Fast and accurate algorithm of 3D magnetic surfaces reconstruction in stellarators

Fast and accurate algorithm of 3D magnetic surfaces reconstruction in stellarators

An algorithm of calculation of approximating functions, which establish the one-to-one correspondence between the real coordinate mesh with arbitrary step and magnetic flux label in the whole plasma volume, was developed. It allows one to calculate flux label along the lines of sight of the applie...

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Дата:2018
Автори: Filippov, V., Grekov, D., Olefir, V.
Формат: Стаття
Мова:English
Опубліковано: Національний науковий центр «Харківський фізико-технічний інститут» НАН України 2018
Назва видання:Вопросы атомной науки и техники
Теми:
Магнитное удержание
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/148813
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Цитувати:Fast and accurate algorithm of 3D magnetic surfaces Reconstruction in stellarators / V. Filippov, D. Grekov, V. Olefir // Вопросы атомной науки и техники. — 2018. — № 6. — С. 27-30. — Бібліогр.: 2 назв. — англ.

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Digital Library of Periodicals of National Academy of Sciences of Ukraine
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spelling irk-123456789-1488132019-02-19T01:28:09Z Fast and accurate algorithm of 3D magnetic surfaces reconstruction in stellarators Filippov, V. Grekov, D. Olefir, V. Магнитное удержание An algorithm of calculation of approximating functions, which establish the one-to-one correspondence between the real coordinate mesh with arbitrary step and magnetic flux label in the whole plasma volume, was developed. It allows one to calculate flux label along the lines of sight of the applied diagnostics and to define the probes locations and RF antenna position in relation to the last closed magnetic surface. Moreover, now it is possible to provide fast visualization of magnetic configuration during the experiment. Наведено швидкий та точний алгоритм обчислення функцій, що апроксимують магнітні поверхні торсатрона. Вони здійснюють взаємно однозначну відповідність між просторовими координатами в об’ємі плазми та позначкою магнітних поверхонь. За допомогою цих функцій можна обчислювати значення позначки магнітних поверхонь вздовж ліній зондування різних діагностик, визначати розташування зондів та положення ВЧ-антен відносно останньої замкнутої магнітної поверхні. Більш того, стає можливим забезпечити швидку візуалізацію магнітної конфігурації торсатрона під час проведення експерименту Представлен быстрый и точный алгоритм вычисления аппроксимирующих функций, которые устанавливают взаимно однозначное соответствие между пространственными координатами в объеме плазмы и меткой магнитных поверхностей. С помощью этих функций можно вычислять значение метки магнитных поверхностей вдоль линий зондирования различных диагностик, определять положение зондов и позицию ВЧ-антенн относительно крайней замкнутой магнитной поверхности. Более того, стало возможным обеспечить быструю визуализацию магнитной конфигурации торсатрона во время проведения эксперимента. 2018 Article Fast and accurate algorithm of 3D magnetic surfaces Reconstruction in stellarators / V. Filippov, D. Grekov, V. Olefir // Вопросы атомной науки и техники. — 2018. — № 6. — С. 27-30. — Бібліогр.: 2 назв. — англ. 1562-6016 PACS: 52.27.Ny http://dspace.nbuv.gov.ua/handle/123456789/148813 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
topic Магнитное удержание
Магнитное удержание
spellingShingle Магнитное удержание
Магнитное удержание
Filippov, V.
Grekov, D.
Olefir, V.
Fast and accurate algorithm of 3D magnetic surfaces reconstruction in stellarators
Вопросы атомной науки и техники
description An algorithm of calculation of approximating functions, which establish the one-to-one correspondence between the real coordinate mesh with arbitrary step and magnetic flux label in the whole plasma volume, was developed. It allows one to calculate flux label along the lines of sight of the applied diagnostics and to define the probes locations and RF antenna position in relation to the last closed magnetic surface. Moreover, now it is possible to provide fast visualization of magnetic configuration during the experiment.
format Article
author Filippov, V.
Grekov, D.
Olefir, V.
author_facet Filippov, V.
Grekov, D.
