Course detail
Numerical Computations with Partial Differential Equations
FEKT-DTE2Acad. year: 2016/2017
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.
Language of instruction
Number of ECTS credits
Mode of study
Guarantor
Learning outcomes of the course unit
Prerequisites
Co-requisites
Planned learning activities and teaching methods
Assesment methods and criteria linked to learning outcomes
Course curriculum
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.
Work placements
Aims
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
Recommended optional programme components
Prerequisites and corequisites
Basic literature
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)
Recommended reading
SIAM Journal on Control and Optimization, ročník 1996 a výše (EN)
Classification of course in study plans
- Programme EKT-PK Doctoral
branch PK-TEE , 1 year of study, summer semester, elective specialised
branch PK-MET , 1 year of study, summer semester, elective specialised
branch PK-FEN , 1 year of study, summer semester, elective specialised
branch PK-SEE , 1 year of study, summer semester, elective specialised
branch PK-KAM , 1 year of study, summer semester, elective specialised
branch PP-BEB , 1 year of study, summer semester, elective specialised
branch PK-MVE , 1 year of study, summer semester, elective specialised
branch PK-EST , 1 year of study, summer semester, elective specialised
branch PK-TLI , 1 year of study, summer semester, elective specialised - Programme EKT-PP Doctoral
branch PP-TLI , 1 year of study, summer semester, elective specialised
branch DP-TEE , 1 year of study, summer semester, elective specialised
branch PP-SEE , 1 year of study, summer semester, elective specialised
branch PP-KAM , 1 year of study, summer semester, elective specialised
branch PP-BEB , 1 year of study, summer semester, elective specialised
branch PP-MVE , 1 year of study, summer semester, elective specialised
branch PP-EST , 1 year of study, summer semester, elective specialised
branch PP-FEN , 1 year of study, summer semester, elective specialised
branch PP-MET , 1 year of study, summer semester, elective specialised
Type of course unit
Seminar
Teacher / Lecturer
Syllabus
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.