Research Division of Mathematical Theory and Applications of Electromagnetic Fields

Staff

Research Projects

• Inverse Boundary, Scattering and Spectral Problems
• Electromagnetic field computing and absorbing boundary conditions
• Microstrip Structures
• Super Conducting Structures in Particle Accelerators and Multipacting

In electromagnetic and acoustic inverse problems the main question is to determine the material parameters of a body from the data measured on the surface of the body or from the scattered data. This research in our institute is carried out in collaboration with Helsinki University of Technology (Prof. E. Somersalo), University of Oulu (Prof. L. Päivärinta), University of Rochester (Prof. A. Uhlmann), Rensselaer Polytechnic Institute (Prof. M. Cheney) and University of Loughborough (Prof. Y. Kurylev). The research has been theoretically oriented.

In the 2-dimensional anisotropic problem Sylvester's and Nachmans's (non)uniqueness result has been extended to conductivities in W^1,p (p > 2) assuming that the conductivity is equal to 1 near the boundary [1].

In the 2-dimensional inverse scattering problem of the Schrödinger equation it has been shown [2] that the local singularities of the unknown potential can be recovered from the backscattered data using the Born approximation.

In the inverse spectral problem of the Maxwell equations, it was shown how the electromagnetic parameters of a body can be recovered from the spectrum of the Maxwell operator and the boundary values of the eigenfunctions [3]. As a new result it is shown that the inverse can also be carried out from a data which is easier to measure [4,5]. The inverse spectral problem has also been studied on a Riemannian manifold [6,7].

In the acoustic inverse problem of an inhomogeneous half-space, the uniqueness of the solution has been proved [8]. In the inverse problem of the impedance tomography it was shown that the Maxwell inverse problem reduces to that of conductivity as the frequency tends to zero [9].

[1]	P. Ola and A. S. Nachman: Reconstruction of rough anisotropic conductivities in two dimensions. - revised manuscript, 1998.
[2]	P. Ola, L. Päivärinta and V. Serov: Recovering singularities from backscattering in two dimensions. - manuscript, 1998.
[3]	M. Lassas: Non-selfadjoint inverse spectral problems and their applications to random bodies. Doctoral thesis, Ann. Acad. Sci. Fenn., Math., Dissertations, 103, pp. 1 - 108, 1995.
[4]	M. Lassas: Inverse boundary spectral problem for non-selfadjoint Maxwell's equations with incomplete data. To appear in Communications in Partial Differential Equations.
[5]	M. Lassas: The essential spectrum of non-selfadjoint Maxwell operator in a bounded domain. To appear in Journal of Mathematical Analysis and Applications.
[6]	Y. V. Kurylev and M. Lassas: The multidimensional Gel'fand inverse problem for non-selfadjoint operators. Inverse Problems 13, pp. 1495 - 1501, 1997.
[7]	Y. V. Kurylev and M. Lassas: Abel-Lidskii basis in non-selfadjoint inverse boundary problem. To appear in Zap. Nauchn. Semin. POMI (in Russian), translation to appear in Soviet Mathematics.
[8]	M. Lassas, M. Cheney and G. Uhlmann; Uniqueness for a wave propagation inverse problem in a half space. To appear in Inverse Problems.
[9]	M. Lassas: Impedance imaging problem as a low frequency limit. Inverse problems 13, pp. 1503 - 1518, 1997.

On this research area computational methods are developed using surface integral equations [1-3], finite element method and absorbing boundary conditions. In particular, the field computing of resonators has been studied [4], and the numerical effectiveness of various surface integral equation formulations have been investigated [5-6]. The results have been applied to the multipacting analysis of a particle accelerator.

The surface integral equation methods have also been applied to computing the fields due to to the eddy currents induced in the hull of a ship [7], to computing the fields of an electrode mine sweeper with a given bottom topology [8], and to modeling the inductive heating of copper [9].

The FEM computing has been applied to the field computing in the impedance tomography [10,11]. Also the grid generating has been studied, and among other things, a grid generator based on the sphere packing method has been constructed. Absorbing boundary conditions and the PML (perfectly matched layer) methods have been investigated [12,13].

