Course Details

Numerical methods for the variational problems

Academic Year 2024/25

DA66 course is part of 7 study plans

D-K-C-SI (N) / VHS Winter Semester 2nd year

D-K-C-SI (N) / MGS Winter Semester 2nd year

D-K-C-SI (N) / PST Winter Semester 2nd year

D-K-C-SI (N) / FMI Winter Semester 2nd year

D-K-C-SI (N) / KDS Winter Semester 2nd year

D-K-C-GK / GAK Winter Semester 2nd year

D-K-E-SI (N) / PST Winter Semester 2nd year

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.

Credits

10 credits

Language of instruction

Czech

Semester

winter

Course Guarantor

Institute

Forms and criteria of assessment

examination

Entry Knowledge

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.

Aims

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.

Basic Literature

BOUCHALA J.: Variační metody. VŠB-TU Ostrava 2012. (cs)
BLAHETA R. Matematické modelování a metoda konečných prvků. ZČU v Plzni 2012. (cs)

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 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.