Finite Formulation o Eletromagneti Field Enzo TONTI Dept.Civil Engin., Univ. o Trieste, Piazzale Europa 1, 34127 Trieste, Italia. e-mail: tonti@univ.trieste.it Otober 16, 2000 Abstrat This paper shows that the equations o eletromagnetism an be diretly obtained in a inite (=disrete) orm, i.e. without going throught the dierential ormulation. This inite ormulation is a natural extension o the network theory to eletromagneti ield and it is suitable or omputational eletromagnetis. 1. Introdution Computational eletromagnetism requires the transormation o Maxwell's dierential equations into algebrai equations. This is obtained using one o the many disretization methods suh as Finite Dierene Method (FDM); Finite Dierene in Time Domain (FDTD); Finite Element Method (FEM); Boundary Element Method (BEM); Edge Element Method (EEM); et. Even in Finite Volume Method (FVM) and in Finite Integration Theory (FIT), whih both use use an integral ormulation, one use integrals o ield untions the last being indispensable ingredients o the dierential ormulation. A question arises: is it possible to express the laws o eletromagnetism diretly by a set o algebrai equations, instead o obtaining them rom a disretization proess applied to dierential equations? The answer is: Yes, it is possible, it is easy and it an be immediately used or omputation. We are introduing an alternative to dierential ormulation. In order to display ield laws in a inite ormulation, we must make use o two dierent lassiiations o physial quantities. 1.1. Coniguration, soure and energy variables A irst lassiiation is based on the role that every physial variable plays in a theory. This leads to three lasses o variables: oniguration, soure and energy variables as shown in Table 1. In every physial ield we an ind: Coniguration variables that desribe the oniguration o the ield or o the system. These variables are linked one another by operations o sum, dierene, limit, derivative and integral. Soure variables that desribe the soures o the ield or the ores ating on the system. These variables are linked one another by operations o sum, dierene, limit, derivative and integral. Energy variables that are obtained as the produt o a oniguration variable by a soure one. These variables are linked one another by operations o sum, dierene, limit, derivative and integral. Table 1: The three lasses o variables o eletromagnetism. CONFIGURATION VARIABLES: gauge untion χ eletri voltage (impulse) U, ( U ) e.m.. (impulse) E, ( E ) eletri ield vetor E magneti lux Φ eletri potential (impulse) V, ( V ) magneti vetor potential A magneti indution B SOURCE VARIABLES: eletri harge ontent eletri harge low eletri urrent density J eletri lux Ψ eletri indution D magneti ield strength H magneti voltage (impulse) U m,( Um ) m.m.. (impulse) F m,( Fm ) magneti salar potential V m dieletri polarization P magnetization M ENERGY VARIABLES: work, heat W, eletri energy density w e magneti energy density w m Poynting vetor S eletromagneti momentum G momentum density g eletromagneti ation A 1
1.2. Global and ield variables A seond lassiiation distinguish between global variables and ield variables. Global variables are those ommonly alled integral variables: we avoid the last expression beause it reers to an integration proess perormed on ield variables. We must emphasize that physial measurements deal mainly with em global variables: we an diretly measure voltages, luxes, harge ontent and harge lows, not ield vetors. Field variables are needed in a dierential ormulation beause the very notion o derivative reers to point untions. On the ontrary a global quantity reers to a system, to a spae or time element, suh as lines, suraes, volumes, intervals, i.e. it is a domain untion. Thus a low meter measures the eletri harge that rosses a given surae in a given time interval. A magneti tensiometer measures the magneti voltage impulse reerred to a line and a time interval. The orresponding physial quantities are assoiated with extensive spae and extensive time elements, not only with points and instants One undamental advantage o global variables is that they are ontinuous through the separation surae o two materials while the ield variables are disontinuous. This implies that the dierential ormulation is restrited to regions o material homogeneity. When this is not the ase it is neessary to break the domain in subdomains, one or eah material, and to introdue jump onditions. I we think about the great number o dierent materials whih we an ind in a real devie, we an see that the idealization required by dierential ormulation is too restritive or the present tehnology. This shows that the dierential ormulation imposes derivability onditions on ield untions that are restritive rom the physial point o view. On the opposite a diret inite ormulation based on global variables an hold material disontinuities, i.e. does not add regularity onditions to those requested by the physial nature o the problem. To help the reader austomed to think in terms o traditional ield variables, we irst examine the global variables orresponding to traditional ield untions: ρ J B D E H Φ Ψ U U m (1) These are olleted in Table 2. This table shows that integral variables arise by integration o ield untions on spae and time domains i.e. lines, suraes, volumes and time intervals. The time integral o a physial variable, say U, will be alled its impulse and will be denoted by the orresponding alligraphi letter, say U. It is remarkable that all global oniguration variables have the dimension o a magneti lux and that all global soure variables have the dimension o a harge. The produt o a global oniguration variable and a global soure variable has the dimension o an ation (energy x time). 