As advanced manufacturing techniques such as additive manufacturing become widely available, it is of interest to investigate the potential advantages that arise when designing periodic metamaterials to achieve a specific desired behavior or physical property. Designing the fine scale detailed geometry of periodic metamaterials to achieve a specified behavior falls under the category of notoriously intractable inverse problems. To simplify solving the inverse problem, most relevant works represent metamaterials as periodic single unit cell structures repeated in regular lattices. Such representation simplifies modeling and simulation task but at the cost of possibly limiting the range of physical behaviors that can be achieved through the use of more than one unit cell structures. This article outlines a quasi-periodic representation that utilizes more than a single unit cell to generate periodic metamaterials. Additionally, a hierarchical optimization scheme to optimize the generating function for a quasi-periodic structure using the genetic algorithm (GA) and a barrier function interior point method is also sketched to solve the inverse problem. To demonstrate the utility of the proposed hierarchical optimization framework to solve quasi-periodic metamaterial inverse problem, a problem in which the objective is to minimize the total strain in the structure while subjected to weight and the total-size constraint is considered. We detail the overall computational approach in which geometric representation, optimization algorithms, and finite element analysis are coupled and report preliminary numerical experiments.

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