Course Details

# Numerical methods for the variational problems

Academic Year 2022/23

DA66 course is part of 23 study plans

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D-K-E-SI (N) Winter Semester 1st year

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D-K-C-SI (N) Winter Semester 1st year

D-K-C-SI (N) Winter Semester 1st year

D-K-C-SI (N) Winter Semester 1st year

D-K-C-SI (N) Winter Semester 1st year

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D-P-E-SI (N) Winter Semester 1st year

D-P-E-SI (N) Winter Semester 1st year

D-P-E-SI (N) Winter Semester 1st year

D-P-E-SI (N) Winter Semester 1st year

D-P-E-SI (N) Winter Semester 1st year

D-P-C-SI (N) Winter Semester 1st year

D-P-C-SI (N) Winter Semester 1st year

D-P-C-SI (N) Winter Semester 1st year

D-P-C-SI (N) Winter Semester 1st year

D-P-C-SI (N) Winter Semester 1st year

D-P-C-GK Winter Semester 1st year

D-K-C-GK Winter Semester 1st 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.

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 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.

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.

Language of instruction

Czech

Credits

10 credits

Semester

summer

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 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.