1.3. Cell omplexes It is well known that there is a strit link between physis and geometry. In spite o this, it is not usually stressed that global physial variables are naturally assoiated with spae and time elements, i.e. points, lines, suraes, volumes, instants and intervals. In the dierential ormulation points play a undamental role: ield untions are point untions. In order to assoiate points with numbers we then introdue oordinate systems. In inite ormulation we need to onsider not only points ( P ) but also lines ( L ), suraes ( S) and volumes ( V ). We shall all these entities spae elements. Table 2: Integral (=global) physial variables o eletromagnetism and orresponding ield untions. oniguration variables (unit: weber) v = V dt T = " d L p A L soure variables (unit: oulomb) = ρ dv V = J d S dt T S " U E" d L dt Ψ = " d T L D S S = Φ = B "d S U S m H " d L dt T L = The natural substitute o oordinate systems are ell omplexes. They exhibit verties, edges, aes and ells that are representative o the our spatial elements. One we have introdued a ell omplex we an onsider a dual omplex. I we make use o a simpliial omplex as a primal omplex, then the ommonst hoies are either the baryentres o every simplex or the irumentres (in 2D) and the irumspheres (in 3D). In this paper we onsider only irumentres and irumspheres. Sine the straight line onneting the irumentres o two adjaent simplexes in 2D is orthogonal to the ommon edge the dual polygon thus obtained has its sides orthogonal to the ommon edge. This is alled Voronoi polygon in 2D and Voronoi polyhedron in 3D. The irumentres have the disadvantage that or triangles with obtuse angles they lie outside the triangle. This is inonvenient when the irumentre o one obtuse triangle goes beyond the one o the adjaent triangle with the ommon sides. This is avoided when the primal triangulation satisies the Delaunay ondition. This leads us to onsider only Delaunay-Voronoi omplexes, as we shall do in this paper. While in oordinate systems it is preerable to deal with orthogonal oordinate systems, in a simpliial omplex it is preerable to deal with a Delaunay omplex and its assoiated Voronoi omplex as dual, as shown in Figure 1. Figure 1: Three kinds o dual entities: those o the irst and seond olumns satisy the orthogonality ondition. 2
1.4. Inner and outer orientation The notions o inner and outer orientation o a spae element play a pivotal role in eletromagnetism as well as in all physial theories. Inner orientation. Points an be oriented to be ``soures" or ``sinks". The notion o soure and sink, borrowed rom luid dynamis, an be used to deine an inner orientation o points beause it allows to maintain the notion o inidene number rom lines and points. A line is endowed with inner orientation when we hoose a diretion on the line. A surae is endowed with inner orientation when its boundary has an inner orientation. A volume is endowed with inner orientation when its boundary is so. The our spae elements endowed with inner orientation will be denoted P,L,S,V. Outer orientation. In order to write a balane we need a notion o exterior o a volume, beause we speak o harge ontained inside the volume. This is usually done by ixing outwards or inwards normals to the volume boundary. A surae is endowed with outer orientation when one o its aes has been hosen as positive and the other as negative: this is equivalent to ixing the diretion o an arrow rossing the surae rom the negative to the positive ae. We need the outer orientation o a surae when we onsider a low rossing the surae. A line is endowed with outer orientation when a diretion o rotation around the line has been deined: or example let us think to the rotation o the polarization plane o a ligth beam. A point is endowed with outer orientation when all line segments with their origin in the point have an outer orientation. Let us think, or example, o the sign o the salar magneti potential o a oil at a point: it depends on the diretion o the urrent in the oil. The our spae elements endowed with outer orientation will be denoted P,L,S,V A ell omplex and its dual enjoy a peuliar property: one the verties, edges, aes and ells o the primal omplex have been endowed with inner orientation, then an outer orientation on the ells, aes, edges and verties o its dual is indued. It ollows that a pair ormed by a ell omplex and its dual is the natural rame to exhibit all spae elements and their orientations. 