Hands-on Tutorial on Boundary Element Methods and Calderon Preconditioning
Kristof Cools
University of Nottingham, United Kingdom
Kristof Cools received the MEng degree in applied physics and the Ph.D. degree from Ghent University, Ghent, Belgium, in 2004 and 2008, respectively. He is currently an Associate Professor with the George Green Institute for Electromagnetics Research, University of Nottingham, Nottingham, U.K. His current research interests include the spectral properties of the boundary integral operators of electromagnetics, stable and accurate discretization schemes for frequency and time domain boundary element methods, domain decomposition techniques, and the implementations of algorithms from computational physics for highperformance computing. Dr. Cools was a recipient of the Young Scientist Best Paper Award at the International Conference on Electromagnetics and Advanced Applications in 2008.
Francesco P. Andriulli
Politecnico di Torino, Italy
Francesco P. Andriulli (S’05–M’09–SM’11) received the Laurea degree in electrical engineering from Politecnico di Torino, Turin, Italy, in 2004, the M.Sc. degree in electrical engineering and computer science from the University of Illinois at Chicago, Chicago, IL, USA, in 2004, and the Ph.D. degree in electrical engineering from the University of
Michigan at Ann Arbor, Ann Arbor, MI, USA, in 2008. From 2008 to 2010, he was a Research Associate with Politecnico di Torino. Since 2010, he has been with the École Nationale Supérieure des Télécommunications de Bretagne (TELECOM Bretagne), Brest, France, where he is currently a Full Professor. His research interests include computational electromagnetics with focus on frequency- and time-domain integral equation solvers, well-conditioned formulations, fast solvers, low-frequency electromagnetic analysis, and simulation techniques for antennas, wireless components, microwave circuits, and biomedical applications.
Abstract
Boundary Element Methods have proven to be a highly flexible, efficient, and accurate modelling methodology. When the device under study is built from piecewise homogeneous constituents, and when the designer is especially interested in the dispersion and radiation characteristics of the system, it is preferred over alternatives such as the finite difference method and the finite element method.
Unfortunately the matrix system resulting upon discretisation of the relevant integral equations as prescribed by the boundary element often cannot be solved within the time and using the computational resources available to the designer. The reason is the complex distribution of the system’s eigenvalues and in particular the high condition number.
In this workshop we will demonstrate when these problems occur and how the practitioner can identify them. You will learn how powerful Calderon type preconditioners can be designed by constructing primal and dual finite element spaces and defining appropriate duality forms. Through both simple and complex examples you will have the opportunity to measure the impact of these preconditioners on the condition number of the system matrix, and more important on the number of iterations and time required for the simulation of the device to be completed.
Course Outline
The instructors will use a combination of slides and interactive exercises. All aspects of the subject matter will be demonstrated in the open source package BEAST.jl: Boundary Element Analysis and Simulation Toolkit. BEAST.jl can be used on a cloud service or can be installed on the participants’ laptops. The (very short!) instructions on how to do this for all common operating systems will be communicated to the participants closer to the date of the workshop.
Contents:
- The Boundary Element Method: from integral equation to linear system (20 min.)
- Krylov Solvers, the condition number, and solution time (20 min)
- Calderon preconditioning: Integral Equation Identities and Discretisation (20 min)
- The Buffa-Christiansen elements: definition and construction (30 min)
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- Calderon Preconditioners for integral equations in Electromagnetics (30 min)
- Time Domain Equations: large time step breakdown and DC instabilities (30 min)
- Concluding Remarks, Questions, and Discussion (30 min)
