INHOMOGENEOUS, NON-LINEAR AND ANISOTROPIC SYSTEMS WITH RANDOM MAGNETISATION MAIN DIRECTIONS

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1 dx dt DIFFERENTIAL EQUATIONS AND CONTROL PROCESSES N Electronic Journal reg. N P23275 at diff@osipenko.stu.neva.ru Applications to physics electrotechnics and electronics INHOMOGENEOUS NON-LINEAR AND ANISOTROPIC SYSTEMS WITH RANDOM MAGNETISATION MAIN DIRECTIONS Part two: Minimisation of functional issue and FEM equations for 3D field produced by permanent magnets BERE IOAN Assoc. Prof. Politehnica University Department of Electrotechnics Timisoara Romania ibere@et.utt.ro Abstract The paper presents the minimization of functional issue associated to the finite elements method (FEM) and establish on detail the equations to solve the problem of the three-dimensional (3D) magnetic field in inhomogeneous nonlinear and anisotropic domains with random main directions of magnetization. Numerical computation of the algebraic equations system obtained it could be established the state quantities of the magnetic field or other quantities which are interested in. Key words: permanent magnets FEM inhomogenity nonlinearity anisotropy.

2 4. The minimization of the energy functional On FEM the functional must be minimized to get the equations system which follow to solve the field problem. If the finite elements of 3D discretization mesh are small enough the relation (33) from 1] become m ( ) 2 ( ) 2 ( ) 2 F = A VH xx A VH yy A VH zz z ( )( ) ( )( ) ( )( ) ] A VH VH xy A VH VH yz A VH VH zx v z z m ( ) 2 ( ) 2 ( ) 2 A VH xx A VH yy A VH zz z ( )( ) ( )( ) ( )( ) A VH VH xy A VH VH yz A VH VH zx z z ( ) ( ) ( ) ] K x VH K y VH K z VH v z m N m (B n ) (V H ) (S N ) N where v is the volume of the finite element. (B n ) (V H ) (S N ) (34) The magnetic scalar potential (V H ) unknown function in a current point P (x y z) inside the finite element = 1 m could be written depending on the magnetic scalar potentials V Hi V Hj V Hk V Hl of the i j k l nodes (fig.4) which determine an irregular tetrahedron (finite element). The expression of (V H ) for random may be written 2] under the form (V H ) = 1 6v (c ii c ji x c ki y c li z) V Hi (c ij c jj x c kj y c lj z) V Hj (c ik c jk x c kk y c lk z) V Hk (c il c jl x c kl y c ll z) V Hl ] (35) where: v = ± det (C) /6; v > 0; c rs (r s = i j k l) are the algebraic complements (cofactors) in the matrix (C) of the nodes coordinates which Electronic Journal. 2

3 determine the tetrahedron : 1 x i y i z i (C) = 1 x j y j z j 1 x k y k z k 1 x l y l z l From (35) for the terms of relation (34) results: ( ) VH = 1 (c ji V Hi c jj V Hj c jk V Hk c jl V Hl ) 6v (37) ( ) VH = 1 (c ki V Hi c kj V Hj c kk V Hk c kl V Hl ) 6v (38) ( ) VH = 1 (c li V Hi c lj V Hj c lk V Hk c ll V Hl ) z 6v. (39). (36) z y l i P (x y z) The last two sums of expression (34) may be not detailed in a general treatment because the boundary Fig.4. Finite element conditions for the possible concrete cases are different. In practice we can also have cases when the Neumann conditions are equal to zero what correspond to canceling of the last two sums. With the object of continuing the general analysis of the minimization process of functional we will detail only the first two sums of relation (34) following to take into account for concrete cases the terms corresponding to the Neumann boundary conditions too. With these mentions taking into account the relations ( ) the functional (34) become F = m 1 A xx (c ji V Hi c jj V Hj c jk V Hk c jl V Hl ) 2 A 36v yy(c ki V Hi c kj V Hj c kk V Hk c kl V Hl ) 2 A zz (c liv Hi c lj V Hj c lk V Hk c ll V Hl ) 2 j k x A xy (c jiv Hi c jj V Hj c jk V Hk c jl V Hl ) (c ki V Hi c kj V Hj c kk V Hk c kl V Hl ) A yz (c ki V Hi c kj V Hj c kk V Hk c kl V Hl ) (c li V Hi c lj V Hj c lk V Hk c ll V Hl ) Electronic Journal. 3

