Course detail

Mathematics 5 (M)

FAST-CA003Acad. year: 2022/2023

Interpolation and approximation of functions. Numerical solution of algebraic equations and their systems. Numerical derivatives and quadrature. Variance analysis, regression analysis. Numerical solution of stationary and non-stationary boundary and initial problems for differential equations with applications to civil engineering. Direct, sensitivity and inverse problems.

Language of instruction

Czech

Number of ECTS credits

4

Mode of study

Not applicable.

Guarantor

prof. Ing. Jiří Vala, CSc.

Department

Institute of Mathematics and Descriptive Geometry (MAT)

Learning outcomes of the course unit

Following the aim of the course, students will receive the basic orientaion in numerical and statistical methods needed in material engineering and in related engineering applications.

Prerequisites

Basic knowledge of numerical mathematics, probability and statistics, applied to technical problems.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Not applicable.

Assesment methods and criteria linked to learning outcomes

Not applicable.

Course curriculum

1. Mathematical modelling. Deterministic and stochastic models. Errors in numerice calculations.
2. Lagrangean and Hermitean interpolation of functions. Interpolation functions, especially polynomials and splines.
3. Numerical solution of linear and nonlinear algebraic equations and their systems.
4. Numerical derivatives and quadrature.
5. Formulation and numerical solution of direct problems with differential and integral equations.
6. Finite difference, element and volume methods for stationary problems.
7. Methods of lines and discretization in time (Rothe sequences) for nonstationary problems.
8. Statistical tests, variance analysis, fuzzy models.
9. Linear regression analysis. Least squares method.
10. Nonlinear regression analysis.
11. Sensitivity analysis. Application to uncertainty transfer and estimates of durability of building structures.
12. Inverse analysis. Application to determination of material parameters from experiments.
13. Application to significant engineering problems.

Work placements

Not applicable.

Aims

Students will obtain the basic knowledge of numerical mathematics, probability and statistics, applied to technical problems, especially from material engineering.

Specification of controlled education, way of implementation and compensation for absences

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

Recommended optional programme components

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

DALÍK J.: Numerické metody. CERM Brno 1997. (CS)
VALA J.: Numerická matematika. FAST VUT v Brně 2021. (CS)

Recommended reading

Not applicable.

Classification of course in study plans

  • Programme N-P-C-SI Master's

    branch M , 1 year of study, winter semester, compulsory

  • Programme N-K-C-SI Master's

    branch M , 1 year of study, winter semester, compulsory

  • Programme N-P-E-SI Master's

    branch M , 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. Ing. Jiří Vala, CSc.

Syllabus

1. Mathematical modelling. Deterministic and stochastic models. Errors in numerice calculations. 2. Lagrangean and Hermitean interpolation of functions. Interpolation functions, especially polynomials and splines. 3. Numerical solution of linear and nonlinear algebraic equations and their systems. 4. Numerical derivatives and quadrature. 5. Formulation and numerical solution of direct problems with differential and integral equations. 6. Finite difference, element and volume methods for stationary problems. 7. Methods of lines and discretization in time (Rothe sequences) for nonstationary problems. 8. Statistical tests, variance analysis, fuzzy models. 9. Linear regression analysis. Least squares method. 10. Nonlinear regression analysis. 11. Sensitivity analysis. Application to uncertainty transfer and estimates of durability of building structures. 12. Inverse analysis. Application to determination of material parameters from experiments. 13. Application to significant engineering problems.

Exercise

13 hod., compulsory

Teacher / Lecturer

prof. Ing. Jiří Vala, CSc.

Syllabus

Follows directly particular lectures: 1. Mathematical modelling. Deterministic and stochastic models. Errors in numerice calculations. 2. Lagrangean and Hermitean interpolation of functions. Interpolation functions, especially polynomials and splines. 3. Numerical solution of linear and nonlinear algebraic equations and their systems. 4. Numerical derivatives and quadrature. 5. Formulation and numerical solution of direct problems with differential and integral equations. 6. Finite difference, element and volume methods for stationary problems. 7. Methods of lines and discretization in time (Rothe sequences) for nonstationary problems. 8. Statistical tests, variance analysis, fuzzy models. 9. Linear regression analysis. Least squares method. 10. Nonlinear regression analysis. 11. Sensitivity analysis. Application to uncertainty transfer and estimates of durability of building structures. 12. Inverse analysis. Application to determination of material parameters from experiments. 13. Application to significant engineering problems.