Skip to main content
Skip header

Mathematical Modelling and FEM

* Exchange students do not have to consider this information when selecting suitable courses for an exchange stay.

Course Unit Code470-8743/03
Number of ECTS Credits Allocated4 ECTS credits
Type of Course Unit *Choice-compulsory type A
Level of Course Unit *Second Cycle
Year of Study *First Year
Semester when the Course Unit is deliveredSummer Semester
Mode of DeliveryFace-to-face
Language of InstructionCzech
Prerequisites and Co-Requisites Course succeeds to compulsory courses of previous semester
Name of Lecturer(s)Personal IDName
LUK76doc. Ing. Dalibor Lukáš, Ph.D.
Summary
The course should prepare the students to be able to formulate the boundary value problems arising in mathematical modelling of heat conduction, elasticity and other physical processes. The students should be also able to derive differential and variational formulation of these problems and understand the mathematical principles of their numerical solution, especially by the finite element method. The course will also touch the principles of proper use of mathematical modelling methods for solving engineering problems.
Learning Outcomes of the Course Unit
Students will be able to formulate the boundary value problems arising in mathematical modeling of heat conduction, elasticity, and other phenomena (diffusion, electro and magnetostatics, etc.). It will also be able to derive the differential and variational formulation of the task and numerical solution of the finite element method. They will know the principles of proper use of mathematical models for solving engineering problems.
Course Contents
Mathematical modeling. Purpose and general principles of modeling. Benefits
mathematical modeling. Proper use of mathematical models.
Differential formulation of mathematical models. One-dimensional heat conduction problem and its mathematical formulation. Generalizing the model. The input linearity,
existence and uniqueness of solutions. Discrete input data. One-dimensional task
flexibility and other models. Multivariate models.
Variational formulation of boundary problems. Weak formulation of boundary problems and its relationship to the classical solutions. Energy and energy functional formulation.
Coercivity and boundedness. Uniqueness, continuous dependence of solutions
input data. Existence and smoothness of the solution.
Ritz - Galerkin (RG) method. RG method. Konenčných element method (FEM)
as a special case of the RG method. History MLP.
Algorithm finite element method. Assembling the stiffness matrix and vector
load. Taking into account the boundary conditions. Numerical solution of linear systems algebraic equations. Different types of finite elements.
The accuracy of finite element solutions. Priori estimate of the discretization error.
Convergence, h-and p-version FEM. Posteriori estimates. Network design for MLP
adaptive technology and optimal network.
FEM software and its use for MM. Preprocessing and postprocessing. Commercial
software systems. Solutions particularly difficult and special problems. Principles
Mathematical modeling using FEM.
Recommended or Required Reading
Required Reading:
R. D. Cook: Finite element modelling for stress analysis, J. Wiley, New
York, 1995.
C. Johnson: Numerical solution of partial differential equations by the
finite element method, Cambridge Univ. Press, 1995
D. Braess: Finite elements. Cambridge University Press, 2001
Mathematical Modeling with Multidisciplinary Applications. Edited by Xin-She
Yang, John Wiley & Sons, Inc., UK, 2013
Recommended Reading:
R. D. Cook: Finite element modelling for stress analysis, J. Wiley, New York, 1995.
C. Johnson: Numerical solution of partial differential equations by the finite element method, Cambridge Univ. Press, 1995
Drábek, P. - Holubová, G.: Parciální diferenciální rovnice. ZČU Plzeň, 2001.
Planned learning activities and teaching methods
Lectures, Tutorials
Assesment methods and criteria
Task TitleTask TypeMaximum Number of Points
(Act. for Subtasks)
Minimum Number of Points for Task Passing
Credit and ExaminationCredit and Examination100 (100)51
        CreditCredit30 10
        ExaminationExamination70 21