Hierarchical lattice: Design strategy and topology characterization

The structure-material integrated design is an art-of-state concept and be enabled by additive manufacturing. The lattice material is classified into structure as well as material because mechanical properties are determined by its topology. However, the lack of a flexible design strategy hinders the lattice achieve the structure-material integrated material candidate. This work suggests the strut-nested based strategies to effectively conduct the hierarchical lattice design. The strut in the larger-scale lattice can be replaced by the smaller-scale lattice structure through the rotation, stretching, and translation operations combining the local and global numbering, thereby complete the multi-scale lattice design. The design skills are well elucidated with custom-developed algorithm; a serious of complex lattices achieve multi-scale design. The influence of hierarchical structures in lattices on a significant parameter, strut length-to-diameter, is identified. Our work offers the alternative strategy to realize the hierarchical lattice design.


Introduction
Lattice materials are composed of the repeated unit cells with the plate or strut element and aim at being stiffer with less materials. [1][2][3][4][5][6][7][8] Hierarchical lattices refer to those lattices whereby each strut of the lattice embeds a set of lattices on a successively smaller scale. [9][10][11][12] The hierarchical lattices are consistent with the porous hierarchical structure, such as the bird nest. 13 Besides, it can increase the surface area for the tissue engineering scaffolds 14 and thermal heat exchanger 15 and vibration isolation. 16 Also, it is an effective topology tuning the deformation mechanisms. 11 The advent of metal additive manufacturing (AM) methods has enabled an unprecedented development of lattices materials, including the hierarchical lattices.
So far, some work has involved in the design of twoscale hierarchical triply periodic minimal surface (TPMS)-based lattices. 17 However, the technology cannot make full use of the available lattice unit cells and fail to extend to strut-based lattice design which forms the motivation of this study.
In the present work, we propose an effective topological strategy to conduct the hierarchical lattice design. Furthermore, we construct couples of hierarchical lattices using the mix combination of bending-dominated lattice based on different element types. The strut length-to The algorithm of strut-nested strategy Firstly, beam structure, including the number and natural coordinate of all the nodes and the numbers of the two endpoints corresponding to each beam) in the smaller-scale is determined. Then, the beam in largescale is replaced by smaller-scale beam. In the replacement procedure, the number of nodes in the connected large-scale beam should be consistent with that in the small-scale beam. After replacing the original beam with a substructure (local secondary structure), we can get the global secondary structure. If we want to change from the global secondary structure to the global tertiary structure, we can recursively repeat the process by considering the current n-level structure as the initial structure and replacing each beam with a local secondary structure to get the n + 1 level structure. The detailed algorithm is presented as follows.
Local-structure nesting to global structure: Mapping from natural to global coordinates When replacing a large-scale beam with a small-scale lattice structure, each node in the small-scale lattice structure needs to be converted from known natural coordinates to global coordinates in order to be nested in the global structure. The mapping of each node of small-scale lattice structure from local coordinates to global coordinates (fromr nat tor glo ) can be achieved by a three-step method: Rotate, scale, and translate. The operations of these three steps are shown in Figure 1, and the key steps are shown as follows.
Rotating relationship: the overall orientation changes from r 0 ! to p 1 ! À p 0 ! . This rotation matrix can be obtained by rotating a specific angle around a specific rotation axis. The axis of rotation and the angle of rotation can be obtained by: (i) In case that r 0 ! and p 1 ! À p 0 ! are not parallel, we can let the rotation axis be perpendicular to both r 0 ! and p 1 ! À p 0 ! , that is, the rotation axis and then construct the rotation matrix R by the rotation axisã and the rotation angle u.
where I is the unit tensor and is the tensor and multiplication operation. (ii) In case of r 0 ! is parallel p 1 ! À p 0 ! , it is impossible to calculate the rotation axis using the cross product, but the rotation matrix can be obtained directly due to the parallelism: if the directions are the same, no rotation is needed if the reverse is true, set the rotation matrix to I. Thus, the rotation matrix R containing the rotation axis and the rotation angle is expressed as: Scaling relationship: the vector length needs to be scaled from r 0 ! in local coordinates to p 1 ! À p 0 ! , and the scaling ratio is = p 1 ! À p 0 ! = r 0 ! .
Translating relationship: The base point of the vector needs to be changed from the origin in local coordinates to the end point of the beam in global coordinates (p 0 ! , i.e. the displacement vectord = p 0 ! ).
Thus, by combining the rotation operation, the scaling operation, and the translation operation, the noder nat in the substructure can be mapped from local to global coordinatesr glõ r glo = uR Ár nat + p 0 ! ð3Þ Sub-structure nesting to the global structure: Local versus global numbering For the substructure, each node and each beam have a corresponding number. When the substructure is nested into the global structure, the local numbers need to be converted into global numbers so that all nodes and beams in the global structure have unique numbers. For the two endpoints in the substructure, since their endpoints need to be connected to the endpoints of other substructures, both substructures need to maintain the same global node number at the endpoint connection. To achieve this, we keep the node numbers of the original parent structure (thick beam) unchanged, and then identify the local numbers of the two endpoints in the substructure by the position coordinates, and replace the local numbers with the node numbers in the parent structure to maintain the endpoint connection relationship between the original parent structure.
In addition, for beam numbering, a dictionary data structure can be used to achieve duplication-free beam numbering. The keys (or indexes) of the dictionary can be used for the node numbers of the beams. Since two nodes can uniquely identify a beam, a sorted tuple of two node numbers can be used as the index of the beam.

Verification and its influence on topology parameters
We construct three-scale lattices based on the customized developed algorithm. The single-scale FCC, two-scale FCC, three-scale FCC, single-scale BCC, two-scale BCC, three-scale BCC, single-scale diamond, two-scale diamond, three-scale diamond are constructed. Our strategy can easily tune the scales in the hierarchical structure, in contrast to that most reported work focuses on two-scale structure ( Figure 2).
The introduction of hierarchy in lattices can adjust the slenderness l=d where l is strut length, and d is the strut length. 11 As Figure 3 shows, the two-scale lattices generally show lower strut length-to-diameter values in case of BCC-BCC hierarchical lattices. For two-scale octet-BCC lattice; in a low relative density, 1.5%, the l=d for single-scale BCC lattice is 16.67, and is reduced to 4.74 for two-scale BCC-BCC lattice. The hierarchical structure can reduce the strut length-to-diameter, thereby tuning the deformation mechanisms for more stretching deformation, 11 and is expected to improve the load-bearing efficiency.

Conclusion
We suggest the strut-nested strategy to achieve the conformal and hierarchical design toward the materialstructure integrated design. The strut-nested strategy applies the rotation, stretching, and translation operations, combining the local versus global numbering, to construct multi-scale hierarchical lattices. Three-scale BCC, FCC, and diamond lattices are constructed. The introduction of hierarchy can tune the slenderness l=d. l=d of two-scale lattice is lower than that of one-scale  lattice at the case of the same relative density. For example, l=d for BCC-Octet truss hierarchical lattice is 2.87 while for single-scale lattice is 6.82 at relative density of 22.5%.

Declaration of conflicting interests
The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding
The author(s) received no financial support for the research, authorship, and/or publication of this article.

Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.