Metamaterials are defined as a family of rationally designed artificial materials which can provide extraordinary effective properties compared with their nature counterparts. This paper proposes a level set based method for topology optimization of both single and multiple-material Negative Poisson's Ratio (NPR) metamaterials. For multi-material topology optimization, the conventional level set method is advanced with a new approach exploiting the reconciled level set (RLS) method. The proposed method simplifies the multi-material topology optimization by evolving each individual material with a single level set function and reconciling the result level set field with the Merriman-Bence-Osher (MBO) operator. The NPR metamaterial design problem is recast as a variational problem, where the effective elastic properties of the spatially periodic microstructure are formulated as the strain energy functionals under uniform displacement boundary conditions. The adjoint variable method is utilized to derive the shape sensitivities by combining the general linear elastic equation with a weak imposition of Dirichlet boundary conditions. The design velocity field is constructed using the steepest descent method and integrated with the level set method. Both single and multiple-material mechanical metamaterials are achieved in 2D and 3D with different Poisson's ratios and volumes. Benchmark designs are fabricated with multi-material 3D printing at high resolution. The effective auxetic properties of the achieved designs are verified through finite element simulations and characterized using experimental tests as well.
Topology optimization has been considered as a promising tool for conceptual design due to its capability of generating innovative design candidates without depending on the designer's intuition and experience. Various optimization methods have been developed through the years, and one of the promising options is the level-set-based topology optimization method. The benefit of this alternative method is that the design is characterized by its clear boundaries. This advantage can avoid postprocessing work in conventional topology optimization process to a large extent and realize direct integration between topology optimization and additive manufacturing (AM). In this paper, practical algorithms and a matlab-based open source framework are developed to seamlessly integrate the level-set-based topology optimization procedure with AM process by converting the design to STereoLithography (STL) files, which is the de facto standard format for three-dimensional (3D) printing. The proposed algorithm and code are evaluated by a proof-of-concept demonstration with 3D printing of both single and multimaterial topology optimization results. The algorithm and the open source framework proposed in this paper will be beneficial to the areas of computational design and AM.
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