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This work is based on some experience acquired during lectures and exercises in classes taught at SISSA Mathematics Area in the Doctoral Programme "Mathematical Analysis, Modelling and Applications", especially in computational mechanics classes, as well as regular courses previously taught at EPF Lausanne and during several summer and winter schools. The book is a companion for master and doctoral degree classes by allowing to go more deeply inside some partial differential equations worked out problems, examples and even exercises, but it is also addressed for researchers who are newcomers in computational mechanics with reduced order modeling.
In order to discuss computational results for the worked out problems presented in this booklet, we will rely on the RBniCS Project. The RBniCS Project contains an implementation in FEniCS of the reduced order modeling techniques (such as certified reduced basis method and Proper Orthogonal Decomposition-Galerkin methods) for parametric problems that will be introduced in this booklet.
Francesco Ballarin received his Ph.D. in Mathematical Models and Methods in Engineering at Politecnico di Milano, Italy in 2015, and is currently assistant professor in Numerical Analysis in the Department of Mathematics and Physics at Università Cattolica del Sacro Cuore, Brescia, Italy. His research focuses on reduced order models for parametrized problems in computational fluid dynamics. He is a passionate developer of open source software, which becomes an integral part of his research.
Leonardo Scandurra received his Ph.D. at Università degli Studi di Catania working on numerical methods for flows with different Mach number in gas dynamics. He contributed to different teaching activities at the HHU in Düsseldorf, where in particular he contributed to introduce a CFD course. He is currently a senior researcher at Engys srl in Trieste as software developer for CFD problems. His main research interests focus on numerical methods applied to statistical convergence assessment and Quantum CFD.
Federico Pichi received his Ph.D. in Mathematical Analysis, Modelling and Applications at SISSA, and he is currently a postdoctoral researcher at EPFL - École Polytechnique Fédérale de Lausanne in the MCSS group of Prof. Jan S. Hesthaven. His research interests include projection-based and data-driven reduced order models in computational science and engineering, with applications to parametrized bifurcating problems. He also develops scientific machine learning approaches bridging numerical analysis and novel architectures.
A short introduction to Reduced Basis Method.- Part I Worked out problems for beginners: steady cases.- Steady heat conduction in a thermal block.- A linear elasticity problem on a square.- Thermal transfer problem in a parametrized geometry.- A transport problem for the 2D Graetz flow.- Heat conduction with Gaussian flux.- Part II Advanced worked out problems:time dependent and nonlinear cases.- Unsteady heat conduction in a thermal block.- Unsteady heat conduction in a thermal block.- A nonlinear parabolic FitzHugh-Nagumo problem.- Part III Real-life worked out problems: engineering applications.- A thermal conduction problem through an extended surface.- An idealized contact problem in linear elasticity with friction.- A linear elasticity application on a beam bridge.- Part IV Fluid dynamics worked out problems.- Navier-Stokes system for a backward-facing step.- Bifurcating Coanda effect in a channel.- Unsteady Navier-Stokes equations for vortexshedding behind a cylinder.- Part V More advanced worked out problems.- Stabilized reduced method for an advection dominated problem.- A parametrized elliptic optimal control problem.- Uncertainty quantification for a stochastic thermal block.
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