| 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 |
|
| 2 |
Vainberg Theorem; Hu-Washizu Functional |
|
| 3 |
Specialized (Simplified) Variational Principles; Hellinger-Reissner Principle; Complementary Energy Principle; Minimum Potential Energy Theorem; Approximation Theory; Rayleigh - Ritz Method |
Assignment 1 Out |
| 4 |
Weighted - Residuals / Galerkin; Principle of Virtual Work; Geometrical Interpretation of Galerkin's Method; Galerkin Weighting; Best Approximation Method; The Finite Element Method |
|
| 5 |
Sobolev Norms; Global Shape Function; Computation of K and fext; Isoparametric Elements |
|
| 6 |
Higher Order Interpolation; Isoparametric Triangular Elements; Numerical Integration; Gauss Quadrature |
Assignment 1 Due
Assignment 2 Out |
| 7 |
Error Estimation, Convergence of Finite Element Approximations; Error Estimates From Interpolation Theory |
|
| 8 |
Linear Elasticity; Numerical Integration Errors; Basic Error Estimates; Conditions for Convergence; Patch Test |
|
| 9 |
Incompressible Elasticity; Hooke's Law; Governing Equations; "B"-Matrix; Volumetric and Deviatoric Components of "Kh" |
|
| 10 |
Constraints Ratio; Variational Principle of Incompressible Elasticity; Saddle Point Problem; Constrained Minimization Problem; Reduced Selective Integration; Penalty Formulation |
|
| 11 |
Assumed Strain Methods; Euler Equations; Mean Dilatation Method; General Expression for Anisotropic Elasticity; Mixed Methods; Discretized Lagrangian |
Assignment 2 Due
Assignment 3 Out |
| 12 |
Finite Elasticity; Metric Changes; State of Stress; Field Equations: Linear Momentum Balance, Angular Momentum Balance, Energy Balance; Nonlinear Elastic Solid |
|
| 13 |
Variational Formulation; Minimum Potential Energy Principle; Finite Element Approximations; Rayleigh - Ritz Method; Galerkin Approach |
|
| 14 |
Newton-Raphson Solution Procedure; Continuation Method; Iteration Process; Computation of Tangent Stiffness; Spatial Formulation |
|
| 15 |
Isoparametric Elements; Commutative Diagram; Tangent Stiffness; Calculation of Tangent Stiffness (continued); Material Frame Indifference; Lagrangian Moduli |
Assignment 3 Due |
| 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 |
|
| 17 |
Time Dependent Problems; Nonlinear Elastodynamics (Hyperbolic); Nonlinear Heat Conduction (Parabolic); Initial Boundary Value Problem (IBVP); Finite Element (semi) Discretization |
Assignment 4 Out |
| 18 |
Constitutive Relations: Fourier Law of Heat Conduction; Finite Element Discretization (Spatial); Time-stepping Algorithms; Newmark Predicators; Newmark Correctors; Convergence Check; Explicit Dynamics |
|
| 19 |
Trapezoidal Rule - Heat Conduction; Trapezoidal Predictor; Equivalent Static Problem; Trapezoidal Correctors; Convergence Check |
|
| 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) |
Assignment 4 Due |
| 21 |
Energy Conservation / Dissipation; Abstract Algorithms; Convergence; Conditions of Convergence; Consistency |
|
| 22 |
Examples: Trapezoidal Rule; Newmark's Algorithm; Stability; Trapezoidal Rule, Scalar Problem |
Assignment 5 Out |
| 23 |
Multidimensional Case; Spectral Radius, Lax Equivalence Theorem |
|
| 24 |
Stability Properties of Trapezoidal Rule; Eigenprojections; Choice of time step; Stability of Newmark's Algorithm; Iron's Bounding Principle |
|
| 25 |
Nonlinear Algorithms; Small-strain Plasticity; Kuhn-Tucker Form; Elastic-plastic Moduli; Isotropic-kinematic Hardening |
|
| 26 |
Time-stepping Algorithms for Constitutive Relations; Numerical Quadrature; Newton-Raphson Solution Procedure; Backward Euler; Geometrical Interpretation; Closest Point Projection Algorithms; J2-isotropic Hardening |
Assignment 5 Due |
