Showing papers on "Finite element method published in 1989"
Book•
01 Jan 1989
TL;DR: In this article, the methodes are numeriques and the fonction de forme reference record created on 2005-11-18, modified on 2016-08-08.
Abstract: Keywords: methodes : numeriques ; fonction de forme Reference Record created on 2005-11-18, modified on 2016-08-08
17,327 citations
Book•
01 Jan 1989TL;DR: In this article, the authors propose a floating frame of reference formulation for large deformation problems in linear algebra, based on reference kinematics and finite element formulation for deformable bodies.
Abstract: 1. Introduction 2. Reference kinematics 3. Analytical techniques 4. Mechanics of deformable bodies 5. Floating frame of reference formulation 6. Finite element formulation 7. Large deformation problem Appendix: Linear algebra References Index.
2,125 citations
TL;DR: This paper presents a class of TVB (total variation bounded) discontinuous Galerkin finite element methods for solving conservation laws ut+Σi=1d(fi(u)xi=0.1d) using a 1-dimensional system as a model, and discusses different implementation techniques and theories analogous to scalar cases proven for linear systems.
1,421 citations
TL;DR: Galerkin/least-squares finite element methods for advective-diffusive equations are presented in this paper, and a convergence analysis and error estimates are presented.
1,323 citations
Book•
09 Mar 1989
TL;DR: In this paper, the finite element method was used to analyze the metal forming process and its properties, including plasticity, viscoplasticity, and plane-strain problems.
Abstract: Introduction Metal forming process Analysis and technology in metal forming Plasticity and viscoplasticity Methods of analysis The finite element method (1) The finite element method (2) Plane-strain problems Axisymmetric isothermal forging Steady state processes of extrusion and drawing Sheet metal forming Thermo-viscoplastic analysis Compaction and forging of porous metals Three dimensional problems Preform design in metal forming Solid formulation, comparison of two formulations, and concluding remarks Index.
1,226 citations
TL;DR: In this paper, a polynomial expansion of the numerical nonlinear structural operator is made according to a response-surface approximation in terms of spatial averages of the design variables, which can be used for the analysis of structural and mechanical systems whose geometrical and material properties have spatial random variability.
Abstract: The present paper introduces and discusses a stochastic finite-element method. It can be used for the analysis of structural and mechanical systems whose geometrical and material properties have spatial random variability. The method utilizes a polynomial expansion of the numerical nonlinear structural operator (for which actual analytical form is unknown). The expansion is made according to a response-surface approximation in terms of spatial averages of the design variables. The polynomial form is then modified by suitable error factors, one for each geometrical or mechanical property. Each error factor is due to the deviations, of the single property, from its spatial average in the different finite elements. The method demands an accurate design of the experiments to be conducted in order to identify the model parameters. A numerical example has been worked out. In this numerical example, the stresses and the strains in a light-water reactor pressurized vessel are computed by a stochastic three-dimensional finite element nonlinear analysis.
598 citations
TL;DR: In this paper, a new method for the solution of problems involving material variability is proposed, which makes use of the Karhunen-Loeve expansion to represent the random material property.
Abstract: A new method for the solution of problems involving material variability is proposed. The material property is modeled as a stochastic process. The method makes use of the Karhunen‐Loeve expansion to represent the random material property. The expansion is a representation of the process in terms of a finite set of uncorrelated random variables. The resulting formulation is compatible with the finite element method. A Neumann expansion scheme is subsequently employed to obtain a convergent expansion of the response process. The response is thus obtained as a homogeneous multivariate polynomial in the uncorrelated random variables. From this representation various statistical quantities may be derived. The usefulness of the proposed method, in terms of accuracy and efficiency, is exemplified by considering a cantilever beam with random rigidity. The derived results pertaining to the second‐order statistics of the response are found in good agreement with those obtained by a Monte Carlo simulation solution ...
496 citations
Book•
12 Oct 1989
TL;DR: This book discusses problems, formulations, Algorithms, and other issues that have not been Considered in the area of discretization of the Discrete Equations, as well as some of the methods used in solving these problems.
