Multi-material Topology Optimization of Lattice Structures using Geometry Projection
Multi-material Topology Optimization of Lattice Structures using Geometry Projection
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DOI：
10.1016/j.cma.2020.112895
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发表时间：
2019-10
期刊：
影响因子：
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通讯作者：
Hesaneh Kazemi;A. Vaziri;Julián A. Norato
中科院分区：
文献类型：
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作者：
Hesaneh Kazemi;A. Vaziri;Julián A. Norato
这项工作提出了一种计算方法的设计架构桁架网格材料，其中每个支柱可以由一组可用的材料之一。我们设计的晶格极值有效属性。按照拓扑优化中的惯例，我们设计了一个周期性的晶格晶胞，并通过数值均匀化得到了有效的性质。每个杆被表示为中轴的圆柱形偏移表面，其由中轴的端点的位置参数化。通过几何投影方法将这些参数平滑地映射到连续密度场上，进行初始和灵敏度分析。每种材料的尺寸变量被赋予每个杆，并在基于密度的拓扑优化中被惩罚，以促进从设计中完全移除杆。在优化过程中，我们允许杆由可用材料的混合物制成。然而，为了确保每个杆件都是由一种材料制成，或者从最佳设计中完全删除，我们施加了优化约束，确保每个尺寸变量为0或1，并且最多有一个材料尺寸变量为1。建议的材料插值方案很容易容纳任何数量的材料。为了获得具有所需材料对称性的晶格，我们仅设计单位晶胞的参考区域，并相对于适当的对称平面反映其几何投影。此外，为了确保在单位单元内或相对于周期性边界反射时杆保持完整，我们对杆施加了无切割约束。我们证明了我们的方法的有效性，通过数值例子的体积和剪切模量最大化和泊松比最小化的两个和三个材料晶格立方对称。
This work presents a computational method for the design of architected truss lattice materials where each strut can be made of one of a set of available materials. We design the lattices to extremize effective properties. As customary in topology optimization, we design a periodic unit cell of the lattice and obtain the effective properties via numerical homogenization. Each bar is represented as a cylindrical offset surface of a medial axis parameterized by the positions of the endpoints of the medial axis. These parameters are smoothly mapped onto a continuous density field for the primal and sensitivity analysis via the geometry projection method. A size variable per material is ascribed to each bar and penalized as in density-based topology optimization to facilitate the entire removal of bars from the design. During the optimization, we allow bars to be made of a mixture of the available materials. However, to ensure each bar is either exclusively made of one material or removed altogether from the optimal design, we impose optimization constraints that ensure each size variable is 0 or 1, and that at most one material size variable is 1. The proposed material interpolation scheme readily accommodates any number of materials. To obtain lattices with desired material symmetries, we design only a reference region of the unit cell and reflect its geometry projection with respect to the appropriate planes of symmetry. Also, to ensure bars remain whole upon reflection inside the unit cell or with respect to the periodic boundaries, we impose a no-cut constraint on the bars. We demonstrate the efficacy of our method via numerical examples of bulk and shear moduli maximization and Poisson’s ratio minimization for two- and three-material lattices with cubic symmetry.