# Correlation Studies of Different Decoupled Two-Scale Simulations for Lattice Structures

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Procedures for Two-Scale Analysis

#### 2.1. Computational Homogenization Method for Plates

_{ij}, ε

_{ij}, and u

_{i}are the microscopic stress, strain, and displacement in the PUC. C

_{ijkl}is the stiffness tensor of the material in the lattice component. ${E}_{\alpha \beta}^{0}$ and ${K}_{\alpha \beta}^{0}$ are the macroscopic in-plane strain and curvature on the reference plane. The microscopic displacement and strain with in-plane periodicity are indicated as ${u}_{i}^{\left(per\right)}$ and ${\epsilon}_{ij}^{\left(per\right)}$. Latin indices i, j, k, and l range from one to three, while Greek indices α, β, γ, and δ take one and two.

_{αβ}and M

_{αβ}are the resultant forces and moments. The area of PUC is $\left|S\right|=\left|{\mathbf{v}}_{1}\times {\mathbf{v}}_{2}\right|$. A

_{αβγδ}, D

_{αβγδ}, and B

_{αβγδ}are the effective extension, bending, and coupling stiffness tensors. These equivalent stiffnesses are obtained in the following procedure.

- (1)
- The microscopic stress in the PUC is calculated by assigning the macroscopic unit strain or curvature (ex. ${E}_{11}^{0}=1,\text{}{E}_{22}^{0}={E}_{12}^{0}={K}_{11}^{0}={K}_{22}^{0}={K}_{12}^{0}=0$) for the microscale problem defined by Equation (1).
- (2)
- The components of effective stiffness tensor are obtained by calculating the macroscopic resultant force and moment with Equation (3) (ex. ${A}_{1111}={N}_{11},\text{}{A}_{1122}={N}_{22},\text{}{A}_{1133}={N}_{12},\text{}{B}_{1111}={M}_{11},\text{}{B}_{1122}={M}_{22},\text{}{B}_{1133}={M}_{12},\text{}$in the case of ${E}_{11}^{0}=1,\text{}{E}_{22}^{0}={E}_{12}^{0}={K}_{11}^{0}={K}_{22}^{0}={K}_{12}^{0}=0$).

#### 2.2. Computational Homogenization Method for Solids

**C**, is calculated.

_{ij}is the resultant stresses, and ${\overline{C}}_{ijkl}$ is the effective 3D stiffness tensor. The independent components of the macrostrain tensor are ${E}_{11},\text{}{E}_{22},\text{}{E}_{33},\text{}{E}_{12},\text{}{E}_{22},\mathrm{a}\mathrm{n}\mathrm{d}\text{}{E}_{31}$ instead of ${E}_{11}^{0},\text{}{E}_{22}^{0},{E}_{12}^{0},{K}_{11}^{0},{K}_{22}^{0},\mathrm{a}\mathrm{n}\mathrm{d}{K}_{12}^{0}$ in Kirchhoff–Love theory.

^{TM}for Ansys

^{®}Workbench

^{TM}[42], which was developed by CYBERNET SYSTEMS Co., Ltd. Of Tokyo, Japan.

## 3. Correlation Study of Different Homogenization Approaches for Lattice-Based Structures

^{3}, respectively. The correlation study was performed by comparing numerical solutions of a detailed model and homogenized models based on the equivalent ABD and C matrices for the plates with the periodically distributed cubic unit cells. The homogenized model based on the equivalent C matrix was further divided into three models. The first one is a model with solid elements, the second one is with shell elements, and the last one is with solsh elements. The solsh element is a shell element type with a wide range of thicknesses offered by Ansys Mechanical [43,44]. These models were simulated by the Abaqus for the detailed and homogenized models with the equivalent ABD matrix and by Ansys Mechanical [42] for the homogenized models based on the equivalent C matrix. Figure 4 shows finite element models with fine solid elements and homogenized elements. For example, approximately 1.3 million hexahedral solid elements were used for the detailed model with b = 1.2 mm to ensure enough accuracy for the study. On the other hand, the homogenized shell models were divided into 3675 quadratic shell or solsh elements, while the homogenized solid model with the equivalent C matrix used 0.24 million hexahedral solid elements. In this study, the tensile, bending, and torsional properties of the plate models were investigated.

