20 August 1986 Stiffness Matrix Partitioning For The Derivation Of Mirror Figure Control Information Based On Incomplete Partial Derivative Data
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Abstract
The usual procedure in finite element structural analysis is to produce a printed or graphical report of strain and/or stress for an object. In the situation where one must derive control information for the correction of static (or very low bandwidth) loads, as may be the case for large astronomical mirrors, it is often impossible to obtain data about the exact loading of the mirror structure. Strain gauges may be placed at the collimating points, to provide information for the adjustment of support forces, but they are not sufficient to sense the mirror's figure. The finite element method not only provides a means of building a working model for an arbitrary mirror structure but also supplies an interpolative approximation to the optically significant strain that is very straightforward, computationally efficient, and particularly useful for the inevitable circumstance of having incomplete data. In addition, the measurement data may be directly related to rotational degrees of freedom in the finite element model, allowing the design of meas-urement hardware to measure local surface tilt rather than optical path displacement. This procedure has the potential of providing a means of correcting the mirror's support system "off-line", that is, between relatively infrequent calibrations using data from a star image analyzer. The various standard finite element codes now commonly used provide a convenient means of constructing the stiffness matrix for a model of a large complex mirror structure. To facilitate development of the efficient iterative use of a finite element interpolation procedure, the stiffness matrix may be removed from the general finite element analysis program, partitioned, and solved with a least squares routine to provide an optical surface error estimate. Looping on this procedure and testing for convergence finally produces the correction desired. Understanding the limits of accuracy for this procedure, based on the kind and number of measurement points and the number of actuators, is crucial to the design of a figure control system.
© (1986) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.
F. B. Ray, F. B. Ray, J. H. Chang, J. H. Chang, } "Stiffness Matrix Partitioning For The Derivation Of Mirror Figure Control Information Based On Incomplete Partial Derivative Data", Proc. SPIE 0628, Advanced Technology Optical Telescopes III, (20 August 1986); doi: 10.1117/12.963564; https://doi.org/10.1117/12.963564
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