• 097724 - MATHEMATICAL AND PHYSICAL MODELING IN ENGINEERING [2]

The main objective of the course is to acquire knowledge of mathematical and physical modeling of physical systems and to develop critical skills for their application in solving engineering problems.

The main goal of part [2] of the course is to make the student understand, both in a theoretical and an application oriented setting, the construction of perturbative approaches with a rigorous application of asymptotic series.

Particular focus is given to the understanding of the boundary layer theory, the construction of  the multi-scale methods, and the analysis of bifurcation problems and wave propagation phenomena in physical system models.

The student is expected:

- to master the mathematical develoment of all the mathematical and physical methods exposed during the course;

- to use dimensional analysis and variational principles to model the governing equations at the physical problem at hand;

- to develop the critical skills for selecting the most adapted method and for providing a rigorous derivation of the required mathematical solution;

- to use widespreadly adopted symbolic programming languages for object and modeling management (e.g. Mathematica);

- to become articulate and fluent in expressing and communicating all the key ideas and proofs of the basic theorems and concepts.

Asymptotic Analysis. Classification of singular points in ordinary differential equations. Irregular singular points: dominant balance method. Introduction to regular and singular perturbation theory.

Asymptotic regimes. Dimensionless Analysis: Buckingham theorem. Examples of dimensionless number in physical system models. Boundary layer theory. Matching conditions: Van Dyke rule and intermediate variable method. Quasilinear problems: applications of Erdelyi theorem. WKB approximation.

Bifurcations and instabilities. Fixed points and their linear stability analysis. Geometrical theory of bifurcation points. Multiple scale expansion and its application to nonlinear oscillators. Introduction to hydrodycamic instability problems. Basic notions of variation calculus. Elastic functionals and Euler-Lagrange equations.

Weakly nonlinear analysis. Applications of weakly nonlinear analysis in linear elastic problems of one-dimensional objects. Application of multiple-scale methods for wave propagation. The theorem of stationary phase and the effect of dispersion. Group velocity. The evolution of a wave envelope. Effect of nonlinearities: solitonic solutions in the Korteweg-deVries equation.

Basic ideas of calculus. Numerical series. Theory of ordinary differential equations and basic notions about partial differential equations. Elementary knowledege of classical Rational Mechanics and Continuum Mechanics (forces, balance equations, Cauchy stress tensor, kinematics of continua).

The exams wil be organized in agreement with the Academic Calendar and the Rules of the School of Engineering.

The verification of knowledge and capabilities acquired by the student is based, for each part, on a written test of about 2h, followed by an oral part.

The two written tests can be taken together during the same exam session, or separately in different exam sessions.

The final grade is based on an overall evaluation of the two parts of the exam.

The verification of knowledge and capabilities acquired by the student in part [2] of the course is based on a written test of about 2h, followed by an oral part.

The written test consists in two exercises based on physical problems, in which the student is expected to provide answers to few itemized questions.

In the oral part the student is expected to answer theoretical questions on course topics, and to verify student's skills at making connections between course topics and previous courses.

The teachers might place lecture notes in the Beep area of the Course.