Applied'&'Computa/onal'Electromagne/cs (ACE) Part/III Introduc8on/to/the/Finite/Element/Technique/for/ Electromagne8c/Modelling

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1 Applied'&'omputa/onal'Electromagne/cs (AE) Part/III Introduc8on/to/the/Finite/Element/Technique/for/ Electromagne8c/Modelling 68

2 lassical and weak formulations Partial differential problem lassical formulation L u = f in % B u = g on ' =!% u + classical solution Weak formulation Notations ( u, v ) = - u(x) v(x) dx, u, v /L 2 (%) % ( u, v ) = - u(x). v(x) dx, u, v /L 2 (%) % ( u, L * v ), ( f, v ) + - Q g ( v) ds =,. v /V(%) u + weak solution ' v + test function ontinuous level : ( system Discrete level : n ( n system 1 numerical solution 69 13

3 lassical and weak formulations Application to the electrostatic problem ' e curl e = div d = d = # e ' d n ( e 2 'e = n. d 2 'd = Electrostatic classical formulation ( d, grad v' ) =,. v' / V(%) with V(%) = { v / H (%) ; v2 'e = } 1 ( div d, v' ) + < n. d, v' > ' =,. v' / V(%) 4 4 div d = n. d 2 'd = Weak formulation of div d = (+ boundary condition) d = # e & e = grad v 3 curl e = ( # grad v, grad v' ) =,. v' / V(%) Electrostatic weak formulation with v 7 14

4 lassical and weak formulations Application to the magnetostatic problem ' h curl h = j div b = b = µ h n ( h 2 'h = n. b 2 'e = ' e Magnetostatic classical formulation ( b, grad 5' ) =,. 5' / 6(%) with 6(%) = { 5 / H (%) ; 52 'h = } 1 ( div b, 5' ) + < n. b, 5' > ' =,. 5' / 6(%) 4 4 div b = n. b 2 'e = Weak formulation of div b = (+ boundary condition) b = µ h & h = h s grad 5 (with curl h s = j) 3 curl h = j ( µ (h s grad 5), grad 5' ) =,. 5' / 6(%) Magnetostatic weak formulation with

5 Magnetostatic formulations Maxwell equations (magnetic - static) curl h = j div b = b = µ h 5 Formulation a Formulation "h" side "b" side 72 26

6 Magnetostatic formulations Basis equations curl h = j b = µ h div b = (h) (m) (b) 5 Formulation Magnetic scalar potential 5 a Formulation Magnetic vector potential a h = h s grad 5 h s given such as curl h s = j (non-unique) div ( µ ( h s grad 5 ) ) = 1 (h) OK (b) OK 8 9 (b) & (m) (h) & (m) & b = curl a curl ( µ 1 curl a ) = j Multivalued potential uts Non-unique potential Gauge condition 73 27

7 Multivalued scalar potential Kernel of the curl (in a domain %) ker ( curl ) = { v : curl v = } grad dom(grad) cod(grad) curl ker(curl) dom(curl) curl. cod(curl) cod ( grad ) ) ker ( curl ) 74 28

8 Multivalued scalar potential - ut curl h = in % irculation of h along path : AB in % OK 8 1? Scalar potential 5 h = grad 5 in % - h. dl = -, grad 5. dl = 5 A, 5 B : AB : AB losed path : AB (A+B) surrounding a conductor (with current I) 1 5 A 5 B = ; I!!! 5 must be discontinuous... through a cut < 5 = I 75 29

9 Vector potential - gauge condition div b = in % OK 8 1? Vector potential a b = curl a in % Non-uniqueness of vector potential a b = curl a = curl ( a + grad = ) Gauge condition ex.: w(r)=r oulomb gauge div a = Gauge a. > = > vector field with non-closed lines linking any 2 points in % 76 3

10 Magnetodynamic formulations h-5 Formulation a * Formulation t-> Formulation Maxwell equations (quasi-stationary) curl h = j curl e =! t b div b = b = µ h j = $ e a-v Formulation "h" side "b" side 77 31

11 Magnetodynamic formulations curl h = j (h) Basis equations b = µ h j = $ e curl e =! t b div b = (b) h-5 Formulation Magnetic field h Magnetic scalar potential 5 h ds % c h = h s grad 5 ds % c curl ($ 1 curl h) +! t (µ h) = div (µ (h s grad 5)) = 1 (h) OK curl h s = j s 9 (b) & 9 in % c & 9 in % c & t-> Formulation Electric vector potential t Magnetic scalar potential > (h) OK 8 curl ($ 1 curl t) +! t (µ (t grad >)) = div (µ (t grad >)) = j = curl t h = t grad > + Gauge 78 32

