Tampereen teknillinen yliopisto. Julkaisu 1209 Tampere University of Technology. Publication 1209 Matti Pellikka Finite Element Method for Electromagnetics on Riemannian Manifolds Topology and Differential Geometry Toolkit Thesis for the degree of Doctor of Science in Technology to be presented with due permission for public examination and criticism in Rakennustalo Building, Auditorium RG202, at Tampere University of Technology, on the 2nd of May 2014, at 12 noon. Tampereen teknillinen yliopisto - Tampere University of Technology Tampere 2014 ISBN 978-952-15-3279-5 (printed) ISBN 978-952-15-3285-6 (PDF) ISSN 1459-2045 Abstract This thesis applies new branches of mathematics in computational electromagnetics soft- ware. Namely, we consider the application of algebraic topology and differential geometry in finite element modeling. We conclude that from this approach, one can draw benefits to practical electromagnetic modeling. For example, more efficient numerical formulations, field-circuit coupling, and metric and coordinate free modeling techniques. We present efficient methods for homology and cohomology computation of finite element meshes together with their software implementation. The presented homology and cohomology solver is a part of finite element mesh generator Gmsh. Therefore, its use can be easily incorporated into finite element modeling workflow. We demonstrate the use of homology and cohomology computation results in static and quasistatic electromagnetic field problems. We describe finite element formulations which can be used in lumped parameter extraction from field problems and which can be naturally coupled to electronic circuit problems. Importantly, cohomology computa- tion enables the use of magnetic scalar potential in eddy current problems without any topological restrictions, leading to more efficient and robust field computations. Lastly, we present a finite element programming environment, where the language of differential geometry has the main role. We interpret the finite element model as a Rie- mannian manifold, and the fields of interest as differential forms. Using the environment, one can give the computational instructions in metric and coordinate free manner, as the used metric and coordinate system are provided separately. Then, the environment trans- lates the instructions to the actual floating-point operations, which ultimately depend on the used metric and coordinate system. The programming environment implementation builds on top of the Gmsh API. That is, we implement tools from differential geometry which utilize an existing finite element framework. The main contribution of this thesis is the development of these tools to the point where they can be readily expoited in computationally demanding engineering problems. Also, this thesis offers a unified exposition of the needed mathematical concepts and their relation to the electromagnetic field problem formulations. Preface I have made this thesis in the inspirational environment of the electromagnetics research group in the Tampere University of Technology. The thesis topic builds on top of the long-term basic research conducted in our research group, to bring some of its results closer to engineering practice. I have done this work under the supervision of Professor Lauri Kettunen, whose en- thusiasm, sincerity, and farsightedness reflects in the atmosphere of the whole research group, creating a close-knit research community. In addition to Lauri Kettunen, I have been privileged to be guided by two other members of our research group, whose ad- vice and perspective complement each other: University Lecturer Saku Suuriniemi and Senior Research Fellow Timo Tarhasaari. Much of my progress is also due to Professor Christophe Geuzaine from the University of Liège, the main developer of the finite el- ement library Gmsh. In addition to his encouragement, he enabled and supported the implementation of the methods of this thesis to Gmsh. I would also like to than Lasse Söderlund and Maija-Liisa Paasonen for taking care of administrative tasks, as well as Juha Tampio, Antti Stenvall, Valtteri Lahtinen, Erkki Härö, Arto Poutala, and Olli Pekkola for giving feedback to some of the software imple- mentations I have done for this thesis. Other people I’d like to thank for more or less casual interaction during my thesis work include Jukka-Pekka Uusitalo, Teemu Rovio, Pasi Raumonen, Janne Keränen, Tuukka Nieminen, Arttu Rasku, Aki Korpela, Risto Mikkonen, Stefan Kurz, and other present and past personnel of our research group who are responsible for its friendly atmosphere. Lastly, I thank my dear wife Kiti for taking care of the fundamentals on the home front, as well as my son Eevertti for demanding my undivided attention on regular basis; and my daughter Muusa for inspiring me while finalizing this thesis. 