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Shape Optimization of Harmonic Helicity in Toroidal Domains

Author

Listed:
  • Rémi Robin

    (Sorbonne Université, PSL Research University)

  • Robin Roussel

    (Sorbonne Université)

Abstract

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 $$L^2$$ L 2 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.

Suggested Citation

  • Rémi Robin & Robin Roussel, 2025. "Shape Optimization of Harmonic Helicity in Toroidal Domains," Journal of Optimization Theory and Applications, Springer, vol. 204(1), pages 1-43, January.
    • Handle: RePEc:spr:joptap:v:204:y:2025:i:1:d:10.1007_s10957-024-02588-y
    DOI: 10.1007/s10957-024-02588-y
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