^{2}and 1.2 mm, while the width of the plates ranged from 108 mm to 18 mm. The length of the plates was still 315 mm. The five models are shown in Figure 9. The equivalent stiffness properties of the plates were calculated for the homogenized models based on the effective C and ABD matrices. However, in the calculations of the equivalent ABD matrices with multiple cells in the thickness direction, a set of unit cells in the thickness direction are considered as a “representative unit cell”, as shown in Figure 10, since the approach assumes in-plane periodicity. Also, plate models with solid elements were evaluated for homogenized models based on the equivalent C matrix.

## 4. Conclusions

- (1)
- Tensile stiffness can be estimated by both homogenizations with reasonable accuracy even if the assumptions in periodic conditions are violated. Therefore, one can choose both homogenization approaches for cost-effective analysis of lattice structures if a focus is on tensile responses;
- (2)
- Predictions of bending and torsional stiffnesses by both homogenizations can provide enough accuracy within a certain threshold. Also, the validity range would be extended if a unit cell of lattices has a higher filling rate. Hence, these homogenizations can still be used for finite periodic arrays of lattices within the threshold.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**A concept of computational homogenization for the lattice structure. Effective characteristics of a lattice structure constructed with periodic unit cells are evaluated based on computational homogenizations with the representative unit cell. The obtained effective properties are used to predict macroscopic responses of the structure with a homogenized structural model.

**Figure 2.**A concept of computational homogenization for the lattice structures. Effective characteristics of a lattice structure constructed with periodic unit cells are evaluated based on computational homogenizations with the representative unit cell. The obtained effective properties are used to predict macroscopic responses of the structure with a homogenized structural model. (

**a**) A periodic unit cell; (

**b**) a lattice structure constructed with periodic unit cells.

**Figure 3.**A plate model constructed with simple cubic unit cells. The in-plane dimensions of the plate model were 315 mm and 105 mm in the x- and y-directions. The unit cell is bounded by 3 mm × 3 mm × 3 mm.

**Figure 4.**Plate models with different finite elements. The lattice plate is modeled as five different finite element models: the detailed model with hex solid elements, the homogenized model based on the computational homogenization for plates, and the homogenized models based on the computational homogenization for solids. The * symbols represent which software was used.

**Figure 5.**Boundary conditions of the plate models. The cantilevered condition is applied at one edge by fixing six degrees of freedom. The forced loading is applied at the center node on the other end.

**Figure 6.**Deformations of detailed and homogenized models under forced tensile displacement. The maximum error in the homogenized shell model with the equivalent ABD matrix is 2.5%. The maximum error in the homogenized models with the equivalent C matrix is 5.9%. (

**a**) Detailed solid model; (

**b**) homogenized shell model (equivalent ABD matrix); (

**c**) homogenized shell model (equivalent C matrix); (

**d**) homogenized solsh model (equivalent C matrix); (

**e**) homogenized solid model (equivalent C matrix). The color indicates the displacements in the x direction.

**Figure 7.**Deformations of detailed and homogenized models with the equivalent ABD matrix under the forced bending rotation. The maximum error in the homogenized shell model with the equivalent ABD matrix is 2.4%. The maximum error in the homogenized models with the equivalent C matrix is more than 46%. (

**a**) Detailed solid model; (

**b**) homogenized shell model (equivalent ABD matrix); (

**c**) homogenized shell model (equivalent C matrix); (

**d**) homogenized solsh model (equivalent C matrix); (

**e**) homogenized solid model (equivalent C matrix). The color indicates the displacements in the z direction.

**Figure 8.**Deformations of detailed and homogenized models with the equivalent ABD matrix under forced torsional rotation. The maximum error in the homogenized shell model with the equivalent ABD matrix is 2.3%. The maximum error in the homogenized models with the equivalent C matrix is more than 49%. (

