using TopOptHeat Conduction: Conductivity Tree Benchmark
Description
This tutorial demonstrates heat conduction topology optimization using the classic “conductivity tree” benchmark from Bendsøe & Sigmund’s Topology Optimization textbook (§1.3). The problem minimizes thermal compliance (temperature work) subject to a volume constraint.
The setup: distributed heat flux enters the top edge, the bottom edge is held at T = 0 (heat sink), and the sides are insulated. The optimal design forms a branching tree structure that efficiently conducts heat from the top to the bottom boundary — analogous to structural force flow paths.
Setup
Define the Problem
The HeatTree problem sets up the standard benchmark with heat flux on top and fixed temperature at the bottom:
nels = (80, 40) # 80×40 rectangular mesh
k = 1.0 # thermal conductivity
q = 1.0 # heat flux on top boundary
problem = HeatTree(Val{:Linear}, nels, (1.0, 1.0), k; q=q)The boundary conditions:
- Top edge: Distributed heat flux q (Neumann BC)
- Bottom edge: Fixed temperature T = 0 (Dirichlet BC)
- Left/Right edges: Insulated (zero heat flux)
Parameter Settings
V = 0.3 # volume fraction (30% high-conductivity material)
xmin = 1e-3 # minimum density
rmin = 2.0 # density filter radius
x0 = fill(V, getncells(problem)) # start from uniform 30% density
solver = FEASolver(DirectSolver, problem; xmin=xmin, penalty=PowerPenaltyFun(1.0))
comp = ThermalComplianceFun(solver)
filter = DensityFilterFun(solver; rmin=rmin)The 30% volume constraint forces the optimizer to strategically place high- conductivity material for maximum heat dissipation.
Objective and Constraint
obj = x -> comp(filter(PseudoDensities(x))) # thermal compliance
volfrac = VolumeFun(solver)
constr = x -> volfrac(filter(PseudoDensities(x))) - V # volume constraintThermal compliance measures the work done by the heat flux against the temperature field: J = QᵀT. Minimizing it drives heat efficiently toward the cold boundary.
Optimization Setup
model = Model(obj)
addvar!(model, zeros(length(x0)), ones(length(x0)))
add_ineq_constraint!(model, constr)
alg = MMA87()
convcriteria = Nonconvex.KKTCriteria()Continuation Strategy