12 Magnetodynamic formulations curl h = j (h) Basis equations b = µ h j = $ e curl e =! t b div b = (b) a * Formulation Magnetic vector potential a * a-v Formulation Magnetic vector potential a Electric scalar potential v b = curl a * e =! t a * 1 (b) OK (b) OK 8 b = curl a e =! t a grad v curl (µ 1 curl a * ) + $! t a * = j s + Gauge in % c 9 (h) & curl (µ 1 curl a) + $ (! t a + grad v)) = j s + Gauge in % 79 33

13 Electrostatic problem Basis equations curl e = div d = " d = # e Se S e 1 Se 2 ) ( v) " * 1 F e Fe 2 grad e # divd e = d e d curl e curl d (u) 2 F d Fd 1 3 F e div e gradd F d F e F d 3 S d 3 S d 2 Sd 1 S e 3 "e" side e = grad v d = curl u "d" side S d 8 1

14 Electrokinetic problem Basis equations curl e = div j = j = $ e Se S e 1 Se 2 ) ( v) * 1 F e Fe 2 grad e $ div j e = j e j curl e curl j (t) 2 F j F j 1 3 F e div e grad j F j F e F j 3 S j 3 S j 2 S j 1 S e 3 "e" side e = grad v j = curl t "j" side S j 81 11

15 Magnetostatic problem Basis equations curl h = j div b = b = µ h ) * "h" side h = grad 5 b = curl a "b" side 82 34

16 Magnetodynamic problem curl h = j Basis equations b = µ h j = $ e curl e =! t b div b = ) * "h" side h = t grad 5 b = curl a "b" side e =! t a grad v 83 35

17 ontinuous mathematical structure Domain %, Boundary!% = ' h U ' e Basis structure Function spaces F h ) L 2, F 1 h ) L 2, F 2 h ) L 2, F 3 h ) L 2 dom (grad h ) = F h = { 5 / L 2 (%) ; grad 5 / L 2 (%), 52 'h = } dom (curl h ) = F 1 h = { h / L 2 (%) ; curl h / L 2 (%), n? h2 'h = } dom (div h ) = F 2 h = { j / L 2 (%) ; div j / L 2 (%), n. j2 'h = } grad h F h ) F 1 h, curl h F 1 h ) F 2 h, div h F 2 h ) F 3 h Boundary conditions on ' h Sequence F h 1 F h curl & 2 F h div & 3 F h Basis structure Function spaces F e ) L 2, F e 1 ) L 2, F e 2 ) L 2, F e 3 ) L 2 dom (grad e ) = F e = { v / L 2 (%) ; grad v / L 2 (%), v2 'e = } dom (curl e ) = F e 1 = { a / L 2 (%) ; curl a / L 2 (%), n? a2 'e = } dom (div e ) = F 2 e = { b / L 2 (%) ; div b / L 2 (%), n. b2 'e = } grad h F e ) F 1 e, curl e F 1 e ) F 2 e, div e F 2 e ) F 3 e Boundary conditions on ' e Sequence F e 3 div 2 curl F e 1 grad F e F e 84 36

18 Discrete mathematical structure ontinuous problem ontinuous function spaces & domain lassical and weak formulations Discretization Approximation Questions Discrete problem Discrete function spaces piecewise defined in a discrete domain (mesh) lassical & weak formulations &? Properties of the fields &? Finite element method Objective To build a discrete structure as similar as possible as the continuous structure 85 38

19 Discrete mathematical structure Finite element Interpolation in a geometric element of simple shape + f Finite element space Function space & Mesh + f i! i Sequence of finite element spaces Sequence of function spaces & Mesh A + B f i D! i E F G D 86 39

20 Finite elements! Finite element (K, P K, H K )! K = domain of space (tetrahedron, hexahedron, prism)! P K = function space of finite dimension n K, defined in K! H K = set of n K degrees of freedom represented by n K linear functionals 5 i, 1 I i I n K, defined in P K and whose values belong to IR 87 4

21 Finite elements! Unisolvance. u / P K, u is uniquely defined by the degrees of freedom! Interpolation u K = n J K i =1 5 i (u) p i Degrees of freedom Basis functions! Finite element space Union of finite elements (K j, P Kj, H Kj ) such as : " the union of the K j fill the studied domain (+ mesh) " some continuity conditions are satisfied across the element interfaces 88 41

22 Sequence of finite element spaces Tetrahedral (4 nodes) Geometric elements Hexahedra (8 nodes) Prisms (6 nodes) Mesh Nodes i / N Edges i / E Faces i / F Geometric entities Volumes i / V S S 1 S 2 S 3 Sequence of function spaces 89 42

23 Sequence of finite element spaces Functions Properties Functionals Degrees of freedom S S 1 S 2 S 3 {s i, i / N} {s i, i / E} {s i, i / F} {s i, i / V}. i, j /E Point evaluation urve integral Surface integral Volume integral Nodal value irculation along edge Flux across face Volume integral Nodal element Edge element Face element Volume element Bases Finite elements 9 43

