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Numerical Analysis

Reduced Order Methods for Computational Mechanics

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.

Topics in Computational Fluid Dynamics

  • Introduction to CFD, examples.
  • Constitutive laws
  • Incompressible flows.
  • Numerical methods for potential and thermal flows
  • Boundary layer theory
  • Thermodynamics effects, energy equation, enthalpy and entropy
  • Vorticity equations
  • Introduction to turbulence
  • Numerical methods for viscous flows: steady Stokes equations
  • Stabilisation (SUPG) and inf-sup condition

Advanced Programming

The course aims to provide advanced knowledge of both theoretical and practical programming in C++11 and Python3, with particular regard to the principles of object-oriented programming and best practices of software development.

Syllabus:

Numerical Solution of PDEs Using the Finite Element Method

The Finite Element Method Using deal.II This is an intensive course that teaches how to use the finite element library deal.II (www.dealii.org).Prerequisites: you should be familiar with C/C++, and with the Unix command line. We'll cover the basics of Finite Element Methods, and go from solving the Laplace equation on a uniformly refined grid, to solving the same equation using adaptively refined grids, in parallel, on a supercomputer.Lectures will be structured in the following way:

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