## Abstract

The uninflated shape configurations of parabolic and spherical membrane mirrors were calculated by solving the inverse problem, i.e., given the design inflation pressure, the membrane material and geometric properties, what must be the initial uninflated shape such that on inflation to the design pressure, the exact desired surface of revolution is obtained. The resulting first order nonlinear differential equation was numerically integrated using the boundary conditions. The initial uninflated shape was then subjected to a forward transformation using FAIM, a proprietary geometric nonlinear membrane finite element code. FAIM has been validated against exact analytical solutions for both small and extremely large deformations that are up to eight orders of magnitude larger compared with the starting undeflected shape. Simulations reveal that to fabricate a very accurate and precise inflated membrane mirror relative to the design parameters, one must not only accurately measure and input the moduli in both meridional and hoop directions but an accurately measured Poisson’s ratio as well. The code was used to guide the membrane mirror design. For very small aperture diameters, the initial uninflated shape may be fabricated by thermo-forming the membrane. For aperture diameters exceeding one meter however, the membrane mirror is built with discrete gores that are joined together with tapes at the seams. This provided the impetus to write a companion computer code FLATE, to calculate the gore shapes using a slight modification of the solution to the inverse transformation equation to account for the presence of the seam tapes. After the gores were determined, the resulting final inflated shape was calculated and verified using FAIM. Sensitivity analyses can now be carried out to determine the resulting surface shape as a function of the different sources of error: gore width, gore length, perimeter attachment uncertainties, thermal effects, variation of material properties over the membrane continuum and inflation pressure changes. The code has been shown to be more robust than equivalent commercial analytical packages in so far as membrane, cable and space-frame element combinations are concerned. In particular, the analytical and finite element codes were used in the preliminary assessment of a membrane optic for the OASIS Mission (Orbiting Astronomical Satellite for Investigating Stellar Systems) [1]. The OASIS is a 20-meter class space observatory operating at high spectral resolution in the terahertz frequencies. Over its nominal 2-year mission it will probe conditions and search for biogenic molecules on hundreds of protoplanetary disks and other solar system objects.

Original language | English (US) |
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Title of host publication | Astronomical Optics |

Subtitle of host publication | Design, Manufacture, and Test of Space and Ground Systems III |

Editors | Tony B. Hull, Daewook Kim, Pascal Hallibert, Fanny Keller |

Publisher | SPIE |

ISBN (Electronic) | 9781510644786 |

DOIs | |

State | Published - 2021 |

Event | Astronomical Optics: Design, Manufacture, and Test of Space and Ground Systems III 2021 - San Diego, United States Duration: Aug 1 2021 → Aug 5 2021 |

### Publication series

Name | Proceedings of SPIE - The International Society for Optical Engineering |
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Volume | 11820 |

ISSN (Print) | 0277-786X |

ISSN (Electronic) | 1996-756X |

### Conference

Conference | Astronomical Optics: Design, Manufacture, and Test of Space and Ground Systems III 2021 |
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Country/Territory | United States |

City | San Diego |

Period | 8/1/21 → 8/5/21 |

## Keywords

- Finite element analysis
- Geometric nonlinear
- Inflatable membrane mirror
- Inverse shape problem solution

## ASJC Scopus subject areas

- Electronic, Optical and Magnetic Materials
- Condensed Matter Physics
- Computer Science Applications
- Applied Mathematics
- Electrical and Electronic Engineering