470-8743/03 – Mathematical Modelling and FEM (MMMKP)
| Gurantor department | Department of Applied Mathematics | Credits | 4 |
| Subject guarantor | doc. Ing. Dalibor Lukáš, Ph.D. | Subject version guarantor | doc. Ing. Dalibor Lukáš, Ph.D. |
| Study level | undergraduate or graduate | Requirement | Choice-compulsory type A |
| Year | 1 | Semester | summer |
| | Study language | Czech |
| Year of introduction | 2016/2017 | Year of cancellation | |
| Intended for the faculties | FMT, USP, FEI | Intended for study types | Follow-up Master |
Subject aims expressed by acquired skills and competences
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.
Teaching methods
Lectures
Tutorials
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.
Compulsory literature:
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
Recommended literature:
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
Way of continuous check of knowledge in the course of semester
Written and oral exam.
E-learning
Lecture notes as a pdf-file are available at am.vsb.cz.
Other requirements
Additional requirements for students are not.
Prerequisities
Subject has no prerequisities.
Co-requisities
Subject has no co-requisities.
Subject syllabus:
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.
Conditions for subject completion
Occurrence in study plans
Occurrence in special blocks
Assessment of instruction
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