In this paper, particular attention is paid to the impact of finite‐element approximation on uniqueness and to approximations implicit in finite element formulations from the uniqueness requirements standpoint. It is also shown that the flux density is unique without qualifications. The theoretical and numerical uniqueness of the magnetic vector potential in three‐dimensional problems is also given. This analysis is restricted to linear, isotropic media with Dirichlet Boundary conditions. As an interesting consequence of this analysis it is shown that, under usual conditions adopted in obtaining three‐dimensional finite‐element solutions, it is not necessary to specify div Ā in order that Ā be uniquely defined

Davis, W. A.

Demerdash, Nabeel

Mohammed, O. A.

Nehl, T. W.

Popovic, B. D.

English

epublications@Marquette

Marquette Universitye-Publications@MarquetteElectrical and Computer Engineering FacultyResearch and PublicationsElectrical and Computer Engineering, Departmentof1-1-1982On the Uniqueness of Solution of MagnetostaticVector‐potential Problems by Three‐dimensionalFinite‐element MethodsO. A. MohammedVirginia Polytechnic Institute and State UniversityW. A. DavisVirginia Polytechnic Institute and State UniversityB. D. PopovicVirginia Polytechnic Institute and State UniversityT. W. NehlVirginia Polytechnic Institute and State UniversityNabeel DemerdashMarquette University, nabeel.demerdash@marquette.eduAccepted verison. Journal of Applied Physics, Vol. 53, No. 11 (November 1987): 8402-8404. DOI. ©2019 AIP Publishing LLC. Used with permission.N.A. Demerdash was affiliated with Virginia Polytechnic Institute and State University at the time ofpublication. Marquette University e-Publications@Marquette  Electrical Engineering Faculty Research and Publications/College of Engineering  This paper is NOT THE PUBLISHED VERSION; but the author’s final, peer-reviewed manuscript. The published version may be accessed by following the link in the citation below.  Journal of Applied Physics, Vol. 53, No. 11 (1982): 8402-8404. DOI. This article is © American Institute of Physics and permission has been granted for this version to appear in e-Publications@Marquette. American Institute of Physics does not grant permission for this article to be further copied/distributed or hosted elsewhere without the express permission from American Institute of Physics.  On the uniqueness of solution of magnetostatic vector-potential problems by three-dimensional finite-element methods  O. A. Mohammed Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061  W. A. Davis Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061  B. D. Popovic Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061  T. W. Nehl Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061  N. A. Demerdash Marquette University, Milwaukee, Wisconsin 53233 Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061   ABSTRACT In this paper, particular attention is paid to the impact of finite-element approximation on uniqueness and to approximations implicit in finite element formulations from the uniqueness requirements standpoint. It is also shown that the flux density is unique without qualifications. The theoretical and numerical uniqueness of the magnetic vector potential in three-dimensional problems is also given. This analysis is restricted to linear, isotropic media with Dirichlet Boundary conditions. As an interesting consequence of this analysis it is shown that, under usual conditions adopted in obtaining three-dimensional finite-element solutions, it is not necessary to specify div A ���in order that A ���be uniquely defined.   PACS numbers: 41.10.Dg INTRODUCTION  Although for a relatively long time the magnetic vector-potential has been used for obtaining numerical solutions of magnetostatic problems, it seems that the question of uniqueness of the vector-potential itself has not been treated in detail, particularly in con-junction with three--dimensional finite-element solu-tions. (This, of course, is due to the fact that in the end curl A�, and not A ���itself, is needed.) Interest in questions of validity and uniqueness of numerical 3-D finite element (FE) solutions to magnetostatic problems has been intensified as a result of the pre-sentation.and publication of two papers, References [1] and [2], and their accompanying discussions.  This paper addresses the questions of validity and uniqueness of such 3-D FE solutions. For simplicity, we shall restrict consideration to linear, isotropic media with Dirichlet boundary conditions. Both the curl and divergence nature of the magnetic vector potential (mvp) shall be considered.   