4 A zx (c li V Hi c lj V Hj c lk V Hk c ll V Hl ) (c ji V Hi c jj V Hj c jk V Hk c jl V Hl ) ] m { 1 36v A xx (c ji V Hi c jj V Hj c jk V Hk c jl V Hl ) 2 A yy (c ki V Hi c kj V Hj c kk V Hk c kl V Hl ) 2 A zz (c liv Hi c lj V Hj c lk V Hk c ll V Hl ) 2 A xy (c jiv Hi c jj V Hj c jk V Hk c jl V Hl ) (c ki V Hi c kj V Hj c kk V Hk c kl V Hl ) A yz (c ki V Hi c kj V Hj c kk V Hk c kl V Hl ) (c li V Hi c lj V Hj c lk V Hk c ll V Hl ) A zx (c liv Hi c lj V Hj c lk V Hk c ll V Hl ) (c ji V Hi c jj V Hj c jk V Hk c jl V Hl )] 1 6 K x (c jiv Hi c jj V Hj c jk V Hk c jl V Hl )K y (c kiv Hi c kj V Hj c kk V Hk c kl V Hl ) K z (c li V Hi c lj V Hj c lk V Hk c ll V Hl )] }. (40) The expression (40) could be written more concentrated in the following way m m ( F = F F F p) (41) where: F is the general term of the first sum; F is the general term of the second sum; F p is the second general term of the second sum. To minimize the functional we have to cancel its derivatives by magnetic scalar potential of the n nodes from the discretization mesh. Using the concentrated form (41) results m F m F m F p = 0; i = 1 n. (42) The system (42) contains n algebraic equations where the unknowns are the magnetic scalar potentials V Hi (i = 1 n) the 3D discretization mesh of the Electronic Journal. 4

5 studied domain. In particular cases where the studied domain has not permanent magnetization zones (permanent magnets) the terms of the last sum are canceled ( K x = K y = K z = 0). 5. The establish of FEM equations For every node of the discretization mesh it must be known the coordinates (x y z) the adjacent finite elements and the neighboring nodes. In order to establish the rules for writing the equations system two local numbering are introduced: N vi is the number of the neighbor nodes to the node i; M ai is the number of adjacent elements to the node i. The concrete values for N vi and M ai depends on the position of every node i = 1 n in the discretization mesh. For instance for any node i from inside of the tetrahedral mesh N vi = 12 and M ai = 20 3]. The magnetic scalar potential V Hi of the node i appears only in elementary functionless F F and F p which belong to the tetrahedral adjacent to the node i. Thus the equation from the system (42) concerning to any node i from inside of the discretization mesh which has M ai a adjacent elements will be ( a F ) ( a ) F ( a F p ) = 0 (43) where are made the notations CV j = c ji V Hi c jj V Hj c jk V Hk c jl V Hl CV k = c ki V Hi c kj V Hj c kk V Hk c kl V Hl (44) CV l = c li V Hi c lj V Hj c lk V Hk c ll V Hl with which the expressions of elementary functionless are: F 1 = 1 36v 1 (A xx ) 1 (CV j)2 1 ( ) A yy (CV 1 k)2 1 (A zz ) 1 (CV l)2 1 ( ) A xy (CV j) 1 1 (CV k) 1 ( ) A yz (CV k) 1 1 (CV l) ] 1 (A zx ) 1 (CV l) 1 (CV j) F a = 1 36v a (A xx ) a (CV j)2 a ( ) A yy (CV a k)2 a (A zz ) a (CV l)2 a ( ) A xy a (CV j) a (CV k) a ( ) A yz (CV k) a a (CV l) a ] (A zx ) a (CV l) a (CV j) a (45) Electronic Journal. 5

