16. Electromagnetics and vector elements (draft, under construction)

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1 16. Elctromagntics (draft) Introduction Paramtric coordinats Edg Basd (Vctor) Finit Elmnts Whitny vctor lmnts Wak Form Vctor lmnt matrics Summary Exrciss Elctromagntics and vctor lmnts (draft, undr construction) 16.1 Introduction: Th Maxwll's quations govrning lctromagntics consist of a group of four diffrntial quations coupld by multipl variabls. Thy can b vry difficult to solv. Potntial functions ar usually introducd to rduc th numbr of quations and variabls. Maxwll s quation for lctromagntics in trms of a tim harmonic vctor potntial A is: 1 μ A + jω 2 σa = J s (16.1-1) whr μ is th magntic prmability j = 1, ω is th harmonic frquncy, σ is th lctrical conductivity, and J s is th imposd lctric currnt dnsity. Th magntic flux dnsity vctor B is rlatd to th vctor potntial by and to th magntic fild intnsity vctor H by th constitutiv law B = A (16.1-2) B = μh (16.1-3) which is normally a non-linar rlationship. In this sction, th convntional finit lmnt mthod introducd bfor for scalar quations will b calld nod basd finit lmnts. Basd on succsss with scalar diffrntial quations, on choic for a finit lmnt analysis of th Maxwll s quation may b to attmpt to dcompos th vctor fild into its thr Cartsian componnts, A x, A y, and A z (lik th way displacmnt vctors ar tratd is strss analysis). Th scalar basis functions and thus th unknown vctor componnts in th finit lmnt quations for strss analysis ar assignd to th nods of th finit lmnt msh. That mad th displacmnt vctors continuous across lmnt intrfacs, which is physically corrct. Thus It was prviously dmonstratd that nod basd finit lmnts ar suitabl for analyzing scalar filds and displacmnt vctors. That is tru bcaus th govrning intgral form for th displacmnt basd quilibrium of a solid 1

2 automatically satisfis th lmnt intrfac displacmnt continuity conditions quations that rquir tangntial continuity of th magnitud of th scalar unknowns but allow jumps in th normal dg gradints of th scalar solution. In lctromagntics, th intrfac conditions btwn diffrnt matrials tak a diffrnt form that rstricts ithr th tangntial or normal componnts of th vctor A. By th vry natur of nodal basd lmnt intrpolation all of th scalar componnts of a vctor ar continuous on th intr-lmnt boundaris. In othr words, th magnitud and dirction of a displacmnt vctor would b continuous along th intrfac btwn two matrials. That approach violats th lctromagntic vctor boundary conditions if th lmnt boundary happns to b th intrfac btwn two matrials with diffrnt lctromagntic proprtis. An altrnativ to th convntional nod basd finit lmnts is th dg basd finit lmnt or vctor lmnts which ar introducd blow. Vctor lmnts ar just spcial kinds of finit lmnts whos dgrs of frdoms ar th magnituds of vctors assignd to dgs of lmnts rathr than scalars at th nods. Th intrpolation functions ar associatd with th dgs of th finit lmnt, and mor importantly th intrpolation functions ar vctors functions that ar dfind ovr th lmnt intrior so an arbitrary vctor fild can b intrpolatd with th st of vctor intrpolation functions dirctly. Vctor lmnts nforc tangntial continuity on vctors but not normal continuity across lmnt intrfacs. It should b notd that thr ar svral lctromagntic problms, lik axisymmtric solids, whr two of th thr componnts of th vctor potntial ar idntically zro. Thn, th nodal basd finit lmnt mthod givs xcllnt rsults Paramtric coordinats: Lik nod basd lmnts, th vctor lmnts also mploy non-dimnsional paramtric coordinats to dfin th lmnt gomtry and to build th intrior vctor intrpolations. Most of th vast litratur on lctromagntic vctor lmnts utilizs unit coordinats for quadrilatral and hxahdral vctor lmnts and barycntric coordinats for triangular and ttrahdral vctor lmnts. Th lctromagntic litratur usually dnots th unit coordinats as (u, v, w) instad of th (r, s, t) prviously utilizd in this book. That prfrnc is adoptd for this sction, (u, v, w) (r, s, t). Th barycntric coordinats ar dirctly rlatd to th unit coordinats for triangls and ttrahdrals (simplx lmnts). Barycntric coordinats ar also calld ara-coordinats and volum-coordinats in two- and thr-dimnsional application, rspctivly. For compltnss, and to mak it asir to rad th lctromagntics litratur, th concpts of barycntric coordinats will b rviwd hr. Th barycntric coordinat at vrtx-k, say λ k, is unity at vrtx-k and zro at all othr vrtics of th lmnt. But, for simplx lmnts th numbr of vrtics is on largr that th spatial dimnsion, n s. Thrfor, th barycntric coordinats ar not all indpndnt and ar mad complt by th rquirmnt that thir sum at any point in th simplx is unity, and thy ar zro at any vrtx othr than k: (n 1 s +1) k=1 λ k and λ k (u j, v j ) = δ ij (16.2-1) Figur illustrats th dfinition of barycntric coordinats in a triangl. Th ara coordinat λ k is unity at nod k and zro on th dg opposit nod k. In btwn at any point P it is th fraction of th total ara that is formd by th triangl dfind by th point and dg k. Similar dfinitions xist for ttrahdra and lins, but not for quadrilatrals or hxahdra. 2

