Contents as of 12/8/2017 Preface 1. Overview...1 1.1 Introduction...1 1.2 Finite element data...1 1.3 Matrix notation...3 1.4 Matrix partitions...8 1.5 Special finite element matrix notations...9 1.6 Finite element analysis sequence...9 1.7 Layout of this book...10 1.8 Summary and notation...15 2. Polynomial interpolation...1 2.1 Types of interpolation:...1 2.2 Lagrange one-dimensional interpolation...2 2.3 Natural coordinates...15 2.4 Hermite one-dimensional interpolation...16 2.5 Lagrangian quadrilateral elements...19 2.6 Lagrangian triangular elements...22 2.7 Serendipity quadrilaterals*...27 2.8 Hierarchical interpolation...29 2.9 Summary and notation...34 2.10 Exercises...36 3. Numerical integration...1 3.1 One-dimensional quadratures...2 3.2 Two- and three-dimensional quadratures...10 3.3 Exact integrals for straight sided triangles...14 3.4 Reduced and selected integration...16 3.5 Summary...18 3.6 Exercises...20 4. Calculus review...1 4.1 Parametric geometry...2 4.2 Jacobian matrix...4 4.3 Inverse Jacobian...11 4.4 Parametric substitution in integrals... 12 4.5 Integration by parts...13 4.6 Integral of point sources...15 4.7 Axisymmetric integrals...16
4.8 Summary...18 4.9 Exercises...22 5. Terminology from differential equations...1 5.1 Definitions*...2 5.2 Boundary conditions...3 5.3 Adjoint operator*...4 5.4 Three classes of PDEs*...5 5.5 Eigen-problems...6 5.6 Model elliptic PDE...6 5.7 Directionally dependent data...7 5.8 Point singularities*...8 5.8 Summary...9 5.9 Exercises...10 6. Equivalent integral forms...1 6.1 Variational calculus*:...2 6.2 Method of weighted residuals...2 6.3 Common weighting methods...3 6.4 Eigen-problem analysis...4 6.5 Summary...6 6.6 Exercises...6 7. Matrix procedures for finite elements...1 7.1 Introduction...2 7.2 Equation numbers for gather and scatter...3 7.3 Vector subscripts...8 7.4 Partitioning the system equations...11 7.5 Numerically equivalent process...13 7.6 EBC by a penalty method*...15 7.7 Multiple point constraints*...16 7.8 Wilson's static condensation algorithm...20 7.9 Equation factorization...21 7.10 Skyline sparse storage...23 7.10 Summary...27 7.11 Exercises...28 8. Applications of 1-d Lagrange elements...1 8.1 Introduction...2 8.3 Variable source terms...21 8.4 Mixed boundary conditions...36 8.5 Automating Lagrange element solutions...43
8.6 Numerically integrated elements...49 8.7 Symbolic solutions...68 8.8 Symmetry and anti-symmetry...73 8.9 Patch test*...74 8.10 Creating exact solutions*...75 8.11 Summary...76 8.12 Exercises...79 9. Truss analysis...1 9.1 Planar truss...2 9.2 Space truss...12 9.3 Summary...13 9.4 Exercises...14 10. Applications of 1-d Hermite elements...1 10.1 Introduction...1 10.2 General case fourth-order beam equation...3 10.3 Integral form...5 10.4 Element arrays...7 10.5 C1 element models...8 10.6 Classic beams...11 10.7 Structural symmetry...19 10.8 The Rayleigh quotient...21 10.9 Multiple span beams...23 10.10 BOEF without axial load...28 10.11 BOEF with axial load...32 10.12 BOEF with axial load and end transverse force...33 10.15 Summary...35 10.16 Exercises...37 11. Frame analysis...1 11.1 Planar frames...1 11.2 Frame member reactions...7 11.3 Enhanced frame post-processing*...9 11.4 Space frames...12 11.5 Numerically integrated frame members...12 11.6 Summary...14 11.7 Exercises...17 12. Thermal and scalar field analysis...1 12.1 Introduction...1 12.2 General field problem...3
12.3 Common flux components...5 12.4 Galerkin integral form...6 12.5 Galerkin integral form*...8 12.6 Orthotropic two-dimensional fields...9 12.7 Corresponding element and boundary matrices...10 12.8 Numerical evaluation of the Jacobian matrix...11 12.9 Jacobian matrix for equal space dimensions...12 12.10 Physical space > parametric space:...16 12.11 Field flux vector at a point...18 12.12 Evaluation of integrals with constant Jacobian...20 12.13 Symmetry and anti-symmetry...21 12.14 Viscous fluid flow in a channel...29 12.15 Axisymmetric fields...29 12.16 Summary...32 13. Elasticity...1 13.1 Introduction...1 13.2 Linear springs...4 13.3 Mechanical work...6 13.5 Strain energy...9 13.6 Material properties...10 13.7 Simplified elasticity models*...12 13.8 Interpolating displacement vectors...16 13.9 Mechanical work in matrix form...18 13.10 The strain-displacement matrix...19 13.11 Stiffness matrix...22 13.12 Work done by initial strains...22 13.13 Matrix equilibrium equations...23 13.14 Symmetry and anti-symmetry...23 13.15 Plane stress analysis...24 13.16 Axisymmetric stress analysis...24 13.17 Solid stress analysis...24 13.18 Kinetic energy...25 13.19 Summary...27 13.20 Exercises...29 14. Eigen-analysis...1 14.1 Introduction...1 14.2 Finite element eigen-problems...4 14.3 Spring-mass systems...5 14.4 Vibrating string...6 14.5 Torsional vibrations...11 14.6 Beam vibrations...12 14.7 Membrane vibration...15
14.8 Beam-column buckling...20 14.9 Beam frequency with an axial load...26 14.10 Plane-frame modes and frequencies:...29 14.11 Modes and frequencies of 2-d continua...29 14.12 Acoustical vibrations:...31 14.13 Principal stresses...31 14.15 Mohr's circle for eigenvalues*...33 14.16 Time independent Schrödinger equation*...35 14.17 Summary...37 14.18 Exercises...38 15. Transient and dynamic solutions...1 15.1 Introduction to transient systems...1 15.2 Generalized trapezoidal algorithms...2 15.3 Accuracy and control...5 15.4 Introduction to dynamic solutions...9 15.5 Wilson method...10 15.6 Summary...12 15.7 Exercises...12 16. Vector field elements (draft)...1 16.1 Introduction...1 16.2 Parametric coordinates...2 16.3 Edge based (vector) finite elements...4 16.4 Whitney vector elements...5 16.5 Ainsworth vector elements...8 16.5 Weak form...10 16.6 Vector element matrices...11 16.7 Summary...12 16.8 Exercises...12 17. Continuous flux and error estimates...1 17.1 Introduction...1 17.2 Optimal gradient locations...1 17.3 Continuous nodal flux recovery...3 17.4 Additional data arrays for flux averaging*...11 17.5 A one-dimensional flux smoothing example...12 17.6 Error estimates...18 17.7 Mesh h-adaptivity*...19 17.8 Summary...19 17.9 Exercises...19 Appendix