The course instructor worked from these notes when presenting each lecture. After the lecture, the notes for the day were posted online for students to access.

LEC # | TOPICS |
---|---|

1 | Elastic Solids; Legendre Transformation; Isotropy; Equilibrium; Compatibility; Constitutive Relations; Variational Calculus; Example of a Functional: String; Extrema - Calculus of Variations; Local Form of Stationarity Condition (PDF) |

2 | Vainberg Theorem; Hu-Washizu Functional (PDF) |

3 | Specialized (Simplified) Variational Principles; Hellinger-Reissner Principle; Complementary Energy Principle; Minimum Potential Energy Theorem; Approximation Theory; Rayleigh - Ritz Method (PDF) |

4 | Weighted - Residuals / Galerkin; Principle of Virtual Work; Geometrical Interpretation of Galerkin's Method; Galerkin Weighting; Best Approximation Method; The Finite Element Method (PDF) |

5 | Sobolev Norms; Global Shape Function; Computation of K and f^{ext}; Isoparametric Elements (PDF) |

6 | Higher Order Interpolation; Isoparametric Triangular Elements; Numerical Integration; Gauss Quadrature (PDF) |

7 | Error Estimation, Convergence of Finite Element Approximations; Error Estimates From Interpolation Theory (PDF) |

8 | Linear Elasticity; Numerical Integration Errors; Basic Error Estimates; Conditions for Convergence; Patch Test (PDF) |

9 | Incompressible Elasticity; Hooke's Law; Governing Equations; "B"-Matrix; Volumetric and Deviatoric Components of "K_{h}" (PDF) |

10 | Constraints Ratio; Variational Principle of Incompressible Elasticity; Saddle Point Problem; Constrained Minimization Problem; Reduced Selective Integration; Penalty Formulation (PDF) |

11 | Assumed Strain Methods; Euler Equations; Mean Dilatation Method; General Expression for Anisotropic Elasticity; Mixed Methods; Discretized Lagrangian (PDF) |

12 | Finite Elasticity; Metric Changes; State of Stress; Field Equations: Linear Momentum Balance, Angular Momentum Balance, Energy Balance; Nonlinear Elastic Solid (PDF) |

13 | Variational Formulation; Minimum Potential Energy Principle; Finite Element Approximations; Rayleigh - Ritz Method; Galerkin Approach (PDF) |

14 | Newton-Raphson Solution Procedure; Continuation Method; Iteration Process; Computation of Tangent Stiffness; Spatial Formulation (PDF) |

15 | Isoparametric Elements; Commutative Diagram; Tangent Stiffness; Calculation of Tangent Stiffness (continued); Material Frame Indifference; Lagrangian Moduli (PDF) |

16 | Material Formulation; Specific Material Models; Isotropic Elasticity; Stress-strain Relations; Cayley-Hamilton Theorem; Examples of Constitutive Relations for Finite Elasticity; Saint-Venant / Kirchhoff Model; Mooney-Riulin Model; Neo-Hookean Model Extended to Compressible Range; Computation of Tangent Moduli (PDF) |

17 | Time Dependent Problems; Nonlinear Elastodynamics (Hyperbolic); Nonlinear Heat Conduction (Parabolic); Initial Boundary Value Problem (IBVP); Finite Element (semi) Discretization (PDF) |

18 | Constitutive Relations: Fourier Law of Heat Conduction; Finite Element Discretization (Spatial); Time-stepping Algorithms; Newmark Predicators; Newmark Correctors; Convergence Check; Explicit Dynamics (PDF) |

19 | Trapezoidal Rule - Heat Conduction; Trapezoidal Predictor; Equivalent Static Problem; Trapezoidal Correctors; Convergence Check (PDF) |

20 | Connection Between Newmark Algorithm and Multistep Methods; Mass Humping; Consistent Mass; Nodal Quadrature; Row (Column) Sum Method; Algorithms Analysis; General Initial Value Problem (IVP) (PDF) |

21 | Energy Conservation / Dissipation; Abstract Algorithms; Convergence; Conditions of Convergence; Consistency (PDF) |

22 | Examples: Trapezoidal Rule; Newmark's Algorithm; Stability; Trapezoidal Rule, Scalar Problem (PDF) |

23 | Multidimensional Case; Spectral Radius, Lax Equivalence Theorem (PDF) |

24 | Stability Properties of Trapezoidal Rule; Eigenprojections; Choice of time step; Stability of Newmark's Algorithm; Iron's Bounding Principle (PDF - 1.1 MB) |

25 | Nonlinear Algorithms; Small-strain Plasticity; Kuhn-Tucker Form; Elastic-plastic Moduli; Isotropic-kinematic Hardening (PDF) |

26 | Time-stepping Algorithms for Constitutive Relations; Numerical Quadrature; Newton-Raphson Solution Procedure; Backward Euler; Geometrical Interpretation; Closest Point Projection Algorithms; J_{2}-isotropic Hardening (PDF) |