Balance Layout Problem with the Optimized Distances Between Objects
The optimal layout problem for objects in a convex bounded domain is considered subject to constraints on the position of the gravity center and the moments of inertia of the system. The layout is aimed to optimize mutual distances between the objects and
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Balance Layout Problem with the Optimized Distances Between Objects S. Plankovskyy , A. Nikolaev , O. Shypul A. Pankratov , and T. Romanova
, I. Litvinchev
,
Contents 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Problem Formulation and Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Solution Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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7.1 Introduction The layout problems have a wide spectrum of applications in science and technology, e.g., in assembling technological complexes [1], placing equipment on the aircraft board [2], and optimizing 3D printing [3, 4]. The problems take into account various constraints on the placement of objects in the domain (for example, the orientation of objects, the minimal allowed distances between objects, the presence of prohibited zones, balancing constraints [5–12]). In this paper, we consider the problem of finding the layout of objects optimizing mutual distances between the objects and the distances between each object and the frontier of the domain. Constraints on the position of the gravity center and the moments of inertia of the system are also taken into account. Such problems
S. Plankovskyy · A. Nikolaev · O. Shypul National Aerospace University named by M.Y. Zhukovskiy “Kharkiv Aviation Institute”, Kharkiv, Ukraine I. Litvinchev Universidad Autónoma de Nuevo León, San Nicolás de los Garza, Mexico A. Pankratov · T. Romanova () Department of Mathematical Modeling and Optimal Design, Institute for Mechanical Engineering Problems of the National Academy of Sciences of Ukraine, Kharkiv, Ukraine © Springer Nature Switzerland AG 2020 P. Vasant et al. (eds.), Data Analysis and Optimization for Engineering and Computing Problems, EAI/Springer Innovations in Communication and Computing, https://doi.org/10.1007/978-3-030-48149-0_7
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arise, e.g., in adaptive manufacturing complex shape parts [13], in shock wave processes, such as ultrasonic hardening [14], thermopulse treatment by detonable gas mixtures [15].
7.2 Problem Formulation and Mathematical Model Let a collection of three-dimensional arbitrary shaped canonically closed objects Ti ⊂ R3 , i ∈ I = {1, . . . , N} and a convex bounded set (domain) Q = (x, y, z) ∈ R 3 : min {fk (x, y, z )} ≥ 0)} be given, where each fk is a differentiable function. k∈In
Objects Ti , i ∈ IN can be continuou
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