Influence functions of a thin shallow meniscus-shaped mirror.

Thin shallow spherical shell theory is used to derive the general influence function, owing to uniform and/or discrete (actuators) loads, for a thin shallow meniscus-shaped mirror of uniform thickness with a central hole and supported at discrete points. Small elastic deformations are considered. No symmetry on the load distribution constrains the model. Explicit analytical expressions of the set of equations are given for calculating the influence functions. Results agree with the finite element analysis (FEA) to within 1%. When the FEA requires megabytes of RAM memory, the analytical method needs only kilobytes and typically runs 30 times faster. This is a crucial advantage for the iterative optimization of mirror supports such as large passive or active meniscus-shaped primary mirror supports or Cassegrain/Gregorian adaptive secondary actuator configurations. References are given on estimating the shear effects (thick mirror), the thickness variation effect, and the influence of the size of the support pads.