# 通过密度方法进行拓扑优化

2019年 1月 4日

### 拓扑优化和密度方法

\rho \frac{D\mathbf{v}}
{Dt}
= -\mathbf
{\nabla}p+\eta\nabla^2\mathbf{v}-\alpha(\theta_c)\mathbf{v}

### 密度模型功能简介

\theta_f = R_\mathrm{min}^2\mathbf{\nabla}
^2\theta_f + \theta_c.

\theta = \frac{\tanh(\beta(\theta_f-\theta_
{\beta}))+\tanh(\beta\theta_{beta}
)}{\tanh(\beta(1-\theta_
{\beta}))+\tanh(\beta\theta_{beta}
)}

\begin{align}
\theta_p %26= \theta_\mathrm{min}(1-\theta_\mathrm{min})\theta^{p_\textsc{simp}}\\
E_p %26= E\theta_p
\end{align}

“密度模型”功能位于组件 > 定义的拓扑优化下。网格边的长度被用作默认过滤器半径，并且效果很好，但是必须替换为固定值才能产生与网格无关的结果。

\theta_c 边界控制材料体积因子 0\leq\theta_c\leq1
\theta_f 过滤的材料体积因子 \theta_f = R_\mathrm{min}^2\mathbf{\nabla}^2\theta_f + \theta_c
\theta 材料体积因子 \theta = \frac{\tanh(\beta(\theta_f-\theta_{beta}
))+\tanh(\beta\theta_

{\beta})}{\tanh(\beta(1-\theta_{beta}
))+\tanh(\beta\theta_

{\beta}
)}

\theta_p 罚材料体积因子 \theta_p = \theta_\mathrm{min}+(1-\theta_\mathrm{min}
)\theta^{p_\textsc{simp}}
或者 \theta_p = \frac{q_\mathrm{Darcy}(1-\theta)}{q_\mathrm{Darcy}
+\theta}

### 后续操作

#### 更多资源

• 尝试对以下示例模型使用密度函数进行拓扑优化：
• 在 COMSOL 博客中了解有关拓扑优化的更多信息：

### 参考文献

1. B.S. Lazarov and O. Sigmund, “Filters in topology optimization based on Helmholtz‐type differential equations,” International Journal for Numerical Methods in Engineering, vol. 86, no. 6, pp. 765–781, 2011.
2. F. Wang, B.S. Lazarov, and O. Sigmund, “On projection methods, convergence and robust formulations in topology optimization,” Structural and Multidisciplinary Optimization, vol. 43, pp. 767–784, 2011.
3. M.P. Bendsøe, “Optimal shape design as a material distribution problem,” Structural Optimization, vol. 1, pp. 193–202, 1989.

glTF 和 glTF 徽标是 Khronos Group Inc. 的商标。

#### 评论 (11)

2023-03-06

##### Kristian Ejlebjærg Jensen
2023-03-06 COMSOL 员工

The bracket_topology_optimization_stl model demonstrates this.

2023-03-06

##### Kristian Ejlebjærg Jensen
2023-03-06 COMSOL 员工

No, four values are used as shown in the UI picture of “Applying Continuation to Avoid Local Minima”, but you can use more values, if you prefer.

2023-03-07

2023-03-08

##### 屹磊 金
2023-03-20 COMSOL 员工

2023-04-02

##### Kristian Ejlebjærg Jensen
2023-04-03 COMSOL 员工

COMSOL 6.1 仅支持对单个研究进行基于梯度的优化，但如果物理场接口都依赖于稳态求解器，则它们可以通过单个研究步骤求解。 可以使用分离求解器来控制物理场求解的顺序（这与单向耦合特别相关）。

2023-11-09

##### Kristian Ejlebjærg Jensen
2023-11-09 COMSOL 员工

Hi Tom Steven

COMSOL does not have examples of topology optimization for electromagnetic wave propagation, but the provided functionality is similar to what is used in academic literature. Problems with grayscale can be due to bad interpolation and/or the absence of a volume constraint. I suggest studying the academic literature on the subject.

Best regards,

Kristian E. Jensen

Technical Product Manager, Optimization