This course is an introduction to finite element methods on Arbitrary Lagrangian/Eulerian (ALE) coordinates.
No prerequisites are needed, other than a general knowledge of numerical methods for partial differential equations (PDEs) and multivariable calculus.
Exposure to the basics of the finite element method are certainly beneficial but not necessary.
The scope of the course is to introduce students to the finite element method and the simulation of problems in which the computational domains(regions) change in shape over time.

[1] H.J.R. Hughes, “The Finite Element Method: Linear Static and Dynamic Analysis”, Dover 2004.
[2] J. Donea and A. Huerta, “Finite Element Methods for Flow Problems”, Wiley 2003.
[3] G. Scovazzi and T.J.R. Hughes, “Lecture Notes on Continuum Mechanics on Arbitrary Moving Domains”, Sandia National Laboratories Report 2007-6312P.
[4] A. Main, G. Scovazzi, “The shifted boundary method for embedded domain computations. Part I: Poisson and Stokes problems,” Journal of Computational Physics,372, 972-995, 2018.

1) Introduction to the finite element method:
. Fundamental concepts in one dimension (Poisson Equation):
– Strong and weak form of a PDE’s boundary value problem and conditions for their equivalence
– Galerkin approximation, shape functions, test/trial function spaces, best approximation property for elliptic PDEs
– Matrix equations, degrees of freedom
. Finite element data structures:
– Element stiffness matrix and force vector
– Finite element assembly
. Generalizations to two and three dimensions (Poisson Equation):
– Element stiffness matrix and force vector
– Finite element assembly
. Iso-parametric elements and programming concepts (triangles and quadrilaterals)
– Parent vs. physical domain
– Derivative of shape functions
– Numerical integration and Gauss quadratures
– Element stiffness matrix and force vector
2) Arbitrary Lagrangian/Eulerian methods (ALE):
. Some fundamental results from vector calculus
. Maps and kinematics Lagrangian, Eulerian, and arbitrary Lagrangian/Eulerian reference frames
. Transport theorem of Leibnitz and Reynolds in generalized reference frame
. Balance laws and Rankine-Hugoniot conditions
. Conservation laws in generalized coordinates
. Variational formulation in generalized coordinates
. The geometric conservation law and consequences
. Variational equations of mechanics in Arbitrary Lagrangian/Eulerian form
3) A preview of immersed/unfitted numerical methods:
. Finite volume versus finite element methods
. An introduction to a possible course offering in the Fall 2022 “Immersed and unfitted finite element methods, theory and practice”

Real-time online lecture.