Olefir, V.
author_sort Filippov, V.
title Fast and accurate algorithm of 3D magnetic surfaces reconstruction in stellarators
title_short Fast and accurate algorithm of 3D magnetic surfaces reconstruction in stellarators
title_full Fast and accurate algorithm of 3D magnetic surfaces reconstruction in stellarators
title_fullStr Fast and accurate algorithm of 3D magnetic surfaces reconstruction in stellarators
title_full_unstemmed Fast and accurate algorithm of 3D magnetic surfaces reconstruction in stellarators
title_sort fast and accurate algorithm of 3d magnetic surfaces reconstruction in stellarators
publisher Національний науковий центр «Харківський фізико-технічний інститут» НАН України
publishDate 2018
topic_facet Магнитное удержание
url http://dspace.nbuv.gov.ua/handle/123456789/148813
citation_txt Fast and accurate algorithm of 3D magnetic surfaces Reconstruction in stellarators / V. Filippov, D. Grekov, V. Olefir // Вопросы атомной науки и техники. — 2018. — № 6. — С. 27-30. — Бібліогр.: 2 назв. — англ.
series Вопросы атомной науки и техники
work_keys_str_mv AT filippovv fastandaccuratealgorithmof3dmagneticsurfacesreconstructioninstellarators
AT grekovd fastandaccuratealgorithmof3dmagneticsurfacesreconstructioninstellarators
AT olefirv fastandaccuratealgorithmof3dmagneticsurfacesreconstructioninstellarators
first_indexed 2025-07-12T20:19:59Z
last_indexed 2025-07-12T20:19:59Z
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fulltext ISSN 1562-6016. ВАНТ. 2018. №6(118) PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2018, № 6. Series: Plasma Physics (118), p. 27-30. 27 FAST AND ACCURATE ALGORITHM OF 3D MAGNETIC SURFACES RECONSTRUCTION IN STELLARATORS V. Filippov1, D. Grekov1,2, V. Olefir2 1National Science Center “Kharkov Institute of Physics and Technology”, Institute of Plasma Physics, Kharkiv, Ukraine; 2V.N. Karazin Kharkiv National University, Kharkiv, Ukraine E-mail: grekov@ipp.kharkov.ua An algorithm of calculation of approximating functions, which establish the one-to-one correspondence between the real coordinate mesh with arbitrary step and magnetic flux label in the whole plasma volume, was developed. It allows one to calculate flux label along the lines of sight of the applied diagnostics and to define the probes locations and RF antenna position in relation to the last closed magnetic surface. Moreover, now it is possible to provide fast visualization of magnetic configuration during the experiment. PACS: 52.27.Ny INTRODUCTION Calculation of the specific vacuum magnetic config- uration of “Uragan-2M” torsatron using Nemov’s de- composition of magnetic field potentials [1] takes few minutes of PC processor time. Calculations with using Biot – Savart law [2] take even more time. The results of calculation of one configuration (60 toroidal cross- sections) occupy about 250 Mb of memory storage. On- line operation with such data arrays or calculation of magnetic configuration during the experiment is hardly possible. The principal objective of this work was to develop the algorithm of calculation of approximating functions, which establish the one-to-one correspond- ence between the real coordinate r  mesh with arbitrary step and magnetic flux label  in the whole plasma volume. 1. THE ALGORITHM DESCRIPTION Three coordinate systems which can be easily con- nected each other were used in this work. These coordi- nate systems are presented in Fig. 1. The first coordinate system is the cylindrical one with the axis along the main axis of the torus ),,( ZR  for initial calculations of magnetic field lines (Poincare plots). The second coordinate system is the “quasicylindri- cal” one ),,( z , which is connected with the magnetic axis. It is necessary for intermediate calculations. Coor- dinates  and z are defined as     axisaxis ZZzRR  , , (1) here axisR and axisZ are coordinates of the magnetic axis. Fig. 1. Coordinate systems: 1 – vacuum vessel; 2 – magnetic axis; 3 – last closed magnetic surface (LCMS) The third coordinate system ),,( r is quasitoroidal one, which is also connected with the magnetic axis   zzr arctan,22  . (2) As an input parameters, L = 34 magnetic surfaces con- sisting of M = 400 poloidal points with N = 60 toroidal cross-sections were calculated using method [1] and used as example (Fig. 2). It should be noted that the values of the set (L, M, N) may vary. Also, the magnetic axis position in each of N cross-sections must be sup- plied. The dependencies of axisR and axisZ on toroidal angle  are shown in Fig. 3. As can be clearly seen from this figure, these dependencies are perfectly ap- proximated by series of the form     sVR S saxis 4cos 0 ,     sWZ S saxis 4sin 1 . (3) The least squares method (LSA) was used to approx- imate Vs and Ws. As it turned out, the averaged approxi- mation error was of the order of 10-9 сm at S = 7. Fig. 2. Poincare plots in three toroidal cross-sections separated by 8  28 ISSN 1562-6016. ВАНТ. 2018. №6(118) Fig. 3. Dependencies of axisR (solid line) and axisZ (dashed line) on toroidal angle Hereinafter, each toroidal cross-section is indexed by n (1 < n < N). Each magnetic surface is indexed by l (1 < l < L). And each point at magnetic surface is in- dexed by m’ (1 < m’ < M). All subsequent actions relate to each magnetic surface and each toroidal cross- section. Unfortunately, input data nmlR ,,  and nmlZ ,,  are dis- tributed over magnetic surface irregularly in angle  . This makes the accurate approximation of magnetic configuration impossible. In order to overcome this ob- stacle, it is necessary to rearrange input data. First, set nmlR ,,  and nmlZ ,,  was transformed into set nml ,,  and nmlz ,,  . As it is seen from Fig. 4, nml ,,  and nmlz ,,  depend on m almost periodically. This motivated the expansion of nml ,,  and nmlz ,,  into trigonometric se- ries over m. To this purpose, dependencies of nl , on φ were established (Fig. 5) and coefficients of linear re- gressions nl , were defined. Fig. 4. Dependencies of 17,,15 m (solid line) and 17,,15 mz (dashed line) on m Finally, the values nl , were averaged over the toroidal angle N N nll  1 , . These allowed to expand nml ,,  and nmlz ,,  into series         K k l k nl K k l k nlnl kmbkmam 1 , 0 ,, sincos  ,         K k l k nl K k l k nlnl kmdkmcmz 1 , 0 ,, sincos  . (4) Fig. 5. Dependence of poloidal angle 17,15 on toroidal angle φ The value K = 11 was used to find coefficients by LSA. The error of LSA in this case was under 10-2 cm. Then these series were used in order to recalculate sets nml ,, and nmlz ,, at values m , which are separated by constant step M 2 . Values of lm  , corre- sponding to these m , were calculated from Eqs. (2), (4) iteratively with relative error ~ 10-3. New M = 128 was adopted and new poloidal index m = 1.2…128 K (Fig. 6). Fig. 6. New set of input data in toroidal cross-section n = 15 After that set nmlr ,, was calculated using new nml ,, and nmlz ,, . An averaged radius was calculated as  NMrr N n M m nmll     1 1 ,, and flux surface label l was defined as Lll rr . Such definition provide 0 at the magnetic axis and 1 at the LCMS. Fig. 7. Dependence of 50,15 on r ISSN 1562-6016. ВАНТ. 2018. №6(118) 29 Thus, on the rays, which started from magnetic axis in toroidal cross-section n in poloidal direction m , the one-to-one correspondence established between nmr , and nm, :  nmnm rr ,,  and  cnmnm ar,,   , where ca is the vessel radius (see Fig. 7). This corre- spondence was approximated like       I i i c i nmcnm arfar 1 ,, and      I i ii nmcnm Far 1 ,,  . Fig. 8. Coefficients of polynomial decomposition of ψ vs poloidal angle for n = 15(f 1 – solid; f 2 – dashed; f 3 – dot and dashed lines) As the calculations showed, I = 3 is sufficient for these representations (Fig. 8). Then, coefficients i nmf , (see Fig. 8) and i nmF , were expanded into series over  :     jgjgf J j ji n J j ji n i nm sin~cos 1 , 0 , ,    ,     jGjGF J j ji n J j ji n i nm sin ~ cos 1 , 0 , ,    . (5) At last, expansions over φ were fulfilled:                    K k k kji H ji G K k k kji H ji G K k k kji h ji g K k k kji h ji g 1 4sin ,,~ )( ,~ 0 4cos ,, )( , 1 4sin ,,~ )( ,~ 0 4cos ,, )( ,     (6) In these expansions J = 15 and K = 10 were accept- ed. Combining together all expansions, following ex- pressions were obtained Another verification of the ac- curacy of the approximation was carried out by calculat- ing   B  . By definition, δ must be equal to zero. In fulfilled calculations, it is of the order of 10-3 (Fig. 10).                                                                                        