1.5. Cell omplex in time Let us onsider a given interval on the time axis and divide it into small time intervals, as shown in Figure 2. A primal instant I is oriented as sink, suh as spae points. A primal interval T will be Figure 2: Cell omplex on time axis and its dual. endowed with an inner orientation, i.e. it is oriented towards inreasing time. I we hoose an instant inside every time interval we obtain a dual instant I that is authomatially endowed with an outer orientation. The interval T between two dual instants is a dual interval and it is automatially endowed with an outer orientation. In this ashion, every instant o the primal omplex there orresponds an interval o the dual and every interval o the primal orresponds an instant o the dual. Thus we have the orrespondene I T and I T and this is a duality map. 1.6. Global variables and spae-time elements From the analysis o a great number o physial variables o lassial ields we an iner [3], [4] that FIRST PRINCIPLE: Global oniguration variables are assoiated with spae and time elements whih are endowed with inner orientation while global soure variables are assoiated with spae and time elements whih are endowed with outer orientation. This priniple gives the reason or whih dierential orms are used in eletromagnetism [5]. The reason or assoiating soure variables with outer orientation is that soure variables are used in balane equations and a balane requires a volume with an outer orientation (outwards or inwards normals). In short: oniguration variables inner orientation soure variables outer orientation This priniple oers a rational riterion or whih Table 3: Global variables o eletromagnetism. global physial variable symbol V TP, eletri potential impulse [ ] eletri voltage impulse U [ TL, ] e.m.. impulse E [ TL, ] magneti lux Φ [ IS, ] eletrokineti momentum p [ IL, ] magneti potential impulse V m TP, magneti voltage impulse U m [ TL, ] m.m.. impulse F m [ TL, ] eletri lux Ψ, IS eletri harge ontent IV, eletri harge low, TS global variables o every physial theory an be assoiated to spae and time elements and is useul in omputational eletromagnetism. Figure 3 shows this assoiation or physial variables o eletromagnetism. It is important to note that eah one o the six variables o Eq.(1) admits an operational deinition [7]. We an say that the role o the dual omplex is to orm 3
a reerene struture to whih soure variables an be reerred. The spae and time assoiation o global eletromagneti variables is summarized in Table 3. Figure 3: Global physial variables are assoiated with elements o a ell omplex and its dual. 1.7. Physial laws and spae-time elements. The irst priniple states that all global physial variables reer to oriented spae and time elements. From the analysis o a great number o physial variables o lassial ields one an iner [3], [4] the SECOND PRINCIPLE: In every physial theory there are physial laws that link global variables reerred to an oriented spae-time element, say Ω, with others reerred to its oriented boundary, say Ω The ield laws o eletromagnetism satisy this priniple.this priniple motivates the role o exterior dierential on dierential orms [5]. 1.8. Field laws in global orm Experiments lead us to iner the ollowing our laws o eletromagnetism (reer to Figure 4): Figure 4: Field laws are assoiated with spae and time elements. The eletromotive ore impulse E reerred to the boundary S o a surae endowed with inner orientation during a time interval T is opposite to the magneti lux Φ variation aross the surae S in the same interval (Faraday). The magneti lux Φ reerred to the boundary V o a volume V endowed with inner orientation vanishes at any instant I (Gauss). The magnetomotive ore impulse F m reerred to the boundary S o a surae endowed with outer orientation in a time interval T is equal to the sum o the eletri harge low aross the surae S in that time interval and the eletri lux Ψ variation aross the surae S in suh interval (Ampère-Maxwell). The eletri lux Ψ aross the boundary V o a volume endowed with outer orientation at any instant I is equal to the eletri harge ontained inside the volume V at suh instant (Gauss). In ormulae: + E [ T, S] = Φ[ I, S] Φ[ I, S] Φ [ I, V] = 0 + Fm[ T, S] = Ψ [ I, S] Ψ [ I, S] + [ T, S] Ψ [ I, V ] = [ I, V ] (2) Equations (2) are the our laws o eletromagnetism in the inite ormulation we were searhing or [7]. These algebrai equations enjoy the ollowing properties: they link physial variables o the same kind, i.e. oniguration variables with oniguration variables and soure variables with soure variables; they are valid in whatever medium and then they are ree rom any material parameter; they are valid or whatever surae and whatever volume and then they are valid both in the large and in the small; they do not involve metrial notions, i.e. lengths, areas, measures o volumes and durations