Abstract: Discretization of the Primitive Variable Formulation: A Primitive Variable Formulation. The Finite Element Problem and the Div-St abi lity Condition. Finite Element Spaces. Alternate Weak Forms, Boundary Conditions and Numerical Integration. Penalty Methods. Solution of the Discrete Equations: Newton's Method and Other Iterative Methods. Solving the Linear Systems. Solution Methods for Large Reynolds Numbers. Time Dependent Problems: A Weak Formulation and Spatial Discretizations. Time Discretizations. The Streamfunction-Vorticity Formulation: Algorithms for the Streamfunction-Vorticity Equations. Solution Techniques for Multiply Connected Domains. The Streamfunction Formulation: Algorithms for Determining Streamfunction Approximations. Eigenvalue Problems Connected with Stability Studies for Viscous Flows: Energy Stability Analysis of Viscous Flows. Linearized Stability Analysis of Stationary Viscous Flows. Exterior Problems: Truncated Domain-Artificial Boundary Condition Methods. Nonlinear Constitutive Relations: A Ladyzhenskaya Model and Algebraic Turbulence Models. Bingham Fluids. Electromagnetically or Thermally Coupled Flows: Flows of Liquid Metals. The Boussinesq Equations. Remarks on Some Topics That Have Not Been Considered: Problems, Formulations, Algorithms, and Other Issues That Have Not Been Considered. Bibliography. Glossary of Symbols. Index.
491 citations
01 Jan 1989
TL;DR: The theoretical foundations and numerical implementation of spectral element methods for the incompressible Navier-Stokes equations are presented, considering the construction and analysis of optimal-order spectral element discretizations for elliptic and saddle (Stokes) problems.
Abstract: Spectral element methods are high-order weighted-residual techniques for partial differential equations that combine the geometric flexibility of finite element techniques with the rapid convergence rate of spectral schemes. The theoretical foundations and numerical implementation of spectral element methods for the incompressible Navier-Stokes equations are presented, considering the construction and analysis of optimal-order spectral element discretizations for elliptic and saddle (Stokes) problems, as well as the efficient solution of the resulting discrete equations by rapidly convergent tensor-product-based iterative procedures. Several examples of spectral element simulation of moderate Reynolds number unsteady flow in complex geometry are presented.
454 citations
TL;DR: In this article, a discrete canonical, singularity-free mapping between the five and the six degree of freedom formulation is constructed by exploiting the geometric connection between the orthogonal group (SO(3)) and the unit sphere (S2).
Abstract: Computational aspects of a linear stress resultant (classical) shell theory, obtained by systematic linearization of the geometrically exact nonlinear theory, considered in Part I of this work, are examined in detail. In particular, finite element interpolations for the reference director field and the linearized rotation field are constructed such that the underlying geometric structure of the continuum theory is preserved exactly by the discrete approximation. A discrete canonical, singularity-free mapping between the five and the six degree of freedom formulation is constructed by exploiting the geometric connection between the orthogonal group (SO(3)) and the unit sphere (S2). The proposed numerical treatment of the membrane and bending fields, based on a mixed Hellinger-Reissner formulation,provides excellent results for the 4-node bilinear isoparametric element. As an example, convergent results are obtained for rather coarse meshes in fairly demanding, singularity-dominated, problems such as the classical rhombic plate test. The proposed theory and finite element implementation are evaluated through an extensive set of benchmark problems. The results obtained with the present approach exactly match previous solutions obtained with state-of-the-art implementations based on the so-called degenerated solid approach.
450 citations
TL;DR: In this paper, a numerical scheme for crack modelling by means of continuous displacement fields is presented, where a crack is modelled as a limiting case of two singular lines (with continuous displacements, but discontinuous displacement gradients across them) which tend to coincide with each other.
Abstract: A numerical scheme for crack modelling by means of continuous displacement fields is presented. In two-dimensional problems a crack is modelled as a limiting case of two singular lines (with continuous displacements, but discontinuous displacement gradients across them) which tend to coincide with each other. An analysis of the energy dissipated inside the band bounded by both lines allows one to obtain an expression for the characteristic length as the ratio between the energy dissipated per unit surface area (fracture energy) and the energy dissipated per unit volume (specific energy) at a point. The application of these mathematical expressions to the finite element discretized medium allow one to obtain a general spatial and directional expression for the characteristic length which guarantees the objectivity of the results with respect to the size of the finite element mesh. The numerical results presented show the reliability of the proposed expressions.
TL;DR: Numerical solution methods surveyed here will be of much use to practicing computational/finite element/structural engineers working in the area of dynamics of structures.
TL;DR: A portable, power operated, hand cultivator comprising a frame having a motor supported thereon which oscillates two or more generally vertically disposed cultivator tines extending downwardly from the frame.
TL;DR: Two a posteriori error estimators for the mini-element discretization of the Stokes equations are presented, based on a suitable evaluation of the residual of the finite element solution, which are globally upper and locally lower bounds for the error of the infinite element discretized.
Abstract: We present two a posteriori error estimators for the mini-element discretization of the Stokes equations. One is based on a suitable evaluation of the residual of the finite element solution. The other one is based on the solution of suitable local Stokes problems involving the residual of the finite element solution. Both estimators are globally upper and locally lower bounds for the error of the finite element discretization. Numerical examples show their efficiency both in estimating the error and in controlling an automatic, self-adaptive mesh-refinement process. The methods presented here can easily be generalized to the Navier-Stokes equations and to other discretization schemes.
TL;DR: In this paper, five different techniques for a posteriori error estimation of adaptive finite element methods for linear elliptic boundary value problems are presented, referred to as the residual estimation method, the duality method, subdomain residual method, a method based on interpolation theory, and a post-processing method.
TL;DR: In this paper, a formulation a elements finis absolument stabilisee for le probleme de Stokes is presented, based on the estimations d'erreurs optimales optimales dans la norme L 2 for l'approximation des champs de vitesses et de pressions.
Abstract: On presente une formulation a elements finis absolument stabilisee pour le probleme de Stokes On etablit des estimations d'erreurs optimales dans la norme L 2 pour l'approximation des champs de vitesses et de pressions
TL;DR: In this paper, a technique for computing rigorous upper bounds on limit loads under conditions of plane strain is described, which assumes a perfectly plastic soil model and employs finite elements in conjunction with the upper bound theorem of classical plasticity theory.
Abstract: This paper describes a technique for computing rigorous upper bounds on limit loads under conditions of plane strain. The method assumes a perfectly plastic soil model, which is either purely cohesive or cohesive-frictional, and employs finite elements in conjunction with the upper bound theorem of classical plasticity theory.
The computational procedure uses three-noded triangular elements with the unknown velocities as the nodal variables. An additional set of unknowns, the plastic multiplier rates, is associated with each element. Kinematically admissible velocity discontinuities are permitted along specified planes within the grid. The finite element formulation of the upper bound theorem leads to a classical linear programming problem where the objective function, which is to be minimized, corresponds to the dissipated power and is expressed in terms of the velocities and plastic multiplier rates. The unknowns are subject to a set of linear constraints arising from the imposition of the flow rule and velocity boundary conditions. It is shown that the upper bound optimization problem may be solved efficiently by applying an active set algorithm to the dual linear programming problem.
Since the computed velocity field satisfies all the conditions of the upper bound theorem, the corresponding limit load is a strict upper bound on the true limit load. Other advantages include the ability to deal with complicated loading, complex geometry and a variety of boundary conditions. Several examples are given to illustrate the effectiveness of the procedure.
TL;DR: In this article, a simple Lagrangian-Eulerian formulation of finite element programs is presented, where an operator split separates the Lagrangians and Eulerian processes, allowing a finite element program to be extended to this formulation with little difficulty.
Abstract: A simple arbitrary Lagrangian-Eulerian formulation is presented. An operator split separates the Lagrangian and Eulerian processes, allowing a Lagrangian finite element program to be extended to this formulation with little difficulty. Example problems illustrate the strengths and weaknesses of the formulation.
TL;DR: In this article, a model for the elastic-plastic finite element simulation of fatigue crack growth and crack closure is presented and evaluated, showing that careful attention must be given to a series of critical decisions about mesh and model design if the analysis is to be reliable.
Book•
06 Feb 1989
TL;DR: Algorithms for FEM on the differentation of stiffness and mass matrices and force vectors subgradient method for convex linearly constrained optimization description of the sequential quadratic programming (SQP) algorithm on theDifferentiability of a projection on a convex set in Hilbert space.
Abstract: Preliminaries - Green's formula abstract setting of optimal shape design problem and its approximation shape optimization of systems governed by unilateral boundary value state problem - scalar case approximation of the optimal shape design problems by finite elements - scalar case numerical realization shape optimization in unilateral boundary value problems with "flux" cost functional optimal shape design in contact problems - elastic case shape optimization of elastic/perfectly plastic bodies in contact on the design of the optimal covering supported by an obstacle state constrained optimal control problems and their approximations FE-grid optimization concluding remarks on references on optimal shape design and related topics. Appendices: algorithms for FEM on the differentation of stiffness and mass matrices and force vectors subgradient method for convex linearly constrained optimization description of the sequential quadratic programming (SQP) algorithm on the differentiability of a projection on a convex set in Hilbert space.
