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

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.