- Original Article
- Open Access
Design of Lattice Structures Using Local Relative Density Mapping Method
© The Author(s) 2018
- Received: 7 June 2016
- Accepted: 28 September 2018
- Published: 25 October 2018
In order to solve the problem of substantial computational resources of lattice structure during optimization, a local relative density mapping (LRDM) method is proposed. The proposed method uses solid isotropic microstructures with penalization to optimize a model at the macroscopic scale. The local relative density information is obtained from the topology optimization result. The contour lines of an optimized model are extracted using a density contour approach, and the triangular mesh is generated using a mesh generator. A local mapping relationship between the elements’ relative density and the struts’ relative cross-sectional area is established to automatically determine the diameter of each individual strut in the lattice structures. The proposed LRDM method can be applied to local finite element meshes and local density elements, but it is also suitable for global ones. In addition, some cases are considered in order to test the effectiveness of the LRDM method. The results show that the solution time of the LRDM is lower than the RDM method by approximately 50%. The proposed method provides instructions for the design of more complex lattice structures.
- Lattice structures
- Local relative density mapping
- Topology optimization
- Additive manufacturing
Additive manufacturing (AM), also known as 3D printing, refers to a process by which a 3D digital model is used to build a part or product by depositing material in layers. The advantage of AM is that it can provide designers with great opportunities to maximize the performance of their designed products through the synthesis of shapes, sizes, hierarchical structures, and material composition . To an extent, AM can not only reduce material costs but also speed up novel and/or conceptual designs, especially with the use of foam, honeycomb structures, and lattice structures. The area of lattice structures has received considerable attention owing to their excellent properties: they can be designed and used for multiple purposes, such as weight reduction, heat transfer, energy absorption, thermal and protection [2–6]. The lattice structure with a relative density of 10% is approximately three times stronger than foam . However, it is a challenging task to design lattice structures owing to their geometric complexities and prohibitive computational costs in the design process. In the past few decades, several approaches were put forward for the design of lattice structures or mesoscale truss structures with a strut diameter in the range of 0.1 mm to 10 mm, and these methods can be roughly classified into two categories: solid modeling techniques and optimization techniques.
Wang and Rosen developed a computer-aided design tool for designing truss structures that could easily be incorporated into 3D printed parts . Wallach et al.  designed a 3D periodic truss structure using the linear array command. They analyzed the elastic moduli as well as the uniaxial and shear strengths of the truss structures. Wang et al.  proposed a parametric modeling method for truss-like structures. Based on a unit cell approach, a conformal lattice structure is designed to enhance the performance of the cellular structure that was developed by Wang . For large truss-like cellular structures, computational complexities can cause difficulties in CAD modeling, and therefore, Wang et al.  introduced a hybrid geometric modeling method. From the above analysis, we can see that solid modeling methods are primarily used for generating truss/lattice structures without considering the optimization problem. Moreover, optimization techniques can also be an effective tool for designing truss/lattice structures. Typical optimization approaches, such as the ground truss approach and homogenization method, synthesize the topology and geometry of the truss structures [12, 13]. Chu et al.  presented a synthesis method using particle swarm optimization and least-squares minimization for designing components comprised of cellular structures. The optimization approaches of truss structures are limited by the computational demands of a great number of design variables and/or by the memory limitations of computers [15–18].
Therefore, one of the key problems is how to optimize lattice structures such that the potential for mass reduction and computational costs can be fully realized in the design process. A non-iterative size, matching, and scaling (SMS) method was proposed by Graf et al. . It eliminates the need for time-consuming optimization by using a combination of a solid-body finite element analysis and a pre-defined lattice configuration to generate a structure’s lattice topology. However, the non-iterative SMS approach lacked a truly systematic methodology . To counter the traditional manufacturing limitation, Chang et al.  presented a new SMS method in order to take advantage of the potential of additive manufacturing. To address the drawbacks of the SMS approach in determining the appropriate diameters for the structure’s struts, a new augmented SMS method that incorporates conformal lattice structure construction methods was presented by Nguyen et al. . They studied the process of generation of lattice structures for a complex shape. However, the design of lattice structures using various SMS methods is only applicable in the case of targeted loading. These methods cannot be used in the case of multiple loading conditions. In addition, a relative density mapping (RDM) method was developed to obtain lattice structures that are capable of handling multi-loading conditions without additional computational costs . However, there exist some problems with the principles of the RDM method. Firstly, one of assumptions made is that the relative cross-section diameter of each strut is dependent on the contribution of all the relevant density elements. The distance calculation requires much computational resources due to its exponential time complexity. Moreover, the relative density elements include several 0 values, which do not contribution for the strut. Furthermore, the RDM method does not guarantee the structural strength of the generated lattice structures during the mapping process.