A fixed penalty often leaves intermediate (gray) densities. We use a continuation strategy that ramps the penalty exponent through 1.0, 2.0, and 3.0, with exponentially decaying KKT tolerances so each subproblem is solved accurately before the penalty increases:
ps = [1.0, 2.0, 3.0]
tols = [1e-2, 1e-3, 1e-4]
x = x0Run Optimization
Each continuation step:
- Set the penalty exponent
p - Set the convergence tolerance
tol - Solve the subproblem with MMA87
- Warm-start the next step from the current solution
for j in eachindex(ps)
global x, convcriteria, res
p = ps[j]
tol = tols[j]
setpenalty!(solver, p)
options = MMAOptions(; tol=Tolerance(; kkt=tol), maxiter=1000, convcriteria)
res = optimize(model, alg, x; options)
global x = res.minimizer
println("Step $j: penalty=$p, tol=$tol, obj=$(obj(x))")
end[ Info: iter obj Δobj violation kkt_residual [ Info: 0 1.1e+04 Inf 0.0e+00 1.3e+01 [ Info: 1 1.1e+04 4.5e-06 0.0e+00 4.5e-06 [ Info: 2 1.1e+04 1.1e-09 0.0e+00 4.5e-06 Step 1: penalty=1.0, tol=0.01, obj=10641.83572115931 [ Info: iter obj Δobj violation kkt_residual [ Info: 0 3.5e+04 Inf 0.0e+00 8.4e+01 [ Info: 1 3.5e+04 2.5e-05 0.0e+00 4.5e-06 [ Info: 2 3.5e+04 4.4e-11 0.0e+00 4.5e-06 Step 2: penalty=2.0, tol=0.001, obj=35199.64800801904 [ Info: iter obj Δobj violation kkt_residual [ Info: 0 1.1e+05 Inf 0.0e+00 4.0e+02 [ Info: 1 1.1e+05 5.5e-03 0.0e+00 5.5e-03 [ Info: 2 1.1e+05 4.5e-10 0.0e+00 5.5e-03 [ Info: 3 1.1e+05 3.0e-09 0.0e+00 5.5e-03 [ Info: 4 1.1e+05 7.6e-04 0.0e+00 4.8e-03 [ Info: 5 1.1e+05 6.3e-06 0.0e+00 4.8e-03 [ Info: 6 1.1e+05 1.2e-03 0.0e+00 3.5e-03 [ Info: 7 1.1e+05 1.6e-03 0.0e+00 5.1e-03 [ Info: 8 1.1e+05 2.2e-03 0.0e+00 2.9e-03 [ Info: 9 1.1e+05 4.0e-04 0.0e+00 2.5e-03 [ Info: 10 1.1e+05 1.9e-04 0.0e+00 2.3e-03 [ Info: 11 1.1e+05 1.4e-05 0.0e+00 2.3e-03 [ Info: 12 1.1e+05 1.7e-04 0.0e+00 2.1e-03 [ Info: 13 1.1e+05 1.5e-04 0.0e+00 2.3e-03 [ Info: 14 1.1e+05 3.0e-04 0.0e+00 2.6e-03 [ Info: 15 1.1e+05 2.0e-03 0.0e+00 4.6e-03 [ Info: 16 1.1e+05 1.5e-03 0.0e+00 3.1e-03 [ Info: 17 1.1e+05 3.1e-03 0.0e+00 3.2e-03 [ Info: 18 1.1e+05 3.5e-03 0.0e+00 1.2e-02 [ Info: 19 1.1e+05 2.3e-03 0.0e+00 4.7e-02 [ Info: 20 1.1e+05 3.2e-03 0.0e+00 2.2e-01 [ Info: 21 1.1e+05 1.8e-02 0.0e+00 1.2e+00 [ Info: 22 1.1e+05 