3 Figur Ara (barycntric) coordinats in a simplx triangl Figur shows a planar paramtric curv rprsnting an dg of a curvd quadrilatral whr th lmnt location and shap ar dfind by th position vctor, R (u, v, w) = x(u, v, w)i + y(u, v, w)j + z(u, v, w)k (16.2-3) at ach of its ight nods. Figur shows that curvd quadrilatral with a numbr of tangnt vctors on ach of its dgs. Th componnts of th position vctor ar intrpolatd using th usual scalar functions which wr covrd in Chaptr 2: n n x(u, v, w) = j=1 H j (u, v, w)x j = H(u, v, w)x (16.2-4) whr x dnots th input x-coordinats of all th scalar nods on lmnt-. Of cours, th y- and z-coordinats ar intrpolatd with th sam scalar functions, H(u, v, w). For an ight-nod quadrilatral in unit coordinats, 0 u, v 1, with th scalar nods that dfin th lmnt shap ordrd as in Fig th wll-known scalar intrpolations ar H 1 (u, v) = (1 2u 2v)(1 u)(1 v) H 2 (u, v) = (1 2u + 2v)(v 1)u H 3 (u, v) = uv(2u + 2v 3) H 4 (u, v) = v(u 1)(2u 2v + 1) H 5 (u, v) = (v 1)((2u 1) 2 1) H 6 (u, v) = u(1 (2v 1) 2 ) H 7 (u, v) = v(1 (2u 1) 2 ) H 8 (u, v) = (u 1)((2v 1) 2 1) (16.2-5) and for th straight sidd four-nod quadrilatral, or a rctangl, thy ar simply H(u, v) = [(1 u v + uv) (u uv) (uv) (v uv)] = [(1 u)(1 v) u(1 v) (uv) v(1 u)] (16.2-6) Of cours, thos scalar functions vanish on all dgs that do not includ th scalar nod point. 3

4 Figur A planar paramtric curv with a (non-unit) tangnt vctor Figur Curvd quadrilatral vctor lmnt with two tangnt vctors pr dg 16.3 Edg Basd (Vctor) Finit Elmnts: Th concpt of vctor dg lmnts was introducd by Ndlc in Thy ar dsignd to nforc continuity of th tangntial componnt, along an dg shard with anothr lmnt, but not th normal componnt. Thus, it can satisfy a rquird intrfac condition if th dg is on th intrfac btwn two matrials with diffrnt lctromagntic proprtis. Th simplst of ths is th thr dg triangular lmnt dvlopd by Whitny which is covrd in dtail blow. Edg lmnts ar still a subjct of rsarch and no standard approach has bn found for dvloping a nw lmnt. Most of th nwr attmpts us intrnal dgrs of frdom to incras th polynomial approximation of th vctor. Howvr, thos intrnal dgrs of frdom ar condnsd out at th lmnt lvl (s Sction 7.8) and only th dg dgrs of frdom ar assmbld and usd to comput th solution of th problm. Post-procssing of th solution allows rcovry of th intrnal dgrs of frdom so othr quantitis of intrst can b calculatd mor accuratly. Evry vctor lmnt is basd on dfining on or mor unit vctors tangnt on ach of its dgs and intrpolating btwn thm to dfin th intrior vctor potntial approximation that allows a discontinuity in th normal componnt if th dg lis on matrials with diffrnt lctromagntic proprtis. Thr ar svral ways to dfin such a vctor on an dg, including th gomtric on givn abov. Th tangnt vctor on th j-th dg of th lmnt will b of th form t u(u) j or t v (v)j or t w(w) j sinc on any dg only on paramtric coordinat varis. Within lmnt th dg vctor intrpolation, N j, which is also calld th vctor basis function, for th j-th dgr of frdom, A j. Each tangnt vctor, t j, on an dg is multiplid by a 4

