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), material models macro, micro and nanoscopic; photonics, nanoelectronics, biophotonics, plasma etc. Each topic is illustrated by practical examples in the ANSYS and MATLAB environment.

Language of instruction

English

Number of ECTS credits

4

Mode of study

Not applicable.

Výsledky učení předmětu

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.

Prerekvizity

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

Korekvizity

Not applicable.

Plánované vzdělávací činnosti a výukové metody

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.

Způsob a kritéria hodnocení

Total number of points 100.

Osnovy výuky

1. Introduction to the functional analysis, differential operators, survey of the partial differential equations, boundary and initial conditions.
2. Finite difference methods (FDM). Finite element methods (FEM). – introduction. Discretization of a region into the finite elements. Approximation of fields from node or edge values.
3. Forward problem. Setup of equations for nodal and edge values by the Galerkin method.
4. Application of the Galerkin method to the static and quasistatic fields (Poisson’s and Helmholtz’s equation).
5. 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, non-stationary time and frequency domains.
6.-7. Coupled problems, models with respect to theory of relativity, stochastic models.
8. The field optimization problem. Survey of the deterministic methods. The local and global minima.
9. Unconstrained problems – gradient method, method of the steepest descent, Newton’s methods, stochastic models, magnetohydrodynamics and relativistic approach to model description.
10. Stochastic modeling together with FEM, microscopic approach to FEM application, nanometric geometry, models, effects, phenomena.
11. 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.
12. A survey on inverse problems in tomography.
13. Methods and models of modeling of an atomic and subatomic levels, nanoelectronics, periodic structures, structural modeling, photonics, biophotonics.

Pracovní stáže

Not applicable.

Učební cíle

To understand the fundamentals of the PDR numerical solution for application in electrical engineering.
Get acquainted with new applications using MKP and MKD in optimization and inverse tasks.

Vymezení kontrolované výuky a způsob jejího provádění a formy nahrazování zameškané výuky

The content and forms of instruction in the evaluated course are specified by the lecturer responsible for the course.

Doporučené volitelné složky programu

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

J.A.Stratton, Electromagnetic Theory, McGraw-Hill Book Company, New York and London, 1941, https://archive.org/details/electromagnetict031016mbp/page/n637 (EN)
Sadiku, M.: Electromagnetics (second edition), CRC Press, 2001 (EN)

SIAM Journal on Control and Optimization, ročník 2013 a výše (EN)
IEEE Transactions on Magnetics, ročník 2012 a výše (EN)

Classification of course in study plans

• Programme DPA-KAM Doctoral, any year of study, summer semester, compulsory-optional
• Programme DPA-EKT Doctoral, any year of study, summer semester, compulsory-optional
• Programme DPA-MET Doctoral, any year of study, summer semester, compulsory-optional
• Programme DPA-SEE Doctoral, any year of study, summer semester, compulsory-optional
• Programme DPA-TLI Doctoral, any year of study, summer semester, compulsory-optional
• Programme DPA-TEE Doctoral, any year of study, summer semester, compulsory

#### Type of course unit

Seminar

39 hours, optionally

Teacher / Lecturer

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.