6.6e-01 0.0e+00 7.5e+00 [ Info: 23 1.1e+05 3.4e+01 0.0e+00 5.4e+01 [ Info: 24 1.1e+05 1.9e+03 0.0e+00 2.6e+02 [ Info: 25 8.6e+04 2.6e+04 0.0e+00 3.3e+03 [ Info: 26 5.6e+04 3.0e+04 0.0e+00 9.8e+03 [ Info: 27 4.3e+04 1.3e+04 0.0e+00 4.4e+03 [ Info: 28 4.5e+04 1.5e+03 0.0e+00 4.6e+03 [ Info: 29 3.8e+04 6.3e+03 0.0e+00 7.8e+03 [ Info: 30 3.2e+04 6.6e+03 0.0e+00 3.9e+03 [ Info: 31 2.8e+04 3.8e+03 0.0e+00 2.6e+03 [ Info: 32 2.3e+04 4.4e+03 0.0e+00 9.8e+02 [ Info: 33 2.1e+04 2.4e+03 0.0e+00 2.9e+02 [ Info: 34 2.0e+04 1.1e+03 0.0e+00 1.6e+02 [ Info: 35 1.9e+04 6.3e+02 0.0e+00 1.1e+02 [ Info: 36 1.9e+04 3.6e+02 0.0e+00 7.9e+01 [ Info: 37 1.9e+04 2.4e+02 0.0e+00 7.3e+01 [ Info: 38 1.9e+04 1.7e+02 0.0e+00 5.7e+01 [ Info: 39 1.8e+04 1.2e+02 0.0e+00 5.2e+01 [ Info: 40 1.8e+04 9.0e+01 0.0e+00 5.2e+01 [ Info: 41 1.8e+04 8.2e+01 0.0e+00 5.2e+01 [ Info: 42 1.8e+04 7.1e+01 0.0e+00 5.2e+01 [ Info: 43 1.8e+04 7.3e+01 0.0e+00 5.2e+01 [ Info: 44 1.8e+04 7.0e+01 0.0e+00 1.3e+01 [ Info: 45 1.8e+04 5.2e+01 0.0e+00 1.1e+01 [ Info: 46 1.8e+04 3.8e+01 0.0e+00 1.3e+01 [ Info: 47 1.8e+04 3.1e+01 0.0e+00 1.2e+01 [ Info: 48 1.8e+04 2.8e+01 0.0e+00 1.6e+01 [ Info: 49 1.8e+04 3.1e+01 0.0e+00 1.9e+01 [ Info: 50 1.8e+04 3.1e+01 0.0e+00 2.1e+01 [ Info: 51 1.8e+04 2.1e+01 0.0e+00 9.8e+00 [ Info: 52 1.8e+04 1.4e+01 0.0e+00 1.1e+01 [ Info: 53 1.8e+04 1.0e+01 0.0e+00 1.1e+01 [ Info: 54 1.8e+04 6.5e+00 0.0e+00 6.9e+00 [ Info: 55 1.8e+04 5.1e+00 0.0e+00 7.2e+00 [ Info: 56 1.8e+04 5.9e+00 0.0e+00 7.5e+00 [ Info: 57 1.8e+04 6.1e+00 0.0e+00 4.6e+00 [ Info: 58 1.8e+04 5.8e+00 0.0e+00 4.9e+00 [ Info: 59 1.8e+04 6.5e+00 0.0e+00 5.3e+00 [ Info: 60 1.8e+04 4.4e+00 0.0e+00 5.1e+00 [ Info: 61 1.8e+04 3.3e+00 0.0e+00 5.3e+00 [ Info: 62 1.8e+04 3.8e+00 0.0e+00 5.8e+00 [ Info: 63 1.8e+04 4.3e+00 0.0e+00 6.4e+00 [ Info: 64 1.8e+04 4.0e+00 0.0e+00 7.0e+00 [ Info: 65 1.8e+04 4.5e+00 0.0e+00 6.0e+00 [ Info: 66 1.8e+04 3.2e+00 0.0e+00 7.4e+00 [ Info: 67 1.8e+04 2.4e+00 0.0e+00 5.0e+00 [ Info: 68 1.8e+04 4.8e+00 0.0e+00 4.0e+00 [ Info: 69 1.8e+04 3.0e+00 0.0e+00 3.3e+00 [ Info: 70 1.8e+04 2.2e+00 0.0e+00 3.7e+00 [ Info: 71 1.8e+04 2.7e+00 0.0e+00 3.9e+00 [ Info: 72 1.8e+04 3.0e+00 0.0e+00 4.0e+00 [ Info: 73 1.8e+04 3.2e+00 