Optimization with geometric constraints: work toward a stellarator systems code

Stellarators designs usually employ complex geometries for the magnets, vacuum vessel, and other components. While stellarator optimization codes often use simplified representations—filaments for coils, or cylinders for ports—a parametric design framework should be capable of using more complex or general geometries. In the final design, the different physical components must not interfere (overlap) with each other. This leads to a challenging optimization problem: stellarator plasma and magnet geometries are often parameterized with tens to hundreds of variables, where gradient-based optimization methods are essential, but interference is a discontinuous function.

There are methods to model interference in a way that presents as a differentiable constraint. The DCOL algorithm [1] allows differentiable collision detection between convex primitives. In principle, this algorithm can be used for complex shapes by decomposing them into primitives. We present our results using DCOL on a test problem analogous to the problem of shaping a stellarator coil around ports or obstructions. We encountered difficulty solving the problem in a robust way with off-the-shelf gradient-based optimizers. We will describe these difficulties and potential paths forward.

[1] K. Tracy, T.A. Howell, and Z. Manchester, 2023, http://arxiv.org/abs/2207.00669