are not required. These properties show that ield equations do not require ininitesimal spae elements and then they are not responsible or dierential ormulation. These our laws an be expressed without reourse to ell omplexes. They are the inite ormulation o ield laws orresponding to Maxwell's equations. Loal ormulation. In order to obtain a set o algebrai equations we must introdue a ell omplex and its dual. All elements must be labelled. Let l, s be the edges and aes o the primal omplex respetively; l and s the same or the dual omplex;,d,,d κ h, the inidene numbers. When equations (2) are applied to the orresponding ells o the two omplexes, we obtain a loal orm o the ield equations o the eletromagneti ield in a inite setting. We an write E τ n+1, l tn+1, s tn, s [ ] + Φ Φ = 0 d κ Φ tn, s = 0 m n, l t n+1, s + t F τ Ψ Ψ n, s = τ n, s d hψ t n, s = t n, v h [ τ ] (3) 4
For omputational purposes it is useul to perorm the n ollowing hanges o symbols: Φ tn, s Φ ; n 1/ 2 n, + Ψ t s Ψ et. In partiular the two evolution equations an be written as ( =, see [7]) n+ 1 n n + 1/ 2 n n + 1/ 2 n 1/ 2 n = + ( m ) ( ) Φ Φ E = Ψ Ψ F This gives rise to the leaprog algorithm. (4) 1.9. Constitutive laws The equations that link the soure variables with oniguration ones are onstitutive or material equations. In a region o uniorm ield the three onstitutive equations o eletromagnetism in inite orm and or orthogonal duals s l are [ τ, ] Ψ t n, s E n l = s τ n l Φ tn, s m n, F τ l = µ s n l τ [ τ n, s ] 1 E [ τ n, l] E [ τ n+ 1, l] = σ + τ n s 2 τ n+ 1 l τ n+ 1 l (5) in whih τ n, τ n,l,l, s,s, are the extensions o the orresponding elements. To explain the partiular orm o Ohm's law let us remark that while the eletri harge is reerred to dual intervals, the eletri tension impulse E is reerred to primal ones. These equations are valid i ells are ubes or they belong to a Delaunay-Voronoi omplex, as shown in Figure 1. In these ases 1-ells o the dual are orthogonal to the primal 2-ells and vieversa. The orthogonality ondition is not neessary, and also baryenters may be used to deine the dual omplexes [2]. The main properties o onstitutive laws are: They are valid in regions in whih the ield is uniorm beause they are tested under suh onditions. They link a variable reerred to a p ell o a omplex with the dual ( n p) ell o the dual omplex. This geometrial property is not apparent in dierential ormulation. They ontain material parameters. They require metrial notions suh as length, areas, volumes and orthogonality. While ield equations in inite orm desribe the orresponding physial laws exatly, the onstitutive ones in inite orm desribe the orresponding physial laws approximately beause they are validated in regions o uniorm ield. Combining the two set o equations, the ield and the onstitutive ones, we obtain a set o algebrai equations. At this point we have two alternatives: 1. To perorm the limit proess on the dimensions o the ells in order to obtain a uniorm ield at the limit. This is the traditional way that leads to dierential ormulation. 2. To make the approximation that the ield is uniorm inside every ell. In this way we obtain a inite ormulation. The last hoie is the one pursued in this paper. 1.10. Comparison with other methods. Sine the distintion between the two lasses o physial variables is not ommonly done and sine it is not reognized that two kind o orientations are needed, it ollows that the dierential ormulation uses only one kind o ininitesimal ells. It ollows that FEM, arising rom a disretization o dierential equations, ignores the need o two dual omplexes and thereore uses only one omplex ormed by the elements. In 1966 Yee [10], using a artesian omplex and an appropriate hoie o points at whih the various ield omponents are to be evaluated, opened the way to the introdution o a pair o dual omplexes alled by Weiland [8], the eletri and magneti grids. Nevertheless the two omplexes were not justiied by physial onsiderations but only by omputational advantages. In the realm o dierential orms the two kinds o orientation give rise to normal and twisted orms [1, p.183]. 1.11. Conlusion It is possible to express both ield and onstitutive laws o eletromagnetism in a inite (i.e. algebrai) orm starting diretly rom experimental ats, i.e. avoiding the passage through the dierential ormulation. This provide a set o algebrai equations that an be diretly used in omputational eletromagnetism. The unique approximation lies in the hypotesis that inside every simplex the ield is uniorm. I we use an adaptive simpliial omplex we an redue the size o the simplexes in the regions where global variables undergo large variations. Doing so, the hypotesis o uniormity inside every simplex an be under ontrol. The inite ormulation an easily handle inhomogeneous and anisotropi materials (this is o partiular interest when dealing with absorbing boundary onditions) and an easily handle onentrated soures beause ininities do not appear. 5
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