Course detail

# Numerical Computations with Partial Differential Equations

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). Each topic is illustrated by practical examples in the ANSYS and MATLAB environment.

Learning outcomes of the course unit

To acquire theoretical knowledge as well as practical application of the FEM and FDM together with the ability to program corresponding forward and inverse problems.

Prerequisites

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

Co-requisites

Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Fotokopie nejnovějších výzkumných zpráv ze serverů zahraničních špičkových pracovišť ke studované problematice budou dodávány průběžně (EN)
Rektorys Karel: Přehled užité matematiky I, II. Prometheus, 1995 (CS)
IEEE Transactions on Magnetics, ročník 1996 a výše (EN)
Dědek, L., Dědková J.: Elektromagnetismus. Skripta VUTIUM Brno, 2000 (CS)
Inverse Problems. IoP Electronic Journals, http://www.iop.org/EJ/journal/IP http://www.inverse-problems.com/ (EN)
SIAM Journal on Control and Optimization, ročník 1996 a výše (EN)
metoda hladinových množin http://www.math.ucla.edu/applied/cam/index.html (EN)
Bossavit Alain.: Computational Electromagnetism – Variational formulations, complementarity, edge elements. Academic Press, 1998 (EN)
Inverse Problems. IoP Electronic Journals, http://www.iop.org/EJ/journal/IP (EN)
Sadiku Mathew: Electromagnetics (second edition), CRC Press, 2001 (EN)
Chari, M, V. K., Salon S. J.: Numerical Methods in Electromagnetism. Academic Press, 2000 (EN)

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 (Moodle) system.

Assesment methods and criteria linked to learning outcomes

Total number of points 100.

Language of instruction

Czech

Work placements

Not applicable.

Course curriculum

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.

Aims

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.

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.

Classification of course in study plans

• Programme EKT-PK Doctoral

branch PK-BEB , 1. year of study, summer semester, 4 credits, optional specialized

• Programme EKT-PP Doctoral

branch PP-BEB , 1. year of study, summer semester, 4 credits, optional specialized
branch PP-KAM , 1. year of study, summer semester, 4 credits, optional specialized

• Programme EKT-PK Doctoral

branch PK-KAM , 1. year of study, summer semester, 4 credits, optional specialized
branch PK-EST , 1. year of study, summer semester, 4 credits, optional specialized

• Programme EKT-PP Doctoral

branch PP-EST , 1. year of study, summer semester, 4 credits, optional specialized
branch PP-MVE , 1. year of study, summer semester, 4 credits, optional specialized

• Programme EKT-PK Doctoral

branch PK-MVE , 1. year of study, summer semester, 4 credits, optional specialized

• Programme EKT-PP Doctoral

branch PP-MET , 1. year of study, summer semester, 4 credits, optional specialized

• Programme EKT-PK Doctoral

branch PK-MET , 1. year of study, summer semester, 4 credits, optional specialized

• Programme EKT-PP Doctoral

branch PP-FEN , 1. year of study, summer semester, 4 credits, optional specialized

• Programme EKT-PK Doctoral

branch PK-FEN , 1. year of study, summer semester, 4 credits, optional specialized
branch PK-SEE , 1. year of study, summer semester, 4 credits, optional specialized

• Programme EKT-PP Doctoral

branch PP-SEE , 1. year of study, summer semester, 4 credits, optional specialized

• Programme EKT-PK Doctoral

branch PK-TLI , 1. year of study, summer semester, 4 credits, optional specialized

• Programme EKT-PP Doctoral

branch PP-TLI , 1. year of study, summer semester, 4 credits, optional specialized

• Programme EKT-PK Doctoral

branch PK-TEE , 1. year of study, summer semester, 4 credits, optional specialized

• Programme EKT-PP Doctoral

branch PP-TEE , 1. year of study, summer semester, 4 credits, optional specialized

#### Type of course unit

Seminar

39 hours, optionally

Teacher / Lecturer