To further reduce the computational costs and to improve the performance of the lattice structures,an LRDM method is presented. First, the SIMP method is used to gain a topology optimization result at the macroscopic scale. The relative density contour is then extracted using the density contour approach and a triangular mesh is generated using a mesh generation algorithm. Finally, a new mapping relationship between the local relative densities and local finite element mesh (FEM) is established to generate the lattice structures. The rest of this paper is organized as follows. We first describe a mesh generating process in Section 2. In Section 3, we present the LRDM method, and in Section 4, some case studies are used to verify the effectiveness and efficiency of the proposed method. The conclusions drawn are presented in the final section.
The LRDM method uses the triangular FEM because it is difficult to approximate the part’s surface for the unit-cell type mesh used in the RDM method. In order to generate the mesh configuration in the optimized region of the topology optimization result, three following steps are required to be taken.
2.1 Generating Relative Density Point Cloud
2.2 Density Contour Approach
2.3 Triangular Mesh Generation
3.1 Formulation of LRDM Method
The main difference between RDM and LRDM is that the RDM process starts with two inputs: the information from the topology optimization and a coarse FEM, while the LRDM process starts with only one input: the topology optimization. Secondly, the key principles of RDM and LRDM are essentially different. The RDM method depends on the relative densities of all the elements and all the FEMs in the design region, while the LRDM method only requires the relative densities of the local elements and local FEM. Thirdly, different formulas are used to calculate the strut relative area in each method. A weighted average is used in the RDM method while a geometric mean is used in the LRDM. Furthermore, the formulas used to calculate the scaling factor Ar and the vector min(Ar) are also different.
3.2 LRDM Method Validation
2D and 3D beams
Loading magnitude (N)
Elasticity modulus (MPa)
In order to precisely test the solution time of the RDM and LRDM methods, the same unit-cell type is used to generate the 2D lattice structures. The second case (Table 1) is the same as that in Ref. , which is a cantilevered beam loaded at the middle of the right tip using the same topology optimization results. The lattice structures are generated using the RDM method with the lower relative cross-sectional value of 0.01, the size of the element is 2.5 mm×2.5 mm. While the LRDM method uses the same unit-cell type, R=6.5 mm and k=1.0.
The solution times for the RDM method are 30 s for generating the lattice structure and only 11 s for generating the lattice structure using the LRDM method. The RDM method requires more time to calculate the distance of all the relative density elements from each strut in the FEM. As for the LRDM method, it only calculates some of the relative density elements that are located in the range of a circle of radius 6.5 mm with its center at the midpoint of the strut. The RDM method produced the smallest tip displacement of 0.635 mm. The LRDM method result had a displacement of 0.613 mm, which is only 3.5% lower than that obtained with the RDM method. The results show that the LRDM method is more efficient with respect to computational costs as compared with the RDM method for almost the same structural strength. Here, we note that our method cannot only be applied to the global FEM but is also suitable for the local FEM.
Test results of computation accuracy and convergence for RDM and LRDM methods
Mesh size (mm)
Target volume (mm3)
Optimized volume (mm3)
1 × 1
2 × 2
3 × 3
Table 2 shows that the average computation accuracy of RDM is 2.05% higher than that of LRDM. One possible reason is that the total volume of the generated lattice structure must meet the volume constraint in the RDM method. The average iterative number for RDM is 43.62% higher than that for LRDM. The main reason for this is the use of the local relative density information and the triangular mesh.
