voxelDesignTool.md
-
Amira Abdel-Rahman authoredAmira Abdel-Rahman authored
Voxel Design Tool
This is automated workflow for the inverse design of voxel units that exhibit a desired deformation given input loads. This is the first step for automated to generate a family of voxels similar to DM3 in order to use later for larger scale optimization.
I formulated the problem as a constrained optimization problem, where, given a dense start domain of fully connected (or locally connected) truss/frame elements, the objective is to minimize the volume (sum of elements' areas) of the structure while respecting the equilibrium and desired deformation. A SIMP penalty pushes the area of members to be either be very big or very small. I derived the gradients for the objective function and the constraints and using MMA (Method of Moving Asymptotes) to do the search.
Constrained Optimization Formulation for Compliant Mechanisms Design
\begin{aligned}
& \underset{\rho^e}{\text{minimize}}
& & V(\rho^e)=\sum_{e \in \Omega } \rho^e \upsilon^e \\
& \text{subject to}
& & K (\rho^e)d-F=0 \\
& & & g= L_i^T d\leq d_{max,i} \ \ \ \forall i \in 1,..,m \\
& & &\rho^e_{min} \leq \rho^e \ \ \ \forall \ e
\end{aligned}
Using the adjoint method:
The gradient of objective function:
f= \sum_{e \in \Omega } \rho^e \upsilon^e \\
\frac{\delta f}{\delta \rho_e}=\sum_{e \in \Omega } \upsilon^e
For each constraint i \forall (1,..,m):
g_a=L_i^T d - d_{max,i} -\lambda_i^T (Kd-F) \\
\frac{\delta g_A}{\delta \rho_e}=L_i^T \frac{\delta d}{\delta \rho_e} -\lambda_i^T(\frac{\delta K(\rho_e)}{\delta \rho_e } d +K \frac{\delta d}{\delta \rho_e}) \\
= - \lambda_i^T \frac{\delta K(\rho_e)}{\delta \rho_e }d + (L_i^T-\lambda_i^TK) \frac{\delta d}{\delta \rho_e}\\
\therefore K\lambda_i=L_i \\
\therefore \frac{\delta g_A}{\delta \rho_e}= - \lambda_i^T \frac{\delta K(\rho_e)}{\delta \rho_e } d \\
SIMP Penalty (to push the elements to be either 0 or 1)
K^e=((\rho^e)^\eta +\rho_{min}^e )K_0^e \\
\frac{\delta f}{\delta \rho^e}=\eta(\rho^e)^{\eta-1}K_0^e
Results
2D Optimization
3D Optimization






Initial Voxel Results (Truss Elements):
- Boundary Conditions and Search Domain:

- Positive Poisson Ratio


- Auxetic


- Chiral


- Shear

Positive Frame Elements:

Next Steps
- Clean Results
- Desired Elasticity Tensor
- Take Advantage of symmetry for more refined search
- Batch Search