Course Details

Numerical methods for the variational problems

Academic Year 2023/24

DAB036 course is part of 23 study plans

​ DPC-S Winter Semester 2nd year

​ DPC-M Winter Semester 2nd year

​ DPC-K Winter Semester 2nd year

​ DPC-GK Winter Semester 2nd year

​ DPC-E Winter Semester 2nd year

​ DPA-V Winter Semester 2nd year

​ DPA-S Winter Semester 2nd year

​ DPA-M Winter Semester 2nd year

​ DPA-K Winter Semester 2nd year

​ DPA-GK Winter Semester 2nd year

​ DPA-E Winter Semester 2nd year

​ DKC-V Winter Semester 2nd year

​ DKC-S Winter Semester 2nd year

​ DKC-M Winter Semester 2nd year

​ DKC-K Winter Semester 2nd year

​ DKC-GK Winter Semester 2nd year

​ DKC-E Winter Semester 2nd year

​ DKA-V Winter Semester 2nd year

​ DKA-S Winter Semester 2nd year

​ DKA-M Winter Semester 2nd year

​ DKA-K Winter Semester 2nd year

​ DKA-GK Winter Semester 2nd year

​ DKA-E Winter Semester 2nd year

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

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.

Syllabus

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.

Prerequisites

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

Language of instruction

Czech

Credits

10 credits

Semester

winter

Forms and criteria of assessment

examination

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

Extent and forms are specified by guarantor’s regulation updated for every academic year.

Offered to foreign students

Not to offer

Course on BUT site

https://www.vut.cz/en/students/courses/detail/272136

Lecture

13 weeks, 3 hours/week, elective

Syllabus

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.