Course detail

Numerical Computations with Partial Differential Equations

FEKT-DTE2Acad. year: 2017/2018

The content of the seminar consists of two related units. The first part deals with the numerical solution of the partial differential equations (PDE), exploiting the Finite Difference method (FDM) and the Finite Element Method. The following PDE are solved by these methods: Laplace’s, Poisson’s, Helmholtz’s, parabolic, and hyperbolic one. The boundary and initial condition as well as the material parameters and source distribution is supposed to be known (forward problem). The connections between the field quantities and the connected circuits as well as the coupled problems are discussed to the end of this part.
The above mentioned FDM and FEM solutions are applied in the second part of the seminar to the evaluation of material parameters of the PDE’s implementing them as a part of the loop of different iterative processes. As the initial values are chosen either some measured data or starting data. The numerical methods utilizing PDE are used for the solution of the optimization problems (finding optimal dimensions or materiel characteristics) and inverse problems (different variants of a tomography known as the Electrical Impedance Tomography, the NMR tomography, the Ultrasound tomography), photonics, nanoelectronics, biophotonics, plasma etc.. Each topic is illustrated by practical examples in the ANSYS, HFSS and MATLAB environment.

Language of instruction

Czech

Number of ECTS credits

Mode of study

Not applicable.

Guarantor

prof. Ing. Pavel Fiala, Ph.D.

Department

Department of Theoretical and Experimental Electrical Engineering (UTEE)

Learning outcomes of the course unit

Student acquires theoretical knowledge as well as skills in practical application of the FEM and FDM, processing the input of the numerical models and their application together with the ability to solve forward, inverse, macroscopic, microscopic, stochastic tasks independently.

Prerequisites

Mathematical calculus, Physics, Electromagnetism on the level of MSc, numerical modeling.

Co-requisites

Not applicable.

Planned learning activities and teaching methods

Teaching methods depend on the type of course unit as specified in the article 7 of BUT Rules for Studies and Examinations. Teaching methods include lectures combined with seminars. Course is taking advantage of e-learning.

Assesment methods and criteria linked to learning outcomes

Total number of points 100. Discussion on the topic of the study.

Course curriculum

Introduction to Functional Analysis, Differential Operators, Overview of Partial Differential Equations, Boundary and Initial Conditions.
Finite Difference Method (MKD). Finite Element Method (FEM) - Introduction. Discretization of the area to the finite elements. Approximation of fields from
the nodal or edge values.
Forward problem. Setup of equations for nodal and edge values by the Galerkin method.
Application of Galerkin's method to static and quasi-static fields (Poisson and Helmholtz equations).
The coupling of MKP and MKD for time-domain variables (diffusion and wave equations). Coupling field equations with circuits described with concentrated parameters, non-stationary time and frequency domains.
Coupled problems, models with respect the theory of relativity, stochastic models.
Field optimization tasks. Overview of deterministic methods. Local and global optimum.
Unconstrained problems – gradient method, method of the steepest descent, Newton’s methods, stochastic models, magnetohydrodynamics, and relativistic approach to model description.
Stochastic modeling in conjunction with FEM, microscopic approach to FEM application, NANO-geometry, models, effects, phenomena.
Inverse tasks for elliptical equations. The smallest square method. Deterministic regularization methods, Survey of Layer Set Methods for inverse tasks and
optimal design.
Using inverse tasks in tomography.
Methods and models of modeling of atomic and subatomic levels, nanoelectronics, periodic structures, structural modeling, photonics, biophotonics.
Note: Practical examples using the ANSYS, HFSS and MATLAB environment will be a part of each point of the curriculum.

Work placements

Not applicable.

Aims

To understand the fundamentals of the PDE numerical solution for application in electrical engineering.
Get acquainted with new applications using FEM and FDM in optimization and inverse tasks.

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

The content and forms of instruction in the evaluated course are specified by the lecturer responsible for the course. It is advisable for the student to have a written assignment processed.

Recommended optional programme components

Numerical modeling courses using ANSYS Classic, ANSYS Maxwell, ANSYS HFSS.

Prerequisites and corequisites

Not applicable.

Basic literature

Bossavit Alain.: Computational Electromagnetism – Variational formulations, complementarity, edge elements. Academic Press, 1998 (EN)
Dědek, L., Dědková J.: Elektromagnetismus. Skripta VUTIUM Brno, 2000 (CS)
Chari, M, V. K., Salon S. J.: Numerical Methods in Electromagnetism. Academic Press, 2000 (EN)
Rektorys Karel: Přehled užité matematiky I, II. Prometheus, 1995 (CS)
Sadiku Mathew: Electromagnetics (second edition), CRC Press, 2001 (EN)

Type of course unit

Seminar

39 hod., optionally

Teacher / Lecturer

prof. Ing. Pavel Fiala, Ph.D.

Syllabus

Introduction to the functional analysis, differential operators, survey of the partial differential equations. Boundary and initial conditions. Finite difference methods (FDM).
Finite element methods (FEM). – introduction. Discretization of a region into the finite elements. Approximation of the field from the nodal or edge values.
Forward problem. Setup of equations for nodal and edge values by the Galerkin method.
Application of the Galerkin method to the static and quasistatic fields (Poisson’s and Helmholtz’s equation).
Application of FEM and FDM on the time variable problems (the diffusion and wave equation).
Connection of the field region with the lumped parameter circuit. Coupled problems.
The field optimization problem. Survey of the deterministic methods. The local and global minima.
Unconstrained problems – gradient method, method of the steepest descent, Newton’s methods.
Constrained optimization problems together with FEM.
Inverse problems for the elliptic equations. The Least Square method. Deterministic regularization methods.
A survey on level set methods for inverse problems and optimal design.
A survey on inverse problems in tomography.
A note: Practical examples using the ANSYS and MATLAB environment will be a part of each point of the curriculum.

VUT

Faculties

University Institutes

Parts

Numerical Computations with Partial Differential Equations

Type of course unit