STATEMENT OF THE PROBLEM  In this work, one is interested in obtaining a solution to the magnetostatic form of Maxwell's equations given by  ∇  ×  H � =  J ̅ (1a)  ∇  •  B � =  0  (1b)  νB�  =  H�   (1c)  where ν is the reluctivity of the medium.  It is well known that B �  may be expressed as [3]  B � =  ∇ × A�  (2)  where  A�  is the magnetic vector potential (mvp), The mvp is a solution to  νX (ν∇ × A�)  = J ̅  (3)  It is assumed here that A ���is the exact solution which satisfies the given boundary conditions. The object of the following section is to consider the uniqueness of A� obtained in the solution of (3).  THE UNIQUENESS OF THE CURL (∇ × A� =  B� ) If A�1 and A�2 are both solutions to (3) with σA�  =  A�1  −  A�2, then it follows that  ∇ × ν (∇ × δA)  =  0.    (4)  To the define integral the uniqueness statement, we first consider the integral of 𝜈𝜈|𝛻𝛻 × 𝛿𝛿A�|2. One obtains we first the consider following through integration by parts:  ∫ 𝜈𝜈|∇ × 𝛿𝛿A�|2dv V = ∫ [𝛿𝛿A� ∙ ∇ × 𝜈𝜈(∇ × 𝛿𝛿A�)]dv V + ∮ [𝛿𝛿A� × ν(∇ × A�)] S ∙ ds̅   (5)  where S is the given Dirichlet boundary at which n� × A� =  0. Based on this boundary condition, the surintegral in (5) vanishes. This integral would also vanish for the Neumann condition. Hence, upon substituting (4) into (5) one obtains  ∫ 𝜈𝜈|∇ × 𝛿𝛿A�|2dv V = 0  (6)  which, for �ositive v, requires that   ∇ × δA� = 0   (7)  and by the Helmholtz theorem [4]  δA� ≈ ∇∅  (8)  Thus, we see that specifying tangential A� on the boundary surface requires A� to be unique to within the gradient of a potential.   The same results may be obtained for the finite element solution to (3). We begin with the following energy functional which was previously used in refer-ences [1] and [2]:  ρ(U�) = ∫ �(U� − A�) ⋅ ∇ × �𝜈𝜈 × (U� − A�)�� V dv  (9)  where U� is the approximate solution of (3).  Expanding U� as  U� = �𝜀𝜀𝑖𝑖U�𝑖𝑖N𝑖𝑖=1  with N approximation functions U�𝑖𝑖 we may write the first variation of ρ as follows:  𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝜕𝑖𝑖= −∫ �U�𝑖𝑖 ⋅ ∇ × �𝜈𝜈∇ × (U� − A�)� + (U� − A�) ⋅ ∇ × (𝜈𝜈∇ × U�𝑖𝑖)� V dv = 0  (11)  Integrating (11) by parts yields  ∫ [2U�𝑖𝑖 ⋅ (∇ × (𝜈𝜈∇ × U�) − J)̅] V dv − ∮ �(U� − A�) × (𝜈𝜈∇ − U�𝑖𝑖) − U�𝑖𝑖 × �𝜈𝜈∇ × (U� − A�)�� S ⋅ ds̅ = 0 (12)  which gives the Euler equation  ∇ × (𝜈𝜈∇ × U�) = J ̅  (13)  For computational purposes, it is convenient to rewrite (12 ) with Dirichlet conditions as  2∫ [𝜈𝜈∇ × U�𝑖𝑖 ∙ ∇ × U� − U�𝑖𝑖 ∙ J]̅ V dv = 0  (14)  for 𝑖𝑖 =  1,2,••• ,𝑁𝑁. For linear approximation functions, the curls in (14) are piecewise constants requiring only simple integrals. If A�1 and A�2 are two solutions to U� in the numerical problem, then based on (10) and (14) it is easily shown that  ∫ [𝜈𝜈|∇ × (A�1 − A�2)|2] V dv = 0  (15)  and thus A� must also be unique to within V0 for the numerical problem. However-'- since the curl of ∇∅ is zero, the solution for ∇ × A�, that is the flux density B�, is unique with no qualifications.  THE DIVERGENCE OF THE VECTOR POTENTIAL  In obtaining B� as the curl of A� as in the previous section, the divergence of A� is not required and the solution for B� has been shown above to be unique. However, in the numerical aspect of solving (14) this non-uniqueness of A� by ∇∅ will create a singular numerical process for obtaining A� which must be solved by constraining the problem in some way. One method would be to add (∇ ∙ U� − ∇ ∙ A�)2 to the functional integrand and specify the gauge ∇ ∙ A� as suggested by Van Bladel [5] to satisfy the sufficiency requirements of the Helmholtz equation. A second method would be to solve the resultant matrix equation for A� by pseudoinverse methods [6]. A third method which has been used successfully in references [l] and [2] is to specify n� ∙ A� (or (n� ∙ ∇)(n� ∙ A�) at the boundary surface. It has been found that this last technique gives good results for piecewise linear, continuous approximation functions. It is the purpose of this section to show that this uniqueness may be predicted a priori and thus a specific gauge need not be im-posed.   A simplistic view of this uniqueness may be obtained in a numerical form by counting the number of unknowns and equations (independence assumed). For the finite-element formulation we obtain three equations and three unknowns at each interior node. To complete the problem, we require boundary conditions on all three components of the mvp at boundary nodes. This is consistent with the additional constraint suggested above.   Let us consider A�1 and A�2 to be solutions of (12) with Dirichlet boundary conditions on A� (not just n� × A�). We wish to show that the difference, ∇∅, between A�1 and A�2 must vanish in V for piecewise linear, continuous bases (using tetrahedral elements).  