5 scalar function, say h j (u, v, w) that is non-zro in th lmnt intrior. Th physical dirction of th tangnt vctor can b dfind in ithr th paramtric coordinats or physical coordinats. Thus, th form of th vctor intrpolation for an dg is: N j (u, v, w) = hj (u, v, w) t j (16.3-1) Finally, that vctor intrpolation product is multiplid by an unknown constant, say A j whos magnitud is to b dtrmind through th solution of Maxwll s quations ovr th solution domain. In othr words, ach tangnt vctor contributs th vctor portion A j N j (u, v, w) to th total vctor, A, on that dg and on th full intrior of th lmnt. By summing ovr all of th dgs of th lmnt th complt approximation of th vctor potntial solution A within th lmnt is obtaind: n A (u, v, w) = t N j (u, v, w) n j=1 Aj = t j=1 h j (u, v, w) t j A j (16.3-2) whr n t dnots th numbr of tangnt dg vctors pr lmnt. This can also b writtn symbolically, lik (16.3-2), as a matrix product to dfin th vctor potntial insid th lmnt: A (u, v, w) N (u, v, w)a (16.3-3) (3 1) = (3 n t )(n t 1) whr th gathrd valus of th magnituds of ach dg intrpolation ar in A and whr th intrior vctor intrpolation functions associatd with ach lmnt dg ar in N (u, v, w). For xampl, for th quadrilatral in Fig with only on unit tangnt vctor pr dg th intrior intrpolation could b pickd to b N (u, v) = [H 5 (u, v)t 1 H 7 (u, v)t 2 H 8 (u, v)t 3 H 6 (u, v)t 4], A = { }. (16.3-4) A 4 Th total approximation is obtaind by summing ovr all of th vctor lmnts in th msh. Th numbr of unknowns to b found is th numbr of dgs in th msh tims th numbr of tangnt vctors pr dg. A tangnt vctor at a point is th sam for all lmnts sharing that dg curv. Thus, som tangnt vctors in an unstructurd msh will hav to hav thir sign changd in th assmbly procss. Th systm unknowns (dgrs of frdom) ar th cofficints associatd with ach (assmbld) dg vctor, t j Whitny vctor lmnts: Th first dg lmnts, or vctor lmnts, wr dvlopd for simplx lmnts in barycntric coordinats and ar known as Whitny lmnts. On th dg of a straight sidd simplx lmnt in physical spac conncting nods j and k th first Whitny vctor intrpolation for lmnt is A 1 A 2 A 3 N jk (λ) = l jk [λ j λ k λ k λ j ]. (16.4-1) 5

6 For simplx lmnts it happns that th barycntric coordinat of a nod is xactly th sam as th scalar intrpolation function for that nod. For a triangular lmnt λ j (u, v) H j (u, v), thus λ j = λ j u t u + λ j v t v = H j(u,v) u t u + H j(u,v) v t v (16.4-2) whr t u is a unit vctor along th u-axis, as illustratd in th top of Fig Rcall that th scalar nod intrpolations for th linar triangl in unit coordinats ar H(u, v) = [(1 u v) (u) (v)]. Substituting thos intrpolation functions into (16.4-2) for dg 1-2 givs: N 12 (u, v) = l 12 {H 1 (u, v) [ H 2 (u,v) u H 2 (u, v) [ H 1 (u,v) u t u + H 2 (u,v) t v] v t u + H 1 (u,v) t v]} (16.4-3) v N 12 (u, v) = l 12 {(1 u v)[(1)t u + (0)t v] (u)[( 1)t u + ( 1)t v]} or simply: N 12 (u, v) = l 12 {(1 v)t u + ut v}. (16.4-4) Figur Local gradint and unit tangnt vctor on a simplx First, xamin how this vctor fild varis on ach dg as sktchd at th bottom of Fig On dg 1-2 v = 0 so N 12 (u, 0) = l 12 {t u + ut v}. On that dg th vctor has th dsird tangntial componnt combind with a linarly incrasing normal componnt. On dg 1-3 6

