Aberration Correction under Force Constraints for a 2.4-m Centrally Obscured Primary: A Modal Least-Squares Assessment of Active-Support Layouts

We evaluate alternative active-support layouts for a relatively thick, high-stiffness Zerodur glass-ceramic meniscus primary mirror by jointly accounting for optical correction and the mechanical force demanded by the actuators. The mirror’s response in optical path difference is spanned by structural free-vibration modes. For each layout we form a force-realized modal basis via a two-step static procedure that maps modal amplitudes to actuator group forces and back to surface sag on an annular pupil. Annular Zernike targets covering tip/tilt, classical astigmatism, coma, trefoil, quadratic astigmatism, and third- and fifth-order spherical aberration are fitted by least squares. From the fitted coefficients we recover per-group force demand and its peak value relative to the admissible range set by mirror weight divided by the number of supports. Performance is reported with two complementary measures: the pupil-averaged residual between the realized and target wavefront, and the directional gain along the target. Assuming linear superposition within the allowed force window, we estimate for each aberration the maximum correctable amplitude at which any force limit is first reached. Across the tested layouts, low-order terms are corrected to the full 550\,\mathrmnm root-mean-square target with negligible residual while keeping peak forces within limits. Higher-order performance is constrained by either force capacity or weak directional gain. The workflow provides a layout-agnostic, reproducible basis for ranking designs by optical benefit and mechanical cost under realistic load constraints.