0.0e+00 4.3e+00 [ Info: 74 1.8e+04 3.6e+00 0.0e+00 5.0e+00 [ Info: 75 1.8e+04 3.9e+00 0.0e+00 4.6e+00 [ Info: 76 1.8e+04 3.3e+00 0.0e+00 3.6e+00 [ Info: 77 1.8e+04 3.4e+00 0.0e+00 4.3e+00 [ Info: 78 1.8e+04 4.7e+00 0.0e+00 4.9e+00 [ Info: 79 1.8e+04 6.7e+00 0.0e+00 5.7e+00 [ Info: 80 1.8e+04 6.9e+00 0.0e+00 6.3e+00 [ Info: 81 1.8e+04 7.4e+00 0.0e+00 7.6e+00 [ Info: 82 1.8e+04 7.5e+00 0.0e+00 8.4e+00 [ Info: 83 1.8e+04 1.3e+01 0.0e+00 9.3e+00 [ Info: 84 1.8e+04 2.1e+01 0.0e+00 9.5e+00 [ Info: 85 1.8e+04 1.4e+01 0.0e+00 6.2e+00 [ Info: 86 1.8e+04 1.2e+01 0.0e+00 6.2e+00 [ Info: 87 1.8e+04 1.3e+01 0.0e+00 9.0e+00 [ Info: 88 1.8e+04 1.7e+01 0.0e+00 1.1e+01 [ Info: 89 1.8e+04 2.0e+01 0.0e+00 1.4e+01 [ Info: 90 1.8e+04 2.2e+01 0.0e+00 1.6e+01 [ Info: 91 1.8e+04 1.7e+01 0.0e+00 5.4e+00 [ Info: 92 1.8e+04 8.0e+00 0.0e+00 5.7e+00 [ Info: 93 1.8e+04 6.6e+00 0.0e+00 5.6e+00 [ Info: 94 1.7e+04 6.8e+00 0.0e+00 5.6e+00 [ Info: 95 1.7e+04 4.6e+00 0.0e+00 5.0e+00 [ Info: 96 1.7e+04 3.8e+00 0.0e+00 5.0e+00 [ Info: 97 1.7e+04 3.6e+00 0.0e+00 5.1e+00 [ Info: 98 1.7e+04 4.3e+00 0.0e+00 5.2e+00 [ Info: 99 1.7e+04 5.1e+00 0.0e+00 5.5e+00 [ Info: 100 1.7e+04 5.2e+00 0.0e+00 4.3e+00 [ Info: 101 1.7e+04 4.5e+00 0.0e+00 4.4e+00 [ Info: 102 1.7e+04 4.1e+00 0.0e+00 4.5e+00 [ Info: 103 1.7e+04 2.9e+00 0.0e+00 3.8e+00 [ Info: 104 1.7e+04 2.1e+00 0.0e+00 2.7e+00 [ Info: 105 1.7e+04 1.7e+00 0.0e+00 2.8e+00 [ Info: 106 1.7e+04 1.4e+00 0.0e+00 3.3e+00 [ Info: 107 1.7e+04 1.6e-02 0.0e+00 1.1e+01 [ Info: 108 1.7e+04 1.5e+00 0.0e+00 9.6e+00 [ Info: 109 1.7e+04 9.2e+00 0.0e+00 3.3e+01 [ Info: 110 1.7e+04 1.5e+01 0.0e+00 2.1e+01 [ Info: 111 1.7e+04 8.6e+00 0.0e+00 5.5e+01 [ Info: 112 1.7e+04 6.6e+00 0.0e+00 2.6e+01 [ Info: 113 1.7e+04 3.5e+01 0.0e+00 1.6e+01 [ Info: 114 1.7e+04 1.7e+01 0.0e+00 8.8e+00 [ Info: 115 1.7e+04 3.3e+00 0.0e+00 3.7e+00 [ Info: 116 1.7e+04 8.3e-01 0.0e+00 3.3e+00 [ Info: 117 1.7e+04 8.3e-01 0.0e+00 3.1e+00 [ Info: 118 1.7e+04 8.7e-01 0.0e+00 3.1e+00 [ Info: 119 1.7e+04 9.9e-01 0.0e+00 3.0e+00 [ Info: 120 1.7e+04 1.0e+00 0.0e+00 2.8e+00 [ Info: 121 1.7e+04 8.2e-01 0.0e+00 2.6e+00 [ Info: 122 1.7e+04 6.6e-01 0.0e+00 2.5e+00 [ Info: 123 1.7e+04 4.2e-01 0.0e+00 2.7e+00 [ Info: 124 1.7e+04 4.5e-01 0.0e+00 2.8e+00 [ Info: 125 1.7e+04 5.3e-01 0.0e+00 2.9e+00 [ Info: 126 1.7e+04 6.3e-01 0.0e+00 3.0e+00 [ Info: 127 1.7e+04 7.5e-01 0.0e+00 3.1e+00 [ Info: 128 1.7e+04 8.9e-01 0.0e+00 3.2e+00 [ Info: 129 1.7e+04 1.1e+00 0.0e+00 3.4e+00 [ Info: 130 1.7e+04 1.4e+00 0.0e+00 3.9e+00 [ Info: 131 1.7e+04 1.9e+00 0.0e+00 4.6e+00 [ Info: 132 1.7e+04 2.7e+00 0.0e+00 5.3e+00 [ Info: 133 1.7e+04 2.7e+00 0.0e+00 5.8e+00 [ Info: 134 1.7e+04 3.3e+00 0.0e+00 6.2e+00 [ Info: 135 1.7e+04 4.0e+00 0.0e+00 5.4e+00 [ Info: 136 1.7e+04 2.9e+00 0.0e+00 4.4e+00 [ Info: 137 1.7e+04 1.6e+00 0.0e+00 4.5e+00 [ Info: 138 1.7e+04 1.2e+00 0.0e+00 4.2e+00 [ Info: 139 1.7e+04 1.2e+00 0.0e+00 4.3e+00 [ Info: 140 1.7e+04 1.5e+00 0.0e+00 4.5e+00 [ Info: 141 1.7e+04 1.7e+00 0.0e+00 1.9e+00 [ Info: 142 1.7e+04 1.1e+00 0.0e+00 2.2e+00 [ Info: 143 1.7e+04 1.0e+00 0.0e+00 2.1e+00 [ Info: 144 1.7e+04 1.3e+00 0.0e+00 2.2e+00 [ Info: 145 1.7e+04 1.7e+00 0.0e+00 2.6e+00 [ Info: 146 1.7e+04 1.9e+00 0.0e+00 3.2e+00 [ Info: 147 1.7e+04 1.8e+00 0.0e+00 3.5e+00 [ Info: 148 1.7e+04 2.0e+00 0.0e+00 3.8e+00 [ Info: 149 1.7e+04 1.7e+00 0.0e+00 2.2e+00 [ Info: 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1.5e+00 [ Info: 173 1.7e+04 9.6e-01 0.0e+00 2.2e+00 [ Info: 174 1.7e+04 1.5e+00 0.0e+00 3.8e+00 [ Info: 175 1.7e+04 3.0e+00 0.0e+00 6.1e+00 [ Info: 176 1.7e+04 7.3e+00 0.0e+00 9.1e+00 [ Info: 177 1.7e+04 8.2e+00 0.0e+00 2.9e+00 [ Info: 178 1.7e+04 2.0e+00 0.0e+00 2.2e+00 [ Info: 179 1.7e+04 8.2e-01 0.0e+00 2.2e+00 [ Info: 180 1.7e+04 8.0e-01 0.0e+00 2.3e+00 [ Info: 181 1.7e+04 1.0e+00 0.0e+00 2.4e+00 [ Info: 182 1.7e+04 1.3e+00 0.0e+00 2.7e+00 [ Info: 183 1.7e+04 1.2e+00 0.0e+00 2.2e+00 [ Info: 184 1.7e+04 1.4e+00 0.0e+00 2.5e+00 [ Info: 185 1.7e+04 1.6e+00 0.0e+00 2.9e+00 [ Info: 186 1.7e+04 1.4e+00 0.0e+00 2.3e+00 [ Info: 187 1.7e+04 1.5e+00 0.0e+00 2.7e+00 [ Info: 188 1.7e+04 2.0e+00 0.0e+00 3.4e+00 [ Info: 189 1.7e+04 3.4e+00 0.0e+00 4.8e+00 [ Info: 190 1.7e+04 5.7e+00 0.0e+00 6.5e+00 [ Info: 191 1.7e+04 7.6e+00 0.0e+00 7.1e+00 [ Info: 192 1.7e+04 3.7e+00 0.0e+00 1.8e+00 [ Info: 193 1.7e+04 9.6e-01 0.0e+00 1.7e+00 [ Info: 194 1.7e+04 7.8e-01 0.0e+00 2.1e+00 [ Info: 195 1.7e+04 9.0e-01 