PERIODO: MAGGIO - GIUGNO 2018

Professor: Guglielmo Scovazzi, Civil & Environmental Engineering
Department, Duke University, USA

Brief description: The course presents a detailed introduction to two of the most common methods for the resolution of continuum mechanics problems with complex geometries:1) the finite element method on arbitrary coordinates frames and 2) immersed boundary/geometries methods.

1) Arbitrary Lagrangian/Eulerian formulations (ALE):
- Some fundamental results of vector calculus
- Maps and kinematics for Lagrangian, Eulerian and arbitrary frames of reference
- Leibnitz and Reynolds transport theorems in generalized coordinates
- Balance laws
- Rankine-Hugoniot conditions
- Conservation laws in generalized coordinates
- Variational formulations in generalized coordinates
- The geometric conservation law and consequences
2) Immersed finite element methods:
- Fundamental concepts, motivation, brief history
- Characterization of immersed geometries, computational geometry
- Brief description of immersed approaches with finite differences and finite volumes methods
- Introduction and analysis of immersed FEM: stability, convergence
- Condition number of the linear system and problems related to small cut cells
- Applications and future perspectives
References:
[1] J. Donea & A. Huerta, "Finite Element Methods for Flow Problems", Wiley 2003.
[2] G. Scovazzi & T.J.R. Hughes, "Lecture Notes on Continuum Mechanics on Arbitrary
Moving Domains", Sandia National Laboratories Report 2007-6312P.
[3] A. Main, G. Scovazzi, "The shifted boundary method for embedded domain computations.
Part I: Poisson and Stokes problems," in press in Journal of Computational Physics, 2018,
https://doi.org/10.1016/j.jcp.2017.10.026 .
[4] A. Main, G. Scovazzi, "The shifted boundary method for embedded domain computations.
Part II: Advection-diffusion and Navier-Stokes equations," in press in Journal of
Computational Physics, 2018.