Strong and Weak Formulations of Electromagnetic Problems Patrick

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

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

Formulations of Electromagnetic Problems Electrostatics Electrokinetics Maxwell equations Magnetostatics Magnetodynamics 3

Electromagnetic models v Electrostatics All phenomena are described by Maxwell equations ♦ 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 4

Maxwell equations curl h = j + ¶t d Ampère equation curl e = – ¶t b Faraday equation div b = 0 div d = rv Conservation equations Principles of electromagnetism Physical fields and sources h b j magnetic field (A/m) magnetic flux density (T) current density (A/m 2) e electric field (V/m) d electric flux density (C/m 2) rv charge density (C/m 3) 5

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

Electrostatics Basis equations curl e = 0 div d = r d = e e e d r e Type of electrostatic structure & boundary conditions n ´ e | G 0 e = 0 n × d | G 0 d = 0 electric field (V/m) electric flux density (C/m 2) electric charge density (C/m 3) dielectric permittivity (F/m) Electric scalar potential formulation div e grad v = – r with e = – grad v • Formulation for W 0 Exterior region Wc, i Conductors Wd, j Dielectric • the exterior region W 0 • the dielectric regions Wd, j • In each conducting region Wc, i : v = vi ® v = vi on Gc, i 7

Electrostatic 8

Electrokinetics Basis equations curl e = 0 div j = 0 j = s e e j s Type of electrokinetic structure & boundary conditions G 0 j n ´ e | G 0 e = 0 n × j | G 0 j = 0 electric field (V/m) electric current density (C/m 2) electric conductivity (W– 1 m– 1) G 0 e, 0 Wc e=? , j=? V = v 1 – v 0 Electric scalar potential formulation div s grad v = 0 G 0 e, 1 Wc Conducting region with e = – grad v • Formulation for • the conducting region Wc • On each electrode G 0 e, i : v = vi ® v = vi on G 0 e, i 9

Magnetostatics Type of studied configuration Equations curl h = j Ampère equation div b = 0 Magnetic conservation equation Constitutive relations b = m h + bs j = js W Magnetic relation Ohm law & source current Studied domain Wm Magnetic domain Ws Inductor 10

Magnetodynamics Type of studied configuration Equations curl h = j Ampère equation curl e = – ¶t b Faraday equation div b = 0 Magnetic conservation equation Constitutive relations b = m h + bs j = s e + js W Magnetic relation Ohm law & source current Studied domain Wp Passive conductor and/or magnetic domain Wa Active conductor Ws Inductor 11

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

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

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

Magnetodynamics - Forces 15

Magnetodynamics - Forces 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! 16

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

Continuous mathematical structure Domain W, Boundary ¶W = Gh U Ge Basis structure Function spaces Fh 0 Ì L 2, Fh 1 Ì L 2, Fh 2 Ì L 2, Fh 3 Ì L 2 dom (gradh) = Fh 0 = { f Î L 2(W) ; grad f Î L 2(W) , f½Gh = 0 } dom (curlh) = Fh 1 = { h Î L 2(W) ; curl h Î L 2(W) , n Ù h½Gh = 0 } dom (divh) = Fh 2 = { j Î L 2(W) ; div j Î L 2(W) , n. j½Gh = 0 } gradh Fh 0 Ì Fh 1 , curlh Fh 1 Ì Fh 2 , divh Fh 2 Ì Fh 3 Sequence grad Boundary conditions on Gh curl div h h h Fh 0 ¾¾¾ ¾ ® Fh 1 ¾¾¾ ®Fh 2 ¾¾¾ ®Fh 3 Basis structure Function spaces Fe 0 Ì L 2, Fe 1 Ì L 2, Fe 2 Ì L 2, Fe 3 Ì L 2 dom (grade) = Fe 0 = { v Î L 2(W) ; grad v Î L 2(W) , v½Ge = 0 } dom (curle) = Fe 1 = { a Î L 2(W) ; curl a Î L 2(W) , n Ù a½Ge = 0 } dom (dive) = Fe 2 = { b Î L 2(W) ; div b Î L 2(W) , n. b½Ge = 0 } gradh Fe 0 Ì Fe 1 , curle Fe 1 Ì Fe 2 , dive Fe 2 Ì Fe 3 Sequence div Boundary conditions on Ge curl grad e e Fe 3 ¬¾¾ ¾ Fe 2 ¬¾¾e ¾ Fe 1 ¬¾¾ ¾ ¾ Fe 0 18

Electrostatic problem Basis equations curl e = 0 Ì F e 0 S e 0 F e 1 S e 1 F e 2 S e 2 F e 3 S e 3 "e" side grad e Fd 3 r (–v) e curl e div d = r d = e e = d 0 d (u) div e div d curl d grad d 0 e = – grad v d = curl u É Fd 2 S d 3 Fd 1 S d 2 Fd 0 S d 1 S d 0 "d" side 19

Electrokinetic problem Basis equations curl e = 0 Ì F e 0 S e 0 F e 1 S e 1 F e 2 S e 2 F e 3 S e 3 "e" side grad e F j 3 r (–v) e curl e div j = 0 j = s e = j d 0 (t) div e div j curl j grad j 0 e = – grad v j = curl t É F j 2 S j 3 F j 1 S j 2 F j 0 S j 1 S j 0 "j" side 20

