Course Details

Numerical methods for the variational problems

DA66 course is part of 22 study plans

Ph.D. full-t. program nD > PST compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program nD > FMI compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program nD > KDS compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program nD > MGS compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program nD > VHS compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program nDK > PST compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program nDK > KDS compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program nDK > VHS compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program nDK > FMI compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program nDK > MGS compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program nDA > PST compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program nDA > FMI compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program nDA > KDS compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program nDA > MGS compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program nDA > VHS compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program nDKA > PST compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program nDKA > FMI compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program nDKA > KDS compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program nDKA > MGS compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program nDKA > VHS compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program I > GAK compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program IK > GAK compulsory-elective Winter Semester 2nd year 10 credits

1. Introduction to the variatoinal calculus: Examples of functionals, the simplest problem of variational calculus, Euler equation of a functional. 2. Differential problems: Classical and variational formulations of boundary-value differential problems. Discretization of stationary differential problems by the finite-difference, Galerkin Ritz methods. Standard time-discretizations of non-stationary differential problems. 3. Formulation and numerical solution of the heat-conduction problem, the linear elasticity problem, of the linear flow problems, of the Navier-Stokes equations and of selected models of simultaneous moisture and heat distribution in porous media.

Course Guarantor

prof. Ing. Jiří Vala, CSc.

Institute

Institute of Mathematics and Descriptive Geometry

Prerequisites

Basic notions of linear algebra and mathematical analysis, elementary methods for exact solutions of differential equations, methods for approximate solutions of systems of linear and non-linear equations, interpolation and approximation of functions, numerical differentiation and numerical integration.

Planned educational activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations.

Objective

Basics of calculus of variations, numerical methods for variationally formulated differential boundary-value problems. The studied boudary-value problems are mathematical models of processes often occuring in the practice of civil engineers.

Specification of controlled instruction, the form of instruction, and the form of compensation of the absences

Vymezení kontrolované výuky a způsob jejího provádění stanoví každoročně aktualizovaná vyhláška garanta předmětu.

Lecture

3 hours/week, 13 weeks, elective

Syllabus of lectures

1. Functional and its Euler equation, the simlest problem ov calculus of variations.
2. Concrete examples of functionals and related Euler equations. Elementary solutions.
3. Derivation of an elliptic problem for ODE of degree 2, the problems of heat conduction and distribution of polution.
4. Discretization of the elliptic problem for ODE of degree 2 by the standard finite difference method, stability of numerical solutions.
5. Variational (weak) and minimization formulation of the elliptic problem for the elliptic problem for ODE of degree 2.
6. The Ritz and Galerkin methods.
7. Discretization of the elliptic problem for ODE of degree 2 by the finite element method.
8. Discretization of the variational formulation of the elliptic problem for ODE of degree 2 by the finite element method.
9. Discretization of the minimization formulation of the elliptic problem for ODE of degree 2 by the finite element method.
10. Discretization of the variational formulation of the elliptic problem for PDE of degree 2 by the finite element method.
11. Variational formulation and the finite element method for the linear elasticity problem.
12. Navier-Stokes equations and their numerical solution by the particle method.
13. A mathematical model of simultaneous distribution of moisture and heat in porous materials, discretizations.