Course Details

1. 1D Boundary Value Problems
Introduction – Second Order Differential Operators and Boundary value problems – Strong Forms and solution spaces of Continuous functions – Weak formulations – Symmetric formulations and Energy functionals – functional minimization and the Rayleigh-Ritz method – variational formulations – Criteria of Equivalence for strong, weak and functional minimization formulations – Solution spaces for the weak formulation – Petrov Galerkin and Bubnov Galerkin methods.
1D fourth order boundary value problems – strong form – variational formulations.
Weak forms for general boundary value problems
Discrete variational formulations -Ritz method- Weighted Residual Methods – Galerkin Method – Least Square Method – Collocation Method – Sub-Domain Method – Momentun Method. Examples. The Finite Difference Method.

2. 1D Finite Elements
Numerical Solution of 2 point boundary value problems – Ritz method – 2 node Finite elements of linear interpolation – local and global stiffness matrixes and force vectors.
Quadratic elements – local and global stiffness matrixes and force vectors.
Introduction to error estimation techniques and the notion of superconvergence.

3. 2D Boundary Value Problems
The finite Element Method for 2D boundary value problems. Variational formulation for the Laplace and Poisson equations. 3 node triangular and 4 node quadrilateral elements. Lagrange and Serendipity elements. Isoparametric elements.

4. The system of linear elasticity
Displacement field formulation – Minimization for strain energy and the principle of virtual work.

5. Mixed and hybrid formulations for beams
Hellinger-Reissner and Hu-Washizou functionals.- Penalty functionals – Mixed formulations for beams- shear Locking phenomena.

6. Mixed and hybrid formulations for Plates and shells
Hellinger-Reissner and Hu-Washizouv functionals.- Penalty functionals – Mixed formulations for plates and shells – shear Locking phenomena.

7. Mixed and hybrid formulations for 3D elasticity
Hellinger-Reissner and Hu-Washizou functionals.- Penalty functionals – Mixed formulations.

8. Adaptive Finite Elements

9. Finite volumes