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Reduced Order Methods for Computational Mechanics

Lecturer: 
Gianluigi Rozza
Course Type: 
PhD Course
Master Course
Academic Year: 
2018-2019
Period: 
April 2019
Duration: 
20 h
Description: 

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.

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 approzimations 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).

Lecture notes, slides and reading material will be provided (please send an email in advance to gianluigi.rozza@sissa.it to register).

Lectures will cover the material in the book : J. Hesthaven, G. Rozza, B. Stamm 'Certified reduced basis methods and a posteriori error bounds for parametrized PDEs', Springer 2015.

Link to educational software: http://mathlab.sissa.it/rbnics

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

 

 

Location: 
A-005
Next Lectures: 
Monday, April 15, 2019 - 14:30 to 18:00
Tuesday, April 16, 2019 - 10:30 to 12:30
Tuesday, April 16, 2019 - 14:15 to 16:00
Wednesday, April 17, 2019 - 09:30 to 13:00
Wednesday, April 17, 2019 - 14:15 to 16:00
Wednesday, April 17, 2019 - 16:30 to 18:15
Thursday, April 18, 2019 - 09:30 to 13:00
Thursday, April 18, 2019 - 14:15 to 16:00

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