Shape Optimization of Harmonic Helicity in Toroidal Domains
Authors: 
Rémi Robin, Robin RousselAuthors Info & Claims
Rémi Robin, Robin RousselAuthors Info & ClaimsAbstract
In this paper, we introduce a new shape functional defined for toroidal domains that we call harmonic helicity, and study its shape optimization. Given a toroidal domain, we consider its associated harmonic field. The latter is the magnetic field obtained uniquely up to normalization when imposing zero normal trace and zero electrical current inside the domain. We then study the helicity of this field, which is a quantity of interest in magneto-hydrodynamics corresponding to the product of the field with its image by the Biot–Savart operator. To do so, we begin by discussing the appropriate functional framework and an equivalent PDE characterization. We then focus on shape optimization, and we identify the shape gradient of the harmonic helicity. Finally, we study and implement an efficient numerical scheme to compute harmonic helicity and its shape gradient using finite elements exterior calculus.
References
[1]
Alnæs MS et al. Unified form language: a domain-specific language for weak formulations of partial differential equations ACM Trans. Math. Softw. 2014 40 2 91-937
[2]
Alonso-Rodríguez A et al. Finite element approximation of the spectrum of the curl operator in a multiply connected domain Found. Comput. Math. 2018 18 6 1493-1533
[3]
Amestoy PR, Duff IS, Koster J, and L’Excellent J-Y A fully asynchronous multifrontal solver using distributed dynamic scheduling SIAM J. Matrix Anal. Appl. 2001 23 1 15-41
[4]
Amestoy PR, Guermouche A, L’Excellent J-Y, and Pralet S Hybrid scheduling for the parallel solution of linear systems Parallel Comput. 2006 32 2 136-156
[5]
Amrouche C, Bernardi C, Dauge M, and Girault V Vector potential in three-dimensional nonsmooth domains Math. Methods Appl. Sci. 1998 21 823-864
[6]
Arnold D, Falk R, and Winther R Finite element exterior calculus: from Hodge theory to numerical stability Bull. Am. Math. Soc. 2010 47 2 281-354
[7]
Arnold DN, Falk RS, and Winther R Finite element exterior calculus, homological techniques, and applications Acta Numerica 2006 15 1-155
[8]
Arnold VI On the topology of three-dimensional steady flows of an ideal fluid J. Appl. Math. Mech. 1966 30 1 223-226
[9]
V. I. Arnold. The asymptotic Hopf invariant and its applications. In: Vladimir I. Arnold - collected works: Hydrodynamics, bifurcation theory, and algebraic geometry 1965-1972. Ed. by A. B. Givental et al. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014, pp 357–375. 10.1007/978-3-642-31031-7_32
[10]
V. I. Arnold and B. A. Khesin. Topological Methods in Hydrodynamics. Vol. 125. Applied Mathematical Sciences. Cham: Springer International Publishing, 2021. 10.1007/978-3-030-74278-2
[11]
S. Balay, W. Gropp, L. C. McInnes, and B. F. Smith. PETSc, the portable, extensible toolkit for scientific computation. In: Argonne National Laboratory 2.17 (1998)
[12]
M. K. Bevir and J. W. Gray. Relaxation, flux consumption and quasi steady state pinches. In: Proceedings of the Reversed-Field. Pinch Theory Workshop. Los Alamos, 1980
[13]
Buffa A, Costabel M, and Sheen D On traces for (curl, ) in Lipschitz domains J. Math. Anal. Appl. 2002 276 2 845-867
[14]
Cantarella J, DeTurck D, and Gluck H The Biot-Savart operator for application to knot theory, fluid dynamics, and plasma physics J. Math. Phys. 2001 10 1063/1 1329659
[15]
Cantarella J, DeTurck D, Gluck H, and Teytel M Isoperimetric problems for the helicity of vector fields and the Biot-Savart and curl operators J. Math. Phys. 2000 41 8 5615-5641
[16]
J. Cantarella, D. Deturck, H. Gluck, and M. Teytel. Influence of Geometry and Topology on Helicity. In: Magnetic Helicity in Space and Laboratory Plasmas. American Geophysical Union (AGU), 1999, pp 17–24. 10.1029/GM111p0017
[17]
Dewar R and Hudson S Stellarator symmetry Phys. D: Nonlinear Phenomena 1998 112 12 275-280
[18]
A. Enciso and D. Peralta-Salas. Non-existence of axisymmetric optimal domains with smooth boundary for the first curl eigenvalue. In: ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE (2023), pp 311–327. 10.2422/2036-2145.202010_008
[19]
Ern A and Guermond J-L Analysis of the edge finite element approximation of the Maxwell equations with low regularity solutions Comput. Math. Appl. 2018 75 3 918-932
[20]
R. D. Falgout and U. M. Yang. hypre: A Library of High Performance Preconditioners. In: Computational Science — ICCS 2002. Ed. by P. M. A. Sloot, A. G. Hoekstra, C. J. K. Tan, and J. J. Dongarra. Lecture Notes in Computer Science. Berlin, Heidelberg: Springer, 2002, pp 632–641. 10.1007/3-540-47789-6_66
[21]
W. Gerner. Existence of optimal domains for the helicity maximisation problem among domains satisfying a uniform ball condition. 2023. 10.48550/ arXiv:2305.13642 [math-ph]. preprint
[22]
Gerner W Isoperimetric problem for the first curl eigenvalue J. Math. Anal. Appl. 2023 519 126808
[23]
Geuzaine C and Remacle J-F Gmsh: a 3-D finite element mesh generator with built-in pre- and post-processing facilities Int. J. Numer. Methods Eng. 2009 79 11 1309-1331
[24]
A. Henrot and M. Pierre. Shape variation and optimization: a geometrical analysis. Vol. 28. EMS Tracts in Mathematics. Zürich, Switzerland: European Math. Soc (EMS), 2018
[25]
Hiptmair R and Xu J Nodal auxiliary space preconditioning in H(curl) and H(div) Spaces SIAM J. Numer. Anal. 2007 45 6 2483-2509
[26]
L.-M. Imbert-Gerard, E. J. Paul, and A. M. Wright. An Introduction to Stellarators: From magnetic fields to symmetries and optimization. arXiv:1908.05360 [physics]. 2020. 10.48550/ arXiv:1908.05360
[27]
E. Lara, R. Rodríguez, and P. Venegas. Spectral approximation of the curl operator in multiply connected domains. In: Discrete and Continuous Dynamical Systems - S 9.1 (Mon Nov 30 19:00:00 EST 2015), pp 235–253. 10.3934/dcdss.2016.9.235
[28]
MacTaggart D and Valli A Magnetic helicity in multiply connected domains J. Plasma Phys. 2019 85 5 775850501
[29]
Moffatt HK The degree of knottedness of tangled vortex lines J. Fluid Mech. 1969 35 1 117-129
[30]
S. Montiel. The Isoperimetric Problem for the Curl Operator. 2023. arXiv:2307.09556 [math]
[31]
Nedelec JC Mixed finite elements in R3 Numerische Mathematik 1980 35 3 315-341
[32]
Paul EJ, Landreman M, Bader A, and Dorland W An adjoint method for gradient-based optimization of stellarator coil shapes Nuclear Fusion 2018 58 7 076015
[33]
Privat Y, Robin R, and Sigalotti M Optimal shape of stellarators for magnetic confinement fusion J. de Math é matiques Pures et Appliqu é es 2022 163 231-264
[34]
Y. Privat, R. Robin, and M. Sigalotti. Existence of surfaces optimizing geometric and PDE shape functionals under reach constraint. In: Interfaces and Free Boundaries (2024). 10.4171/ifb/523
[35]
P. A. Raviart and J. M. Thomas. A mixed finite element method for 2-nd order elliptic problems. In: Mathematical Aspects of Finite Element Methods. Ed. by I. Galligani and E. Magenes. Lecture Notes in Mathematics. Berlin, Heidelberg: Springer, 1977, pp 292–315. 10.1007/BFb0064470
[36]
Saad Y and Schultz MH GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems SIAM J. Sci. Stat. Comput. 1986 7 3 856-869
[37]
Scroggs MW, Baratta IA, Richardson CN, and Wells GN Basix: a runtime finite element basis evaluation library J. Open Source Softw. 2022 7 73 3982
[38]
Scroggs MW, Dokken JS, Richardson CN, and Wells GN Construction of arbitrary order finite element degree-of-freedom maps on polygonal and polyhedral cell meshes ACM Trans. Math. Softw. 2022 48 2 1-23
[39]
Valli A A variational interpretation of the Biot-Savart operator and the helicity of a bounded domain J. Math. Phys. 2019 60 2 021503
[40]
Virtanen P et al. SciPy 1.0: fundamental algorithms for scientific computing in python Nature Methods 2020 17 261-272
[41]
F. Warmer et al. From W7-X to a HELIAS fusion power plant: On engineering considerations for next-step stellarator devices. In: Fusion Engineering and Design. Proceedings of the 29th Symposium on Fusion Technology (SOFT-29) Prague, Czech Republic, September 5-9, 2016 123 (2017), pp. 47–53. 10.1016/j.fusengdes.2017.05.034
[42]
Zarnstorff MC et al. Physics of the compact advanced stellarator NCSX Plasma Phys. Control. Fusion 2001 43 A237-A249
Index Terms
- Shape Optimization of Harmonic Helicity in Toroidal Domains
Index terms have been assigned to the content through auto-classification.
Recommendations
Shape optimization of corrugated airfoils
The effect of corrugations on the aerodynamic performance of a Mueller C4 airfoil, placed at a $$5^{\circ }$$5 angle of attack and $$Re=10{,}000$$Re=10,000, is investigated. A stabilized finite element method is employed to solve the incompressible flow ...
Handle-aware isolines for scalable shape editing
SIGGRAPH '07: ACM SIGGRAPH 2007 papersHandle-based mesh deformation is essentially a nonlinear problem. To allow scalability, the original deformation problem can be approximately represented by a compact set of control variables. We show the direct relation between the locations of handles ...
Shape Optimization by Free-Form Deformation: Existence Results and Numerical Solution for Stokes Flows
Shape optimization problems governed by PDEs result from many applications in computational fluid dynamics. These problems usually entail very large computational costs and require also a suitable approach for representing and deforming efficiently the ...
Comments
Information & Contributors
Information
Published In
© The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.
Publisher
Plenum Press
United States
Publication History
Published: 29 December 2024
Accepted: 09 October 2024
Received: 29 January 2024
Author Tags
Qualifiers
- Research-article
Funding Sources
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 0Total Downloads
- Downloads (Last 12 months)0
- Downloads (Last 6 weeks)0
Reflects downloads up to 20 Jan 2025