[1]	P. A. Martin and P. Ola: Boundary integral equations for the scattering of electromagnetic waves by a homogeneous dielectric obstacle. Proc. of Royal Soc. of Edinburgh, 123A, 185 - 208, 1993.
[2]	J. Sarvas: Solving the scattering problem by surface integral equations. Rolf Nevanlinna Research reports A8, 1992.
[3]	P. Ola: Remarks on a Transmission Problem. J. Math. Anal. and Appl. 196, 639 - 658, 1995.
[4]	P. Ylä-Oijala and E. Somersalo: Computing of Electromagnetic Fields in Axisymmetric RF Structures with Boundary Integral Equations. manuscript, 1998 (submitted to Journal of Electromagnetic Waves and Applications).
[5]	P. Ylä-Oijala: Applications of the boundary Integral Equation Method to Interior Boundary-value Problems for Maxwell's Equations. Licentiate's Dissertation, Rolf Nevanlinna Institute, Research Reports, C29, May 1998.
[6]	P. Ylä-Oijala: Comparison of Boundary Integral Formulations for Electromagnetic Field Computation in Axisymmetric Resonators. manuscript, 1998.
[7]	J. Sarvas, P. Ylä-Oijala ja H. Mäkiö: Alumiinirunkoisten alusten pyörrevirtakenttien laskeminen. (computing the eddy currents in an aluminium hull of a ship). MATINE raporttisarja A, 1997/3, 1-19.
[8]	J. Sarvas ja S. Vänskä: Pohjatopografian vaikutus elektrodiraivaimen synnyttämiin sähkömagneettisiin kenttiin II. (Computing the electromagnetic field with a given bottom topography for an electrode mine sweeper, part II). Rolf Nevanlinna Institute, Research Reports C 30, 1 - 77.
[9]	J. Sarvas: Kuparinauhan induktiokuumentamisen matemaattisia perusmalleja ja laskentaohjelmia. (mathematical models and computing algorithms for the inductive heating of a copper band). Ibid. Ser. C, no. 28, 1997, pp. 1 - 26.
[10]	S. Järvenpää and E. Somersalo: Inpedance Imaging and Electrode Models. Proceedings of the Conference in Oberwolfach, pp. 65 - 74, 1997.
[11]	S. Järvenpää: A Finite Element Model for the Inverse Conductivity Problem. Rolf Nevanlinna Institute, Research Reports C 25, 1996.
[12]	M. Lassas, E. Sarkola and E. Somersalo: The MEI Method and Double Surface Radiation Conditions. submitted to IEEE, Antennas and Propagation.
[13]	M. Lassas and E. Somersalo: On the existence and the convergence of the solution of the PML equations. To appear in Computing.

This is a joint research project with Nokia Research Center and the aim of the project is to build a computer program package for computing the scattering matrix of a multilayer and multiport microstrip structure [1,2]. This package is a part of a larger circuit design program package APLAC, which is, for instance, used in the research and development of mobile telephones. The two-layer computing program with stripline and coaxial feed ports has already been finished and annexed to APLAC [3]. The results will be extended to multilayer models. The mixed potential integral equation has been used with roof-top and triangular basis functions. Effective ways with the Matrix Pencil methods have been developed for fast computing of Green's functions. To speed up the computing a special grid generator is developed which maximizes the number of congruent pair of the basis triangles.

[1]	J. Sarvas: Electromagnetic field computation in multiport microstrip structures. Proceedings of the Claremont Conference on differential equations in industry, pp. 423 - 430, 1994.
[2]	J. Sarvas, M. Taskinen and S. Järvenpää: Computing the scattering matrix of a multiport and multilayer microstrip patch with the mixed potential integral equation method. PIERS96 Conference, 1996.
[3]	APLAC 7.0 Reference Manual II, pp. 274 - 283, 1996

The multipacting is a harmful phenomenon which may occur in the super conducting structures of a particle accelerator. Electrons emitted from the walls of a resonator may enter resonant trajectories and be multiplied by successive impacts on the walls. This multipacting may lead to a harmful electron avalanche, which one wants to avoid by a proper design of the resonators. Rolf Nevanlinna Institute in collaboration with DESY (Deutsches Elektronen-Synchrotron, Hamburg, Germany) has studied the multipacting analysis and methods to suppress multipacting [1-3]. This has been a part of the international TESLA project. The developed methods have been applied to coaxial and other feed lines and accelerator resonators for various kinds of wave forms [4]. Especially, the electromagnetic field computing methods for structures with ceramic windows have been developed. Various methods for suppressing multipacting have been studied [5]. A fast electron trajectory computing in non-axially symmetric structures has also been developed [6].

[1]	E. Somersalo , P. Ylä-Oijala and D. Proch: Electron multipacting in RF structures. TESLA Reports, 14 - 94.
[2]	E. Somersalo, P. Ylä-Oijala and D. Proch: Analysis of multipacting in coaxial lines. FAE08, IEEE Proceedings, PAC 95, pp. 1500 - 1502, 1996.
[3]	E. Somersalo, P. Ylä-Oijala, D. Proch and J. Sarvas: Computational Methods for Analyzing Electron Multipacting in RF Structures. Particle Accelerators, 1998, in press.
[4]	P. Ylä-Oijala: Analysis of Electron Multipacting in Coaxial Lines with Travelling and Mixed Waves. TESLA Reports, 97 - 20, 1997.
[5]	P. Ylä-Oijala: Suppressing Electron Multipacting in Coaxial Lines by DC Voltage. TESLA Reports 97 - 21, 1997.
[6]	M. Ukkola, K. Tarvainen and J. Sarvas: Numerical Calculation of the 3-dimensional Flight Path of an Electron in an Accelerator. Rolf Nevanlinna Institute, Research Reports C 26, 1996.