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Shape optimization of harmonic helicity in toroidal domains

Remi Robin, Robin Roussel

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 L2 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.

Shape optimization of harmonic helicity in toroidal domains

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 L2 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.
Paper Structure (21 sections, 26 theorems, 137 equations, 6 figures, 1 table)

This paper contains 21 sections, 26 theorems, 137 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Functional spaces
  4. Harmonic fields
  5. Vector potentials and Bevir--Gray formula
  6. Harmonic helicity and its shape derivative
  7. Pullback of the De Rham complex on a fixed domain
  8. Differentiability of the PDEs
  9. Proof of \ref{['th:shape_derivative']}
  10. Approximation by finite element exterior calculus
  11. Classical finite elements exterior calculus families
  12. Discretization of the De Rham complex
  13. Numerical convergence of the harmonic helicity
  14. Numerical implementation and results
  15. Specificity of simulations for stellarators
  16. ...and 6 more sections

Key Result

Proposition 1

here exists a unique solution to the following problem. Find $(B_\mathrm{div}\,, u_\mathrm{div}\,) \in H_0(\mathrm{div}\,, \Omega) \times L^2_0(\Omega)$ such that, for all $(\tau, v) \in H_0(\mathrm{div}\,, \Omega) \times L^2(\Omega)$ we have Furthermore, we have $B_\mathrm{div}\, = B$ defined in eq:B_curl.

Figures (6)

  • Figure 1: Illustration of the curves $\gamma$ and $\gamma'$, and the surfaces $\Sigma$ and $\Sigma'$
  • Figure 2: One section (i.e. one third) of NCSX plasma. In the upper plot, we represent the function $u_h$ of \ref{['eq:VF_B_h_curl']}. Its gradient $B_h^\mathrm{curl}\,$ is shown in the middle figure. Note that the boundary conditions on the left and right cuts are a jump $2\pi/3$, whereas we have Neumann boundary condition on the plasma surface. The bottom figure is a reprentation of $A^2_h$.
  • Figure 3: On the left, we provide better and better approximations of the surface to the mesher. We then plot the characteristic size of an element versus the harmonic helicity. On the right, we mesh with different characteristic size $h$ the same polyhedral approximation of the continuous surface and compare it to a reference solution obtained with second order elements and $h=0.025$. The two lines are merely indicative and their slopes correspond to theoretical first order (blue) and second order (red) convergence rates.
  • Figure 4: One section of NCSX plasma. The plot shows the shape gradient $2A^2_h\cdot B_h^\mathrm{curl}\,$ on the boundary. NCSX plasma configuration has a negative harmonic helicity. Improving our criterion implies following the opposite of the shape gradient.
  • Figure 5: The results of the two optimization problems, BPC (shown in green above) and BVC (shown in blue below). The associated costs are reported in \ref{['tab:res']}. Additionally, the reference NCSX plasma is plotted in transparent orange.
  • ...and 1 more figures

Theorems & Definitions (51)

  • Remark 1
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • Definition 1
  • Proposition 4
  • Corollary 1
  • Theorem 1
  • ...and 41 more