Shape differentiation for Poincare maps of harmonic fields in toroidal domains

TL;DR

This paper analyzes how the Poincare maps of harmonic fields in toroidal domains change with domain shape, providing a shape derivative formula and exploring specific cases like axisymmetric and Diophantine rotation number domains.

Contribution

It introduces a shape differentiability framework for Poincare maps of harmonic fields in toroidal domains and characterizes their sensitivity to domain perturbations.

Findings

Shape derivative of Poincare maps is zero for axisymmetric domains.

In certain domains, the shape derivative can be any smooth function with appropriate perturbations.

General formula for the shape derivative of Poincare maps is derived.

Abstract

In this article, we study Poincare maps of harmonic fields in toroidal domains using a shape variational approach. Given a bounded domain of $\mathbb{R}^3$, we define its harmonic fields as the set of magnetic fields which are curl free and tangent to the boundary. For toroidal domains, this space is one dimensional, and one may thus single out a harmonic field by specifying a degree of freedom, such as the circulation along a toroidal loop. We are then interested in the Poincare maps of such fields restricted to the boundary, which produce diffeomorphisms of the circle. We begin by proving a general shape differentiability result of such Poincare maps in the smooth category, and obtain a general formula for the shape derivative. We then investigate two specific examples of interest; axisymmetric domains, and domains for which the harmonic field has a diophantine rotation number on the boundary. We prove that, in the first case, the shape derivative of the Poincare map is always identically zero, whereas in the second case, assuming an additional condition on the geometry of the domain, the shape derivative of the Poincare map may be any smooth function of the circle by choosing an appropriate perturbation of the domain.