Convexity of the longitudinal variation of third-order resonance driving terms and its application in dynamic aperture optimization

Abstract

The optimization of the dynamic aperture (DA) of a storage ring is typically a non-convex problem with multiple local optima. Recent studies showed that reducing the variation of resonance driving terms (RDTs) along the longitudinal position improves DA very effectively, as the reduction in the longitudinal variation of lower-order RDTs suppresses higher-order nonlinear terms. Therefore, minimizing the longitudinal variation of third-order RDTs is crucial for DA optimization. In this paper, we prove that the longitudinal variation of third-order RDTs, quantified using their RMS value $f_{3,\mathrm{rms}}$ at sextupole locations, is a special convex function. In the space of sextupole strengths, the iso-surfaces of $f_{3,\mathrm{rms}}$ are a series of concentric and coaxial ellipsoidal surfaces, with the central position possessing minimum $f_{3,\mathrm{rms}}$. The scanning results of a storage ring lattice show a strong consistency between the distributions of $f_{3,\mathrm{rms}}$ and DA, indicating that the optimization of DA can be regarded as a roughly approximate convex optimization problem. Based on this, a fast DA optimization method based on particle tracking is developed, where a high-quality initial population for an intelligent algorithm is generated with a Gaussian distribution based on the geometric structure of $f_{3,\mathrm{rms}}$.

Figures (12)

• Figure 1: Magnet layout and optical functions of a FODO lattice cell, one quarter of the simply designed ring with thin-lens quadrupoles and sextupoles. The four families of chromatic sextupoles indicated in the layout are placed at the same locations as quadrupoles.
• Figure 2: Relationship between $f_{3,\mathrm{rms}}$, $N_{\text{particle}}$ and $\overline{d_{r}}$ for the nonlinear solutions of the FODO lattice. The blue solutions within the red ellipse are influenced by the ADTS terms.
• Figure 3: Relationship between $h_{3,\text{ring}}$, $N_{\text{particle}}$ and $\overline{d_{r}}$ for the nonlinear solutions of the FODO lattice.
• Figure 4: A three-dimensional plot depicting a simple convex quadratic function $f(x_{1},x_{2})=x_{1}^{2}+x_{2}^{2}$.
• Figure 5: Magnitude variation of third-order RDTs along the longitudinal position for a nonlinear solution of the FODO lattice.