The method of topological optimization of binary structures (TOBS) was originally developed in the context of optimal distribution of material in mechanical components. This package implements the heuristic for binary nonlinear programming problems.
Begin by installing the base package and what's necessary for the specific problem:
import Pkg
Pkg.add("NonconvexTOBS")
Pkg.add("TopOpt")First, the finite element problem must be built. This is done using TopOpt.jl:
using NonconvexTOBS, TopOpt
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
# Define FEA problem
problem_size = (160, 100) # size of rectangular mesh
x0 = fill(1.0, prod(problem_size)) # initial design
problem = PointLoadCantilever(Val{:Linear}, problem_size, (1.0, 1.0), E, v, f)FEA solver and auxiliary functions need to be defined as well:
solver = FEASolver(Direct, problem; xmin=xmin)
cheqfilter = DensityFilter(solver; rmin=rmin) # filter function
comp = TopOpt.Compliance(solver) # compliance functionThe usual topology optimization problem adresses compliance minimization under volume restriction. Therefore, the objective and the constraint are:
obj(x) = comp(cheqfilter(PseudoDensities(x))) # compliance objective
constr(x) = sum(cheqfilter(PseudoDensities(x))) / length(x) - V # volume fraction constraintFinally, the optimization problem is defined and solved:
# Optimization setup
m = Model(obj) # create optimization model
addvar!(m, zeros(length(x0)), ones(length(x0))) # setup optimization variables
Nonconvex.add_ineq_constraint!(m, constr) # setup volume inequality constraint
options = TOBSOptions() # optimization options with default values
TopOpt.setpenalty!(solver, p)
# Perform TOBS optimization
@time r = Nonconvex.optimize(m, TOBSAlg(), x0; options=options)
# Results
@show obj(r.minimizer)
@show constr(r.minimizer)
topology = r.minimizerVisualizing the results from this example:
