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# An Introduction to the Finite Element Method

01 Jan 1984-

TL;DR: Second-order Differential Equations in One Dimension: Finite Element Models (FEM) as discussed by the authors is a generalization of the second-order differential equation in two dimensions.

Abstract: 1 Introduction 2 Mathematical Preliminaries, Integral Formulations, and Variational Methods 3 Second-order Differential Equations in One Dimension: Finite Element Models 4 Second-order Differential Equations in One Dimension: Applications 5 Beams and Frames 6 Eigenvalue and Time-Dependent Problems 7 Computer Implementation 8 Single-Variable Problems in Two Dimensions 9 Interpolation Functions, Numerical Integration, and Modeling Considerations 10 Flows of Viscous Incompressible Fluids 11 Plane Elasticity 12 Bending of Elastic Plates 13 Computer Implementation of Two-Dimensional Problems 14 Prelude to Advanced Topics

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TL;DR: In this paper, the dynamic thermoelastic response of functionally graded cylinders and plates is studied, and a finite element model of the formulation is developed, where the heat conduction and the thermo-elastic equations are solved for a functionally graded axisymmetric cylinder subjected to thermal loading.

Abstract: The dynamic thermoelastic response of functionally graded cylinders and plates is studied. Thermomechanical coupling is included in the formulation, and a finite element model of the formulation is developed. The heat conduction and the thermoelastic equations are solved for a functionally graded axisymmetric cylinder subjected to thermal loading. In addition, a thermoelastic boundary value problem using the first-order shear deformation plate theory (FSDT) that accounts for the transverse shear strains and the rotations, coupled with a three-dimensional heat conduction equation, is formulated for a functionally graded plate. Both problems are studied by varying the volume fraction of a ceramic and a metal using a power law distribution.

1,196 citations

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TL;DR: An elastic-plastic finite element model for the frictionless contact of a deformable sphere pressed by a rigid flat is presented in this paper, which provides dimensionless expressions for the contact load, contact area and mean contact pressure, covering a large range of interference values from yielding inception to fully plastic regime of the spherical contact zone.

Abstract: An elastic-plastic finite element model for the frictionless contact of a deformable sphere pressed by a rigid flat is presented. The evolution of the elastic-plastic contact with increasing interference is analyzed revealing three distinct stages that range from fully elastic through elastic-plastic to fully plastic contact interface. The model provides dimensionless expressions for the contact load, contact area, and mean contact pressure, covering a large range of interference values from yielding inception to fully plastic regime of the spherical contact zone. Comparison with previous elastic-plastic models that were based on some arbitrary assumptions is made showing large differences. ©2002 ASME

867 citations

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TL;DR: This work shows that in addition to the heterogeneous distribution of blood supply, hindered interstitial transport, and rapid extravascular binding of macromolecules, the elevated interstitial pressure plays an important role in determining the penetration of macROMolecules into tumors.

Abstract: A general theoretical framework for transvascular exchange and extravascular transport of fluid and macromolecules in tumors is developed. The resulting equations are applied to the most simple case of a homogeneous, alymphatic tumor, with no extravascular binding. Numerical simulations show that in a uniformly perfused tumor the elevated interstitial pressure is a major cause for heterogeneous distribution of nonbinding macromolecules, because it (i) reduces the driving force for extravasation of fluid and macromolecules in tumors, (ii) results in nonuniform filtration of fluid and macromolecules from blood vessels, and (iii) leads to experimentally verifiable, radially outward convection which opposes the inward diffusion. The models are used to predict the interstitial pressure, interstitial fluid velocity, and concentration profiles as a function of radial position and tumor size. The model predictions agree with the following experimental data: (i) the interstitial pressure in a tumor is lowest at the periphery of the tumor and increases towards the center; (ii) the radially outward fluid velocity predicted by the fluid transport model is of the same order of magnitude as that measured in tissue-isolated tumors; and (iii) the concentration of macromolecules is higher in the periphery than in the center of tumors at short times postinjection; however, at later times the peripheral concentration is less than the concentration in the center. This work shows that in addition to the heterogeneous distribution of blood supply, hindered interstitial transport, and rapid extravascular binding of macromolecules (e.g., monoclonal antibodies), the elevated interstitial pressure plays an important role in determining the penetration of macromolecules into tumors. If the genetically engineered macromolecules are to fulfill their clinical promise, methods must be developed to overcome these physiological barriers in tumors.

823 citations

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12 Jul 2000TL;DR: Numerical Techniques in Electromagnetics is designed to show the reader how to pose, numerically analyze, and solve electromagnetic (EM) problems using a variety of available numerical methods.

