An introduction to nonlinear finite element analysis : with applications to heat transfer, fluid mechanics, and solid mechanics / J.N. Reddy.

Previous edition: 2004.

Includes bibliographical references (663-678) and index.

Machine-generated contents note: 1. General Introduction and Mathematical Preliminaries -- 1.1. General Comments -- 1.2. Mathematical Models -- 1.3. Numerical Simulations -- 1.4. The Finite Element Method -- 1.5. Non-linear Analysis -- 1.5.1. Introduction -- 1.5.2. Classification of Non-linearities -- 1.6. Review of Vectors and Tensors -- 1.6.1. Preliminary Comments -- 1.6.2. Definition of a Physical Vector -- 1.6.2.1. Vector addition -- 1.6.2.2. Multiplication of a vector by a scalar -- 1.6.3. Scalar and Vector Products -- 1.6.3.1. Scalar product (or "dot" product) -- 1.6.3.2. Vector product -- 1.6.3.3. Plane area as a vector -- 1.6.3.4. Linear independence of vectors -- 1.6.3.5. Components of a vector -- 1.6.4. Summation Convention and Kronecker Delta and Permutation Symbol -- 1.6.4.1. Summation convention -- 1.6.4.2. Kronecker delta symbol -- 1.6.4.3. The permutation symbol -- 1.6.5. Tensors and their Matri -- Note continued: 6.7.4.2. Fully-discretized equations -- 6.7.5. Stability and Accuracy -- 6.7.5.1. Preliminary comments -- 6.7.5.2. Stability criteria -- 6.7.6. Computer Implementation -- 6.7.7. Numerical Examples -- 6.8. Summary -- Problems -- 7. Non-linear Bending of Elastic Plates -- 7.1. Introduction -- 7.2. The Classical Plate Theory -- 7.2.1. Assumptions of the Kinematics -- 7.2.2. Displacement and Strain Fields -- 7.3. Weak Formulation of the CPT -- 7.3.1. Virtual Work Statement -- 7.3.2. Weak Forms -- 7.3.3. Equilibrium Equations -- 7.3.4. Boundary Conditions -- 7.3.4.1. The Kirchhoff free-edge condition -- 7.3.4.2. Typical edge conditions -- 7.3.5. Stress Resultant-Deflection Relations -- 7.4. Finite Element Models of the CPT -- 7.4.1. General Formulation -- 7.4.2. Tangent Stiffness Coefficients -- 7.4.3. Non-Conforming and Conforming Plate Elements -- 7.5. Computer Implementation of the CPT Elements -- 7.5.1. General Remarks -- 7.5.2. Programming Aspects -- 7.5.3. Post-Computation of Stresses -- 7.6. Numerical Examples using the CPT Elements -- 7.6.1. Preliminary Comments -- 7.6.2. Results of Linear Analysis -- 7.6.3. Results of Non-linear Analysis -- 7.7. The First-Order Shear Deformation Plate Theory -- 7.7.1. Introduction -- 7.7.2. Displacement Field -- 7.7.3. Weak Forms using the Principle of Virtual Displacements -- 7.7.4. Governing Equations -- 7.8. Finite Element Models of the FSDT -- 7.8.1. Weak Forms -- 7.8.2. The Finite Element Model -- 7.8.3. Tangent Stiffness Coefficients -- 7.8.4. Shear and Membrane Locking -- 7.9. Computer Implementation and Numerical Results of the FSDT Elements -- 7.9.1. Computer Implementation -- 7.9.2. Results of Linear Analysis -- 7.9.3. Results of Non-linear Analysis -- 7.10. Transient Analysis of the FSDT -- 7.10.1. Equations of Motion -- 7.10.2. The Finite Element Model -- 7.10.3. Time Approximation -- 7.10.4. Numerical Examples -- 7.11. Summary -- Problems -- 8. Non-linear Bending of Elastic Shells -- 8.1. Introduction -- 8.2. Governing Equations -- 8.2.1. Geometric Description -- 8.2.2. General Strain-Displacement Relations -- 8.2.3. Stress Resultants -- 8.2.4. Displacement and Strain Fields -- 8.2.5. Equations of Equilibrium -- 8.2.6. Shell Constitutive Relations -- 8.3. Finite Element Formulation -- 8.3.1. Weak Forms -- 8.3.2. Finite Element Model -- 8.3.3. Linear Analysis -- 8.3.4. Non-linear Analysis -- 8.4. Summary -- Problems -- 9. Finite Element Formulations of Solid Continua -- 9.1. Introduction -- 9.1.1. Background -- 9.1.2. Summary of Definitions and Concepts from Continuum Mechanics -- 9.1.3. Energetically-Conjugate Stresses and Strains -- 9.2. Various Strain and Stress Measures -- 9.2.1. Introduction -- 9.2.2. Notation -- 9.2.3. Conservation of Mass -- 9.2.4. Green-Lagrange Strain Tensors -- 9.2.4.1. Green-Lagrange strain increment tensor -- 9.2.4.2. Updated Green-Lagrange strain tensor -- 9.2.5. Euler-Almansi Strain Tensor -- 9.2.6. Relationships Between Various Stress Tensors -- 9.2.7. Constitutive Equations -- 9.3. Total Lagrangian and Updated Lagrangian Formulations -- 9.3.1. Principle of Virtual Displacements -- 9.3.2. Total Lagrangian Formulation -- 9.3.2.1. Weak form -- 9.3.2.2. Incremental decompositions -- 9.3.2.3. Linearization -- 9.3.3. Updated Lagrangian Formulation -- 9.3.3.1. Weak form -- 9.3.3.2. Incremental decompositions -- 9.3.3.3. Linearization -- 9.3.4. Some Remarks on the Formulations -- 9.4. Finite Element Models of 2-D Continua -- 9.4.1. Introduction -- 9.4.2. Total Lagrangian Formulation -- 9.4.3. Updated Lagrangian Formulation -- 9.4.4. Computer Implementation -- 9.4.5. A Numerical Example -- 9.5. Conventional Continuum Shell Finite Element -- 9.5.1. Introduction -- 9.5.2. Incremental Equations of Motion -- 9.5.3. Finite Element Model of a Continuum -- 9.5.4. Shell Finite Element -- 9.5.5. Numerical Examples -- 9.5.5.1. Simply-supported orthotropic plate under uniform load -- 9.5.5.2. Four-layer (0°/907/907deg;/0°) clamped plate under uniform load -- 9.5.5.3. Cylindrical shell roof under self-weight -- 9.5.5.4. Simply-supported spherical shell under point bad -- 9.5.5.5. Shallow cylindrical shell under point load -- 9.6. A Refined Continuum Shell Finite Element -- 9.6.1. Backgound -- 9.6.2. Representation of Shell Mid-Surface -- 9.6.3. Displacement and Strain Fields -- 9.6.4. Constitutive Relations -- 9.6.4.1. Isotropic and functionally-graded shells -- 9.6.4.2. Laminated composite shells -- 9.6.5. The Principle of Virtual Displacements and its Discretization -- 9.6.6. The Spectral/hp Basis Functions -- 9.6.7. Finite Element Model and Solution of Non-linear Equations -- 9.6.7.1. The Newton procedure -- 9.6.7.2. The cylindrical arc-length procedure -- 9.6.7.3. Element-level static condensation and assembly of elements -- 9.6.8. Numerical Examples -- 9.6.8.1. A cantilevered plate strip under an end transverse load -- 9.6.8.2. Post-buckling of a plate strip under axial compressive load -- 9.6.8.3. An annular plate with a slit under an end transverse load -- 9.6.8.4. A cylindrical panel subjected to a point load -- 9.6.8.5. Pull-out of an open-ended cylindrical shell -- 9.6.8.6. A pinched half-cylindrical shell -- 9.6.8.7. A pinched cylinder with rigid diaphragms -- 9.6.8.8. A pinched hemisphere with an 18° hole -- 9.6.8.9. A pinched composite hyperboloidal shell -- 9.7. Summary -- Problems -- 10. Weak-Form Finite Element Models of Flows of Viscous Incompressible Fluids -- 10.1. Introduction -- 10.2. Governing Equations -- 10.2.1. Introduction -- 10.2.2. Equation of Mass Continuity -- 10.2.3. Equations of Motion -- 10.2.4. Energy Equation -- 10.2.5. Constitutive Equations -- 10.2.6. Boundary Conditions -- 10.3. Summary of Governing Equations -- 10.3.1. Vector Form -- 10.3.2. Cartesian