using NonconvexTOBS, TopOpt
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)TOBS: Topological Optimization of Binary Structures
Description
TOBS (Topological Optimization of Binary Structures) is a heuristic algorithm for binary topology optimization. Unlike SIMP which uses continuous densities, TOBS directly optimizes 0/1 (void/solid) designs by linearizing the objective and constraints, then solving a binary nonlinear program. The default solver is Cbc.jl (COIN-OR Branch and Cut), which determines which binary variables must be flipped in each iteration.
This tutorial demonstrates TOBS on a 2D cantilever beam with 80×50 elements (4,000 design variables). The algorithm efficiently handles the combinatorial challenge of binary optimization through sequential linearization and branch- and-cut solving.
Setup
WGLMakie.activate!(; resize_to=:parent) selects the browser renderer and fills the Quarto output column. Bonito.Page(exportable=true, offline=true) embeds the assets needed by visualize(...; static=true) in the Quarto output, so the visualization does not require a running Julia process.
Problem Definition
We use a large 2D cantilever beam to demonstrate TOBS scalability:
E = 1.0 # Young's modulus
v = 0.3 # Poisson's ratio
f = 1.0 # downward force
rmin = 6.0 # filter radius
xmin = 0.001 # minimum density
V = 0.5 # maximum volume fraction
p = 3.0 # topological optimization penalty
problem_size = (80, 50) # 4,000 elements
x0 = fill(1.0, prod(problem_size)) # start from fully solid design
problem = PointLoadCantilever(problem_size, (1.0, 1.0), E, v, f)The mesh (80×50) tests TOBS ability to handle high-dimensional binary optimization problems.
FEA Solver and Filter
solver = FEASolver(DirectSolver, problem; xmin=xmin)
cheqfilter = DensityFilterFun(solver; rmin=rmin) # mesh-independent filtering
comp = ComplianceFun(solver)The density filter ensures smooth designs and prevents checkerboarding, critical for binary optimization where small features can cause numerical instability.
Objective and Constraint
obj(x) = comp(cheqfilter(PseudoDensities(x))) # compliance objective
constr(x) = sum(cheqfilter(PseudoDensities(x))) / length(x) - V # volume constraintThe objective minimizes compliance (strain energy) while the constraint limits material usage to 50% of the domain volume.
Optimization Setup
m = Model(obj)
addvar!(m, zeros(length(x0)), ones(length(x0))) # bounds: 0 ≤ xᵢ ≤ 1
Nonconvex.add_ineq_constraint!(m, constr)
# TOBS starts from the provided x0 and flips at most
# `movelimit * numVars` variables per iteration, so reaching the 50%
# volume target from 100% solid needs enough iterations to remove half
# the material.
options = TOBSOptions(; maxiter=100, movelimit=0.05)
setpenalty!(solver, p)TOBS uses binary variables (0 or 1) with bound constraints. The penalty exponent p=3.0 penalizes intermediate densities in the FEA solution.
