Course Details
Numerical methods for the variational problems
Academic Year 2022/23
DAB036 course is part of 23 study plans
DPC-GK Winter Semester 1st year
DKC-GK Winter Semester 1st year
DPA-GK Winter Semester 1st year
DKA-GK Winter Semester 1st year
DPC-E Winter Semester 1st year
DKC-E Winter Semester 1st year
DPA-E Winter Semester 1st year
DKA-E Winter Semester 1st year
DKC-S Winter Semester 1st year
DPC-S Winter Semester 1st year
DPA-S Winter Semester 1st year
DKA-S Winter Semester 1st year
DKC-V Winter Semester 1st year
DKA-V Winter Semester 1st year
DPA-V Winter Semester 1st year
DKC-K Winter Semester 1st year
DPC-K Winter Semester 1st year
DKA-K Winter Semester 1st year
DPA-K Winter Semester 1st year
DKC-M Winter Semester 1st year
DPC-M Winter Semester 1st year
DKA-M Winter Semester 1st year
DPA-M Winter Semester 1st 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
Institute
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
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.