1 Strong and Weak Formulations of Electromagnetic Problems Patrick Dular, University of Liège - FNRS, Belgium

2 Content  Formulations of electromagnetic problems ♦Maxwell equations, material relations ♦Electrostatics, electrokinetics, magnetostatics, magnetodynamics ♦Strong and weak formulations v Discretization of electromagnetic problems ♦Finite elements, mesh, constraints ♦Weak finite element formulations

3 Formulations of Electromagnetic Problems Maxwell equations Electrostatics Electrokinetics Magnetostatics Magnetodynamics

4 Electromagnetic models v Electrostatics ♦Distribution of electric field due to static charges and/or levels of electric potential v Electrokinetics ♦Distribution of static electric current in conductors v Electrodynamics ♦Distribution of electric field and electric current in materials (insulating and conducting) v Magnetostatics ♦Distribution of static magnetic field due to magnets and continuous currents v Magnetodynamics ♦Distribution of magnetic field and eddy current due to moving magnets and time variable currents v Wave propagation ♦Propagation of electromagnetic fields All phenomena are described by Maxwell equations

5 Maxwell equations curl h = j +  t d curl e = –  t b div b = 0 div d =  v Maxwell equations Ampère equation Faraday equation Conservation equations Principles of electromagnetism hmagnetic field (A/m)eelectric field (V/m) bmagnetic flux density (T)delectric flux density (C/m 2 ) jcurrent density (A/m 2 )  v charge density (C/m 3 ) Physical fields and sources

6 Constants (linear relations) Functions of the fields (nonlinear materials) Tensors (anisotropic materials) Material constitutive relations b =  h (+ b s ) d =  e (+ d s ) j =  e (+ j s ) Constitutive relations Magnetic relation Dielectric relation Ohm law b s remnant induction,... d s... j s source current in stranded inductor,... Possible sources  magnetic permeability (H/m)  dielectric permittivity (F/m)  electric conductivity (  –1 m –1 ) Characteristics of materials

7 Formulation for the exterior region  0 the dielectric regions  d,j In each conducting region  c,i : v = v i  v = v i on  c,i eelectric field (V/m) delectric flux density (C/m 2 )  electric charge density (C/m 3 )  dielectric permittivity (F/m) Electrostatics Type of electrostatic structure  0 Exterior region  c,i Conductors  d,j Dielectric div  grad v = –  with e = – grad v Electric scalar potential formulation curl e = 0 div d =  d =  e Basis equations n  e |  0e = 0 n  d |  0d = 0 & boundary conditions

8 Electrostatic

9 Formulation for the conducting region  c On each electrode  0e,i : v = v i  v = v i on  0e,i eelectric field (V/m) jelectric current density (C/m 2 )  electric conductivity (  –1 m –1 ) Electrokinetics Type of electrokinetic structure  c Conducting region div  grad v = 0 with e = – grad v Electric scalar potential formulation curl e = 0 div j = 0 j =  e Basis equations n  e |  0e = 0 n  j |  0j = 0 & boundary conditions  0e,0  0e,1  0j e=?, j=? cc V = v 1 – v 0

10 curl h = j div b = 0 Magnetostatics Equations b =  h + b s j = j s Constitutive relations Type of studied configuration Ampère equation Magnetic conservation equation Magnetic relation Ohm law & source current  Studied domain  m Magnetic domain  s Inductor

11 curl h = j curl e = –  t b div b = 0 Magnetodynamics Equations b =  h + b s j =  e + j s Constitutive relations Type of studied configuration Ampère equation Faraday equation Magnetic conservation equation Magnetic relation Ohm law & source current  Studied domain  p Passive conductor and/or magnetic domain  a Active conductor  s Inductor

12 Magnetodynamics Inductor (portion : 1/8th) Stranded inductor - uniform current density (j s ) Massive inductor - non-uniform current density (j)

13 Magnetodynamics - Joule losses Foil winding inductance - current density (in a cross-section) All foils With air gaps, Frequency f = 50 Hz

14 Transverse induction heating (nonlinear physical characteristics, moving plate, global quantities) Eddy current density Temperature distribution Search for OPTIMIZATION of temperature profile Magnetodynamics - Joule losses

15 Magnetodynamics - Forces

16 Magnetic field lines and electromagnetic force (N/m) (8 groups, total current 3200 A) Currents in each of the 8 groups in parallel non-uniformly distributed! Magnetodynamics - Forces

17 Inductive and capacitive effects v Frequency and time domain analyses v Any conformity level Magnetic flux densityElectric field Resistance, inductance and capacitance versus frequency

