# Symmetry Based Material Optimization

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## Abstract

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## 1. Introduction

- Both static and kinematic symmetry behaviors are considered. Under a symmetric force field, we ask the displacement of the static equilibrium status and acceleration in kinematic simulation to be symmetry.
- We find that all the above symmetric vector fields fall in the kernel of a linear operator associated with the symmetry. Therefore, the asymmetric error can be efficiently measured via a simple ${\ell}_{2}$ norm.
- Taking the common assumption used in modal analysis that low frequency modes dominant the behavior, we formulate the physical equation in the above kernel space and take its leading eigenvectors as the training force fields so that the resulting material can be generally applied.

## 2. Related Work

#### 2.1. Symmetry

#### 2.2. Material Optimization

## 3. Symmetry Behavior

#### 3.1. Shape-Symmetry

#### 3.2. Symmetry of Fields

#### 3.3. Symmetric Behavior

#### 3.4. Training Force Fields

## 4. Discretization and Numerical Method

#### Numerical Method

## 5. Results

#### 5.1. Optimized Symmetric Deformation

#### 5.2. Optimized Material Distribution

#### 5.3. Kernel Forces Visualization

#### 5.4. Performance

#### 5.5. Symmetrizing Dynamics

## 6. Conclusion and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Gradients

## Appendix B. Diagonal of the Approximate Hessian

## References

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**Figure 1.**Asymmetric deformations caused by meshing. Even subdividing the meshes, asymmetry still exists, which is largely due to the asymmetrically distribution of badly shaped elements as shown in the close-up figure of the cut section.

**Figure 3.**Deformation of a cuboid model under different forces. We use a linear material with Poisson’s ratio $0.3$ and Young’s modulus ${10}^{4}$. The top and bottom rows show the deformation results under a downward and an outward force, respectively. In both cases, points on the reflection plane are fixed. In the middle columns, we show the deformation result before optimization (w/o opt.) and after optimization (w/ opt.). As for a quantitative evaluation, we also calculate the symmetry deviation per node. After optimization (green), the error distribution among nodes is significantly refined compared to its unoptimized counterpart (red). Before optimization, the averaged nodal symmetric deviation is $0.244$ for the first example and $0.099$ for the second. After optimization, this metric reduces to $0.054$ and $0.009$, respectively.

**Figure 4.**Deformation of a cross model under different forces. We still use a linear material with Poisson’s ratio $0.3$ and Young’s modulus ${10}^{4}$. The top and bottom rows show the deformation results under a downward force and a spin force, respectively. In both cases, points on the innermost layer are fixed (red points in the first column). The histograms count the same metric as Figure 3. Before and after optimization, the first deformation example has an averaged nodal symmetry deviation of $0.173$ and $0.199$, respectively, while they become $0.003$ and $0.042$ for the second example.

**Figure 5.**Visualization of the distribution of symmetric displacement error ${\u03f5}_{s}$ on sampling points in one of the symmetric parts (other parts are shown in white color). The color bar of ${\u03f5}_{s}$ is shown for each model. The value increases as the color varies from blue to red. For all these models, symmetric forces applied to their ends with center parts are fixed.

**Figure 6.**This figure shows the optimized material for the cuboid model (left) and the cross model (right) by visualizing the stiffness scale ${s}_{e}$ on each element. Generally, big elements have a small stiffness scale to compensate their numerical stiffness and vice versa. Compared to applying the reduced method [14] to optimize the material, although being faster in the computations, the resulting material could be undesirably smoothed out and introduce a large error. In this figure, the error is computed as ${\overline{\u03f5}}_{s}=\frac{1}{N}{\sum}_{i=1}^{N}{\int}_{\Omega}{\parallel {\tilde{z}}_{i}\left({x}_{1}\right)\parallel}_{2}^{2}d{x}_{1}$. These two examples have an average nodal symmetry deviation ${\overline{\u03f5}}_{s}$ of 0.245 and 0.259, respectively, without material optimization.