24 Sequence of finite element spaces Base functions ontinuity across element interfaces odomains of the operators S S 1 S 2 S 3 {s i, i / N} value {s i, i / E} tangential component grad S ) S 1 {s i, i / F} normal component curl S 1 ) S 2 {s i, i / V} discontinuity div S 2 ) S 3 S S 1 grad S S 2 curl S 1 S 3 div S 2 onformity Sequence S & S 1 S S

25 Function spaces S et S 3 For each node i / N & scalar field s i (x) = p i (x) / S p i = A B 1 at node i at all other nodes node i p i = 1 p i continuous in % For each Volume v / V & scalar field s v = 1 / vol (v) / S

26 Edge function space S 1 For each edge e ij = {i, j} / E & vector field s e ij = p j grad J J p r, p i grad p r s e / S 1 r/n F,ji r/n F,ij Definition of the set of nodes N F,mn - Illustration of the vector field s e N.B.: In an element : 3 edges/node 93 46

27 Edge function space S 1 Geometric interpretation of the vector field s e p j grad J r/n F, ji p r J J K N s e = F, ji ij p j grad p r, p i grad p r r/n F,ji r/n F,ij J, p i grad p r r/n F,ij K N F,ij 94 47

28 Function space S 2 For each facet f / F & vector field f = f ijk(l) = {i, j, k (, l) } = {q 1, q 2, q 3 (, q 4 ) } s f = a f # N f p q c c=1 J L O L gradn p M r Q? gradn P M J r /N F,qc q c + 1 J r /N F,qc q c, 1 p r O Q P s f / S 2 #N f = 3 & a f = 2 4 & a f = 1 Illustration of the vector field s f 95 48

29 Particular subspaces of S 1 Kernel of the curl operator Applications h / S 1 (%) ; curl h = in % c ) % & h +? Gauged subspace a / S 1 (%) ; b = curl a / S 2 (%) & a +? Gauge a. > = to fix a Ampere equation in a domain % c without current (R % c ) Gauge condition on a vector potential Definition of a generalized source field h s such that curl h s = j s 96 49

30 Kernel of the curl operator ase of simply connected domains % c H ={ h /S 1 (%) ; curl h = in % c } h = J h a s a = h k s k e/e h = grad 5 in % c J + J h l s l k /E c l/e c h l = - h. dl = -,grad 5. dl = 5 al, 5 b l N c, E c E c N c, E c (interface)!% c % c h = l ab J J h = h k s k + 5 n v n k /E c l ab J ( 5 al J h k s k +, 5 b l ) s l k/e c l/e c n/n c with v n = J nj/e c s nj 5 n 5 n' Base of H + basis functions of inner edges of % c nodes of % c, with those of!% c h k 97 5

31 Kernel of the curl operator ase of multiply connected domains H ={ h /S 1 (%) ; curl h = in % c } 5 = 5 cont + 5 disc h = grad 5 in % c (cuts) = 52 ' + eci 52 ' eci = I i h = J 5 disc = I i q i i/ discontinuity of 5 disc J J h k s k + 5 cont J n v n + I i c i k /A c n/n c with edges of % c starting from a node of the cut and located on side '+' but not on the cut c i = J i/ nj/a c n/n ec i j/n c + jsn eci s nj q i defined in % c unit discontinuity across ' eci continuous in a transition layer zero out of this layer Basis of H + basis functions of inner edges of % c nodes of % c cuts of 98 51

32 Gauged subspace of S 1 Gauged space in % b = curl a b f = with a = Tree + set of edges connecting (in %) all the nodes of % without # forming any loop (E) " o-tree + complementary set of the tree (E) J a e s e / S 1 J (%), b = b f s f / S 2 (%) e /E J i(e,f) a e, f /F matrix form: b f = FE a e e /E f /F Face-edge incidence matrix tree # E Gauged space of S 1 (%) S " 1 (%) = {a / S 1 (%) ; a J j =,. j / E} " a = a i s i / S " 1 (%) " i/ " E Basis of S 1 (%) + co-tree edge basis functions (explicit gauge definition) co-tree " E 99 52

33 Mesh of electromagnetic devices! Electromagnetic fields extend to infinity (unbounded domain)! Approximate boundary conditions: " zero fields at finite distance! Rigorous boundary conditions: " "infinite" finite elements (geometrical transformations) " boundary elements (FEM-BEM coupling)! Electromagnetic fields are confined (bounded domain)! Rigorous boundary conditions 1 53

34 Mesh of electromagnetic devices! Electromagnetic fields enter the materials up to a distance depending of physical characteristics and constraints! Skin depth T (T<< if >, $, µ >>)! mesh fine enough near surfaces (material boundaries)! use of surface elements when T & 11 54

35 Mesh of electromagnetic devices! Types of elements! 2D : triangles, quadrangles! 3D : tetrahedra, hexahedra, prisms, pyramids! oupling of volume and surface elements " boundary conditions " thin plates " interfaces between regions " cuts (for making domains simply connected)! Special elements (air gaps between moving pieces,...) 12 55