*!)-A088 230 NAVAL UNDERWATER SYSTEMS CENTER NEW LONDON CT NWL--ETC P/6 20/3AFUNCTIONAL FOR FINITE ELEMENT ANALYSIS OF LOW INTENSITY NAffiE-ETC(U)

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MICROCOPY RESOLUTION TEST CHARTNATIONAL. BUREAU Of STANDARDS 1963-A

NUSC Techncal Report 6239

ILEVEL

.J i A Functional For Finite ElementAnalysis of Low Inten.tyMagnetic Fields Including

Permanent Magnetization Effectslet

iJoh. W. Frye

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DTICS ELECTE1 "AUG 22 1

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This report was prepared under NUSC Project No. A53223, "Finite ElementModeling of Electric and Electromagnetic Fields" (U), Principal Investigator, Rt. 0.aKasper (Code 401), and Associate Principal Investigator, J. W. Frye (code 401),Navy Subproject and Task No. B-COO, 11221; Naval Industrial Funding fromDavid W. Taylor Naval Ship Research and D4evelopment Center, Project Director,W. Andahazy (DTNSRDC, Code 2704); Program Manager, W. L. Welsh

(NAVMAT PM-2-2 1).The Technical Reviewer for this report Was M. Melehy, Code 401 (Professor of

Electrical Engineering at the University of Connecticut).

Reviewed .. il Approveth 17 Jew. IOU

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/Y -,',/ --]REPORT DOCUMENTATION PAGE READ INSTRUCTIONSBE FORE COMPLETING FORM'' E- t'rON NUB£ ER GO3ACSINN. )RE~CIPIENT'S CATALOG HUMOIEN

M-6289SIN T REPORT &Pti RI COVED

A FUNCTIONAL FOR INITE JLEMENT.AALYSIS OF 1OW /oC/

INTENSITY 4AGNETIC JIELDS INCLUDING PERMANENTMAGNETIZATION EFFECTS.

- A.I'O()- 111. CONTRACT O GRANT NUMUIERt()

John rye9. PERFORMING ORGANIZATION NAME ANO AOORESS I0. PROGRAM ELEMENT. PROJECT. TASK

Naval Underwater Systems Center ARA 3 WORK UNIT NUMUERS

New London Laboratory A0,2 2 2New London, CT 06320

II. CONTROLLING OFFICE NAME ANO ADDRESSDavid W. Taylor Naval Ship Research and 17 J 8fl

Development Center 3. NUMBER OF PAS

Annapolis, MD 21402 414. MONITORING AGENCY NAME & AOORESS(If dfleml from Controllin Office) IS. SECURITY CLASS. (ol this report)

Naval Material Command (PM-2-21) UNCLASSIFIEDWashington, DC 20362 OECLASIIP|CATON/OOWNGRAOINGcC~iiij//' I~a.SCHEDULE

16. 0ISTRISUTION STATEMENT (of this Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of the .bat1,ct etered In Block 20. It ditteo t Ino Reprt)

I. SUPPLEMENTARY NOTES

19. KEY WOROS (Contdnue on revers ide It necceewy O Identtly by block mmm9)Magnetic field PermeabilityFinite element Reuanent flux

.* Biot-Savartcalar potential

20. Ag9UACT (Continue an t e dm ad. It noceeomy and Idaentiy by we*& mbwt)

'A functional is presented that is useful in the finite element analysis oflow intensity, static magnetic field solutions. The variational principle iswritten in terms of a scalar potential function and, when minimized, leads tofield equations and boundary conditions that characterize a magnetic field--accounting for the presence of current, permanent magnetization, and incidentmagnetic fields. This variational principle finds applications in the finiteelement analysis of magnetic field problems.

Do 'A1 1473 " S., ,o I Nov dai ,.Io. /C . ,a/" 101014640

TR 6289 1

A Functional for Finite ElementAnalysis of Low IntensityMagnetic Fields Including

Permanent Magnetization Effects

Introduction

The analysis of static magnetic fields using scalar potential functions inassociation with the Biot-Savart law has been reported in the open literature byZienkiewicz et al.1 and Armstrong et al.2 Wikswo3 has indicated how widelydistributed current distributions can be economically treated using this approach.!Although the work of Zienkiewicz et al.' uses a general variational, there is nodiscussion on how a problem with permanent magnetization effects should beapproached.

Solution Approach

If the magnetic fields are of low intensity or do not change rapidly over themagnetized body, the change in magnetic flux as a function of magnetic field in-tensity will be linear; i.e.,

B = Bit + jAjH,(!

where pA is the incremental permeability and BR is the remanent magnetization

measured by drawing a tangent from the operating point on the B-H curve to the H= 0 axis.

