Sliding Interfaces for Eddy Current Simulations Raffael Casagrande - PDF document

Sliding Interfaces for Eddy Current Simulations Raffael Casagrande Master Thesis Supervisor: Prof. Dr. Ralf Hiptmair Zürich, April 2013

Contents Contents i 1. Introduction 1 2. Eddy Current Problem 3 2.1. Coulomb gauged formulation . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2. Temporal gauged formulation . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3. H -formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.4. The eddy current problem in a moving, solid body . . . . . . . . . . . . . 9 3. Discontinuous Galerkin formulation 11 3.1. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.1.1. Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.1.2. Broken spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.1.3. Discrete Friedrichs inequality . . . . . . . . . . . . . . . . . . . . . 17 3.1.4. Trace inequalities and polynomial approximation . . . . . . . . . . 17 3.2. DG formulation of Eddy Current problem . . . . . . . . . . . . . . . . . . 18 3.2.1. Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4. Aspects of the Implementation 32 4.1. Reduction to 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

CONTENTS 4.2. Details of the imlementation. . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.2.1. The mesh data structure . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3. Variational formulation for moving systems . . . . . . . . . . . . . . . . . 40 5. Results 45 5.1. Convergence to analytical solution . . . . . . . . . . . . . . . . . . . . . . 45 5.2. Translational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 5.3. Rotational motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6. Conclusion and Outlook 54 A. Maxwell’s equations in a moving frame of reference 55 B. Documentation of the code 60 C. Acknowledgements 61 Bibliography 62 ii

1. Introduction We present a discontinuous Galerkin (DG) approach for solving the transient eddy cur- rent problem in the presence of moving, rigid bodies which share common interfaces. Such problems are important in the modelling of electromechanical devices. In partic- ular the induced currents can have positive or negative effects and are often essential for the functioning of electric engines. Consider for example the setting depicted in Fig. 1.1. Here multiple bodies are moving with respect to each other and sliding interfaces exist. We attach to each body a separate coordinate frame and mesh so we can solve the equations in Lagrangian variables. Figure 1.1.: An example configuration with 3 solid bodies. Separate coordinate systems are attached to each one. In order to deal with the non-matching grids we use symmetric-interior penalty tech- niques at the interfaces. We use conforming finite element spaces in the interior of each body. Together with the appropriate transformation formulas we then link the different descriptions into one consistent framework. This approach doesn’t need any remeshing and allows us to model the problem from a Lagrangian perspective. In particular no convective terms appear in the formulation and the method is therefore expected to be stable for bodies moving at high speed. In time, we use an implicit Euler discretization which results in a sparse, symmetric- positive-definite linear system that has to be solved in every iteration. The proposed discretization is analyzed in a general 3 dimensional DG framework but only 2 dimen- sional model problems have been implemented and tested. In space, the problem is discretized by means of first order Lagrangian elements as well as first order Nedelec

1. Introduction elements of the first kind. Another commonly used approach to solve the same physical problem is based on mortar- methods (see for instance [14],[15], [4],[8]). These methods have typically symmetric- positive definite matrices as well and need no remeshing. However the implementation is harder because additional degrees of freedom are introduced along the interface. This is especially true for translational settings where the interface itself is changing over time. There exist many other approaches which rely on conforming meshes, but they almost always need a partial or full remeshing in every time step which increases the computational cost heavily (cf. [14, Introduction]). More recently Ferrer [7],[1] has applied the DG theory to the Navier-Stokes equations for treating sliding interfaces. The material is presented in five chapters: We start off in chapter 2 by giving a brief introduction to the eddy current model and its possible formulations. The chapter ends with a discussion of the Galilean transformation properties of the previously presented eddy current formulations. Chapter 3 starts by introducing mathematical tools from the DG world and continues with a presentation of the DG variational formulation. The latter is then analyzed and a proof of convergence is given. Following this rather theo- retical material we present some implementation details in chapter 4. Finally, chapter 5 presents and discusses all the numerical results. The thesis ends with a short summary and some concluding remarks (chapter 6). 2

2. Eddy Current Problem In this chapter we will derive three different formulations of the transient eddy current problem by neglecting the displacement current in Maxwell’s equations. By requiring a unique solution we will derive appropriate initial and boundary conditions for the given problems. In the last part of this chapter we will show that the three formulations are invariant under rotation and translation if the quantities are transformed correctly. It should be mentioned that there are many other ways of treating the eddy current problem. In this short presentation, we focus our attention on the Coulomb gauged potential formulation (unknowns: A , ϕ ), the temporal gauged potential formulation (unknown: A ) and the H formulation (unknown: the magnetic field H ). We refer the reader to Albanese and Rubinacci [2] for a survey of different formulations. The eddy current problem is a simplified version of Maxwell’s Maxwell’s equations equations, Magnetic Gauss’s law: div B = 0 (2.1a) curl E + ∂ B Faraday’s law of induction: ∂t = 0 (2.1b) Electric Gauss’s law: div D = ρ (2.1c) curl H − ∂ D ∂t = j f + j i . Ampère’s law: (2.1d) Here B denotes the magnetic induction ([ Vs / m 2 ]), E the electric field ([ V / m]), D the electric induction ([ N / Vm]), ρ the free charge density ([ C / m 3 ]), H the magnetic field ([ A / m]), j f the free current density ([ A / m 2 ]) and j i the impressed current density ([ A / m 2 ]). The latter is a prescribed current density which can be used to model coils in the computational domain. We consider only linear, isotropic materials with immediate response in this thesis and therefore the constitutive relations j f = σ E B = µ H D = ε E

2. Eddy Current Problem apply. ε , µ and σ are all material dependent properties and are termed permittivity, permeability and conductivity, respectively. They are simple scalars which are only a function of the material. In the following we are going to derive the eddy current problem. For this we need to make the central Assumption 2.1. The displacement current, ∂ D ∂t in Ampères law (2.1d) is zero. This is justified if the electric field is varying only slowly ( Quasistatic simplification ). This assumption is typically justified for very high conductivities respectively if the electric field E is varying only slowly. 2.1. Coulomb gauged formulation Remark. In this chapter we assume that A ∈ [ C ∞ (Ω)] 3 and ϕ ∈ C ∞ (Ω) to simplify the derivations and manipulations. This assumption is relaxed later when we consider the variational formulation in chapter 3. In addition we assume that the domain Ω is simply connected. By virtue of the Helmholtz decomposition (cf. [18]) we can express B and E through the vector potential A ∗ and the scalar potential ϕ ∗ : B = curl A ∗ , (2.2a) E = − grad ϕ ∗ − ∂ A ∗ ∂t . (2.2b) This way the first two Maxwell equations (2.1a),(2.1b) are fulfilled automatically and only Ampères law remains. Expressed in the variables A ∗ and ϕ ∗ it reads as σ∂ A ∗ + curl 1 µ curl A ∗ + σ grad ϕ ∗ = j i . (2.3) ∂t It is well known that this equation doesn’t possess a unique solution. Indeed, we can use for example the gauge transformation ϕ ′∗ = ϕ ∗ − ∂ψ A ′∗ = A ∗ + grad ψ, ∂t , to transform A and ϕ without affecting the observable quantities B and E ( ψ ( x , t ) is an arbitrary function). In practice the Coulomb gauge condition, div A ∗ = 0 (2.4) is often introduced to fix this ambiguity. Together with appropriate boundary conditions this fixes the value of A and ϕ and leads to a well posed problem. We will refer to the system of equations represented by Eq. 2.3 together with Eq. 2.4 as the Coulomb gauged eddy current problem . 4