1161 zy IEEE zyxwvutsrqponmlkjihg TRANSACTIONS ON MAGNETICS, VOL. 33, NO. 2, MARCH 1997 Numerical Techniques for Solving Magnetostatic Problems Involving Motion and Nonuniform Geometry Young-Kil Shin Dept. of Electrical Engineering, Kunsan National University, Kunsan, Chonbuk 573-70 zyxw 1, Korea Yushi Sun and William Lord Dept. of Electrical and Computer Engineering, Iowa State University, Ames, zyxw IA 5001 1, U S.A Abstract-Numerical techniques to solve moving problems with nonuniform geometry are presented. Common features of successful techniques are in good agreement with those of up- winding techniques. They are the evaluation of motional induc- tion term at the old time level and the introduction of the same time dependent artificial term. The unsuccessful time step method is found to be equivalent to the standard Galerkin method which requires mesh refinement. The application of the variable reluctance probe is studied as an example. I. INTRODUCTION As often found in nondestructive evaluation (NE) prob- lems, the geometric shape concerned in this paper is nonuni- form in the direction of motion. Examples of such objects include test specimen with defect, and/or with thickness variation and a tube with support plate. In NDE applications, it is essential for detection probes to be scanned over a test specimen to find more information about the test objective. Since most electromagnetic zyxwvuts (EM) NDE probes do not require contacting the specimen, fast moving inspection is possible even though it causes motional induction. If the probe is assumed to be relatively stationary, the nonuniform object moves in the opposite direction to that of the probe. Then, the geometrical relationship, between the nonuniform mov- ing part and the stationary part, keeps changing with time. This results in continuous changes in the distribution of motional induction currents. Since induced currents set up their own magnetic fields, even the field values at the sta- tionary part change with time. Therefore, this is truly a tran- sient situation as pointed out in [ I] and [2]. This paper chooses a problem of identifying the support plate outside the tube from inside using a Hall plate mounted on the variable reluctance (VR) probe [3]. The governing equation for this axisymmetric, moving problem can, thus, be written as where v, zyxwvutsrq o, and V are the reluctivity.,the conductivity, and Manuscript received March 19, 1996, revised August 23. 1996. Young-Kil Shin, e-mail ykshin@knusunl .kunsan.ac.kr. fax 82-654-466- 2086; Yushi Sun, e-mail suny(@iastate.edu, fax 1-515-294-1 152; William Lord e-mail lord@ee.iastate.edu. the constant velocity of specimen that has only a z compo- nent, respectively. The magnetic vector potential, A, and the applied dc current density, J, has only a I$ component. Since (1) includes a time derivative term, a time step algo- rithm is necessary. In order to find a suitable time step method, some techniques, used in the solution of convective diffusion problems in fluid mechanics, are investigated whether they can be applied to our moving EM problems. Two of them are successful and produce solutions that are very close to the steady state upwinding result. This justifies their use in moving EM problems with nonuniform geometry. In the mean time, we realize that they are closely related to upwinding techniques in terms of the treatment of spurious oscillations and the first order spatial derivative (i.e., mo- tional induction) term. It also agrees well with the theory of electromagnetics.These are explained in this paper. 11. FINDINGS FROM ANALYSIS OF UPWINDING TECHNIQUES In solving the motional induction problems, spurious os- cillations may occur in domain based numerical solutions if the mesh size in the direction of motion exceeds a certain limit. This limit is expressed by the cell magnetic Reynolds number (R,= oWv) to be less than 2 [4]. To remove such oscillations, severe mesh refinement is necessary. However, this increases the burden on the computer resources and may limit the practical use of the numerical method. Upwinding techniques, developed to overcome such diffculties, have been successfully applied to steady state moving problems. However, numerical dissipation associated with these tech- niques prevents them being used with time step methods [5]. Investigation of finite difference upwind scheme zy [6], [7] and finite element upwinding techniques (both the Petrov- Galerkin method [SI, [!I] and upwinding quadrature methods [IO], [ll]) shows that all these techniques are, in effect, applied only to the first order spatial derivative term [12] and they give emphasis to the condition of the upwind posi- tion. In EM NDE problems, the upwind position is the area where the probe already passed, thus, motional induction currents are already formed. The reason for such emphasis is that upwinding techniques are developed for steady state equations. In transient problems, the upwind position corre- sponds to the previous time. This means that the motional 0018-9464/97$10.00 zyxwvut 0 1997 IEEE