Following the usual scalar potential solution to magnetic field problems, wehave

= , +V+, (2)

where ITc is the Biot-Savart law magnetic field strength due to currents, it, is theapplied farfield magnetic field strength, and # is an unknown continuous scalarpotential function. For the purposes of this discussion, I1c and it, are assumed tohave either low amplitudes or to be changing slowly, so that #4 can be taken as aconstant for the problem.

Using the assumed solution of equation (2), we immediately satis one ofMaxwell's equations involving the curl of the magnetic field strength H and thecurrent densityT:

Vx jT- 'J.(3)t+1

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Furthermore, the continuity requirements on the scalar potential functionprovide for the satisfaction of the boundary condition between dissimilar materialsinvolving the magnetic field strength components tangent to the boundary:

nxHI, = xHb., (4)

where"n is a vector normal to the boundary and the symbols I1 and 12 indicate thatthe operation is to be carried out in materials I and 2, respectively.

The relation requiring that the divergence of the flux density must vanish fur-nished the field relation for the determination of the scalar potential function:

V 0BO. (5)

Substitution of equations (1) and (2) into equation (5) gives

" O ATV+ = -V P 1aHC -V 0aAH" -V B'R. (6)

The boundary condition requiring that the normal component of the flux densitybe constant across the boundary must also be satisfied:

neI, = nOB12. (7)

A further boundary condition on + is that it decay to zero as the distance from the

permanently magnetized body becomes large:

+ = 0 as I T- B.s, (8)

where 1 -7" is the distance between the point r at which + is evaluated and thenearest point of the finitely sized permanently magnetized region.

Functional and Finite Element Equations

A functional that, when minimized, leads to the field and boundary conditions ofequations (6) and (7) is given below.

X =f /2 if+ + Ii + 'c) • (u' + moiii +A-H + 2B*)dV. (9)

This functional may be used as the basis of a finite element formulation. In such aformulation the scalar potential function is set equal to the sum of the products ofinterpolation functions with discrete unknowns:

* - ± N 1, (10)

2

. . .. ... .. .. . .. .. ...... . . ... .. ...... . .... . .... .... . . .. .. . -•n [

TR 6289

where Ni = Ni (x, y, z) are the interpolation functions, +j are discrete unknownvalues of potential associated with various discrete points in the problem domain,and n is the number of discrete unknowns.

A discussion of the details of the finite element method is unnecessary here sincesuch details are covered in any standard text on the subject.4

When equation (10) is substituted into the variational principle (9) and thefunctional is minimized with respect to the j unknowns, a set of linear algebraicequations results that can be conveniently represented in matrix form:

[K] {+) + {C + {fl) + {fR) = 0, (1)

where

k= 'NVN. A'VNjdV (12)V

fci= fvNi. PAHcdV (13)

f rV"N, 0 #,H dV (14)

f,, = fN 1 RdV. (iS)V

Equations (12) and (13) are identical to those given by Zienkiewicz et al.'Equation (14) shows the "loading" term relation that accounts for the incidentmagnetic field on the body. If we use this solution approach for a problem with anincident magnetic field, the finite element solution gives us the magnetic fielddisturbance caused by the body directly. Often the magnetic field disturbance is thetopic of interest in such problems. Equation (15) shows the term accounting forpermanent magnetization effects; the distribution of remanent flux density BR mustbe known for this formulation.

Concluons

Equations (11) through (IS) allow determination of the magnetic field around apermeable body. The equations account for the effects of currents, incident fields,and permanent magnetization. Since the heat transfer elements in standard finiteelement programs have the same formulation as that given by equation (12), suchprograms may be used to solve a variety of magnetic field problems. A preprocessormust be developed, however, to calculate the loadings defined by equations (13)through (15), and a postprocessor must be used to perform the additional operationgiven by equation (2) in order to obtain the total field solution.

3

L- .. . . ) ... -. 4

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References

1. 0. C. Zienkiewicz, John Lyness, and Dr. J. Owen, "Three-DimensionalMagnetic Field Determination Using a Scalar Potential - A Finite ElementSolution," lEE Transactions on Magnetics, vol. 13, no. 5, September 1977, pV.1649-1656.

2. A. G. Armstrong, et al., The Solution of 3D Magnetostatic Problems UsingScalar Potentials, Rutherford Laboratory Report RL-78-0888, RutherfordLaboratory, Chilton, Didcot, Oxon, OXI 100X, U.K., September 1978.

3. J. P. Wikswo, Jr., "The Calculation of the Magnetic Field from a CurrentDistribution- Application to Finite Element Techniques," IEEE Transactions onMagnetics, vol. 14, no. 5, September 1978, pp. 1076-1077.

