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The method for solving topology optimization problems using hyper-dual numbers

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Abstract

Air-inflated structure such as an air bed, tents, and floats is a category of a high-strength membrane structure design problem. This problem is described as a surface energy minimization problem to make maximum use of limited space. Such problems can be treated as the topology optimization problems. Some topology optimization problems are design problems involving physical phenomena such as heat and structure. When the topology optimization is applied to such problems, it may be difficult to derive the sensitivity of the variation of the objective function analytically, especially in surface energy minimization problems where the volume of the design space is limited. In this study, we apply a numerical differential method using hyper-dual number to the sensitivity of the variation of the objective function, and then show that the desired optimization shape can be obtained even when analytical sensitivity is difficult to derive. By expanding the design variables of the topology optimization problem to the hyper-dual numbers, the sensitivity required to find the solution to the optimization problem can be determined without calculating the variation. The sensitivity obtained using the hyper-dual numbers does not include rounding and truncation errors. As an example of a problem whose sensitivity is difficult to obtain analytically, under the condition that the volume of the given domain is conserved, the solution to the surface minimization problem, which is formulated as a kind of surface energy minimization problem, is obtained by expanding the design variables to the hyper-dual numbers. The proposed method can be used to find solutions to topology optimization problems involving physical phenomena.

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Acknowledgements

We would like to thank Editage (www.editage.com) for English editing.

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Authors and Affiliations

  1. TOYOTA Central Research and Development Laboratories Incorporated, 41–1 Yokomichi, Nagakute, Aichi, 480–1192, Japan

    Daisuke Murai, Ryuji Omote & Masato Tanaka

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  1. Daisuke Murai
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  2. Ryuji Omote
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  3. Masato Tanaka
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Correspondence to Daisuke Murai.

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Code availability

In this calculation, python code is created to discretize the design variables and objective and constraint functions. If necessary, we can disclose the source code used in the calculation.

Authors’ contributions

All authors contributed to the study conception and design. Material preparation and numerical calculation were performed by Daisuke Murai and Ryuji Omote. The algebraic and transcendental functions using hyper-dual number were formulated by Masato Tanaka. The python program for hyper-dual number using operator overloading was created by Ryuji Omote. The first draft of the manuscript was written by Daisuke Murai and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Murai, D., Omote, R. & Tanaka, M. The method for solving topology optimization problems using hyper-dual numbers. Arch Appl Mech 92, 2813–2824 (2022). https://doi.org/10.1007/s00419-022-02183-5

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