## Topics in Scientific Computing for the Solution of PDEs

## Numerical Methods for PDEs

- Finite Elements
- Elliptic Problems
- Parabolic Problems
- Hyperbolic Problems

HPC Techniques for the solutions of PDEs

- Domain Decomposition
- Reduced Basis Approximations
- Multipole Expansion

## Topics in Computational Fluid Dynamics

- Introduction to CFD, examples.
- Incompressible flows.
- Numerical methods for potential and thermal flows
- Numerical methods for viscous flows: steady Stokes equations
- Discretization techniques for steady and unsteady Navier-Stokes equations.
- Advanced optional topic (1): compressible flows.
- Advanced optional topic (2): fluid and structure interaction.

Material will be provided during classes.

## Topics in the mechanics of soft and bio-materials

Topics in the mechanics of soft and bio-materials

This course aims to provide an introduction to the mechanics of soft materials, of which biological materials are prominent examples. Soft materials are those that can be easily deformed by external stress, electromagnetic fields or even thermal fluctuations: in other words everything that is wet, squishy, sticky, flabby or spongy.

## Introduction to Mechanics of Solids, Fluids, and Biological Systems

- Kinematics of deformable continua
- Eulerian and Lagrangian descriptions of motion
- The balance laws of continuum mechanics
- Conservation of mass
- Balance of linear and angular momentum
- Constitutive Equations
- Fluid dynamics: the Navier Stokes equations
- Solid mechanics: nonlinear and linear elasticity
- Selected topics from the mechanics of biological systems

## Numerical Analysis and Scientific Computing

The research deals with the analysis, development, application of mathematical models for the integration of complex systems. The analysis is conducted using mathematical methods in several fields such as linear algebra, approximation theory, partial differential equations, optimization and control. Solution methods are developed and applied to domains as diverse as (potential and viscous) flow dynamics, (linear and nonlinear) structural analysis, mass transport, heat transfer and in general to multiscale and multiphysics applications. The methods have been integrated into complex multidisciplinary systems.

## Research topics

- CONTROL AND OPTIMIZATION The efficient solution of optimal control or shape optimization problems involving partial differential equations (PDEs) is a problem of interest in computational science and engineering. The goal of an optimal control problem is the minimization/maximization of a given output of interest (expressed by suitable cost functionals) under some constraints, controlling either suitable variables (such as sources, model coefficients or boundary values) or the shape of the domain itself. In the latter case, we deal with shape optimization or optimal shape design problems.
- REDUCED ORDER MODELLING Model order reduction techniques provide an efficient, accurate and reliable way of solving (systems of) parametrized partial differential equations in the many-query or real-time context thanks to offline-online computational splittings, such as (shape) optimization, flow control, characterization, parameter estimation, uncertainty quantification. Our research is mostly based, but not limited to, on certified reduced basis methods and proper orthogonal decomposition for parametrized PDEs.
- FREE SURFACES Techniques to study the position of an interface as a part of the problem itself, when studying the dynamics of a boat, for example.
- FLUID-STRUCTURE INTERACTION Development of efficient algorithms and methods for the coupling between the fluid and structure dynamics finds applications in a large variety of fields dealing with internal or external flows, also at the reduced order level (cardiovascular applications, naval engineering).
- PARALLEL and HIGH PERFORMANCE COMPUTING
- OPEN SOURCE SOFWARE DEVELOPMENT Several open source software libraries are developed and maintained

## Collaborating Institutes

- Politecnico di Milano, MOX, Modeling and Scientific Computing Center
- EPFL, Lausanne, Switzerland
- Massachusetts Institute of Technology, Cambridge, US
- Università di Pavia, Italy
- University of Houston, US
- University of Toronto, Canada
- Laboratoire Jacques Louis Lions, Paris VI, France
- Duke University, Durham, US
- Imperial College, London, UK
- Politecnico di Torino, Italy
- Virginia Tech, Blacksburg, Virginia, US
- Scuola Superiore S.Anna, Pisa, Italy
- University of Cambridge, UK
- University of Sevilla, Spain
- University of Santiago de Compostela, Spain
- RWTH Aachen, Germany
- University of Ghent, Belgium