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Thesis title: Small elastic deformations of plates and shallow spherical shells:
influence functions and application to the optimization of passive or active
supports for thin telescope mirrors and to adaptive mirrors.**

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**Abstract**

Small elastic deformations of thin telescope mirrors are studied using thin shallow spherical shell theory. This theory is consistent with meniscus-shaped mirrors having a focal ratio greater than or equal to 0.5. The general solution is given for the deformation normal to the mirror surface (the influence function), that is created by a uniform or gravitational load, or by a discrete forces distribution of any symmetry. The linear system of equations is described for the calculation of the constants which define the influence and stress functions for a given load distribution. Explicit expressions of the influence function are then given for the caseof a flat mirror (infinite radius of curvature).

These theoretical results are used for the optimization of passive or active mirror supports. The efficiency of several optimized support topologies is calculated. It is shown that if a zenith-dependant paraboloidal deformation of the mirror is tolerated, the support efficiency may be improved by up to 40%. A method for optimizing an active support is also presented: this couples an optimization of combined actuator influence functions that fit low order aberrations (or modes) with a minimization of the gravity influence function. It is shown that the range of active correction is up to 50% better than that obtainable using classical mirror cells, and polishing specifications can thereby be lowered to allow cost reductions. This technique can be extended to the design of adaptive mirrors, in which case gravitational effects are neglected.