**Figure 7.**Another setting of fixed points for the cuboid model. In this experiment, the sampling points colored in red on the end of the model are fixed (

**a**), and the optimized material is shown in (

**b**). We use a linear material with Poisson’s ratio $0.3$ and Young’s modulus ${10}^{4}$. In this case, we apply a downward force to the whole model, and the result shows that after optimization (

**d**), the deformation result is also more symmetric than before (

**c**).

**Figure 8.**Visualization of the first eight kernel vectors of ${\mathbf{S}}_{1}$ (i.e., the training forces). The kernel vectors associated with smaller eigenvalues are smoother and dominate the physical behavior.

**Figure 9.**Statistic of the energy value and gradient (Frobenius norm) for the material optimization of two models. In the curve graph, the energy values and gradient norms are normalized by their initial values and taken by the logarithm, so that the magnitude change of their values could be better observed. For both models, the energy values quickly drop in a few iterations.

**Figure 10.**We use a linear material with Poisson’s ratio $0.3$ and Young’s modulus ${10}^{4}$. From two different camera views (top and bottom two rows, respectively), we can clearly see that the symmetry of the dynamics is preserved much better after optimizing both the elastic stiffness and mass scalings. Reflected in the numerical metrics, the symmetry error calculated by Equation (7) is reduced greatly along the entire simulation as depicted by the bottom chart, in which the blue curve is for the optimized result, while the green one is for the unoptimized result.

**Figure 11.**After optimizing the stiffness and mass scaling sequentially (bottom row), the vibration amplitude and phase of each component are increasingly consistent compared to those without optimization (top row) and optimizing elastic stiffness only (middle row). The colored curves record the vertical displacements of three marker nodes (blue dots on the model) selected on each wing with the same color.

**Table 1.**The time statistics of the material optimization for more models. The second and third rows show the number of vertices $\left|\mathcal{X}\right|$ and elements $\left|\mathcal{E}\right|$, respectively. Constraint num is the number of symmetry constraints. Kernel dim is the dimension of the kernel of ${\mathbf{S}}_{1}$. Training records the time of generating $\left\{{\mathbf{f}}_{i}\right\}$, which includes the cost of computing the kernel of ${\mathbf{S}}_{1}$ and solving the general eigenvalue problem in Equation (25). Optimization records the time of the numerical optimization in Equation (28). The last row displays the total running time of our implementation.

Mode | ||||||
---|---|---|---|---|---|---|

Vertex num | 879 | 1922 | 3944 | 497 | 675 | 3187 |

Tet num | 2835 | 6444 | 12,508 | 1169 | 1951 | 9603 |

Constraint num | 921 | 1386 | 5982 | 3069 | 2556 | 5502 |

Kernel dim | 1011 | 1461 | 5538 | 18 | 21 | 2640 |

Training | 0.73 s | 1.27 s | 43.26 s | 18.07 ms | 6.11 ms | 33.83 s |

Optimization | 11.79 s | 33.66 s | 83.34 s | 6.92 s | 8.42 s | 60.75 s |

Total | 12.98 s | 35.21 s | 136.28 s | 7.13 s | 8.81 s | 99.54 s |

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**MDPI and ACS Style**

Shi, Z.; Lin, J.; Chen, J.; Jin, Y.; Huang, J.
Symmetry Based Material Optimization. *Symmetry* **2021**, *13*, 315.
https://doi.org/10.3390/sym13020315

**AMA Style**

Shi Z, Lin J, Chen J, Jin Y, Huang J.
Symmetry Based Material Optimization. *Symmetry*. 2021; 13(2):315.
https://doi.org/10.3390/sym13020315

**Chicago/Turabian Style**

Shi, Zeyun, Jinkeng Lin, Jiong Chen, Yao Jin, and Jin Huang.
2021. "Symmetry Based Material Optimization" *Symmetry* 13, no. 2: 315.
https://doi.org/10.3390/sym13020315