Finite elements for plates, C1 finite elements | ||
No. | Date | Topics |
---|---|---|
1 | 1/18 | Elasticity and elastic plates, variational, weak, and strong form of the clamped Kirchhoff plate model |
2 | 1/20 | Simply-supported plate, Babuška's paradox, Hermite elements in 1D |
3 | 1/23 | C1 finite elements; Hermite cubic and Hermite quintic (Argyris) elements; approximation properties of Argyris elements |
4 | 1/25 | Reduced Hermite cubic element (Bell's triangle); HCT element |
Nonconforming finite elements | ||
5 | 1/27 | Nonconforming elements, P1 nonconforming elements, convergence theory |
6 | 1/30 | Consistency error bound for P1 nonconforming elements |
7 | 2/1 | The Morley nonconforming plate finite element |
Mixed finite elements | ||
8 | 2/3 | Mixed finite elements for the Poisson equation |
9 | 2/6 | Boundary conditions in the mixed formulation, the Stokes equations, the structure of saddle point problems |
10 | 2/8 | Duality in Hilbert and Banach spaces, ranges and annihilators |
11 | 2/10 | Closed Range Theorem, Brezzi's theorem |
12 | 2/13 | Proof of Brezzi's theorem and well-posedness of saddle point problems |
13 | 2/15 | Stability of Galerkin methods for saddle point problems |
14 | 2/17 | The lowest order Raviart-Thomas space, P1- |
15 | 2/20 | The Raviart-Thomas projection and stability of the mixed method |
16 | 2/22 | Higher order Raviart-Thomas spaces (Pr-); unisolvences |
17 | 2/24 | Stability of higher order Raviart-Thomas spaces (Pr-); improved error estimates; the BDM family |
The Stokes equations | ||
18 | 2/27 | Finite elements for the Stokes equations, some numerical results |
19 | 2/29 | Fortin operators; stability of the P2-P0 Stokes elements |
20 | 3/2 | The mini element and its analysis; other stable Stokes elements in 2- and 3-D |
Elasticity | ||
21 | 3/5 | Introduction to elasticity: displacement, stress, strain, constitutive equations, elastic moduli, isotropy |
3/7 | no class | |
3/9 | Midterm | |
22 | 3/19 | Boundary value problems of linear elasticity, weak formulations |
23 | 3/21 | Korn's inequality, coercivity, Galerkin methods and estimates, numerical examples |
24 | 3/23 | The pure traction problem, rigid motions, compliance tensor, trace-free (deviatoric)/pure-trace decomposition, incompressibility and near incompressibility; the mixed (stress-displacement) formulational of elasticity |
3/26 | no class | |
25 | 3/28 | The Johnson-Mercier elements for mixed elasticity |
26 | 3/30 | Airy stress function, compatibility of strain, the elasticity complex |
27 | 4/2 | Completion of unisolvence proof and stability analysis for Johnson-Mercier |
4/4 | no class | |
28 | 4/6 | The AW mixed elements for elasticity, unisolvence |
29 | 4/9 | Mixed finite elements for nearly incompressible and incompressible elasticity, weak symmetry |
Ten lectures on Finite Element Exterior Calculus
lectures will be 75 minutes |
||
30 | 4/11 | Introduction to finite element exterior calculus; basic homological algebra: chain complexes, chain maps, and homology; simplicial homology and de Rham cohomology |
4/13 | no class | |
31 | 4/16 | Unbounded operators on Hilbert space, graph norm, closed operators, adjoints |
32 | 4/18 | Duality between null space and range, closed range theorem, examples with common function spaces |
33 | 4/20 | Hilbert complexes, dual complex, three key properties of closed Hilbert complexes: harmonic forms, Hodge decomposition, Poincaré inequality; the L2 de Rham complex |
34 | 4/23 | The abstract Hodge Laplacian, equivalence of strong, primal weak, and mixed weak formulations; well-posedness |
35 | 4/25 | Galerkin methods for the abstract Hodge Laplacian; three key properties: approximation, subcomplexes, bounded cochain projects; three major conclusions: preservation of cohomology, approximation of harmonic forms, uniform Poincaré inequality |
36 | 4/27 | Stability and convergence of Galerkin methods for the abstract Hodge Laplacian, basic error estimate, improved error estimates |
37 | 4/30 | Exterior algebra and calculus: algebraic forms, differential forms, the de Rham complex |
38 | 5/2 | The Koszul complex and the homotopy formula; the spaces Pr and Pr- spaces, unisolvence, finite element de Rham subcomplexes |
39 | 5/4 | Bounded cochain projections, applications |