Computation of stationary 3D halo currents in fusion devices with accuracy control. (English) Zbl 1351.82070
Summary: This paper addresses the calculation of the resistive distribution of halo currents in three-dimensional structures of large magnetic confinement fusion machines. A Neumann electrokinetic problem is solved on a geometry so complicated that complementarity is used to monitor the discretization error. An irrotational electric field is obtained by a geometric formulation based on the electric scalar potential, whereas three geometric formulations are compared to obtain a solenoidal current density: a formulation based on the electric vector potential and two geometric formulations inspired from mixed and mixed-hybrid Finite Elements. The electric vector potential formulation is usually considered impractical since an enormous computing power is wasted by the topological pre-processing it requires. To solve this challenging problem, we present novel algorithms based on lazy cohomology generators that enable to save orders of magnitude computational time with respect to all other state-of-the-art solutions proposed in literature. Believing that our results are useful in other fields of scientific computing, the proposed algorithm is presented as a detailed pseudocode in such a way that it can be easily implemented.
MSC:
| 82C80 | Numerical methods of time-dependent statistical mechanics (MSC2010) |
| 82D10 | Statistical mechanics of plasmas |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
Keywords:
magnetic confinement fusion; halo currents; Neumann boundary value problem; elliptic partial differential equations; complementarity; mixed-hybrid formulation; lazy cohomology generatorsReferences:
| [1] | Park, W.; Belova, E. V.; Fu, G. Y.; Tang, X. Z.; Strauss, H. R.; Sugiyama, L. E., Plasma simulation studies using multilevel physics models, Phys. Plasmas, 6, 1796 (1999) |
| [2] | Albanese, R.; Bettini, P.; Marconato, N.; Furno Palumbo, M.; Peruzzo, S.; Rubinacci, G.; Specogna, R.; Ventre, S.; Villone, F., Numerical modeling of 3d halo current path in iter structures, Fusion Eng. Des., 88, 529-532 (2013) |
| [3] | Albanese, R.; Rubinacci, G.; Villone, F., An integral computational model for crack simulation and detection via eddy currents, J. Comput. Phys., 152, 2, 736-755 (1999) · Zbl 1017.74502 |
| [4] | Bettini, P.; Specogna, R., Lazy cohomology generators enable the use of complementarity for computing halo current resistive distribution in fusion reactors, IEEE Trans. Magn., 50, 7012004 (2014) |
| [5] | Strang, G.; Fix, G. J., An Analysis of the Finite Element Method (1973), Prentice-Hall: Prentice-Hall Englewood Cliffs, NJ · Zbl 0278.65116 |
| [6] | Synge, J. L., The Hypercircle in Mathematical Physics (1957), Cambridge University Press: Cambridge University Press New York · Zbl 0079.13802 |
| [7] | Rikabi, J. A.H.; Bryant, C. F.; Freeman, E. M., An error-based approach to complementary formulations of static field solutions, Int. J. Numer. Methods Eng., 26, 9, 1963-1987 (1988) · Zbl 0662.73073 |
| [8] | Rikabi, J. A.H., Complementary solutions of direct current flow problems, IEEE Trans. Magn., 29, 1, 98-107 (1993) |
| [9] | Bossavit, A., Computational Electromagnetism (1998), Academic Press · Zbl 0945.78001 |
| [10] | Specogna, R., Complementary geometric formulations for electrostatics, Int. J. Numer. Methods Eng., 86, 1041-1068 (2011) · Zbl 1235.78038 |
| [11] | Specogna, R., Extraction of VLSI multiconductor transmission line parameters by complementarity, IEEE Trans. Very Large Scale Integr. (VLSI) Syst., 22, 1, 146-154 (2014) |
| [12] | Gross, P. W.; Kotiuga, P. R., Electromagnetic Theory and Computation: A Topological Approach (2004), Cambridge University Press · Zbl 1096.78001 |
| [13] | Munkres, J. R., Elements of Algebraic Topology (1984), Perseus Books: Perseus Books Cambridge, MA · Zbl 0673.55001 |