TL;DR: A new Navier-Stokes algorithm for use on unstructured triangular meshes is presented, which can be shown to be equivalent to a finite-volume approximation for regular equilateral triangular meshes.
Abstract: A Navier-Stokes algorithm for use on unstructured triangular meshes is presented. Spatial discretization of the governing equations is achieved using a finite element Galerkin approximation, which can be shown to be equivalent to a finite volume approximation for regular equilateral triangular meshes. Integration steady-state is performed using a multistage time-stepping scheme, and convergence is accelerated by means of implicit residual smoothing and an unstructured multigrid algorithm. Directional scaling of the artificial dissipation and the implicit residual smoothing operator is achieved for unstructured meshes by considering local mesh stretching vectors at each point. The accuracy of the scheme for highly stretched triangular meshes is validated by comparing computed flat-plate laminar boundary layer results with the well known similarity solution, and by comparing laminar airfoil results with those obtained from various well-established structured quadrilateral-mesh codes. The convergence efficiency of the present method is also shown to be competitive with those demonstrated by structured quadrilateral-mesh algorithms.
TL;DR: In this paper, the classical and shear deformation theories up to the third-order are presented in a single theory, and results of linear and non-linear bending, natural vibration and stability of composite laminates are presented for various boundary conditions and lamination schemes.
Abstract: Finite element models of the continuum-based theories and two-dimensional plate/shell theories used in the analysis of composite laminates are reviewed. The classical and shear deformation theories up to the third-order are presented in a single theory. Results of linear and non-linear bending, natural vibration and stability of composite laminates are presented for various boundary conditions and lamination schemes. Computational modelling issues related to composite laminates, such as locking, symmetry considerations, boundary conditions, and geometric non-linearity effects on displacements, buckling loads and frequencies are discussed. It is shown that the use of quarter plate models can introduce significant errors into the solution of certain laminates, the non-linear effects are important even at small ratio of the transverse deflection to the thickness of antisymmetric laminates with pinned edges, and that the conventional eigenvalue approach for the determination of buckling loads of composite laminates can be overly conservative.
TL;DR: In this article, a simple finite element method for the Reissner-Mindlin plate model in the primitive variables is presented and analyzed, which uses nonconforming linear finite elements for the transverse displacement and conforming linear infinite elements enriched by bubbles for the rotation, with the computation of the element stiffness matrix modified by the inclusion of a simple elementwise averaging.
Abstract: A simple finite element method for the Reissner–Mindlin plate model in the prim-itive variables is presented and analyzed The method uses nonconforming linear finite elements for the transverse displacement and conforming linear finite elements enriched by bubbles for the rotation, with the computation of the element stiffness matrix modified by the inclusion of a simple elementwise averaging It is proved that the method converges with optimal order uniformly with respect to thickness
TL;DR: In this paper, a semi-discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary, fourth order in space, Cahn-Hilliard equation.
Abstract: A semi-discrete finite element method requiring only continuous element is presented for the approximation of the solution of the evolutionary, fourth order in space, Cahn-Hilliard equation. Optimal order error bounds are derived in various norms for an implementation which uses mass lumping. The continuous problem has an energy based Lyapunov functional. It is proved that this property holds for the discrete problem.
TL;DR: Some refined estimates for the approximation of the eigenvalues and eigenvectors of selfadjoint eigenvalue problems by finite element or Galerkin methods by Hilbert space are established.
Abstract: : This paper establishes some refined estimates for the approximation of the eigenvalues and eigenvectors of selfadjoint eigenvalue problems by finite element or, more generally, Galerkin methods. Suppose lambda is an eigenvalue of multiplicity q of a selfajoint problem and let M(lambda) denote the space of eigenvectors corresponding to lambda. Keywords: Hilbert space.
TL;DR: In this paper, a numerical method is presented for obtaining the values of K* ≥ 1,K ≥ 2, K ≥ 3 and K ≥ 4 in the elasticity solution at the tip of an interface crack in general states of stress.