Let us consider an element at the boundary surface. Since ∇∅ is linear and must vanish at the boundary, one can write  ∇∅ = C1�n�1 ∙ (r − rs)�  (16)  Where n�1 is the boundary surface normal and r̅s is on the surface. Taking the curls of ∇∅ we obtain  ∇ × (∇∅) = n�1 × C�1  (17)  which must be identically zero. Thus C�1 must e parallel to n�1 and we may write (16) as  ∇∅ = c1[n�1 ∙ (r̅ − r̅s)]n�2  (19)  If these elements fall along an edge such that  n�1 ≠ n�2  (20)  then c and c must be zero since ∇∅ must be continuous at the 1 adjoining 2 faces. Hence, these two tetrahedrons may be deleted from ∇ to determine ∇∅. This result may be generalized to elements with adjacent edges, but non-adjacent faces. Since for a closed, finite volume there must always be two such tetrahedrons, all tetra-hedrons may be deleted to obtain ∇∅ = 0. Thus the ∇ ∙ A� in the solution is unique (though we have not determined the value). This result may be extended to the problem with Neumann boundary conditions. NUMERICAL COMPARISONS  The air-cored coil described in reference (2] has been solved using the discritization grids given in that reference. The solution is obtained here with the boundary conditions A� = 0 on the outermost surface (same as in reference (2]), and also with the condition on the normal component of A replaced by setting the normal derivative to zero (Neumann condition). Table (1) shows these results for arbitrarily chosen tetra-hedral elements in the given volume. The results for the curl and the divergence of the vector potential are also shown for both of the above cases in that table. As can be easily noticed, the values (single precision on an IBM 370) of the curl of the vector potential A� are the same in both cases, hence the curl is unique. However, there is no fixed pattern for the divergence of A�, and the value of the divergence of A� is inconsequential to the flux densities as expected.  Table (1) Values of Curl (lines /sq. inch) and Divergence of the Vector Potential at Arbitrary Points  Pt.  Dirichlet Boundaries    Neumann Boundaries     Bx By Bz ∇ ∙ A� Bx By Bz ∇ ∙ A� 1 64.4 0.0 2055. 2.7 64.3 0.0 2058. 6.5 2 40.4 57.8 2159. -7.7 40.5 57.5 2161. -0.7 3 22.0 -2.1 -67. -2.8 22.5 -2.6 -64. -15.1 4 69.3 -6.9 -190. 1.3 68.9 -7.5 -190. -2.2 5 2.3 17.8 -50 -13.4 2.5 17.7 -50. 1.0 6 220.0 22.5 -274. -22.7 220.2 22.4 -271. -2.3 7 48.4 19.7 -45. -0.9 48.9 18.3 -44. -7.0 8 87.7 548.5 -166. -4.4 87.6 547.9 -165. -22.5 9 -102.3 26.1 310. 85.7 -102.4 25.8 310. 90.8 10 89.7 95.3 -113. 2.7 89.6 95.1 -114. 9.1 CONCLUSION  A theoretical proof of uniqueness for a three dimensional finite element magnetostatic field solution has been given for a class of 3-D problems. It has been shown that the flux density solution is unique with no qualifications. The theoretical and numerical uniqueness of the magnetic vector potential for linear finite-elements have also been derived, where the derivation has been restricted to linear, isotropic media with Dirichlet boundary conditions. The consequence of this analysis is that under usual con-ditions assumed in obtaining a three-dimensional finite-element solution, it is not necessary to explicitly, specify the gauge ∇ ∙ A� in order to uniquely define the vector potential A�. REFERENCES [1] Demerdash, N. A., Nehl, T. W., Fouad, F. A. and Mohammed, O. A., "Three Dimensional Finite Element Vector Potential Formulation of Magnetic Fields in Electrical Apparatus", IEEE Transactions on Power Power Apparatus and Systems, Vol. PAS-100, No. 8, August 1981 pp. 4104-4111. [2] Demerdash, N. A., Nehl, T. W., Mohammed, 0. A. and Fouad, F. A., "Experimental Verification and Application of the Three Dimensional Finite Element Magnetic Vector Potential Method in Electrical Apparatus", IEEE Transactions on Power Apparatus and Systems, Vol. PAS-100, No. 8, August 1981 pp. 4112-4122. [3] Stratton, J. A., Electromagnetic Thoery, McGrawHill, NY, 1941, pp. 23ff.  [4] Collin, R. E., Field Theory of Guided Waves, McGraw-Hill, NY, 1960, pp. 564-568. [5] Van Bladel, J., Electromagnetic Fields, McGrawHill, NY, 1964, pp. 150-152.  [6] Rao, C. R. and S. K. Mitra, Generalized Inverse of Matrices and Its Applications, Wiley, NY, 1971. 

On the Uniqueness of Solution of Magnetostatic Vector‐potential Problems by Three‐dimensional Finite‐element Methods

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On the Uniqueness of Solution of Magnetostatic Vector‐potential Problems by Three‐dimensional Finite‐element Methods

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