18 dom (grad e )= F e 0 = { v  L 2 (  ) ; grad v  L 2 (  ),v   e = 0 } dom (curl e )= F e 1 = { a  L 2 (  ) ; curl a  L 2 (  ),n  a   e = 0 } dom (div e )= F e 2 = { b  L 2 (  ) ; div b  L 2 (  ),n. b   e = 0 } Boundary conditions on  e dom (grad h )= F h 0 = {   L 2 (  ) ; grad   L 2 (  ),   h = 0 } dom (curl h )= F h 1 = { h  L 2 (  ) ; curl h  L 2 (  ),n  h   h = 0 } dom (div h )= F h 2 = { j  L 2 (  ) ; div j  L 2 (  ),n. j   h = 0 } Boundary conditions on  h Function spaces F e 0  L 2, F e 1  L 2, F e 2  L 2, F e 3  L 2 Function spaces F h 0  L 2, F h 1  L 2, F h 2  L 2, F h 3  L 2 Basis structure Continuous mathematical structure grad h F h 0  F h 1, curl h F h 1  F h 2, div h F h 2  F h 3 F h 0 grad h  F h 1 curl h  F h 2 div h  F h 3 Sequence grad h F e 0  F e 1, curl e F e 1  F e 2, div e F e 2  F e 3 F e 3 div e  F e 2 curl e  F e 1 grad e  F e 0 Sequence Domain , Boundary  =  h U  e

19 "e" side"d" side Electrostatic problem Basis equations curl e = 0 div d =  d =  e e = – grad vd = curl u   e =d d  (u) grad curl div e e e e 0 0 (–v) F e 0 F e 1 F e 2 F e 3 div curl grad d d d F d 3 F d 2 F d 1 F d 0 e S 0 S e 1 S e 2 S e 3 S d 3 S d 2 S d 1 S d 0

20 "e" side"j" side Electrokinetic problem Basis equations curl e = 0div j = 0 j =  e e = – grad vj = curl t   e =j d  (t) grad curl div e e e e 0 0 (–v) F e 0 F e 1 F e 2 F e 3 div curl grad j j j F j 3 F j 2 F j 1 F j 0 e S 0 S e 1 S e 2 S e 3 S j 3 S j 2 S j 1 S j 0

21 "h" side"b" side Magnetostatic problem Basis equations curl h = jdiv b = 0 b =  h h = – grad  b = curl a 

22 "h" side"b" side Magnetodynamic problem h = t – grad  b = curl a  curl h = j curl e = –  t b div b = 0 b =  h Basis equations j =  e e = –  t a – grad v

23 Discretization of Electromagnetic Problems Nodal, edge, face and volume finite elements

24 Discrete mathematical structure Continuous function spaces & domain Classical and weak formulations Continuous problem Finite element method Discrete function spaces piecewise defined in a discrete domain (mesh) Discrete problem DiscretizationApproximation Classical & weak formulations  ? Properties of the fields  ? Questions To build a discrete structure as similar as possible as the continuous structure Objective

25 Discrete mathematical structure Sequence of finite element spaces Sequence of function spaces & Mesh Finite element space Function space & Mesh    f i i   f i i         Finite element Interpolation in a geometric element of simple shape + f

26 Finite elements  Finite element (K, P K,  K ) ♦K =domain of space (tetrahedron, hexahedron, prism) ♦P K = function space of finite dimension n K, defined in K ♦  K = set of n K degrees of freedom represented by n K linear functionals  i, 1  i  n K, defined in P K and whose values belong to IR

27 Finite elements v Unisolvance  u  P K, u is uniquely defined by the degrees of freedom v Interpolation v Finite element space Union of finite elements (K j, P Kj,  Kj ) such as :  the union of the K j fill the studied domain (  mesh) m some continuity conditions are satisfied across the element interfaces Basis functions Degrees of freedom u K  i (u)p i i  1 n K 

28 Sequence of finite element spaces Geometric elements Tetrahedral (4 nodes) Hexahedra (8 nodes) Prisms (6 nodes) Mesh Geometric entities Nodes i  N Edges i  E Faces i  F Volumes i  V Sequence of function spaces S0S0 S1S1 S2S2 S3S3

29 Sequence of finite element spaces {s i, i  N} {s i, i  E} {s i, i  F} {s i, i  V} Bases Finite elements S0S0 S1S1 S2S2 S3S3 Volume element Point evaluation Curve integral Surface integral Volume integral Nodal value Circulation along edge Flux across face Volume integral Functions Functionals Degrees of freedom Properties Face element Edge element Nodal element  i,j  E

30 Sequence of finite element spaces Base functions Continuity across element interfaces Codomains of the operators {s i, i  N} value {s i, i  E} tangential component grad S 0  S 1 {s i, i  F} normal component curl S 1  S 2 {s i, i  V} discontinuity div S 2  S 3 Conformity S 0 grad  S 1 curl  S 2 div  S 3 Sequence S0S0 S1S1 grad S 0 S2S2 curl S 1 S3S3 div S 2 S0S0 S1S1 S2S2 S3S3