1 Contents List of symbols 5 1 Introduction 10 1.1 Background, motivation, and usefulness of the research .......... 12 1.1.1 Homology and cohomology computation .............. 12 1.1.2 Differential geometry and Riemannian manifolds .......... 13 1.2 Survey of recent research ........................... 14 1.3 Original contributions ............................ 14 1.3.1 Development of reduction techniques for homology and cohomology computation .............................. 15 1.3.2 Implementation of homology and cohomology solver ........ 15 1.3.3 Cohomology based formulations of the electromagnetic boundary value problems ............................ 15 1.3.4 Implementation of Riemannian manifold programming interface . 16 1.4 Organization ................................. 16 2 Mathematical concepts 17 2.1 Algebraic structures ............................. 18 2.1.1 Abelian group ............................. 18 2.1.2 Homological algebra ......................... 21 2.1.3 Vector space .............................. 25 2.1.4 Exterior algebra ............................ 27 2.2 Manifold and its cell decomposition ..................... 29 2.2.1 Real coordinate space ......................... 30 2.2.2 Euclidean space ............................ 30 2.2.3 Manifolds ............................... 31 2.2.4 Cell complex of a manifold ...................... 34 2.2.5 Finite elements ............................ 37 2.3 Homology and cohomology of a manifold .................. 38 2.3.1 Chain complexes of a manifold .................... 38 2.3.2 Cochain complexes of a manifold .................. 40 2.3.3 Homology and cohomology of a manifold .............. 40 2.4 Differential forms ............................... 43 2.4.1 The basic construction ........................ 43 2.4.2 Integration .............................. 45 2.4.3 de Rham cohomology ......................... 46 2 2.4.4 Harmonic differential forms ..................... 47 2.4.5 Whitney forms ............................ 48 3 Homology and cohomology computation of finite element meshes 51 3.1 Construction of chain complexes from a finite element mesh ....... 52 3.1.1 Data structures and construction of the cell complex ....... 52 3.1.2 Construction of chain complexes ................... 53 3.2 Reduction of chain complexes ........................ 54 3.2.1 Reduction algorithms ......................... 54 3.2.2 Reduction pair ............................ 55 3.2.3 Homology reduction algorithms ................... 57 3.2.4 Cohomology reduction algorithms .................. 61 3.3 Computation of homology and cohomology ................. 62 3.3.1 Smith normal form .......................... 64 3.3.2 Kernel-image problem ........................ 64 3.3.3 Quotient problem ........................... 65 3.3.4 The homology and cohomology computation algorithm ...... 67 3.4 The homology and cohomology solver in Gmsh ............... 67 3.4.1 Homology computation routine ................... 68 3.4.2 Cohomology computation routine .................. 68 3.5 Post-processing of homology and cohomology ................ 69 3.5.1 Basis element representative selection ................ 70 3.5.2 Basis selection ............................. 70 3.5.3 Computation of harmonic representatives .............. 73 3.6 Examples ................................... 74 3.6.1 Example: Solid cube ......................... 74 3.6.2 Example: Closed surfaces ...................... 75 3.6.3 Example: Torus knots ........................ 76 4 Application of cohomology in the finite element method for electromag- netics 79 4.1 Electromagnetic modeling .......................... 80 4.2 Static problems ................................ 81 4.2.1 Electrokinetic problem ........................ 81 4.2.2 Cohomology basis functions ..................... 83 4.2.3 Electric field -conforming formulation ................ 85 4.2.4 Current density -conforming formulation .............. 86 4.2.5 Electrostatic problem ......................... 88 4.2.6 Magnetostatic problem ........................ 89 4.3 Circuit coupled eddy current problem .................... 90 4.3.1 Eddy current problem ........................ 90 4.3.2 Faraday’s law conforming formulation ................ 91 4.3.3 Ampere’s law conforming formulation ................ 92 4.4 Examples ................................... 93 4.4.1 Induced EMF in squirrel cage rotor ................. 93 4.4.2 Mutual inductance magnetostatic problem ............. 94 3 4.4.3 Induction heating eddy current problem .............. 97 5 Finite element imitation of the Riemannian manifold 103 5.1 Motivation ................................... 103 5.2 Imitation of the manifold ........................... 104 5.2.1 Differentials