using TopOpt
using TopOpt.OpenLSTO
using WGLMakie
WGLMakie.activate!(; resize_to=:parent)
using Bonito
if haskey(ENV, "QUARTO_PROJECT_DIR")
Bonito.Page(exportable=true, offline=true)
else
Bonito.browser_display()
end
display_app(app) = display(app)Level-Set Topology Optimization (OpenLSTO)
Description
Level-set topology optimization represents the design boundary implicitly as the zero contour of a signed-distance function and moves it with shape sensitivities, so the boundary stays crisp throughout the optimization (no intermediate densities). This tutorial uses the TopOpt.OpenLSTO submodule, a self-contained Julia translation of OpenLSTO.
We solve two problems:
- Compliance minimization on a cantilever (left edge fixed, downward load at the midpoint of the right edge) under a volume-fraction constraint.
- Stress minimization on an L-beam, minimizing the p-norm of the von Mises stress under a volume-fraction constraint.
A 3D level-set compliance solver (compliance_minimization_3d) with marching-cubes boundary discretization is also available.
Setup
Compliance minimization
compliance_minimization runs the level-set compliance loop. The level-set grid is seeded with a “Swiss cheese” arrangement of circular holes by default (which requires a mesh of at least about 75×75); for a smaller mesh you pass your own LevelSetHole list:
holes = [
LevelSetHole(16, 8, 4), LevelSetHole(32, 8, 4), LevelSetHole(48, 8, 4),
LevelSetHole(24, 16, 4), LevelSetHole(40, 16, 4), LevelSetHole(56, 16, 4),
LevelSetHole(16, 24, 4), LevelSetHole(32, 24, 4), LevelSetHole(48, 24, 4),
]
result = compliance_minimization(; nelx=80, nely=40, holes=holes, verbose=false)The returned LevelSetResult holds the final level set, its discretized boundary, the finite element study and sensitivities, and the per-iteration objective and area-fraction histories:
println("iterations: ", length(result.objectives))
println("final compliance: ", round(result.objectives[end]; digits=4))
println("final area fraction: ", round(result.areas[end]; digits=4))iterations: 300
final compliance: 15.1155
final area fraction: 0.5003Visualize the design
visualize(::LevelSetResult) converts the level-set design to a PointLoadCantilever problem on the same grid and shows the per-cell area fractions as the density field, so the same load/support arrows and static viewer apply as for SIMP results:
app = visualize(result; static=true)
display_app(app)undeformed mesh
load arrows
support arrows
Stress minimization (L-beam)
stress_minimization minimizes the p-norm of the von Mises stress on an L-beam (top edge fixed, downward load on the right edge at 2/5 height):
lresult = stress_minimization(;
nelx=40, nely=40, max_iterations=40,
holes=[LevelSetHole(8, 8, 5), LevelSetHole(8, 20, 5), LevelSetHole(20, 8, 5)],
verbose=false,
)println("iterations: ", length(lresult.objectives))
println("final p-norm stress: ", round(lresult.objectives[end]; digits=4))
println("max von Mises stress: ", round(lresult.sensitivities.von_mises_max; digits=4))iterations: 40
final p-norm stress: 25.8358
max von Mises stress: 16.9205The area fractions of the stress result can be shown with the same viewer. The load and support arrows are read from the L-beam’s own finite element solve, so the load appears at 2/5 height and the supports on the fixed top edge:
app = visualize(lresult; static=true)
display_app(app)undeformed mesh
load arrows
support arrows
Hole nucleation
When the volume constraint is far from being met, compliance_minimization can nucleate new holes (a port of OpenLSTO’s projects/hole_creation) rather than only moving the existing boundary. Enable it with hole_nucleation=true:
hresult = compliance_minimization(;
nelx=40, nely=20,
holes=[LevelSetHole(8, 4, 2), LevelSetHole(16, 4, 2), LevelSetHole(24, 4, 2),
LevelSetHole(12, 8, 2), LevelSetHole(20, 8, 2), LevelSetHole(28, 8, 2),
LevelSetHole(8, 12, 2), LevelSetHole(16, 12, 2), LevelSetHole(24, 12, 2)],
hole_nucleation=true, max_iterations=8, verbose=false,
)println("final compliance: ", round(hresult.objectives[end]; digits=4))final compliance: 19.52943D level-set
A 3D level-set solver discretizes the boundary with marching cubes and drives a trilinear-hex finite element solve; the final surface can be exported as an STL:
r3 = compliance_minimization_3d(; nelx=10, nely=4, nelz=4, max_iterations=4, verbose=false)
stl = write_stl(r3.level_set, joinpath(mktempdir(), "mystlfile.stl"))println("final compliance: ", round(r3.compliances[end]; digits=4))
println("wrote ", r3.level_set.num_triangles, " triangles to ", stl)final compliance: 2173.4913
wrote 248 triangles to /tmp/jl_S8ZN4F/mystlfile.stlBy default only the load region is pinned solid, matching the upstream projects/3d/comp_min.cpp. The optimizer can then erode material from the left-face supports. To keep material on the supports, pass pin_support:
pin_support=:softassigns the support boundary a large sensitivity (analogous to the load pin), keeping the support mostly solid.pin_support=:hardclamps the signed distance on the support face so the contour cannot recede past it, keeping the support fully solid.
r3_hard = compliance_minimization_3d(; nelx=10, nely=4, nelz=4, max_iterations=4, pin_support=:hard, verbose=false)# The left-face (support) layer: cells with x = 0 (10 x 4 x 4 grid).
x0 = [r3_hard.level_set.volumefraction_vector[10j + 40k + 1] for j in 0:3, k in 0:3]
println("hard-pinned support layer mean: ", round(sum(x0) / 16; digits=3))hard-pinned support layer mean: 1.0The 3D design can be visualized with the same viewer as a 3D SIMP result:
app = visualize(r3.level_set; static=true)
display_app(app)Cameraφθxyz
undeformed mesh
load arrows
support arrows
Exporting the design
The OpenLSTO module mirrors OpenLSTO’s own VTK/TXT writers for the level-set function and area fractions:
out = mktempdir()
save_level_set_vtk(result.level_set, 1; output_directory=out)
save_area_fractions_vtk(result.level_set.mesh, 1; output_directory=out)
save_boundary_segments_txt(result.boundary, 1; output_directory=out)println(readdir(out))["area_0001.vtk", "boundary-segments_0001.txt", "level-set_0001.vtk"]