**a**) Detailed solid model; (

**b**) homogenized shell model (equivalent ABD matrix); (

**c**) homogenized shell model (equivalent C matrix); (

**d**) homogenized solsh model (equivalent C matrix); (

**e**) homogenized solid model (equivalent C matrix). The color indicates the displacements in the z direction.

**Figure 9.**Plates with different numbers of unit cell layers for detailed solid models. The numbers of unit cell layers are varied from one to six to change the thickness-wise periodicity, while the cross-sectional areas and the width of lattice beams are fixed. (

**a**) Model 1 with the cross-sectional aspect ratio = 36; (

**b**) Model 2 with the cross-sectional aspect ratio = 9; (

**c**) Model 3 with the cross-sectional aspect ratio = 4; (

**d**) Model 4 with the cross-sectional aspect ratio = 2.25; (

**e**) Model 5 with the cross-sectional aspect ratio = 1.

**Figure 10.**An image of a “representative unit cell” of an equivalent ABD matrix for a plate with multiple cells in the thickness direction. The whole layers of the multiple unit cells, as bounded in the red box, are considered as a “representative unit cell”.

**Figure 11.**Errors of resultant loads on the center node for the homogenized models to the ones for the detailed model with different cell layers under the forced tensile displacement. The homogenized models with both equivalent ABD and C matrices maintain the prediction differences to the detailed models of less than 4%.

**Figure 12.**Errors of resultant moments on the center node for the homogenized models to the ones for the detailed model with different cell layers under the forced bending rotation. The errors of the homogenized models with the equivalent ABD matrices are maintained at less than 1%, while the ones with the equivalent C matrices increase for the higher cross-sectional aspect ratios.

**Figure 13.**Errors of resultant moments on the center node for the homogenized models to the ones for the detailed model with different cell layers under the forced torsional rotation. The errors of the homogenized models with the equivalent ABD matrices increase for the lower aspect ratio, while the ones with the equivalent C matrices increase for the higher cross-sectional aspect ratios.

**Figure 14.**Errors of resultant loads on the center node for the homogenized models to the ones for the detailed model with different widths of lattice beams under forced tensile displacement. The errors of the homogenized models are reduced as the width of the lattice beams increases. (

**a**) ABD matrices; (

**b**) C matrices.

**Figure 15.**Errors of resultant moments on the center node for the homogenized models to the ones for the detailed model with different widths of lattice beams under forced bending rotation. The errors of the homogenized models are mostly reduced as the width of the lattice beams increases. (

**a**) ABD matrices; (

**b**) C matrices.

**Figure 16.**Errors of resultant moments on the center node for the homogenized models to the ones for the detailed model with different widths of lattice beams under the forced torsional rotation. The errors of the homogenized models are mostly reduced as the width of the lattice beams increases. (