6 F 1 = 1 (A xx ) 1 (CV j)2 1 ( ) A yy (CV 1 k)2 1 (A zz ) 1 (CV l)2 1 ( ) A xy (CV j) 1 1 (CV k) 1 ( ) A yz (CV k) 1 1 (CV l) ] 1 (A zx ) 1 (CV l) 1 (CV j) F a = 1 36v a (A xx ) a (CV j)2 a ( ) A yy (CV a k)2 a (A zz ) a (CV l)2 a ( ) A xy a (CV j) a (CV k) a ( ) A yz (CV k) a a (CV l) a ] (A zx ) a (CV l) a (CV j) a 36v 1 F p1 = 1 (K 6 x ) 1 (CV j) 1 ( ) ] K y (CV k) 1 1 (K z ) 1 (CV l) F pa = 1 6 (K x ) a (CV j) a ( K y ) a (CV k) a (K z ) a (CV l) a ]. (46) (47) In the relations ( ) the subscripts ( a) written after the parenthesis show that the quantities ( CV ( j CV k CV l) (V Hi V Hj V Hk V Hl ) c rs (r = j k l; s = i j k l) A xx A yy A zz A xy A yz ) ( A zx A xx A yy A zz A xy A yz A zx) correspond to the finite element with the same index and (v 1 v 2... v a ) are the adjacent tetrahedral volumes of the node i for what the relation is written. The algebraic complements c rs of the matrix (C) written with the nodes coordinates i j k l which determine the finite element are constant quantities for an established discretization mesh. The elementary functionals F and F p are different by zero only for the finite elements from the zone with permanent magnetization where the functionals F are null. If the element is in the zone without permanent magnet F 0 and F = F p = 0. Derive the relations ( ) the general forms of these three kind of terms from functional becomes: F = 1 18v { A xx c 2 jia yyc 2 kia zzc 2 lia xyc ji c ki A yzc ki c li A zxc li c ji ] VHi A xx c jic jj A yy c kic kj A zz c lic lj 1 2 A xy (c jic kj c ki c jj ) 1 2 A yz (c ki c lj c li c kj ) 1 2 A zx (c li c jj c ji c lj ) ] V Hj A xx c jic jk A yy c kic kk A zz c lic lk 1 2 A xy (c jic kk c ki c jk ) Electronic Journal. 6

7 1 2 A yz (c kic lk c li c kk ) 1 2 A zx (c lic jk c ji c lk ) ] V Hk A xx c jic jl A yy c kic kl A zz c lic ll 1 2 A xy (c jic kl c ki c jl ) 1 2 A yz (c ki c ll c li c kl ) 1 2 A zx (c li c jl c ji c ll ) ] V Hl } (48) F = 1 { A xx 18v c2 ji A yy c2 ki A zz c2 li A xy c jic ki A yz c kic li A zx c ] lic ji VHi A xxc ji c jj A yyc ki c kj A zzc li c lj 1 2 A xy (c ji c kj c ki c jj ) 1 2 A yz (c kic lj c li c kj ) 1 2 A zx (c lic jj c ji c lj ) ] V Hj A xxc ji c jk A yyc ki c kk A zzc li c lk 1 2 A xy (c ji c kk c ki c jk ) 1 2 A yz (c kic lk c li c kk ) 1 2 A zx (c lic jk c ji c lk ) ] V Hk A xxc ji c jl A yyc ki c kl A zzc li c ll 1 2 A xy (c ji c kl c ki c jl ) 1 2 A yz (c kic ll c li c kl ) 1 2 A zx (c lic jl c ji c ll ) ] V Hl } (49) F p = 1 V 6 Hi where = a. ( K x c ji K y c ki K z c ) li (50) We notice that for specify equation i of the system (42) the term having the form (50) represents the free terms of the equation because it doesn t contain the magnetic scalar potentials of the nodes (the unknowns of the problem). After the introduction of the terms ( ) in relation (43) results the general form of an equation from the system (42) written for any node i from inside of the discretization mesh N vi C i V Hi C it V Ht = T L i ; i = 1 n (51) t=1 where we note that: a 1 ( ) C i = Axx c 2 18v jia yy c 2 kia zz c 2 lia xy c ji c ki A yz c ki c li A zx c li c ji (52) t t t=1 Electronic Journal. 7

