Partial Differential Equations (PDE) are a formidable tool for describing problems of interest, ranging over a wide set of possible applications.
Partial Differential Equations in Action: from Theory to Practice
Partial Differential Equations (PDE) are a formidable tool for describing problems of interest, ranging over a wide set of possible applications. In the last years, the numerical solution of PDE became part of regular industrial processes, and also of medical as well as environmental applications. PDE of practical interest are a challenging problem from both the theoretical and the practical perspectives. While analytical methods to find the explicit solution are rather limited, a numerical approximation is usually in order to get the quantitative answers needed by the applications. Nevertheless, the accurate and efficient solution of PDE relies on a deep knowledge of analytical methods and theoretical investigations. This course is intended to be an “advanced undergraduate course” in PDE with a pretty specific target: to cover the entire modeling process from the formulation of a specific problem of practical interest to the definition of a numerical solver of the appropriate PDE system, including the processing of available data (images, measures). We target some specific applications, in particular, environmental sciences and computational hemodynamics. Students will be engaged in a pretty interactive stream of information moving from theoretical notions to the solution of a real problem. Consistent with this ultimate goal, the course is split into two parts. PART 1: Introduction to PDE and their numerical solution. Foundations of the Finite Element Method. Advection-Diffusion-Reaction problems and their numerical treatment. Exercises in FreeFem++. PART 2: A complete problem: Patient-Specific Hemodynamics. Image processing (MRI, CT). Numerical modeling of blood flow in arteries. An example in 3D using FreeFem++.
The course will use some open source software like FreeFem++, NetGen, VascularModeling ToolKit (VMTK).
Il corso si voglerà dal 12 marzo al 24 maggio 2018 presso le Aule della Sede IUSS di Palazzo del Broletto.
C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method, Dover Publishing, 2009
Suggested readings (not necessary):
1. A. Quarteroni, Numerical modeling for differential problems, Springer
2. A. Quarteroni, A. Valli, Numerical Approximation of Partial Differential Equations, Springer
3. L. Formaggia, F. Saleri, A. Veneziani, Solving Numerical PDEs: Problems, Applications, Exercises, Springer
4. S. Salsa F. Vegni, A. Zaretti, P. Zunino, PDE: A Primer, Springer
Grading will be based on two projects.
One project at the end of the first part will be devoted to the solution of an environmental problem. The second one, at the end of the first part, will be devoted to the solution of a problem of computational hemodynamics. The latter can be replaced by a project close to topics of interest of the student (to be decided in accordance with the instructor). The projects will be discussed in June 2018.
Department : Science, Technology and Society
Scope : Science and Technology
Period: Semester II
Academic year: 2017-2018