### Abstract

Design of transient thermo-mechanical systems is a challenging task often encountered during the design of high precision machines and instrumentation. Topology optimization can provide valuable insight during the design process, however, for large scale problems the backward time integration required to obtain adjoint sensitivity information is undesired. Previous work has illustrated how the introduction of a reduced modal basis allows to eliminate the backward time integration to obtain the adjoint variables. In order to reduce computational effort further, additional reduction approaches are considered. The focus is specifically on design cases where the relevant heat loads can be expressed or approximated analytically by combinations of harmonic, polynomial or exponential functions of time. Using the method of undetermined coefficients, an exact particular solution is obtained using the full system. Then, the corresponding homogenous solution is expressed using a reduced modal basis, for which a relatively small set of modes is required to obtain an accurate approximation. For the cases where the time component of the heat loads are expressed by the considered analytical functions, the backward time integration is eliminated from the calculation of the design sensitivities, while the forward integration is handled by convolutions.

Original language | English |
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Title of host publication | Advances in Structural and Multidisciplinary Optimization |

Subtitle of host publication | Proceedings of the 12th World Congress of Structural and Multidisciplinary Optimization (WCSMO12) |

Editors | A Schumacher, T Vietor, S Fiebig, K-U Bletzinger, K Maute |

Place of Publication | Cham, Switzerland |

Publisher | Springer |

Pages | 1027-1041 |

ISBN (Electronic) | 978-3-319-67988-4 |

ISBN (Print) | 978-3-319-67987-7 |

DOIs | |

Publication status | Published - 2017 |

Event | WCSMO 2017: 12th World Congress of Structural and Multidisciplinary Optimisation - Braunschweig, Germany Duration: 5 Jun 2017 → 9 Jun 2017 |

### Conference

Conference | WCSMO 2017: 12th World Congress of Structural and Multidisciplinary Optimisation |
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Country | Germany |

City | Braunschweig |

Period | 5/06/17 → 9/06/17 |

### Keywords

- Thermo-mechanical
- Model order reduction
- Analytical solution
- Adjoint sensitivities
- Transient problems

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## Cite this

van der Kolk, M., Hooijkamp, E., Langelaar, M., & van Keulen, F. (2017). Using exact particular solutions and modal reduction in topology optimization of transient thermo-mechanical problems. In A. Schumacher, T. Vietor, S. Fiebig, K-U. Bletzinger, & K. Maute (Eds.),

*Advances in Structural and Multidisciplinary Optimization : Proceedings of the 12th World Congress of Structural and Multidisciplinary Optimization (WCSMO12)*(pp. 1027-1041). Springer. https://doi.org/10.1007/978-3-319-67988-4_78