A problem of determining the electrostatic field formed by a set of charged electrodes has been considered. The details of the approximate solving of the Dirichlet problem have been given for the Laplace's equation in a substantially spatial formulation based on the use of the model order reduction method. The mathematical models have been improved and the problem of calculating the electrostatic field has been simplified, taking into account the present symmetry of electrodes positioning in electronic optics systems. For the eighth-order abstract group, three independent structures of the corresponding class of systems have been identified. The application domain of the model order reduction method based on finite-group theory for numerically solving integral equations has been extended by transforming the initial boundary-value problem not containing symmetry groups into two problems. The boundary surface of one of them has a finite symmetry group and the other allows for a sufficiently simple numerical solution. This simplification of the problem is aimed at improving the accuracy of computational methods, eliminating sources of instability of these methods, and speeding up computations. To confirm the efficiency of the proposed algorithm, a model problem of calculating the electrostatic field of a quadrupole lens has been considered. The example of its solving demonstrates all the advantages of the developed computational algorithm. A number of numerical experiments have been conducted. The electrostatic field of the corresponding planar approximations has been calculated to verify the validity of the obtained results.
- Salerno M., Landoni P., Verganti R. Designing foresight studies for Nanoscience and Nanotechnology (NST) future developments. Technological Forecasting and Social Change. 75 (8), 1202--1223 (2008).
- Nikulainen T., Palmberg C. Transferring science-based technologies to industry --– Does nanotechnology make a difference? Technovation. 30 (1), 3--11 (2007).
- Antonyuk V. S., Timchik G. S., Vercanova O. V., et. al. Microscopy in nanotechnology: monograph. Kyiv, NTUU "KPI" (2014), (in Ukrainian).
- Okayawa S. A new type of quadrupole lens for electron-beam lithography. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment. 298 (1--3), 488--495 (1990).
- Garkusha І. P., Gorbaruk І. І., Kurinnuj V. P., et. al. General course of physics. Collection of tasks. Kyiv, Technics (2003), (in Ukrainian).
- Romanenko A. V., Ostudin B. A. Calculation of the motion of a charged particle in an electrostatic field, taking into account the spatial charge. Theoretical Electrical Engineering. 59, 87--103 (2002), (in Ukrainian).
- Hawkes P., Kasper E., Kirkland A. Principles of Electron Optics. Applied Geometrical Optics. Vol. 2. 2nd Edition, Academic Press (2017).
- Abramovich S. N., Zavyalov N. V., Zvenigorod A. G., Ignatiev I. G., Magilin D. V., Mel'nik K. I., Ponomarev A. G. Optimization of the probe-forming system of a nuclear scanning microprobe based on the EGP-10 electrostatic charge-exchange accelerator. Journal of Technical Physics. 75, 6--12 (2005), (in Russian).
- Gladkov V. S., Guchenko O. A., Shesterikov O. V. Doslidzhennya experimental installation of the installation of purified transformer oil. Bulletin of NTU "KPI". 39, 46--50 (2009), (in Ukrainian).
- Serre J.-P. Linear representations of finite groups. Moscow, RHD (2003), (in Russian).
- Mochurad L., Harasym Y., Ostudin B. Maximal Using of Specifics of Some Boundary Problems in Potential Theory After Their Numerical Analysis. International Journal of Computing. 8, 149--156 (2009).
- Mochurad L. Method of reduction model for calculation of electrostatic fields of electronic optics systems. Radio Electronics Computer Science Control. 48 (1), 29--40 (2019).
- Sergienko A. B. Digital signal processing: studies, allowance. SPb., Peter (2003), (in Russian).
- Melnik I. V. Using the method of integral equations for numerical simulation of the optical properties of diode systems of a high-voltage glow discharge with an anode plasma. Proceedings of the Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine. 6, 80--85 (2001).
- Chapko R., Tomas Johansson B., Savka Yu. On the use of an integral approach for the numerical solution of a Cauchy problem for Laplace equation in a doubly conneted planar domain. Inverse Problems in Science and Engineering. 22 (1), 130--149 (2014).
- Mochurad L. I., Pukach P. Ya. An Efficient Approach to Calculating the Electrostatic Field of a Quadrupole. Edition of Kherson National Technical University. 62, 155--165 (2017), (in Ukrainian).
- Zakharov Y., Safronov S., Tarasov P. The Abelian groups of finite order in numeral analysis of linear boundary tasks of the theory of potential. Journal of computational mathematics and mathematical physics. 32, 40--58 (1992), (in Russian).
- Mochurad L. I., Ostudin B. A. Flat variant of substantially spatial problem of electrostatics and some aspects of its solution, related to specifics of input information. Journal of Numerical and Applied Mathematics. 2 (105), 98--110 (2011).
- Verlan A. F., Dyachuk A. A., Palagin V. V. Methods of mathematical reduction of models dynamic systems. Kiev, Naukova Dumka (2019), (In Russian).
- Balandin M. Yu., Shurina E. P. The Methods for Solving High-dimensional SLAE. Novosibirsk, NSTU (2000), (in Russian).
- Vladimirov V. S. Equation of mathematical physics. Мoscow, Science (1981).