Run TOBS Optimization
@time r = Nonconvex.optimize(m, TOBSAlg(), x0; options=options)┌ Warning: Subproblem is infeasible. Temporarily relaxing the subproblem. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:188 ┌ Warning: Subproblem is infeasible. Temporarily relaxing the subproblem. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:188 ┌ Warning: Subproblem is infeasible. Temporarily relaxing the subproblem. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:188 ┌ Warning: Subproblem is infeasible. Temporarily relaxing the subproblem. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:188 ┌ Warning: Subproblem infeasible 5 times. Switching to feasibility restoration. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:180 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 ┌ Warning: Restoration subproblem still infeasible. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:184 [ Info: Restoration step succeeded; resuming normal iterations. [ Info: iter = 26, obj = 809.072, constr_vio_norm = 0.5, er = 1.0 [ Info: iter = 27, obj = 361.466, constr_vio_norm = 0.0, er = 1.074 [ Info: iter = 28, obj = 239.047, constr_vio_norm = 0.0, er = 0.978 [ Info: iter = 29, obj = 454.029, constr_vio_norm = 0.0, er = 0.855 [ Info: iter = 30, obj = 768.805, constr_vio_norm = 0.0, er = 0.725 [ Info: iter = 31, obj = 330.695, constr_vio_norm = 0.0, er = 0.792 [ Info: iter = 32, obj = 817.734, constr_vio_norm = 0.0, er = 0.75 [ Info: iter = 33, obj = 338.705, constr_vio_norm = 0.0, er = 0.804 [ Info: iter = 34, obj = 272.602, constr_vio_norm = 0.0, er = 0.769 [ Info: iter = 35, obj = 310.148, constr_vio_norm = 0.0, er = 0.727 [ Info: iter = 36, obj = 166.062, constr_vio_norm = 0.0, er = 0.731 [ Info: iter = 37, obj = 141.06, constr_vio_norm = 0.0, er = 0.716 [ Info: iter = 38, obj = 106.67, constr_vio_norm = 0.0, er = 0.708 [ Info: iter = 39, obj = 86.677, constr_vio_norm = 0.0, er = 0.7 [ Info: iter = 40, obj = 66.907, constr_vio_norm = 0.0, er = 0.695 [ Info: iter = 41, obj = 59.58, constr_vio_norm = 0.0, er = 0.688 [ Info: iter = 42, obj = 54.326, constr_vio_norm = 0.0, er = 0.682 [ Info: iter = 43, obj = 59.914, constr_vio_norm = 0.0, er = 0.676 [ Info: iter = 44, obj = 62.165, constr_vio_norm = 0.0, er = 0.668 [ Info: iter = 45, obj = 55.3, constr_vio_norm = 0.0, er = 0.518 [ Info: iter = 46, obj = 66.246, constr_vio_norm = 0.0, er = 0.507 [ Info: iter = 47, obj = 58.809, constr_vio_norm = 0.0, er = 0.515 [ Info: iter = 48, obj = 60.656, constr_vio_norm = 0.0, er = 0.487 [ Info: iter = 49, obj = 56.427, constr_vio_norm = 0.0, er = 0.458 [ Info: iter = 50, obj = 75.74, constr_vio_norm = 0.0, er = 0.426 [ Info: iter = 51, obj = 57.229, constr_vio_norm = 0.0, er = 0.308 [ Info: iter = 52, obj = 58.394, constr_vio_norm = 0.0, er = 0.198 [ Info: iter = 53, obj = 57.648, constr_vio_norm = 0.0, er = 0.193 [ Info: iter = 54, obj = 54.221, constr_vio_norm = 0.0, er = 0.197 [ Info: iter = 55, obj = 61.091, constr_vio_norm = 0.0, er = 0.137 [ Info: iter = 56, obj = 56.038, constr_vio_norm = 0.0, er = 0.134 [ Info: iter = 57, obj = 55.375, constr_vio_norm = 0.0, er = 0.116 [ Info: iter = 58, obj = 54.493, constr_vio_norm = 0.0, er = 0.105 [ Info: iter = 59, obj = 56.613, constr_vio_norm = 0.0, er = 0.093 [ Info: iter = 60, obj = 54.75, constr_vio_norm = 0.0, er = 0.089 [ Info: iter = 61, obj = 55.841, constr_vio_norm = 0.0, er = 0.086 [ Info: iter = 62, obj = 54.76, constr_vio_norm = 0.0, er = 0.082 [ Info: iter = 63, obj = 54.086, constr_vio_norm = 0.0, er = 0.081 [ Info: iter = 64, obj = 53.862, constr_vio_norm = 0.0, er = 0.076 [ Info: iter = 65, obj = 55.267, constr_vio_norm = 