7 u = 0 so N 12 (0, v) = l 12 {(1 v)t u + 0 } so that dg th vctor fild has only a linarly dcrasing normal componnt, and finally on dg 2-3 u + v = 1 and N 12 (u, v(u)) = l 12 {(1 (1 u))t u + ut v} = l 12 u{t u + t v} which givs only a linarly incrasing normal componnt. Th complt vctor fild dfind by (16.4-4) is shown, at uniformly spacd sampling points, in th lft box of Fig Figur Thr Whitny dg intrpolations on a triangl, N 12, N 23, N 13 In a similar fashion, th othr two dg vctor intrpolation functions ar found to b: N 23 (u, v) = l 23 {ut v vt u} (16.4-5) N 13 (u, v) = l 13 {(1 u)t v + vt u}. (16.4-6) Thos vctor filds ar shown in Fig in th middl and right boxs, rspctivly. That figur may giv som insight into how th Whitny intrpolations wr dvlopd. Not that for ach dg th vctors appar ar undrgoing a countr-clockwis rigid body rotation about th opposit cornr. That mans that ach of th thr functions has th vry dsirabl proprty (in lctromagntics) of bing divrgnc fr. For xampl, in th paramtric lmnt = u t u + v t v so N 12 = ( u t u + t v) l v 12 {(1 v)t u + ut v} = 0. Summing ovr all of th thr dg functions, as in (16.3-3), givs A = 0. Undr an affin transformation to an arbitrary straight sidd triangl in physical spac this dsirabl condition rmains tru. Gnrally, ths dg vctor constructions crat a constant tangntial componnt along on dg combind with a proportional, linarly incrasing, normal componnt on all dgs. Assmbly of ths lmnts mans that th scond lmnt joining th dg must adapt th sam sns (sign) as th first tangnt vctor on that dg. That mans than in th assmbly (scattr) procss som lmnt matrics ar subtractd rathr that addd to th systm matrics. Th assmbly of two lmnts sharing dg 1-2 is sktchd in Fig

8 Figur Assmbld dg contributions from dg 1-2 For nodal bas lmnts a sufficint, but not ncssary, condition for convrgnc of th solution with msh rfinmnt is that a constant scalar solution can b intrpolatd xactly. A ncssary condition for convrgnc is for th lmnt to pass th patch tst. On might xpct similar conditions for vctor solutions. For xampl, if th solution is a constant vctor you might xpct th vctor intrpolation to yild that vctor vrywhr intrior th lmnt. Th Whitny triangular lmnt only xactly intrpolats th constant vctor solution whn th lmnt is a right triangl and whn th vctor is prpndicular to on of th thr dgs Wak Form: Th quivalnt Galrkin wak form of (16.1-1) for vctor lmnts is ( 1 μ A ) A d + jω 2 σa A d = J s A d. To rduc th scond drivativs in th first intgral mploy th vctor algbra idntity b a = (a b ) + a b and Grn s Thorm th first intgral convrts to ( 1 μ A ) ( A )d so th wak form, bfor applying boundary conditions, is + ( 1 μ A ) A n dγ Γ ( 1 μ A ) ( A )d + jωσa A d = J s A d + ( 1 μ A ) A n dγ Γ On boundary sgmnt Γ D th ncssary condition is A n = 0 (16.5-1) which mans that th tangntial componnt of A is zro on that sgmnt and that also implis that B n = 0 on that boundary. On a non-ovrlapping boundary sgmnt Γ N th boundary condition is 8