**a**) ABD matrices; (

**b**) C matrices.

**Table 1.**Resultant loads on the center node of the detailed and homogenized models under the forced tensile.

Model | b, mm | Resultant Load, N |
---|---|---|

Detailed | 1.2 | 512.32 |

Shell (ABD matrix) | 515.35 (0.59%) | |

Shell (C matrix) | 527.83 (3.03%) | |

Solsh (C matrix) | 527.88 (3.04%) | |

Solid (C matrix) | 527.85 (3.03%) | |

Detailed | 1.6 | 939.60 |

Shell (ABD matrix) | 963.06 (2.50%) | |

Shell (C matrix) | 994.68 (5.86%) | |

Solsh (C matrix) | 994.84 (5.88%) | |

Solid (C matrix) | 994.74 (5.87%) | |

Detailed | 2.0 | 1575.1 |

Shell (ABD matrix) | 1585.6 (0.67%) | |

Shell (C matrix) | 1625.4 (3.19%) | |

Solsh (C matrix) | 1626.0 (3.23%) | |

Solid (C matrix) | 1625.7 (3.21%) |

**Table 2.**Resultant moments on the center node of the detailed and homogenized model with the equivalent ABD.

Model | b, mm | Resultant Moment, N·mm |
---|---|---|

Detailed | 1.2 | 12.903 |

Shell (ABD matrix) | 13.013 (0.85%) | |

Shell (C matrix) | 6.9361 (46.24%) | |

Solsh (C matrix) | 6.9204 (46.37%) | |

Solid (C matrix) | 6.9189 (46.38%) | |

Detailed | 1.6 | 20.038 |

Shell (ABD matrix) | 20.520 (2.41%) | |

Shell (C matrix) | 13.133 (34.46%) | |

Solsh (C matrix) | 13.065 (34.80%) | |

Solid (C matrix) | 13.133 (34.46%) | |

Detailed | 2.0 | 27.704 |

Shell (ABD matrix) | 28.234 (1.91%) | |

Shell (C matrix) | 21.630 (21.92%) | |

Solsh (C matrix) | 21.417 (22.69%) | |

Solid (C matrix) | 21.633 (21.91%) |

**Table 3.**Resultant moments on the center node of the detailed and homogenized model with the equivalent ABD matrix under the forced torsional rotation.

Model | b, mm | Resultant Moment, N·mm |
---|---|---|

Detailed | 1.2 | 9.0123 |

Shell (ABD matrix) | 9.2158 (2.26%) | |

Shell (C matrix) | 3.7271 (58.64%) | |

Solsh (C matrix) | 3.6902 (59.05%) | |

Solid (C matrix) | 3.6699 (59.28%) | |

Detailed | 1.6 | 23.019 |

Shell (ABD matrix) | 23.442 (1.84%) | |

Shell (C matrix) | 11.645 (49.41%) | |

Solsh (C matrix) | 11.541 (49.86%) | |

Solid (C matrix) | 11.473 (50.16%) | |

Detailed | 2.0 | 41.934 |

Shell (ABD matrix) | 41.549 (0.92%) | |

Shell (C matrix) | 25.644 (38.85%) | |

Solsh (C matrix) | 25.424 (39.37%) | |

Solid (C matrix) | 25.266 (39.75%) |

**Table 4.**Resultant loads on the center node of the detailed and homogenized models with multiple cell layers under the forced tensile displacement.

b, mm | Width, mm | Thickness, mm | Resultant Load, N | ||
---|---|---|---|---|---|

Detailed | Shell (ABD Matrix) | Solid (C Matrix) | |||

1.2 | 108 | 3 | 526.996 | 530.083 (0.59%) | 545.720 (3.55%) |

54 | 6 | 533.166 | 535.813 (0.50%) | 545.540 (2.32%) | |

36 | 9 | 535.078 | 537.723 (0.49%) | 545.490 (1.95%) | |

27 | 12 | 535.843 | 538.677 (0.53%) | 545.470 (1.80%) | |

18 | 18 | 536.225 | 539.631 (0.64%) | 545.460 (1.72%) | |

1.6 | 108 | 3 | 987.637 | 990.646 (0.30%) | 1023.210 (3.60%) |

54 | 6 | 999.829 | 1002.53 (0.27%) | 1022.384 (2.26%) | |

36 | 9 | 1003.39 | 1006.47 (0.31%) | 1022.151 (1.87%) | |

27 | 12 | 1004.82 | 1008.44 (0.36%) | 1022.061 (1.72%) | |

18 | 18 | 1005.54 | 1010.41 (0.48%) | 1022.017 (1.64%) | |

2.0 | 108 | 3 | 1633.99 | 1631.08 (−0.18%) | 1672.289 (2.34%) |

54 | 6 | 1647.24 | 1642.77 (−0.27%) | 1669.681 (1.36%) | |

36 | 9 | 1651.11 | 1646.60 (−0.27%) | 1668.957 (1.08%) | |

27 | 12 | 1652.69 | 1648.52 (−0.25%) | 1668.685 (0.97%) | |

18 | 18 | 1653.49 | 1650.44 (−0.18%) | 1668.556 (0.91%) |

**Table 5.**Resultant moments on the center node of the detailed and homogenized models with multiple cell layers under the forced bending rotation.

b, mm | Width, mm | Thickness, mm | Resultant Moment, N·mm | ||
---|---|---|---|---|---|