4.1 2D Cantilever Beam
The reason why different proportions for case 2 and case 3 is that the same FEM is used to map the topology optimization result to the lattice structure with RDM and LRDM methods respectively in case 2. The various FEMs are adopted to generate different lattice structures with RDM and LRDM in case 3. Thus, the various FEMs may result in different proportions of the LRDM method to some extent.
4.2 3D Cantilever Beam
It should be noted that the solution time for the lattice structure generated using the RDM method remains almost unchanged as the volume fraction of the topology optimization increases. In the case of the LRDM method, the solution time of the generated lattice structure increases gradually as the volume fraction increases in the topology optimization, but the solution time is still less than that of the RDM method. The reason for this phenomenon is that the computational cost is relate to the number of FEMs and relative density elements. In the case of the RDM method, the number of FEMs is fixed once the initial design area is set; however, the number of FEMs increases with the increase in the volume fraction of the LRDM method during the mapping process.
In order to test the structural strength of the lattice structures using the two models from Figure 7d, e, we print them with the help of a MakerBot Replicator 2 printer using fused deposition modeling technology. We use a plastic PC-ABS material with a yield strength of 4.1×107 N/m2. An electromechanical universal testing machine (HUALONG WDW-100) is used to evaluate the strength of the printed lattice structures. We ran compression tests for the two printed lattice structures. The 2460.467 g lattice structure that was tested using the RDM method could resist a force of 983.7 N; the other lattice structure tested using the LRDM method, failed at an applied force of 1153.5 N with a mass of 2450.074 g. Therefore, the calculated strength-to-weight ratio of the LRDM method is 14.83% higher than that of the RDM method. Moreover, we can calculate the strength using a simple formula based on the volumetric density . The volume fraction of the lattice structures optimized with the RDM and LRDM are 0.3045% and 0.3568%, respectively. The theoretical calculation results are 14.65% higher than that obtained with the RDM, which almost the same as the experiment results.
4.3 Micro Jet Engine Bracket
The average iterative number of RDM is 43.62% higher than that of LRDM, and the average computation accuracy of RDM is 2.05% higher than that of LRDM for the same two-dimensional mesh.
As compared to the continuum topology optimization, the LRDM method can save approximately 9.8% 3D printing material for the same 3D model.
It is no contributions to the strut’s relative cross-sectional diameter to the relative density elements that are beyond a certain distance away from the strut for RDM and LRDM method.
The solution time of the lattice structures generated using the LRDM method is approximately 50% lower than that of the RDM method.
G-HS and S-KJ was in charge of the whole trial; F-LZ and Y-DW wrote the manuscript; HX and L-FQ assisted with sampling and laboratory analyses. All authors read and approved the final manuscript.
Guo-Hua Song, born in 1984, is currently an engineer at Beijing Xinghang Mechanical and Electrical Equipment Co. Ltd, China. His main research interests include structure design, additive manufacturing and topology optimization.
Shi-Kai Jing, born in 1975, is currently a lecturer and a master candidate supervisor at School of Mechanical Engineering, Beijing Institute of Technology, China. He received his PhD degree from Northwestern Polytechnical University, Xi’an, in 2005. His research interests include product data management, knowledge management, design theory and method, and manufacturing services technology.
Fang-Lei Zhao, born in 1991, is currently a master candidate at School of Mechanical Engineering, Beijing Institute of Technology, China. His main research interests include additive manufacturing and functionally graded materials.
Ye-Dong Wang, born in 1993, is currently a master candidate at School of Mechanical Engineering, Beijing Institute of Technology, China. His main research interests include additive manufacturing and functionally graded materials.
Hao Xing, born in 1991, is currently a master candidate at School of Mechanical Engineering, Beijing Institute of Technology, China. His main research interests include additive manufacturing and functionally graded materials.
Long-Fei Qie, born in 1989, is currently a PhD candidate at School of Mechanical Engineering, Beijing Institute of Technology, China. His main research interests include additive manufacturing and topology optimization.
The authors declare that they have no competing interests.
Supported by National Hi-tech Research and Development Program of China (863 Program, Grant No. 2015BAF04B00), and China Aerospace Science and Technology Corporation Program of China (CASIC Program, Grant No. 461717).
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