh accuracy

Abstract: Quadrilateral elements for plane stress and strain are formulated so as to achieve high coarse-mesh accuracy, particularly for bending problems and incompressible materials. The Hu-Washizu variational principle is used in conjunction with an orthogonal projection so that the resulting element stiffness requires no matrix inversions. Results are given for several problems, including distorted meshes, and the elements exhibit excellent performance for coarse meshes.

A Finite Element/Lagrange Approach to Modeling Lightweight Flexible Manipulators

Abstract: This paper presents a finite element/Lagrangian approach for the mathematical modeling of lightweight flexible manipulators. Each link of the manipulator is treated as an assemblage of a finite number of elements for each of which kinetic and potential energies are derived. These elemental kinetic and potential energies are then suitably combined to derive the dynamic model for the system. It is contended that satisfactory modeling and analysis of the manipulator dynamics can lead to the use of advanced control techniques to solve some of the problems associated with the flexure of otherwise attractive lightweight manipulator arms. Detailed model development and simulation results for the case of a two-link manipulator system are presented.