Detailed | Shell (ABD Matrix) | Solid (C Matrix) | |||

1.2 | 108 | 3 | 13.2743 | 13.3873 (0.85%) | 7.154 (46.11%) |

54 | 6 | 33.9294 | 34.189 (0.77%) | 28.579 (15.77%) | |

36 | 9 | 68.6652 | 69.1156 (0.66%) | 64.272 (6.40%) | |

27 | 12 | 117.475 | 118.291 (0.69%) | 114.240 (2.75%) | |

18 | 18 | 257.200 | 259.103 (0.74%) | 257.020 (0.07%) | |

1.6 | 108 | 3 | 20.9876 | 21.1148 (0.61%) | 13.439 (35.97%) |

54 | 6 | 59.6814 | 59.9601 (0.47%) | 53.598 (10.19%) | |

36 | 9 | 124.748 | 125.31 (0.45%) | 120.463 (3.43%) | |

27 | 12 | 216.184 | 217.178 (0.46%) | 214.066 (0.98%) | |

18 | 18 | 477.997 | 480.607 (0.55%) | 481.534 (0.74%) | |

2.0 | 108 | 3 | 28.8551 | 29.0628 (0.72%) | 22.034 (23.64%) |

54 | 6 | 92.8346 | 92.9382 (0.11%) | 87.637 (5.60%) | |

36 | 9 | 199.988 | 199.98 (0.00%) | 196.772 (1.61%) | |

27 | 12 | 350.387 | 350.214 (0.05%) | 349.517 (0.25%) | |

18 | 18 | 780.946 | 780.662 (0.04%) | 786.025 (0.65%) |

**Table 6.**Resultant moments on the center node of the detailed and homogenized models with multiple cell layers under forced torsional rotation.

b, mm | Width, mm | Thickness, mm | Resultant Moment, N·mm | ||
---|---|---|---|---|---|

Detailed | Shell (ABD Matrix) | Solid (C Matrix) | |||

1.2 | 108 | 3 | 9.37778 | 9.58021 (2.16%) | 3.873 (58.70%) |

54 | 6 | 17.9976 | 18.8126 (4.53%) | 13.402 (25.53%) | |

36 | 9 | 30.1893 | 33.3146 (10.35%) | 26.474 (12.31%) | |

27 | 12 | 42.0466 | 50.8146 (20.85%) | 39.482 (6.10%) | |

18 | 18 | 52.3985 | 83.9939 (60.30%) | 51.423 (1.86%) | |

1.6 | 108 | 3 | 23.9114 | 24.2967 (1.61%) | 11.901 (50.23%) |

54 | 6 | 52.9161 | 54.6308 (3.24%) | 42.151 (20.34%) | |

36 | 9 | 92.9082 | 99.7746 (7.39%) | 83.874 (9.72%) | |

27 | 12 | 132.062 | 151.731 (14.89%) | 125.723 (4.80%) | |

18 | 18 | 166.669 | 236.039 (41.62%) | 164.490 (1.31%) | |

2.0 | 108 | 3 | 42.5185 | 43.017 (1.17%) | 26.130 (38.54%) |

54 | 6 | 109.803 | 111.973 (1.98%) | 93.362 (14.97%) | |

36 | 9 | 201.765 | 212.075 (5.11%) | 187.237 (7.20%) | |

27 | 12 | 292.105 | 323.915 (10.89%) | 281.315 (3.69%) | |

18 | 18 | 372.514 | 486.983 (30.73%) | 368.809 (0.99%) |

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## Share and Cite

**MDPI and ACS Style**

Tsushima, N.; Higuchi, R.; Yamamoto, K.
Correlation Studies of Different Decoupled Two-Scale Simulations for Lattice Structures. *Aerospace* **2023**, *10*, 723.
https://doi.org/10.3390/aerospace10080723

**AMA Style**

Tsushima N, Higuchi R, Yamamoto K.
Correlation Studies of Different Decoupled Two-Scale Simulations for Lattice Structures. *Aerospace*. 2023; 10(8):723.
https://doi.org/10.3390/aerospace10080723

**Chicago/Turabian Style**

Tsushima, Natsuki, Ryo Higuchi, and Koji Yamamoto.
2023. "Correlation Studies of Different Decoupled Two-Scale Simulations for Lattice Structures" *Aerospace* 10, no. 8: 723.
https://doi.org/10.3390/aerospace10080723