Abstract: A numerical method is presented for obtaining the values of K*
1,K
*
II and K*
III in the elasticity solution at the tip of an interface crack in general states of stress. The basis of the method is an evaluation of theJ-integral by the virtual crack extension method. Individual stress intensities can then be obtained from further calculations ofJ perturbed by small increments of the stress intensity factors. The calculations are carried out by the finite element method but minimal extra computations are required compared to those for the boundary value problem. Very accurate results are presented for a crack in the bimaterial interface and compared with other methods of evaluating the stress intensity factors. In particular, a comparison is made with stress intensity factors obtained by computingJ by the virtual crack extension method but separating the modes by using the ratio of displacements on the crack surface. Both techniques work well with fine finite element meshes but the results suggest that the method that relies entirely on J-integral evaluations can be used to give reliable results for coarse meshes.
TL;DR: It is shown that this approach permits a flexible balance among iterative solver, local error estimator, and local mesh refinement device—the main components of an adaptive PDE code, making the method particularly attractive in view of parallel computing.
Abstract: This paper presents the mathematical concepts underlying the new adaptive finite element code KASKADE, which, in its present form, applies to linear scalar second-order 2D elliptic problems on general domains. The starting point for this new development is the recent work on hierarchical finite element bases by H. Yserentant ( Numer. Math. 49 , 379–412 (1986)). It is shown that this approach permits a flexible balance among iterative solver, local error estimator, and local mesh refinement device—the main components of an adaptive PDE code. Without use of standard multigrid techniques, the same kind of computational complexity is achieved—independent of any uniformity restrictions on the applied meshes. In addition, the method is extremely simple and all computations are purely local, making the method particularly attractive in view of parallel computing. The algorithmic approach is illustrated by a well-known critical test problem.
Book•
01 Oct 1989
TL;DR: In this article, a single-degree-of-freedom (SDF) dynamic system is considered, and the effect of different degrees of freedom on the dynamics of the system is investigated.
Abstract: TABLE OF CONTENTS PREFACE 1 INTRODUCTION 1.1 Objectives of the Study of Structural Dynamics 1.2 Importance of Vibration Analysis 1.3 Nature of Exciting Forces 1.4 Mathematical Modeling of Dynamic Systems 1.5 Systems of Units 1.6 Organization of the Text PART I 2 FORMULATION OF THE EQUATIONS OF MOTION: SINGLE-DEGREE-OF-FREEDOM SYSTEMS 2.1 Introduction 2.2 Inertia Forces 2.3 Resultants of Inertia Forces on a Rigid Body 2.4 Spring Forces 2.5 Damping Forces 2.6 Principle of Virtual Displacement 2.7 Formulation of the Equations of Motion 2.8 Modeling of Multi Degree-of-Freedom Discrete Parameter System 2.9 Effect of Gravity Load 2.10 Axial Force Effect 2.11 Effect of Support Motion 3 FORMULATION OF THE EQUATIONS OF MOTION: MULTI-DEGREE-OF-FREEDOM SYSTEMS 3.1 Introduction 3.2 Principal Forces in Multi Degree-of-freedom Dynamic System 3.3 Formulation of the Equations of Motion 3.4 Transformation of Coordinates 3.5 Static Condensation of Stiffness matrix 3.6 Application of Ritz Method to Discrete Systems 4 PRINCIPLES OF ANALYTICAL MECHANICS 4.1 Introduction 4.2 Generalized coordinates 4.3 Constraints 4.4 Virtual Work 4.5 Generalized Forces 4.6 Conservative Forces and Potential Energy 4.7 Work Function 4.8 Lagrangian Multipliers 4.9 Virtual Work Equation For Dynamical Systems 4.10 Hamilton's Equation 4.11 Lagrange's Equation 4.12 Constraint Conditions and Lagrangian Multipliers 4.13 Lagrange's Equations for Discrete Multi-Degree-of-Freedom