31 v Electromagnetic fields extend to infinity (unbounded domain) ♦Approximate boundary conditions: m zero fields at finite distance ♦Rigorous boundary conditions: m "infinite" finite elements (geometrical transformations) m boundary elements (FEM-BEM coupling) v Electromagnetic fields are confined (bounded domain) ♦Rigorous boundary conditions Mesh of electromagnetic devices

32 v Electromagnetic fields enter the materials up to a distance depending of physical characteristics and constraints ♦Skin depth  (  >) ♦mesh fine enough near surfaces (material boundaries) ♦use of surface elements when   0 Mesh of electromagnetic devices

33 v Types of elements ♦2D : triangles, quadrangles ♦3D : tetrahedra, hexahedra, prisms, pyramids ♦Coupling of volume and surface elements m boundary conditions m thin plates m interfaces between regions m cuts (for making domains simply connected) ♦Special elements (air gaps between moving pieces,...) Mesh of electromagnetic devices

34 u  weak solution uvvvv(,L * )  (f,)  Q g ()ds    0,  V(  ) Classical and weak formulations u  classical solution L u = fin  B u = gon  =  Classical formulation Partial differential problem Weak formulation (u,v)  u(x)v(x)dx  ,u,v  L 2 (  ) (u,v)  u(x).v(x)dx  ,u,v  L 2 (  ) Notations v  test function Continuous level :    system Discrete level :n  n system  numerical solution

35 Constraints in partial differential problems v Local constraints (on local fields) ♦Boundary conditions m i.e., conditions on local fields on the boundary of the studied domain ♦Interface conditions m e.g., coupling of fields between sub-domains v Global constraints (functional on fields) ♦Flux or circulations of fields to be fixed m e.g., current, voltage, m.m.f., charge, etc. ♦Flux or circulations of fields to be connected m e.g., circuit coupling Weak formulations for finite element models Essential and natural constraints, i.e., strongly and weakly satisfied

36 Constraints in electromagnetic systems v Coupling of scalar potentials with vector fields ♦e.g., in h-  and a-v formulations v Gauge condition on vector potentials ♦e.g., magnetic vector potential a, source magnetic field h s v Coupling between source and reaction fields ♦e.g., source magnetic field h s in the h-  formulation, source electric scalar potential v s in the a-v formulation v Coupling of local and global quantities ♦e.g., currents and voltages in h-  and a-v formulations (massive, stranded and foil inductors) v Interface conditions on thin regions ♦i.e., discontinuities of either tangential or normal components Interest for a “correct” discrete form of these constraints Sequence of finite element spaces

37 Strong and weak formulations curl h = j s div b =  s b =  h Constitutive relation Equations Boundary conditions (BCs) in  h = h s  grad , with curl h s = j s b = b s + curl a, with div b s =  s Scalar potential  Vector potential a Strongly satisfies

38 Strong formulations curl e = 0, div j = 0, j =  e, e =  grad v or j = curl u Electrokinetics curl e = 0, div d =  s, d =  e, e =  grad v or d = d s + curl u Electrostatics curl h = j s, div b = 0, b =  h, h = h s  grad  or b = curl a Magnetostatics curl h = j, curl e = –  t b, div b = 0, b =  h, j =  e + j s,... Magnetodynamics

39 Grad-div weak formulation grad-div Green formula integration in  and divergence theorem grad-div scalar potential  weak formulation

40 Curl-curl weak formulation curl-curl Green formula integration in  and divergence theorem curl-curl vector potential a weak formulation

41 Grad-div weak formulation 1 Use of hierarchal TF  p ' in the weak formulation Error indicator: lack of fulfillment of WF... can be used as a source for a local FE problem (naturally limited to the FE support of each TF) to calculate the higher order correction b p to be given to the actual solution b for satisfying the WF... solution of : or Local FE problems 1

42 A posteriori error estimation (1/2) V Higher order hierarchal correction v p ( 2nd order, BFs and TFs on edges) Electric scalar potential v (1st order) Large local correction  Large error Coarse mesh Fine mesh Electric field Field discontinuity directly Electrokinetic / electrostatic problem

43 Curl-curl weak formulation 2 Use of hierarchal TF a p ' in the weak formulation Error indicator: lack of fulfillment of WF... also used as a source to calculate the higher order correction h p of h... solution of : Local FE problems 2

44 A posteriori error estimation (2/2) Magnetostatic problemMagnetodynamic problem Magnetic vector potential a (1st order) Higher order hierarchal correction a p ( 2nd order, BFs and TFs on faces) Coarse mesh Fine mesh skin depth Large local correction  Large error Conductive coreMagnetic core V