Course Details

Numerical methods for the variational problems

DAB036 course is part of 24 study plans

Ph.D. full-t. program DPC-M compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program DPC-K compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program DPC-V compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program DPC-E compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program DPC-S compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program DPC-GK compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program DKC-S compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program DPA-S compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program DKC-V compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program DPA-V compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program DKC-M compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program DPA-M compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program DKC-K compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program DPA-K compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program DKC-E compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program DPA-E compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program DKC-GK compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. full-t. program DPA-GK compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program DKA-GK compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program DKA-S compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program DKA-M compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program DKA-K compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program DKA-V compulsory-elective Winter Semester 2nd year 10 credits

Ph.D. combi. program DKA-E compulsory-elective Winter Semester 2nd year 10 credits

Introduction to the variatoinal calculus, analysis of initial and boundary problems for ordinary and partial differential equations, selected applications to civil engineering.

Course Guarantor

prof. Ing. Jiří Vala, CSc.

Institute

Institute of Mathematics and Descriptive Geometry

Learning outcomes

The aim is the deeper understanding of relations between mathematical and numerica analysis and engineering problems, using the modern variational approaches.

Prerequisites

Mathematical and numerical analysis at the level of the course DA61.

Corequisites

Not required.

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.

Forms and criteria of assessment

Evaluation of work on specified tasks, final examination.

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 of 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.