Mathematical Modelling of the Continuum (FSI-9MMK)

The aim of the course is to acquaint students with mathematical modeling using partial differential equations of a wider range of engineering problems for the continuum: elasticity, conduction, convection, linear and nonlinear models and coupled problems. To teach students to formulate basic problems, including initial, boundary and possibly other conditions, to know where the sources of errors are. To prepare them for a critical approach to the use of computer systems such as MATLAB, ANSYS, etc.

Vectors and matrices, differential and integral calculus of several variables, ordinary differential equations. Appropriate completion of the course 9RF1 Equations of Mathematical Physics.

The concept of continuum and its description. Coordinates, quantities and problem formulation. Mathematical means: differential equations, classical, generalized and approximate solutions. Spaces of integrable functions and integral functionals.

Derivation of conduction, linear and nonlinear elasticity equations. Elastic, viscous and plastic behavior. Heterogeneous material modelling, homogenization and coupled problems.

Fluid mechanics, derivation of transfer equations and Navier-Stokes equations. Coupled problems: flow and thermal phenomena.

Existence, uniqueness and stability of generalized solutions. Conditions for the existence of a minimum of an integral functional. Basic numerical methods: Finite Element Method and Finite Volume Method, adaptive methods.

The exam consists of a practical and a theoretical part. Practical part: mathematical formulation of a specific engineering problem. Theoretical part: 3-5 questions from the subject matter. In case of absence, the student must make up for the missed material by self-study of literature.

Lectures

1. The concept of continuum and its description, coordinates and quantities, types of problems.


2. Mathematical means: differential equations, classical and generalized solutions.


3. Laws of conservation and constitutional relations. Linear problems, derivation of the equation of heat conduction in a body, formulation of the initial boundary value problem.


4. Description of deformation and stress in a body, linear and nonlinear elasticity, Piol transformation. Pooled tasks.


5. Minimization of integral functional, generalized convexity conditions.


6. Heterogeneous material: transition conditions, homogenization.


7. Models of elastic, viscous and plastic material – hysteresis.


8. Fluid modelling: coordinates, quantities. Derivation of mass-heat transfer equations and Navier's Stokes equations.


9. Generalized formulation of flow equations.

10. Finite element and finite volume method, adaptive methods.

M. Brdička, L. Samek, B. Sopko: Mechanika kontinua, Academia, Praha 2000.

J. Nečas, I. Hlaváček: Úvod do matematické teorie pružných a pružně-plastických těles, SNTL Praha 1983

P. G. Ciarlet: Mathematical elasticity Volume I: Three-dimensional elasticity, North-Holland, Amsterdam 1988 

A. Kufner, O. John, S. Fučík: Function spaces, Academia, Prague 1977.

A. Ženíšek: Sobolev Spaces and Their Applications in the Finite Element Method. VUTIUM, Brno, 2005.

J. Franců: Parciální diferenciální rovnice, CERM, Brno 2011

J. Franců: Moderní metody řešení diferenciálních rovnic, CERM, Brno 2006