Systems 4.14 Rayleigh's Dissipation Function PART II 5 FREE VIBRATION RESPONSE: SINGLE-DEGREE-OF-FREEDOM SYSTEM 5.1 Introduction 5.2 Undamped Free Vibration 5.3 Free Vibrations with Viscous Damping 5.4 Damped Free vibration with Hysteretic Damping 5.5 Damped Free vibration with Coulomb Damping 6 FORCED HARMONIC VIBRATIONS: SINGLE-DEGREE-OF-FREEDOM SYSTEM 6.1 Introduction 6.2 Procedures for the Solution of Forced Vibration Equation 6.3 Undamped Harmonic Vibration 6.4 Resonant Response of an Undamped System 6.5 Damped Harmonic Vibration 6.6 Complex Frequency Response 6.7 Resonant Response of a Damped System 6.8 Rotating Unbalanced Force 6.9 Transmitted Motion due to Support Movement 6.10 Transmissibility and Vibration Isolation 6.11 Vibration Measuring Instruments 6.12 Energy Dissipated in Viscous Damping 6.13 Hysteretic Damping 6.14 Complex Stiffness 6.15 Coulomb Damping 6.16 Measurement of Damping 7 RESPONSE TO GENERAL DYNAMIC LOADING AND TRANSIENT RESPONSE 7.1 Introduction 7.2 Response to an Impulsive force 7.3 Response to General Dynamic Loading 7.4 Response to a Step Function Load 7.5 Response to a Ramp Function Load 7.6 Response to a Step Function Load With Rise Time 7.7 Response to Shock Loading 7.8 Response to a Ground Motion Pulse 7.9 Analysis of Response by the Phase Plane Diagram 8 ANALYSIS OF SINGLE-DEGREE-OF-FREEDOM SYSTEMS: APPROXIMATE AND NUMERICAL METHODS 8.1 Introduction 8.2 Conservation of Energy 8.3 Application of Rayleigh Method to Multi Degree of Freedom Systems 8.4 Improved Rayleigh Method 8.5 Selection of an Appropriate Vibration Shape 8.6 Systems with Distributed Mass and Stiffness: Analysis of Internal Forces 8.7 Numerical Evaluation of Duhamel's Integral 8.8 Direct Integration of the Equations of Motion 8.9 Integration Based on Piece-wise Linear Representation of the Excitation 8.10 Derivation of General Formulae 8.11 Constant Acceleration Method 8.12 Newmark's beta Method 8.13 Wilson-theta Method 8.14 Methods Based on Difference Expressions 8.15 Errors involved in Numerical Integration 8.16 Stability of the Integration Method 8.17 Selection of a Numerical Integration Method 8.18 Selection of Time Step 9 ANALYSIS OF RESPONSE IN THE FREQUENCY DOMAIN 9.1 Transform Methods of Analysis 9.2 Fourier Series Representation of a Periodic Function 9.3 Response to a Periodically Applied Load 9.4 Exponential Form of Fourier Series 9.5 Complex Frequency Response Function 9.6 Fourier Integral Representation of a Nonperiodic Load 9.7 Response to a Nonperiodic Load 9.8 Convolution Integral and Convolution Theorem 9.9 Discrete Fourier Transform 9.10 Discrete Convolution and Discrete Convolution Theorem 9.11 Comparison of Continuous and Discrete Fourier Transforms 9.12 Application of Discrete Inverse Transform 9.13 Comparison Between Continuous and Discrete Convolution 9.14 Discrete Convolution of an Infnite and a Finite duration Waveform 9.15 Corrective Response Superposition Methods 9.16 Exponential Window Method 9.17 The Fast Fourier Transform 9.18 Theoretical Background to Fast Fourier Transform 9.19 Computing Speed of FFT Convolution 9.16 Exponential Window Method 9.17 The Fast Fourier Transform 9.18 Theoretical Background to Fast Fourier Transform 9.19 Computing Speed of FFT Convolution PART III 10 FREE VIBRATION RESPONSE: MULTI-DEGREE-OF-FREEDOM SYSTEM 10.1 Introduction 10.2 Standard Eigenvalue Problem 10.3 Linearized Eigenvalue Problem and its Properties 10.4 Expansion Theorem 10.5 Rayleigh Quotient 10.6 Solution of the Undamped Free-Vibration Problem 10.7 Mode Superposition Analysis of Free-Vibration Response 10.8 Solution of the Damped Free-Vibration Problem 10.9 Additional Orthogonality Conditions 10.10 Damping Orthogonality 11 NUMERICAL SOLUTION OF THE EIGENPROBLEM 11.1 Introduction 11.2 Properties of Standard Eigenvalues and Eigenvectors 11.3 Transformation of a Linearized Eigenvalue Problem to the Standard Form 11.4 Transformation Methods 11.5 Iteration Methods 11.6 Determinant Search Method 11.7 Numerical Solution of Complex