Computational mechanics by reduced order methods

Lecturer:

Gianluigi Rozza

Course Type:

PhD Course

Academic Year:

2023-2024

Period:

May

Duration:

20 h

Description:

Mathematics Area, PhD in Mathematical Analysis, Modelling and Applications (AMMA)

Master in High Performance Computing (MHPC)

Lectures Prof Gianluigi Rozza, Tutorials coordinated by Dr Michele Girfoglio, Dr Federico Pichi and Dr Ivan Prusak.

Learning outcomes and objectives

The course aims to provide the basic aspects of numerical approximation and efficient solution of parametrized PDEs for computational mechanics problem (heat and mass transfer, linear elasticity, viscous and potential flows) using reduced order methods.

Description

In this course we present reduced basis (RB) approximation and associated a posteriori error estimation for rapid and reliable solution of parametrized partial differential equations (PDEs). The focus is on rapidly convergent Galerkin approximations on a subspace spanned by "snapshots'"; rigorous and sharp a posteriori error estimators for the outputs/quantities of interest; efficient selection of quasi-optimal samples in general parameter domains; and Offline-Online computational procedures for rapid calculation in the many-query and real-time contexts. We develop the RB methodology for a wide range of (coercive and non-coercive) elliptic and parabolic PDEs with several examples drawn from heat transfer, elasticity and fracture, acoustics, and fluid dynamics. We introduce the concept of affine and non-affine parametric dependence, some elements of approximation and algebraic stability. Finally, we consider application of RB techniques to parameter estimation, optimization, optimal control, and a comparison with other reduced order techniques, like Proper Orthogonal Decomposition. Some tutorials are prepared for the course based on FEniCS and Python within the training/educational library RBniCS (open-source based on python and FEniCS).

Please contact gianluigi.rozza@sissa.it if you need more info. If you plan to attend in order to provide you links to lectures, exercise sessions as well as to material and software instructions, please send an email to confirm your participation in advance (no later than May 3, 2024).

Topics/Syllabus

Introduction to RB methods, offline-online computing, elliptic coercive affine problems
Parameters space exploration, sampling, Greedy algorithm, POD
Residual based a posteriori error bounds and stability factors
Primal-Dual Approximation
Time dependent problems: POD-greedy sampling
Non-coercive problems
Approximation of coercivity and inf-sup parametrized constants
Geometrical parametrization
Reference worked problems
Examples of Applications in CFD and flow control
Tutorials

Software for the exercise sessions

We will employ the RBniCS package, which is an open-source code developed at SISSA mathLab (and at the Catholic University of the Sacred Heart) in the framework of the AROMA-CFD ERC CoG project. The first half of the exercise sessions will be dedicated to proving the necessary basic knowledge and components of FeniCS and RBniCS libraries for simple problems which is necessary for understanding more complex problems treated in the second part. The tutorials will run on the cloud service provided by Google Colab, so that no local installation is required. See our tutorials page for more information on the RBniCS library and on how to use the Google Colab platform.

Research Group:

Mathematical Analysis, Modelling, and Applications

mathLab

MsC Course

A companion digital collection of worked out problems is on argos.sissa.it (registration required)

References:

https://link.springer.com/book/10.1007/978-3-319-22470-1

https://link.springer.com/book/9783031498916

https://www.degruyter.com/serial/mor-b/html

Lecture details: