TU CE GSC CE People at the Graduate School CE Alumni Todorka Banova Research Project of Todorka Banova

Systematic Data Extraction in High-Frequency Electromagnetic Fields

The field of quantum chaos encompasses the study of the manifestations of classical chaos in the properties of the corresponding quantum or more generally, wave-dynamical system (nuclei, atoms, quantum dots, and electromagnetic or acoustic resonators). Prototypes are billiards of arbitrary shape. In its interior a point-like particle moves freely and is reflected specularly at the boundaries. Depending on the shape its properties could exhibit chaotic dynamics.

Within this work, microwave resonators with chaotic characteristics are simulated and the eigenfrequencies that are needed for its analysis are computed. Accordingly, the eigenfrequency analysis for determining the statistical properties requires many (in order of thousands) eigenfrequencies to be calculated and the accurate determination of the eigenfrequencies has a crucial significance. Moreover, considering that the problem is to compute a large number of eigenfrequencies, they can be often located in different ranges, i.e. left-most, right-most or interior portions of the spectrum could be sought. In many scientific and engineering applications, as well as, in computational science, this results in solving one of the fundamental problems, the large-scale eigenvalue problem. Below are listed just a few of the applications areas where eigenvalue calculations arise: acceleration of charged particles, structural dynamics, electrical networks, Markov chain techniques, combustion processes, chemical reactions, macro-economics, control theory, etc. The above-mentioned realistic applications frequently challenge the limit of both computer hardware and numerical algorithms, as the involved matrices are of large scale.

Various types of numerical methods for eigenvalue determination, i.e., Jacobi–Davidson, Arnoldi, Lanczos, and Krylov–Schur, are available in different software packages: MATLAB, SLEPc, PRIMME library, Trilinos, and so on. Despite the fact that many algorithms for eigenvalue determination exist, accompanied by the numerical models that are becoming increasingly more sophisticated, not as many are specifically adapted for computing thousands of eigen pairs for matrices with dimension in excess of several millions. Thus, some numerical methods might result in an extremely time consuming simulation, along with a slow convergence and huge memory requirements. Particularly, structures with complicated geometry require a large number of grid points to achieve accurate simulation results. Finally, computing a large number of interior eigenvalues remains one of the most difficult problems in computational linear algebra today.

Here, a fast approach for an accurate eigenfrequency determination is addressed, based on a finite element computation of electromagnetic fields for a superconducting cavity and further employment of the Lanczos method for the eigenvalue determination. The Lanczos algorithm with its variations is very attractive for the project necessities. The major practical advantage of this method is the tridiagonal reduction of the eigenvalue problem that yields minimal storage requirements, as do the associated algorithms for its eigenvalue and eigenvector computations. In addition, the Lanczos method takes a significant advantage over its competitors, which concentrate on individual frequency sample per iteration. Therefore, based on the project requirements, the investigations comprise efficient, robust, and accurate computations of many desired eigenfrequencies by employing a proper numerical solution of the electromagnetic problem and an efficient implementation of the Lanczos method to solve the large-scale eigenvalue problem.

Moreover, within this work an approach for extraction of resonant frequencies given the output from time-domain computations of closed resonators is covered. The proposed approach uses the advantage that one single time-domain simulation can provide the whole response of an electromagnetic system in a wide frequency band. In addition, due to the fact that the time-domain computations in the field of electromagnetics are already highly developed and considerably more efficient, as well as the fact that the transient solver contained in CST Microwave Studio uses a high degree of parallelization provided with modern graphics processing units (GPUs) feature the simulation can be dramatically accelerated. Therefore, the time-domain responses for a wide frequency band are easily and quickly obtained. In this way, significant reduction in computation time is achieved and therefore, the high interest leads to time-domain computations for electromagnetic problems. Namely, upon excitation of the cavity, the electric field intensity is recorded at different detection probes inside the cavity. Thereafter, Fourier analysis of the recorded signals is performed and by means of fitting techniques with the theoretical cavity response model (in support of the applied excitation) the requested eigenfrequencies are extracted by finding the optimal model parameters in least square sense.

Eigenvalues, eigenvectors, finite element method, signal processing.

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