0.0, er = 0.068 [ Info: iter = 66, obj = 53.519, constr_vio_norm = 0.0, er = 0.064 [ Info: iter = 67, obj = 52.932, constr_vio_norm = 0.0, er = 0.063 [ Info: iter = 68, obj = 54.472, constr_vio_norm = 0.0, er = 0.061 [ Info: iter = 69, obj = 52.223, constr_vio_norm = 0.0, er = 0.046 [ Info: iter = 70, obj = 54.184, constr_vio_norm = 0.0, er = 0.032 [ Info: iter = 71, obj = 53.12, constr_vio_norm = 0.0, er = 0.032 [ Info: iter = 72, obj = 52.42, constr_vio_norm = 0.0, er = 0.032 [ Info: iter = 73, obj = 51.369, constr_vio_norm = 0.0, er = 0.03 [ Info: iter = 74, obj = 53.005, constr_vio_norm = 0.0, er = 0.025 [ Info: iter = 75, obj = 51.081, constr_vio_norm = 0.0, er = 0.023 [ Info: iter = 76, obj = 52.7, constr_vio_norm = 0.0, er = 0.024 [ Info: iter = 77, obj = 51.812, constr_vio_norm = 0.0, er = 0.024 [ Info: iter = 78, obj = 51.395, constr_vio_norm = 0.0, er = 0.022 [ Info: iter = 79, obj = 51.155, constr_vio_norm = 0.0, er = 0.021 [ Info: iter = 80, obj = 51.575, constr_vio_norm = 0.0, er = 0.02 [ Info: iter = 81, obj = 50.433, constr_vio_norm = 0.0, er = 0.02 [ Info: iter = 82, obj = 51.525, constr_vio_norm = 0.0, er = 0.021 [ Info: iter = 83, obj = 50.483, constr_vio_norm = 0.0, er = 0.022 [ Info: iter = 84, obj = 50.175, constr_vio_norm = 0.0, er = 0.021 [ Info: iter = 85, obj = 50.879, constr_vio_norm = 0.0, er = 0.02 [ Info: iter = 86, obj = 49.823, constr_vio_norm = 0.0, er = 0.02 [ Info: iter = 87, obj = 49.726, constr_vio_norm = 0.0, er = 0.019 [ Info: iter = 88, obj = 50.471, constr_vio_norm = 0.0, er = 0.018 [ Info: iter = 89, obj = 49.206, constr_vio_norm = 0.0, er = 0.017 [ Info: iter = 90, obj = 49.296, constr_vio_norm = 0.0, er = 0.016 [ Info: iter = 91, obj = 50.773, constr_vio_norm = 0.0, er = 0.017 [ Info: iter = 92, obj = 49.12, constr_vio_norm = 0.0, er = 0.018 [ Info: iter = 93, obj = 49.421, constr_vio_norm = 0.0, er = 0.016 [ Info: iter = 94, obj = 50.139, constr_vio_norm = 0.0, er = 0.015 [ Info: iter = 95, obj = 48.936, constr_vio_norm = 0.0, er = 0.015 [ Info: iter = 96, obj = 49.153, constr_vio_norm = 0.0, er = 0.014 [ Info: iter = 97, obj = 50.623, constr_vio_norm = 0.0, er = 0.015 [ Info: iter = 98, obj = 49.132, constr_vio_norm = 0.0, er = 0.016 [ Info: iter = 99, obj = 49.335, constr_vio_norm = 0.0, er = 0.016 ┌ Warning: No feasible solution was found during the optimization. Returning the best design found. └ @ NonconvexTOBS ~/.julia/packages/NonconvexTOBS/P1EYO/src/NonconvexTOBS.jl:221 139.384376 seconds (93.24 M allocations: 6.223 GiB, 3.26% gc time, 16.12% compilation time: 16% of which was recompilation)
NonconvexTOBS.TOBSResult{Vector{Float64}, Float64, Float64}([1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0 … 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0], 51.39511269241478, 0.01627173951423671)The TOBS algorithm iterates:
- Linearize objective and constraints around current design
- Solve binary subproblem with Cbc.jl
- Determine which variables to flip (0→1 or 1→0)
- Update design and repeat
Results
@show obj(r.minimizer)
@show constr(r.minimizer)obj(r.minimizer) = 51.39511269241478
constr(r.minimizer) = 7.319359018875815e-87.319359018875815e-8The final design should be nearly binary (values close to 0 or 1) with satisfied volume constraint. TOBS typically produces crisp black/white designs without intermediate densities.
Visualization
topology = r.minimizer
fig = visualize(problem; static=true, topology=topology)
display_app(fig)undeformed mesh
load arrows
support arrows
The visualization shows the final binary structure — TOBS naturally produces clean 0/1 designs without the gray regions common in SIMP results.