I i i cnm I i i ccnm J j j K k k kji H J j j K k k kji Har ar J j j K k k kji h J j j K k k kji har 1 , 1 , )( 1 sin 1 4sin ,,~ 0 cos 0 4cos ,, ),,( )( 1 sin 1 4sin ,,~ 0 cos 0 4cos ,, ),,(   , . (7) As the result, for the specific magnetic configuration  r   or  r  completely defined by set of I · J · K constants, which allow appropriate reconstructions dur- ing less then 1 s PC time. 2. VERIFICATION OF THE ALGORITHM In order to check the accuracy of the proposed algo- rithm, the relative error in the calculations of  nmr , was defined for several values of m and n, see Fig. 9 for example. The relative error of calculations turned out to be less then 10-3. Fig. 9. The relative error of calculations of r Fig. 10. The relative error in equality 0 B  for prescribed values of ψ CONCLUSIONS On an example of the U-2M torsatron it has been shown that application of the developed in this paper algorithm to the specific magnetic configuration allows one to define completely  car or  r by set of I · J ·K constants, which give appropriate reconstruc- tions during less then 1 s PC time and occupy about 10 Kb of memory storage. The small volume of con- sumed memory gives the possibility to calculate in ad- vance the decompositions of the big number of the 30 ISSN 1562-6016. ВАНТ. 2018. №6(118) magnetic configurations. Then these decompositions may be used for: - calculations of  along the along the lines of sight of the applied diagnostics; - definition of probes locations and RF antenna po- sition in relation to the last closed magnetic surface; - procuring of the fast visualization of magnetic configuration; - definition of local parameters in transport or wave codes, for example, in modeling of slow wave propaga- tion in Wendelstein 7X. Fig. 11. Profile of the rotational transform This algorithm may be used also for calculations of the input parameters – rotational transform profile (Fig. 11) and Furies decomposition of the LCMS - for VMEC code (Fig. 12). Fig. 12. Reconstructed LCMS of torsatron U-2M ACKNOWLEDGEMENTS This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission. REFERENCES 1. V.N. Kaljuzhnyj, V.V. Nemov // Voprosy Atomnoj Nauki i Tekhn. Ser. “Termoyadern. Sintez”. 1985, № 2, p. 35 (in Russian). 2. N.T. Besedin, G.G. Lesnyakov, I.M. Pankratov // Voprosy Atomnoj Nauki i Tekhn. Ser. “Termoyadern. Sintez”. 1991, № 1, p. 48 (in Russian). Article received 02.10.2018 БЫСТРЫЙ И ТОЧНЫЙ АЛГОРИТМ ТРЕХМЕРНОЙ РЕКОНСТРУКЦИИ МАГНИТНЫХ ПОВЕРХНОСТЕЙ В СТЕЛЛАРАТОРАХ В. Филиппов, Д. Греков, В. Олефир Представлен быстрый и точный алгоритм вычисления аппроксимирующих функций, которые устанавли- вают взаимно однозначное соответствие между пространственными координатами в объеме плазмы и мет- кой магнитных поверхностей. С помощью этих функций можно вычислять значение метки магнитных по- верхностей вдоль линий зондирования различных диагностик, определять положение зондов и позицию ВЧ-антенн относительно крайней замкнутой магнитной поверхности. Более того, стало возможным обеспе- чить быструю визуализацию магнитной конфигурации торсатрона во время проведения эксперимента. ШВИДКИЙ ТА ТОЧНИЙ АЛГОРИТМ ТРИВИМІРНОЇ РЕКОНСТРУКЦІЇ МАГНІТНИХ ПОВЕРХОНЬ У СТЕЛАРАТОРАХ В. Філіппов, Д. Греков, В. Олефір Наведено швидкий та точний алгоритм обчислення функцій, що апроксимують магнітні поверхні торсат- рона. Вони здійснюють взаємно однозначну відповідність між просторовими координатами в об’ємі плазми та позначкою магнітних поверхонь. За допомогою цих функцій можна обчислювати значення позначки маг- нітних поверхонь вздовж ліній зондування різних діагностик, визначати розташування зондів та положення ВЧ-антен відносно останньої замкнутої магнітної поверхні. Більш того, стає можливим забезпечити швидку візуалізацію магнітної конфігурації торсатрона під час проведення експерименту.

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