Mixed-integer second-order cone optimization for composite discrete ply-angle and thickness topology optimization proble
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Mixed‑integer second‑order cone optimization for composite discrete ply‑angle and thickness topology optimization problems Sicheng He, et al. [full author details at the end of the article] Received: 16 November 2019 / Revised: 28 September 2020 / Accepted: 8 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Discrete variable topology optimization problems are usually solved by using solid isotropic material with penalization (SIMP), genetic algorithms (GA), or mixed-integer nonlinear optimization (MINLO). In this paper, we propose formulating discrete ply-angle and thickness topology optimization problems as a mixed-integer second-order cone optimization (MISOCO) problem. Unlike SIMP and GA methods, MISOCO efficiently finds the problem’s globally optimal solution. Furthermore, in contrast with existing MISOCO formulations of discrete ply-angle optimization problems, our reformulations allow the structure to change topology, consider the more realistic Tsai–Wu stress yield criteria constraint, and eliminate checkerboard patterns using simple linear constraints. We address two types of discrete ply-angle and thickness problems: a structural mass minimization problem and a compliance optimization problem where the objective is to maximize the structural stiffness. For each element, one first chooses if the element is present or not in the structure. One can then choose the element’s ply-angle and thickness from a finite set of possibilities for the former case. The discrete design space for ply-angle and thickness is a result of manufacturing limitations. To improve the problem’s MISOCO solution approach, we develop valid inequality constraints to tighten the continuous relaxation of the MISOCO reformulation. We compare the performance of various MISOCO solvers: Gurobi, CPLEX, and MOSEK to solve the MISOCO reformulation. We also use BARON to solve the original MINLO formulations of the problems. Our results show that solving the MISOCO problem’s formulation using MOSEK is the most efficient solution approach. Keywords Laminated composites · Discrete ply-angle · Topology optimization · Global optimization · Branch-bound-and-cut (B&B&C) · Mixed-integer secondorder cone optimization (MISOCO)
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1 Introduction The laminated composite optimization problem with ply-angle design variables is a classic structural optimization problem. In this problem, a structural performance metric, such as compliance or mass, is optimized with respect to the plyangles in a laminated plate. This design problem can be formulated as one of two main types of optimization problems, depending on whether the ply angles are considered to be continuous or discrete: (a) A continuous optimization problem in which the ply-angles are continuous variables; (b) A mixed-integer optimization problem in which the ply-angles are picked from a discrete set. Ghiasi et al. (2010) and Nikbakt et al. (2018) provide additional details on these types of problems. The continuous ply-ang
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