Completed notes of the course Numerical Methods

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LEZIONE 1 – 13.09 GENERAL CONCEPTS Why do we study this course for PDE. The final objective of this curse is to give you a feeling of what the MM based on PDE are helpful for the description of phenomena that you will study. All fields of engineering are about PDE. This course will help you in how to approach at new model, and how to use the computer for your help. You have to understand that with mathematical tools you can represent all of your interest. This is the purpose of this course. We have more tools to analyze hard problems. All the models of continuum mechanics are based on 2 fundamental ingredients: the conservation principles (of mass, momentum, energy, charge…), which are theoretically derived, that you have to translate in mathematical models, and the constitutive laws where you reconnect, experimentally determined (diffusion, friction, chemical kinetics…) and specific of all various fields. This is called phenomenological approach, because you observe the phenomena and try to learn his laws through this approach. Just a few examples. 1. SOLID MECHANICS: ELASTICITY You just take mass and momentum conservation into 3 dimensions for an object, and you combine it with laws about deformation. The stress is proportional to the deformation, so, if you pull the stress increases like a coil and writing this in a form of 3 dimensional setting (infinitesimal calculus), you obtain the information of solid in 3D, and this is fundamental for the design. 2. FLUID MECHANICS: NAVIER-STOKES is the principle of conservation of mass for a fluid that is called computational fluid dynamics (CFD). 3. FLUID STRUCTURE INTERACTION In this picture you can put all things together: the interaction of a solid with a fluid. The combination of mechanics is being the dance under the…