Abstract: Numerical Techniques in Electromagnetics is designed to show the reader how to pose, numerically analyze, and solve electromagnetic (EM) problems. It gives them the ability to expand their problem-solving skills using a variety of available numerical methods. Topics covered include fundamental concepts in EM; numerical methods; finite difference methods; variational methods, including moment methods and finite element methods; transmission-line matrix or modeling (TLM); and Monte Carlo methods. The simplicity of presentation of topics throughout the book makes this an ideal text for teaching or self-study by senior undergraduates, graduate students, and practicing engineers.

662 citations

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TL;DR: A general two-dimensional shear deformation theory of laminated composite plates is presented in this article, which accounts for a desired degree of approximation of the displacements through the laminate thickness.

Abstract: A general two-dimensional shear deformation theory of laminated composite plates is presented. The theory account for a desired degree of approximation of the displacements through the laminate thickness. As special cases, the classical, first-order (Reissner–Mindlin) and other shear deformation theories available in the literature can be deduced from the present theory.

658 citations

##### References

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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

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01 Jan 1974

TL;DR: In this article, the authors present a formal notation for one-dimensional elements in structural dynamics and vibrational properties of a structural system, including the following: 1. Isoparametric Elements.

Abstract: Notation. Introduction. One-Dimensional Elements, Computational Procedures. Basic Elements. Formulation Techniques: Variational Methods. Formulation Techniques: Galerkin and Other Weighted Residual Methods. Isoparametric Elements. Isoparametric Triangles and Tetrahedra. Coordinate Transformation and Selected Analysis Options. Error, Error Estimation, and Convergence. Modeling Considerations and Software Use. Finite Elements in Structural Dynamics and Vibrations. Heat Transfer and Selected Fluid Problems. Constaints: Penalty Forms, Locking, and Constraint Counting. Solid of Revolution. Plate Bending. Shells. Nonlinearity: An Introduction. Stress Stiffness and Buckling. Appendix A: Matrices: Selected Definition and Manipulations. Appendix B: Simultaneous Algebraic Equations. Appendix C: Eigenvalues and Eigenvectors. References. Index.

6,126 citations

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01 Jan 1982

TL;DR: In this paper, the Finite Element Method is used to derive a system equation from a set of finite element vectors and matrices and then to solve the problem of finding the solution.

Abstract: 1. Overview of the Finite Element Method, 2. Discretization of the Domain, 3. Interpolation Models, 4. Higher Order and Isoparametric Elements, 5. Derivation of Element Matrices and Vectors, 6. Assembly of Element Matrices and Vectors and Derivation of System Equations, 7. Numerical Solution of Finite Element Equations, 8. Basic Equations and Solution Procedure, 9. Analysis of Trusses, Beams and Frames, 10. Analysis of Plates, 11. Analysis of Three-Dimensional Problems, 12. Dynamic Analysis, 13. Formulation and Solution Procedure, 14. One-Dimensional Problems, 15. Two-Dimensional Problems, 16. Three-Dimensional Problems, 17. Basic Equations of Fluid Mechanics, 18. Inviscid and Incompressible Flows, 19. Viscous and Non-Newtonian Flows, 20. Solution of Quasi-Harmonic Equations, 21. Solution of Helmhotz Equation, 22. Solution of Reynolds Equation, Appendix-A Green Greass Theorem.

1,247 citations

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01 Jan 1984

TL;DR: A review of the equations of MECHANICS can be found in this paper, where the Ritz Method and Weighted-Residual Methods are used to approximate the distance between two points.

Abstract: A REVIEW OF THE EQUATIONS OF MECHANICS. Introduction. Kinetics. Kinematics. Thermodynamic Principles. Constitutive Equations. Boundary--Value Problems of Mechanics. Equations of Bars, Beams, Torsion, and Plane Elasticity. ENERGY AND VARIATIONAL PRINCIPLES. Preliminary Concepts. Calculus of Variations. Virtual Work and Energy Principles. Stationary Variational Principles. Hamiltona s Principle. Energy Theorems of Structural Mechanics. VARIATIONAL METHODS OF APPROXIMATION. Some Preliminaries. The Ritz Method. Weighted--Residual Methods. The Finite--Element Method. THEORY AND ANALYSIS OF PLATES AND SHELLS. Classical Theory of Plates. Shear Deformation Theories of Plates. Laminated Composite Plates. Theory of Shells. Finite--Element Analysis of Plates and Shells. Bibliography. Answers to Selected Exercises. Index.

347 citations