ALM Cantilever

A cantilever optimised under Additive Layer Manufacturing (ALM) constraints. The MNA continuous formulation arranges design variables in \(N_y = 10\) horizontal layers, each with \(N_p = 5\) printed columns. Linear overhang constraints enforce a 45° self-support angle between adjacent layers.

Running

ggp optimize --preset alm_cantilever

To save the density image:

ggp optimize --preset alm_cantilever --output-dir docs/_static

The preset uses the AMJax FEM solver by default (fem_solver: amjax).

Result

Optimized ALM cantilever topology — Optimized density field after 150 MMA iterations (40 % volume fraction, 45° overhang limit).

Problem details

Parameter	Value
Domain	60 × 30
Mesh	60 × 30 quads
Formulation	ALM 2D (MNA continuous)
Layers \(N_y\)	10
Columns \(N_p\)	5
Design variables	\(2 \times 10 \times 5 + 2 \times 5 + 2 = 112\)
Volume fraction	0.40
Overhang angle	45°
Algorithm	MMA
Max iterations	150

Variable Structure

Each design variable vector has the layout:

\[\mathbf{x} = [\underbrace{X_{c,0,0},\,L_{0,0},\;\ldots\;,X_{c,9,4},\,L_{9,4}}_{2 N_y N_p},\; \underbrace{h_0,\ldots,h_4}_{N_p},\; \underbrace{M_{c,0},\ldots,M_{c,4}}_{N_p},\; y_0,\;\theta_0]\]

\(h_j\) — normalised total print height of column \(j\)
\(M_{c,j}\) — membership density for the full column
\(y_0\), \(\theta_0\) — global printing-plane offset and rotation (both 0 here)

Overhang constraint (per layer interface, per column):

\[ \begin{align}\begin{aligned}X_{c,k+1,j} + \tfrac{L_{k+1,j}}{2} - X_{c,k,j} - \tfrac{L_{k,j}}{2} &\le \Delta h \tan(45°)\\X_{c,k,j} - \tfrac{L_{k,j}}{2} - X_{c,k+1,j} + \tfrac{L_{k+1,j}}{2} &\le \Delta h \tan(45°)\end{aligned}\end{align} \]

To add a bridge-length constraint (limiting lateral drift from the base layer):

constraints:
  - name: overhang
    type: ineq
    bound: 0.0
    params:
      alpha_deg: 45.0
      bridge_length: 12.5