Cantilever discrete design

This third example is a new variation of the cantilever design problem with a three fields optimization, i.e. with the raw, the filtered and the projected density parameter fields.

Projected density parameter field is obtained by projecting the filtered field using a smooth Heaviside function. This strategy could be use to obtain discrete 0-1 design at the end of the optimization by carring a multi-steps optimization process and increasing the steepness of the smooth Heaviside function at each step.

 We first create the optimization problem

import numpy as np

from HydrOpTop.Functions.verification import Mechanical_Compliance
from HydrOpTop.Functions import Volume_Percentage
from HydrOpTop.Materials import SIMP
from HydrOpTop.Filters import Density_Filter, Volume_Preserving_Heaviside_Filter
from HydrOpTop.Crafter import Steady_State_Crafter
from HydrOpTop.Solvers import Linear_Elasticity_2D

#create PFLOTRAN simulation object
sim = Linear_Elasticity_2D("cantilever")

#get cell ids in the region to optimize and parametrize permeability
#same name than in pflotran input file
perm = SIMP(cell_ids_to_parametrize="all", property_name="YOUNG_MODULUS", bounds=[0, 2000], power=3)

#define cost function
cf = Mechanical_Compliance(ids_to_consider="everywhere")

#define maximum volume constrains
max_vol = Volume_Percentage("parametrized_cell")
max_vol.constraint_tol = 0.5

#define filter
dfilter = Density_Filter(0.3)
hfilter = Volume_Preserving_Heaviside_Filter(0.5, 1, max_vol)

#craft optimization problem
#i.e. create function to optimize, initiate IO array in classes...
crafted_problem = Steady_State_Crafter(cf, sim, [perm], [max_vol], filters=[dfilter, hfilter]) #apply first density filter (dfilter) and then the Heaviside filter (hfilter)
crafted_problem.IO.output_every_iteration(2)
crafted_problem.IO.define_output_format("vtu")

#initialize optimizer
p = np.zeros(crafted_problem.get_problem_size(),dtype='f8') + 0.2

The optimization is carried out in several step, starting from a smooth Heavyside density filter and increasing the steepness parameter. As this can be difficult to converge for high step parameter, we increase it gradually to to reach a discrete distribution of material.

p_opt = crafted_problem.optimize(optimizer="nlopt-mma", action="minimize",
                                 max_it=50, ftol=0.0001, initial_guess=p)
hfilter.update_stepness(2)
p_opt = crafted_problem.optimize(optimizer="nlopt-mma", action="minimize",
                                 max_it=20, initial_guess=p_opt.p_opt)
hfilter.update_stepness(4)
p_opt = crafted_problem.optimize(optimizer="nlopt-mma", action="minimize",
                                 max_it=10, initial_guess=p_opt.p_opt)
hfilter.update_stepness(8)
p_opt = crafted_problem.optimize(optimizer="nlopt-mma", action="minimize",
                                 max_it=10, initial_guess=p_opt.p_opt)
hfilter.update_stepness(20)
p_opt = crafted_problem.optimize(optimizer="nlopt-mma", action="minimize",
                                 max_it=5, initial_guess=p_opt.p_opt)

crafted_problem.IO.write_fields_to_file([p_opt.p_opt_filtered], "./out.vtu", ["Filtered_density"])
crafted_problem.IO.plot_convergence_history()