| [14] | Pellikka, M.; Suuriniemi, S.; Kettunen, L., Powerful heuristics and basis selection bring computational homology to engineers, IEEE Trans. Magn., 47, 1226-1229 (2011) |
| [15] | Dłotko, P.; Specogna, R.; Trevisan, F., Automatic generation of cuts on large-sized meshes for \(t-ω\) geometric eddy-current formulation, Comput. Methods Appl. Mech. Eng., 198, 3765-3781 (2009) · Zbl 1230.78038 |
| [16] | Dłotko, P.; Specogna, R., Efficient cohomology computation for electromagnetic modeling, Comput. Model. Eng. Sci., 60, 247-278 (2010) |
| [17] | Dłotko, P.; Specogna, R., Physics inspired algorithms for (co)homology computations of three-dimensional combinatorial manifolds with boundary, Comput. Phys. Commun., 184, 2257-2266 (2013) · Zbl 07866089 |
| [18] | Dłotko, P.; Specogna, R., Lazy cohomology generators: a breakthrough in (co)homology computations for CEM, IEEE Trans. Magn., 50, 7014204 (2014) |
| [19] | Brezzi, F.; Fortin, M., Mixed and Hybrid Finite Element Methods (1991), Springer Verlag: Springer Verlag Heidelberg, Germany · Zbl 0788.73002 |
| [20] | Boffi, D.; Brezzi, F.; Fortin, M., Mixed Finite Element Methods and Applications, Springer Ser. Comput. Math., vol. 44 (2013), Heidelberg, Germany · Zbl 1277.65092 |
| [21] | Tonti, E., On the formal structure of physical theories, Monogr. Ital. Natl. Res. Counc. (1975) |
| [22] | Codecasa, L.; Specogna, R.; Trevisan, F., A new set of basis functions for the discrete geometric approach, J. Comput. Phys., 229, 7401-7410 (2010) · Zbl 1196.78027 |
| [23] | Codecasa, L.; Specogna, R.; Trevisan, F., Base functions and discrete constitutive relations for staggered polyhedral grids, Comput. Methods Appl. Mech. Eng., 198, 1117-1123 (2009) · Zbl 1229.78025 |
| [24] | Kettunen, L.; Forsman, K.; Bossavit, A., Formulation of the eddy current problem in multiply connected regions in terms of \(h\), Int. J. Numer. Methods Eng., 41, 5, 935-954 (1998) · Zbl 0902.65075 |
| [25] | Tarhasaari, T.; Kettunen, L.; Bossavit, Alain, Some realizations of a discrete hodge operator: a reinterpretation of finite element techniques [for em field analysis], IEEE Trans. Magn., 35, 3, 1494-1497 (1999) |
| [26] | Cormen, T. H.; Leiserson, C. E.; Rivest, R. L.; Stein, C., Introduction to Algorithms (2001), The MIT Press · Zbl 1047.68161 |
| [27] | Hiptmair, R.; Ostrowski, J., Generators of \(h_1(\gamma_h, Z)\) for triangulated surfaces: construction and classification, SIAM J. Comput., 31, 5, 1405-1423 (2002) · Zbl 1001.05046 |
| [28] | Alonso Rodriguez, A.; Bertolazzi, E.; Ghiloni, R.; Valli, A., Construction of a finite element basis of the first de Rham cohomology group and numerical solution of 3d magnetostatic problems, SIAM J. Numer. Anal., 51, 4, 2380-2402 (2013) · Zbl 1284.78012 |
| [29] | Biro, O.; Pack, S.; Richter, K., On the use of the magnetic vector potential in the nodal and edge finite element analysis of 3d magnetostatic problems, IEEE Trans. Magn., 32, 3, 651-654 (1996) |
| [30] | Dłotko, P.; Specogna, R., Critical analysis of the spanning tree techniques, SIAM J. Numer. Anal., 48, 1601-1624 (2010) · Zbl 1219.78142 |
| [31] | Courant, R., Variational methods for the solution of problems of equilibrium and vibration, Bull. Am. Math. Soc., 69, 1-23 (1943) · Zbl 0810.65100 |
| [32] | Benzi, M.; Golub, G. H.; Liesen, J., Numerical solution of saddle point problems, Acta Numer., 14, 5, 1-137 (2005) · Zbl 1115.65034 |
| [33] | Bossavit, A., Mixed-hybrid methods in magnetostatics: complementarity in one stroke, IEEE Trans. Magn., 39, 3, 1099-1102 (2003) |
| [34] | Paccagnella, R.; Strauss, H. R.; Breslau, J., 3D MHD VDE and disruptions simulations of tokamaks plasmas including some ITER scenarios, Nucl. Fusion, 49, 035003 (2009) |
| [35] | Notay, Y., AGMG software and documentation (2012), see |
| [36] | Notay, Y., Aggregation-based algebraic multigrid for convection-diffusion equations, SIAM J. Sci. Comput., 34, 2288-2316 (2012) · Zbl 1250.76139 |
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