9 H n = h (16.5-2) whr H magntic fild intnsity vctor, and h is th known valu on that boundary sgmnt. Including ths boundary conditions th Galrkin wak form is ( 1 μ A ) ( A )d + jωσa A d = J s A d + h A dγ (16.5-3) Γ N??( a x b) x a. n = Dividing th domain into a vctor lmnt msh maks th domain intgrals bcom th sum of th lmnt intgrals, and thus dfins th lmnt matrics for ach vctor lmnt Vctor lmnt matrics: Dividing th domain and its boundaris into a msh of vctor finit lmnts lads to th systm quations (K + M)A = (c s + c h ) which ar assmbld from th lmnt matrics xtractd from th abov wak form: K ij = 1 ( N μ i ) ( N j ) d (16.6-1) M ij = jω σ N i N j d (16.6-2) c i = J s N i d (16.6-3) c b b i = b(h n ) N Γ i dγ. (16.6-4) N Excpt for th gnralizd mass matrix, M, du to lctrical conductivity, ths arrays ar fully intgratd using numrical quadraturs. Th Hlmholtz thorm stats that a vctor fild is uniquly dtrmind only if both its curl and divrgnc, A, ar spcifid. Thrfor, th divrgnc of th magntic vctor potntial should b spcifid by a so-calld gaug condition. Th gaug condition choic may b spcifid frly without affcting th physical problm. Usually, th divrgnc fr condition (th wll-known Coulomb gaug) is imposd, i.. A = 0. (16.6-5) Advancd vctor intrpolations, N j, hav bn dvlopd in th litratur that satisfy th divrgnc fr stat. In such cass th lmnt conductivity matrix, M, is also intgratd fully. Othrwis, (16.6-5) bcoms a constraint on th algbraic systm, which can caus th solution to lock and to giv rsults with maninglss physical drivativs. Th liklihood of a solution locking du to such a constraint can b prdictd by doing a constraint count on a singl lmnt is addd into a uniform structurd msh. Howvr, th dg vctor intrpolation functions usd hr ar not divrgnc fr and xprinc with such constraints shows that a numrical trick is ndd to avoid a systm that locks. That trick is to mploy rducd intgration to valuat th matrix in (16.6-2). Th minimum numbr of quadratur points rquird th valuat th volum of th lmnt,, is usd to comput M. That rndrs M and th assmbld systm M matrix rank dficint, but K and th combind systm matrix (K + M) ar of full rank and can b solvd to yild th dg vctor magnituds, A, that dos not lock th vctor potntial distribution in th domain. 9

10 16.7 Ainsworth lmnts: 16.7 Summary 16.8 Exrciss Indx Ainsworth lmnts, 10 ara-coordinats, 2 assmbly, 5 barycntric coordinats, 2 Coulomb gaug, 10 curl, 10 displacmnt vctor, 1 divrgnc fr, 8 dg basd lmnts, 2 dg tangnt vctor, 5 lctric currnt dnsity, 1 lctrical conductivity, 1 lctromagntics, 1 lmnt conductivity matrix, 10 Exrciss, 10 Galrkin mthod, 9 harmonic frquncy, 1 Hlmholtz thorm, 10 intrfac continuity conditions, 2 intrnal DOF, 4 jumps, 2 magntic fild intnsity, 1 magntic flux dnsity, 1 magntic prmability, 1 magntic vctor potntial, 10 Maxwll's quations, 1 nodal bas lmnts, 8 numrical intgration, 10 rank dficint, 10 rducd intgration, 10 scalar filds, 2 scattr, 8 sign chang, 5 simplx lmnt, 6 Summary, 10 tangntial continuity, 2 tangntial vctor componnts, 2 tim harmonic, 1 unstructurd msh, 5 vctor lmnts, 2, 4 vctor intrpolation, 5 vctor potntial, 1 vctors functions, 2 volum-coordinats, 2 Whitny dg intrpolations, 7 Whitny lmnts, 6 10

3 Finite Element Parametric Geometry

3 Finite Element Parametric Geometry 3 Finit Elmnt Paramtric Gomtry 3. Introduction Th intgral of a matrix is th matrix containing th intgral of ach and vry on of its original componnts. Practical finit lmnt analysis rquirs intgrating matrics,