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**Solution procedures for three-dimensional eddy current problems.**
*(English)*
Zbl 0563.35054

The authors consider scattering of time-harmonic electro-magnetic waves by three-dimensional metallic objects, the conductivity of which is such that displacement currents in the metal are negligible. This generalizes earlier work of S. I. Hariharan and the first author [J. Comput. Phys. 45, 80-99 (1982; Zbl 0478.65080)] on a class of two-dimensional problems.

The electric and magnetic vectors are expressed as asymptotic series in inverse powers of a parameter that is large when the product of frequency and conductivity is large. The first terms in these series correspond to the perfect conductor approximation, and may be determined by solving an exterior boundary value problem. A new integral equation procedure, which makes it relatively easy to calculate the tangential magnetic field on the air-metal interface, is developed. This procedure is analogous to one used to solve scalar problems by means of a simple layer and integral equations of the first kind. Once the perfect conductivity problem has been solved, the remaining coefficients in the series may be found recursively, by solving a sequence of problems of this same kind.

The paper is quite technical. Some familiarity with pseudo-differential operators and Sobolev space theory is required. The asymptotic procedure is described formally; a verification of its asymptotic nature for the two-dimensional problem cited above will be given elsewhere. There is an assumption about the frequency of the waves in order to ensure uniqueness and to avoid difficulties associated with eigenfrequencies of a related interior problem.

The electric and magnetic vectors are expressed as asymptotic series in inverse powers of a parameter that is large when the product of frequency and conductivity is large. The first terms in these series correspond to the perfect conductor approximation, and may be determined by solving an exterior boundary value problem. A new integral equation procedure, which makes it relatively easy to calculate the tangential magnetic field on the air-metal interface, is developed. This procedure is analogous to one used to solve scalar problems by means of a simple layer and integral equations of the first kind. Once the perfect conductivity problem has been solved, the remaining coefficients in the series may be found recursively, by solving a sequence of problems of this same kind.

The paper is quite technical. Some familiarity with pseudo-differential operators and Sobolev space theory is required. The asymptotic procedure is described formally; a verification of its asymptotic nature for the two-dimensional problem cited above will be given elsewhere. There is an assumption about the frequency of the waves in order to ensure uniqueness and to avoid difficulties associated with eigenfrequencies of a related interior problem.

Reviewer: R.Millar

### MSC:

35P25 | Scattering theory for PDEs |

78A45 | Diffraction, scattering |

35C20 | Asymptotic expansions of solutions to PDEs |

35S15 | Boundary value problems for PDEs with pseudodifferential operators |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

### Keywords:

eddy current; scattering; time-harmonic electro-magnetic waves; integral equation procedure; perfect conductivity; pseudo-differential operators; Sobolev space; asymptotic procedure### Citations:

Zbl 0478.65080
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XMLCite

\textit{R. C. MacCamy} and \textit{E. Stephan}, J. Math. Anal. Appl. 101, 348--379 (1984; Zbl 0563.35054)

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### References:

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