Eigenvalue Problem 11.8 Semi-definite or Unrestrained Systems 11.9 Selection of a Method for the Determination of Eigenvalues 12 FORCED DYNAMIC RESPONSE: MULTI-DEGREE-OF-FREEDOM SYSTEMS 12.1 Introduction 12.2 Normal Coordinate Transformation 12.3 Summary of Mode Superposition Method 12.4 Complex Frequency Response 12.5 Vibration Absorbers 12.6 Effect of Support Excitation 12.7 Forced Vibration of Unrestrained System 13 ANALYSIS OF MULTI-DEGREE-OF-FREEDOM SYSTEMS: APPROXIMATE AND NUMERICAL METHODS 13.1 Introduction 13.2 Rayleigh-Ritz Method 13.3 Application of Ritz Method to Forced Vibration Response 13.4 Direct Integration of the Equations of Motion 13.5 Analysis in the Frequency Domain PART IV 14 FORMULATION OF THE EQUATIONS OF MOTION: CONTINUOUS SYSTEMS 14.1 Introduction 14.2 Transverse Vibrations of a Beam 14.3 Transverse Vibrations of a Beam: Variational Formulation 14.4 Effect of Damping Resistance on Transverse Vibrations of a Beam 14.5 Effect of Shear Deformation and Rotatory Inertia on the Flexural Vibrations of a Beam 14.6 Axial Vibrations of a Bar 14.7 Torsional Vibrations of a Bar 14.8 Transverse Vibrations of a String 14.9 Transverse Vibration of a Shear Beam 14.10 Transverse Vibrations of a Beam Excited by Support Motion 14.11 Effect of Axial Force on Transverse Vibrations of a Beam 15 CONTINUOUS SYSTEMS: FREE VIBRATION RESPONSE 15.1 Introduction 15.2 Eigenvalue Problem for the Transverse Vibrations of a Beam 15.3 General Eigenvalue Problem for a Continuous System 15.4 Expansion Theorem 15.5 Frequencies and Mode Shapes for Lateral Vibrations of a Beam 15.6 Effect of Shear Deformation and Rotatory Inertia on the Frequencies of Flexural Vibrations 15.7 Frequencies and Mode Shapes for the Axial Vibrations of a Bar 15.8 Frequencies and Mode Shapes for the Transverse Vibration of a String 15.9 Boundary Conditions Containing the 15.10 Free-Vibration Response of a Continuous System 15.11 Undamped Free Transverse Vibrations of a Beam 15.12 Damped Free Transverse Vibrations of a Beam 16 CONTINUOUS SYSTEMS: FORCED-VIBRATION RESPONSE 16.1 Introduction 16.2 Normal Coordinate Transformation: General Case of an Undamped System 16.3 Forced Lateral Vibration of a Beam 16.4 Transverse Vibrations of a Beam Under Traveling Load 16.5 Forced Axial Vibrations of a Uniform Bar 16.6 Normal Coordinate Transformation, Damped Case 17 WAVE PROPAGATION ANALYSIS 17.1 Introduction 17.2 The Phenomenon of Wave Propagation 17.3 Harmonic Waves 17.4 One Dimensional Wave Equation and its Solution 17.5 Propagation of Waves in Systems of Finite Extent 17.6 Reection and Refraction of Waves at a Discontinuity in the System Properties 17.7 Characteristics of the Wave Equation 17.8 Wave Dispersion PART V 18 FINITE ELEMENT METHOD 18.1 Introduction 18.2 Formulation of the Finite Element Equations 18.3 Selection of Shape Functions 18.4 Advantages of the Finite Element Method 18.5 Element Shapes 18.6 One-dimensional Bar Element 18.7 Flexural Vibrations of a Beam 18.8 Stress-strain Relationship for a Continuum 18.9 Triangular Element in Plane Stress and Plane Strain 18.10 Natural Coordinates 19 COMPONENT MODE SYNTHESIS 19.1 Introduction 19.2 Fixed Interface Methods 19.3 Free Interface Method 19.4 Hybrid Method 20 ANALYSIS OF NONLINEAR RESPONSE 20.1 Introduction 20.2 Single-degree-of-freedom System 20.3 Errors involved in Numerical Integration of Nonlinear Systems 20.4 Multiple Degree-of-freedom System ANSWERS TO SELECTED PROBLEMS INDEX
TL;DR: A number of well-known optimal interpolation results are generalized in the case where the functions to be interpolated are on the one hand not very smooth, and on the other are defined on curved domains.
Abstract: This paper is devoted to a general theory of approximation of functions in finite-element spaces. In particular, the case is considered where the functions to be interpolated are on the one hand not very smooth, and on the other are defined on curved domains. Thus, a number of well-known optimal interpolation results are generalized.