8 C it = w t=1 1 18v t Axx c ji c jj A yy c ki c kj A zz c li c lj 1 2 A xy (c ji c kj c ki c jj ) 1 2 A yz (c ki c lj c li c kj ) 1 2 A zx (c li c jj c ji c lj ) ] t (53) T L i = 1 6 a t=1 ( K xc ji K y c ki K z c li )t. (54) In relations (52 53) obtained by writing more concentrated the relations (48 49) the terms (A xx ) t (A yy ) t (A zz ) t (A xy ) t (A yz ) t (A zx ) t contain the components of the tensors of the ferromagnetic materials permeability (µ u µ v µ w ) of the air gap (µ u = µ v = µ w = µ 0 ) or of the permanent magnet (µ pu µ pv µ pw ) ( s.1]) as the finite element t adjacent to the node i for which is written the relation (51) is situated in the ferromagnetic material zone in air gap or in permanent magnet. The current index t from the expressions ( ) refers to the finite element = t which could particularized for all the nodes i = 1 n of the discretization mesh identifying the elements (1 M ai ) from around of each node. The quantity w from the relation (53) represent the number of finite element (of the M ai adjacent to the node i) to which any neighbor node (of the N vi neighbor to the node i) is adjacent. For the analyzed case (i an inner node) w = 5 3]. The coefficients c rs from the relations ( ) are obtained by identifying the nodes j k and l of each finite element around the common node i for what the relation (51) is written. Therefore for each node i with unknown magnetic scalar potential an equation having the form (51) could be written. In relation with the concrete position of each node i the number of the neighbor nodes N vi respectively of the adjacent elements M ai of the node i gets particularly values. For the nodes on the boundary surface with non-null Neumann conditions of the studied domain in equation (43) we introduce supplementary particular terms correspond to these conditions (the last two sums from energetic functional (34)). 6. Conclusions If we write the equation (51) for all the nodes i = 1 n with unknowns magnetic scalar potentials of the 3D discretization mesh in which the field problem is analyzed a compatible system of algebraic equations is obtained. The values of the magnetic scalar potentials are gotten by numerical computation of Electronic Journal. 8

9 the system. After that we can get (s.eq ) the components of the magnetic field intensity for each finite element = 1 m: ( ) VH (H x ) = = 1 (c ji V Hj c jj V Hj c jk V Hk c jl V Hl ) 6v (55) ( ) VH (H y ) = = 1 (c ki V Hi c kj V Hj c kk V Hk c kl V Hl ) 6v (56) ( ) VH (H z ) = = 1 (c li V Hi c lj V Hj c lk V Hk c ll V Hl ) z 6v. (57) The components of the magnetic flux density B are determined from H and the tensors of magnetic permeability in the following way: in air-gap B = µ 0 H ; in ferromagnetic materials B = µ H ; in permanent magnets B = µ p H. In conclusion by numerical computation of equation system (51) we get H and B ( = 1 m) in each finite element of the studied domain which is 3D inhomogeneous nonlinear and anisotrop. After we know H and B we can determine other quantities which we are interested in such as: magnetic voltage magnetic flux through certain surfaces forces (torque) applied on coils or electric conductors placed within the air-gap and so on. References 1] I.Bere Inhomogeneous non-linear and anisotropic systems with random magnetisation main directions. Part one: The useful form of functional 3D for computation the field created by permanent magnets Electronic journal Differential equations and control processes N pp(26-35). ( 2] I.Bere Relative permeability of nonlinear and anisotrop permanent magnets Proceedings of the scientific communications A. Vlaicu University Arad 1996 pp(9-14). 3] K.H.Bachmann a.a. Kleine Enzyklopadie der Mathematik Ed. Tehnica Bucuresti 1980 pp( ). Electronic Journal. 9