Wednesday, May 6th, 2020. Time: 12 noon
ONLINE (Session link: https://meet.google.com/qfj-jtsz-mry)
|Seminar video on Youtube|
Reduced Order Models (ROM) in computational mechanics aim at solving problems approximating the solution in spaces of very low dimension.
The idea is to solve first the Full Order Model (FOM) in a high-fidelity space, of high dimension, extracting the main features of the solution and, from these, construct the basis of the ROM space. In this talk we shall concentrate on the case in which the FOM is solved by means of a Finite Element (FE) method and the ROM is obtained from a Proper Orthogonal Decomposition (POD) of a series of 'snapshots', i.e., high-fidelity solutions obtained for example at different time instants or for different values of a parameter of the problem to be solved. This way, the ROM solution can be considered to belong to a subspace of the FOM FE space, but defined on the same FE mesh.
The Variational Multi-scale (VMS) concept applied to the approximation of boundary value problem is quite simple. The idea is to split the unknown into the resolvable component, in our case living in the FE space, and a remainder, called sug-grid scale (SGS). After setting a problem for the SGS, this problem is what is in fact approximated somehow, so that the SGS can be expressed in terms of the FE solution. When the resulting expression is inserted into the equation projected into the FE space, one ends up with a problem for the FE unknown with enhanced stability problems. This idea is mainly used for the space approximation, being finite differences the most common option in time.
The purpose of this talk is to explain why the VMS strategy can be applied quite naturally to the ROM approximation when this is based in a FE method to approximate flow problems. In particular, it is explained how the SGS can be computed in a VMS-based ROM, and how this SGS can be used as an a posterior error estimation.
Ramon Codina is Professor at the Departament de Resistència de Materials i Estructures a l'Enginyeria (RMEE), in the Universitat Politècnica de Catalunya (UPC). He teaches Structural Mechanics and Continuum Mechanics and his research is concerned with numerical methods in engineering and applied sciences, with particular emphasis on finite element methods in fluid mechanics.