Magnetostatic problem Basis equations curl h = j div b = 0 b = m h Ì "h" side É h = – grad f b = curl a "b" side 21

Magnetodynamic problem Basis equations curl h = j b = m h j = s e Ì "h" side curl e = – ¶t b div b = 0 É h = t – grad f b = curl a e = – ¶t a – grad v "b" side 22

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

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

Discrete mathematical structure Finite element Interpolation in a geometric element of simple shape + f Finite element space Function space & Mesh + Uf i i Sequence of finite element spaces Sequence of function spaces & Mesh + ì í ïî U i ü fi ý ïþ 25

Finite elements v Finite element (K, PK, SK) ♦ K= domain of space (tetrahedron, hexahedron, prism) ♦ PK = function space of finite dimension n. K, defined in K ♦ SK = set of n. K degrees of freedom represented by n. K linear functionals fi, 1 £ i £ n. K, defined in PK and whose values belong to IR 26

Finite elements v Unisolvance " u Î PK , u is uniquely defined by the degrees of freedom v Interpolation Degrees of freedom n. K u. K = å f (u) p i i i =1 v Basis functions Finite element space Union of finite elements (Kj, PKj, SKj) such as : m the union of the Kj fill the studied domain (º mesh) m some continuity conditions are satisfied across the element interfaces 27

Sequence of finite element spaces Geometric elements Tetrahedral Hexahedra Prisms (4 nodes) (8 nodes) (6 nodes) Mesh Nodes Edges Faces Volumes i Î N i Î E i Î F i Î V Geometric entities S 0 S 1 S 2 S 3 Sequence of function spaces 28

Sequence of finite element spaces Functions S 0 {si , i Î N} S 1 {si , i Î E} S 2 {si , i Î F} S 3 {si , i Î V} Properties Bases " i, j ÎE Functionals Point evaluation Curve integral Surface integral Volume integral Degrees of freedom Nodal value Circulation along edge Flux across face Volume integral Nodal element Edge element Face element Volume element Finite elements 29

Sequence of finite element spaces S 0 S 1 S 2 S 3 Base functions Continuity across element interfaces {si , i Î N} value {si , i Î E} tangential component {si , i Î F} normal component {si , i Î V} discontinuity Codomains of the operators S 0 Conformity grad S 0 Ì S 1 grad S 0 S 2 curl S 1 S 3 div S 2 curl S 1 Ì S 2 div S 2 Ì S 3 Sequence grad curl div S 0 ¾¾¾® S 1 ¾¾ ¾® S 2 ¾¾ ¾® S 3 30

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

Mesh of electromagnetic devices v Electromagnetic fields enter the materials up to a distance depending of physical characteristics and constraints ♦ Skin depth d (d<< if w, s, m >>) ♦ mesh fine enough near surfaces (material boundaries) ♦ use of surface elements when d ® 0 32

Mesh of electromagnetic devices v Types of elements ♦ 2 D : triangles, quadrangles ♦ 3 D : 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, . . . ) 33

Classical and weak formulations Partial differential problem Classical formulation L u = f B u = g in W on G = ¶W u º classical solution Notations ò ( u , v ) = u( x). v( x) dx ò ( u, v ) = u( x) v( x) dx , u, v ÎL 2 (W) W Weak formulation ò ( u , L* v ) ( f , v ) + Q g ( v ) ds = 0 , " v ÎV(W) G u º weak solution v º test function Continuous level : ¥ ´ ¥ system Discrete level : n ´ n system Þ numerical solution 34

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 v e. g. , coupling of fields between sub-domains 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 35

Constraints in electromagnetic systems v Coupling of scalar potentials with vector fields ♦ e. g. , in h-f and a-v formulations v Gauge condition on vector potentials ♦ e. g. , magnetic vector potential a, source magnetic field hs v Coupling between source and reaction fields ♦ e. g. , source magnetic field hs in the h-f formulation, source electric scalar potential vs in the a-v formulation v Coupling of local and global quantities ♦ e. g. , currents and voltages in h-f 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 36

Strong and weak formulations Equations curl h = js in W Strongly satisfies Scalar potential f h = hs grad f , with curl hs = js div b = rs Constitutive relation Stron gly sat isfi b = m h es Vector potential a b = bs + curl a , with div bs = rs Boundary conditions (BCs) 37

Strong formulations Electrokinetics curl e = 0 , div j = 0 , j = s e , e = grad v or j = curl u Electrostatics curl e = 0 , div d = rs , d = e e , e = grad v or d = ds + curl u Magnetostatics curl h = js , div b = 0 , b = m h , h = hs grad f or b = curl a Magnetodynamics curl h = j , curl e = – ¶t b , div b = 0 , b = m h , j = s e + js , . . . 38

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

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

Grad-div weak formulation Use of hierarchal TF fp' in the weak formulation 1 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 bp to be given to the actual solution b for satisfying the WF. . . solution of : or 1 Local FE problems 41

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

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

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