4. 0. C. Zienkiewicz, The Finite Element Method in Engineering Science,McGraw-Hill, London, 1971.

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AMMSSON forNTIS White SftftDOC Butf seo 0UNANN3OV#VEBJUSTI CtION................... I

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7tR 6289

Initial Distribution ListNO. OF

ADDRESSEE COPIES

ONR, Code 427, 483, 412-8, 480,410, Earth Sciences Division (T. Quinn) 6NRL, (J. Davis, W. Meyers, R. Dinger, F. Kelly, D. Forester,

Code 645 1, J. Clement, Code 6454) 6NAVELECSYSCOMHQ, CodeO03, PME-1 17, -117-21, -117-213,

-1 17-213A, -1 17-215 6NELC (R. Moler, H. Hughes, R. Pappert, Code 3300) 3DTNSRDC/Annapolis, (W. Andahazy, D. Everstine, F. Baker,

Code 2704; B. Hood, D. Nixon, Code 2782; E. Bieberich, Code 2813) 6DThSRDC/Cardarock (R. Knutson, Code 1548, J. Stinson,

Code 1102.2, M. Hurwitz, Code 1844) 3NAVSURFWPNCEN, (K. Bishop, M. Lackey, Jr., W. Menzel,

E. Peizer, Code WE-12), R. Brown, J. Cunningham, Jr., M. Kraichrnan,G. Usher, Code WR-43) 8

NAVCOASTSYSLAB, (C. Stewart, Code 721, K. Allen, Code 773, M. Wynn,W. Wynn, Code 792) 4

NAVSEC, (C. Butler, G. Kahier, D. Muegge, Code 6157 B) 3NAVFACENGSYSCOM, (W. Sherwood, Rt. McIntyre, A. Sutherland,

Code FPO-1IC7) 3NAVAIR, (L. Goertzen, Code AIR-0632 B) 1NAVAIRDEVCEN, (J. Duke, R. Gasser, E. Greeley, A. Ochadlick,

L. Ott, W. Payton, W. Schmidt, Code 2022) 7NISC (G. Batts, Code 20, J. Erdmann, Code 43, M. Koontz, Code OW 17) 3NOSC, (C. Ramstedt, Code 407) 1NAVPGSCOL, (R. Fossum, Code 06) 1U.S. Naval Academy/Annapolis (C. Schneider) IGTE Sylvania/Needham, MA (G. Pucillo, D. Esten, R. Warshamer,

D. Boots, R. Row) 5Lockheed/Palo Alto, CA (J. Reagan, W. Imhof, T. Larsen) 3Lawrence Livermore Labs/Livermore; CA (J. Lytle, E. Miller, L. Martin) 3Raytheon Co. /Norwood, MA (J. deBettencourt) IU/Nebraska/Lincoln, NB (E. Bahar) INewmont Exploration Ltd./Danbury, CT (A. Brant) IIITRI/Chicago, IL (J. Bridges) IStanford U/Stanford, CA (F. Crawford)IU/Colorado/Boulder, CO (D. Chang) ISRI/Menlo Park, CA (L. Dolphin, Jr., A. Fraser-Smith, J. Chown,

Rt. Honey, M. Morgan) 5Colorado School of Mines/Golden, CO (R. Geyer, 0. Keller) 2U/Arizona/Tucson, AZ (D. Hastings) IU/Michigan/Ann Arbor, MI (R. Hiatt) IU/Washington/Seattle, WA (A. Ishimaru)I

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Initial Distribution List (Cont'd)NO. OF

ADDRESSEE COPIES

U/Wisconsin/Madison, WI (R. King) IU/Wyoming/Laramie, WY (J. Lindsay, Jr.) IU/Illinois/Urbana, IL (R. Mittra) 1U/Kansas/Lawrence, KS (R. Moore) IWashington State Univ./Puilman, WA (R. Olsen) 1N. Carolina State Univ./Raleigh, NC (R. Rhodes) 1Ohio State Univ./Columbus, OH (J. Richmond) IMIT Lincoln Lab./Lexington, MA (J. Ruze, D. White, J. Evans,

A. Griffiths, L. Ricardi)5Purdue Univ./Lafayette, IN (W. Weeks) IU/Pennsylvania/Philadelphia, PA (R. Showers) 1EBDIV. General Dynamics/Groton, CT (R. Clark, L. Conk,

H. Hemond, G. McCue, D. Odryna) 5Science Application Inc./McLean, VA (J. Czika) 1IJHU/APL, Silver Spring, MD (W. Chambers, P. Fueschel, L. Hart, H. Ko) 4DTIC 12

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