Component Form -- 10.4. Velocity-Pressure Finite Element Model -- 10.4.1. Weak Forms -- 10.4.2. Semi-discrete Finite Element Model -- 10.4.3. Fully-Discretized Finite Element Model -- 10.5. Penalty Finite Element Models -- 10.5.1. Introduction -- 10.5.2. Penalty Function Method -- 10.5.3. Reduced Integration Penalty Model -- 10.5.4. Consistent Penalty Model -- 10.6. Computational Aspects - 10.6.1. Properties of the Finite Element Equations -- 10.6.2. Choice of Elements -- 10.6.3. Evaluation of Element Matrices in Penalty Models -- 10.6.4. Post-Computation of Pressure and Stresses -- 10.7. Computer Implementation -- 10.7.1. Mixed Model -- 10.7.2. Penalty Model -- 10.7.3. Transient Analysis -- 10.8. Numerical Examples -- 10.8.1. Preliminary Comments -- 10.8.2. Linear Problems -- 10.8.3. Non-linear Problems -- 10.8.4. Transient Analysis -- 10.9. Non-Newtonian Fluids -- 10.9.1. Introduction -- 10.9.2. Governing Equations in Cylindrical Coordinates -- 10.9.3. Power-Law Fluids -- 10.9.4. White-Metzner Fluids -- 10.9.5. Numerical Examples -- 10.10. Coupled Fluid Flow and Heat Transfer -- 10.10.1. Finite Element Models -- 10.10.2. Numerical Examples -- 10.10.2.1. Heated cavity -- 10.10.2.2. Solar receiver -- 10.11. Summary -- Problems -- 11. Least-Squares Finite Element Models of Flows of Viscous Incompressible Fluids -- 11.1. Introduction -- 11.2. Least-Squares Finite Element Formulation -- 11.2.1. The Navier-Stokes Equations of Incompressible Fluids -- 11.2.2. Numerical Examples -- 11.2.2.1. Low Reynolds number flow past a circular cylinder -- 11.2.2.2. Steady flow over a backward facing step -- 11.2.2.3. Lid-driven cavity flow -- 11.3. A Least-Squares Finite Element Model with Enhanced Element-Level Mass Conservation -- 11.3.1. Introduction -- 11.3.2. Unsteady Flows -- 11.3.2.1. The velocity-pressure-vorticity first-order system -- 11.3.2.2. Temporal discretization -- 11.3.2.3. The standard L2-norm based Least-Squares model -- 11.3.2.4. A modified L2-norm based Least-Squares model with improved element-level mass conservation -- 11.3.3. Numerical Examples: verification Problems -- 11.3.3.1. Steady Kovasznay flow -- 11.3.3.2. Steady flow in a 1 X 2 rectangular cavity -- 11.3.3.3. Steady flow past a large cylinder in a narrow channel -- 11.3.3.4. Unsteady flow past a circular cylinder -- 11.3.3.5. Unsteady flow past a large cylinder in a narrow channel -- 11.4. Summary and Future Direction -- Problems -- Appendix 1 Solution Procedures for Linear Equations -- A1.1. Introduction -- A1.2. Direct Methods -- A1.2.1. Preliminary Comments -- A1.2.2. Symmetric Solver -- A1.2.3. Unsymmetric Solver -- A1.3. Iterative Methods -- Appendix 2 Solution Procedures for Non-linear Equations -- A2.1. Introduction -- A2.2. The Picard Iteration Method -- A2.3. The Newton Iteration Method -- A2.4. The Riks and Modified Risk Methods.

This book presents the theory and computer implementation of the finite element method as applied to nonlinear problems of heat transfer and similar field problems, fluid mechanics (flows of incompressible fluids), and solid mechanics (elasticity, beams and plates). Both geometric as well as material nonlinearities are considered, and static and transient (i.e. time-dependent) responses are studied. Although there exist a number of books on nonlinear finite elements that serve asgood references for engineers who are familiar with the subject and wish to learn advanced topics or the latest deve