Teaching notes:
Finite Element Analysis: Fundamental concepts and techniques of primal finite element methods. Method of weighted residuals, Galerkin's method and variational equations. Linear eliptic boundary value problems in one, two and three space dimensions; applications in structural, solid and fluid mechanics and heat transfer. Properties of standard element families and numerically integrated elements. Implementation of the finite element method using Matlab, assembly of equations, and element routines. Lagrange multiplier and penalty methods for treatment of constraints. The mathematical theory of finite elements.
- [Problem Session 1]: Formulation of variational problems for PDEs.
- [Problem Session 2]: Vector space of functions and clarification of concepts.
- [Problem Session 3]: Solving (discretized) variational problems with shape functions.
- [Problem Session 4]: Local-to-global map and assembly with shape functions.
- [Problem Session 5]: Solving variational problems with LG matrix.
- [Problem Session 6]: Global assembly in 2D with P1 elements.
- [Firedrake Tutorial]: Solving a 2D Poisson equation.
- [FEniCS Tutorial]: Solving a 2D Poisson equation.
- [Problem Session 8]: Norms and convergence (numerical analysis).
- [Problem Session 9]: (final review) Heat transfer in 2D using P1 elements.
- [Course Summary] | [Teaching Evaluation]
Mechanics: Elasticity and Inelasticity: Introduction to the theory of elasticity, plasticity and fracture and their applications. Elasticity: stress function approach to solve 2D problems and Green's function in 3D; applications to contact problems. Plasticity: yield surface, associative flow rule, strain hardening models; and applications to plastic bending, torsion and pressure vessels. Fracture: linear elastic fracture mechanics, J-integral, plastic zone in front of crack tip; applications to brittle fracture and fatigue crack growth. Computer programming in Matlab is used to aid analytic derivation and numerical solutions.
- [Problem Session 1]: General solution strategies for elasticity problems.
- [Problem Session 2]: Euler-Bernoulli beam theory and stress functions.
- [Problem Session 4]: Biharmonic equation in polar coordinates and Michell solutions.
- [Problem Session 6]: (midterm review) Basic equations, beams, Green's functions, contact, wedge & crack.
- [Problem Session 7]: Plasticity with loading paths under tension and shear.
- [Problem Session 8]: Flow rule, fracture mechanics, and the J-integral.
- [Problem Session 9]: Linear elastic fracture mechanics and crack opening displacement.
- [Final Review] | [Teaching Evaluation]
Course and learning notes:
- Foundations of Solid Mechanics. [PDF]
- Elasticity & Inelasticity. [PDF]
- Finite Element Analysis. [PDF]
- Statistical Mechanics. [PDF]
- Functional Analysis. [PDF] [Final Presentation]
- Nonlinear Finite Element Analysis. [PDF]
- Partial Differential Equations. [PDF]
- Defects & Disorders in Materials. [PDF]
- Linear Algebra. [PDF]
- Atomistic Modeling. [PDF]
- Inverse Problems. [PDF]
- Principles of Large-Scale Machine Learning. [PDF]
- Engineering Thermodynamics. [PDF]
- Computational Fluid Dynamics. [PDF]
- Density Functional Theory. [Notebooks 1; Notebooks 2]
- Design Optimization. [PDF]
- Mathematical Modeling. [PDF]