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.

翻译：本文引入了一个定义在环形域上的新形状泛函，称之为调和螺度，并研究了其形状优化问题。给定一个环形域，我们考虑其关联的调和场——该磁场在满足零法向迹且域内无电流的条件下唯一确定（可归一化）。随后研究了该场的螺度，这是磁流体动力学中表征L2内积与Biot-Savart算子像之间关系的重要量。为此，我们首先讨论了合适的泛函框架及等价偏微分方程刻画，其次聚焦于形状优化并导出了调和螺度的形状梯度。最后，我们研究并实现了一种基于有限元外微分的有效数值方案，用于计算调和螺度及其形状梯度。