0.0e+00 2.6e+00 [ Info: 196 1.7e+04 9.5e-01 0.0e+00 3.0e+00 [ Info: 197 1.7e+04 7.3e-01 0.0e+00 3.3e+00 [ Info: 198 1.7e+04 6.2e-01 0.0e+00 3.5e+00 [ Info: 199 1.7e+04 7.0e-01 0.0e+00 3.7e+00 [ Info: 200 1.7e+04 7.6e-01 0.0e+00 3.9e+00 [ Info: 201 1.7e+04 9.1e-01 0.0e+00 4.1e+00 [ Info: 202 1.7e+04 1.1e+00 0.0e+00 4.3e+00 [ Info: 203 1.7e+04 1.2e+00 0.0e+00 4.4e+00 [ Info: 204 1.7e+04 6.4e-01 0.0e+00 4.7e+00 [ Info: 205 1.7e+04 3.2e+00 0.0e+00 1.8e+01 [ Info: 206 1.7e+04 4.2e+00 0.0e+00 1.7e+01 [ Info: 207 1.7e+04 9.9e+00 0.0e+00 3.6e+01 [ Info: 208 1.7e+04 7.3e+00 0.0e+00 2.2e+01 [ Info: 209 1.7e+04 1.1e+01 0.0e+00 2.0e+01 [ Info: 210 1.7e+04 1.1e+01 0.0e+00 1.4e+01 [ Info: 211 1.7e+04 4.9e+00 0.0e+00 3.4e+00 [ Info: 212 1.7e+04 5.5e-01 0.0e+00 3.4e+00 [ Info: 213 1.7e+04 2.0e-01 0.0e+00 1.7e+00 [ Info: 214 1.7e+04 2.1e-01 0.0e+00 1.9e+00 [ Info: 215 1.7e+04 2.4e-01 0.0e+00 2.0e+00 [ Info: 216 1.7e+04 3.2e-01 0.0e+00 2.1e+00 [ Info: 217 1.7e+04 4.4e-01 0.0e+00 2.2e+00 [ Info: 218 1.7e+04 6.5e-01 0.0e+00 2.2e+00 [ Info: 219 1.7e+04 5.8e-01 0.0e+00 2.2e+00 [ Info: 220 1.7e+04 6.1e-01 0.0e+00 2.4e+00 [ Info: 221 1.7e+04 6.3e-01 0.0e+00 2.5e+00 [ Info: 222 1.7e+04 8.1e-01 0.0e+00 2.7e+00 [ Info: 223 1.7e+04 1.1e+00 0.0e+00 2.5e+00 [ Info: 224 1.7e+04 6.3e-01 0.0e+00 2.5e+00 [ Info: 225 1.7e+04 6.1e-01 0.0e+00 2.5e+00 [ Info: 226 1.7e+04 7.2e-01 0.0e+00 2.6e+00 [ Info: 227 1.7e+04 7.7e-01 0.0e+00 2.9e+00 [ Info: 228 1.7e+04 8.6e-01 0.0e+00 3.0e+00 [ Info: 229 1.7e+04 8.9e-01 0.0e+00 3.2e+00 [ Info: 230 1.7e+04 1.2e+00 0.0e+00 3.5e+00 [ Info: 231 1.7e+04 1.9e+00 0.0e+00 3.7e+00 [ Info: 232 1.7e+04 2.7e+00 0.0e+00 4.2e+00 [ Info: 233 1.7e+04 3.5e+00 0.0e+00 5.1e+00 [ Info: 234 1.7e+04 3.2e+00 0.0e+00 3.7e+00 [ Info: 235 1.7e+04 1.9e+00 0.0e+00 3.1e+00 [ Info: 236 1.7e+04 1.7e+00 0.0e+00 2.8e+00 [ Info: 237 1.7e+04 1.2e+00 0.0e+00 2.7e+00 [ Info: 238 1.7e+04 5.6e-01 0.0e+00 2.4e+00 [ Info: 239 1.7e+04 4.4e-01 0.0e+00 2.7e+00 [ Info: 240 1.7e+04 5.1e-01 0.0e+00 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1.4e-02 0.0e+00 3.1e-01 [ Info: 264 1.7e+04 1.7e-02 0.0e+00 3.6e-01 [ Info: 265 1.7e+04 1.3e-02 0.0e+00 2.1e-01 [ Info: 266 1.7e+04 4.6e-03 0.0e+00 1.4e-01 [ Info: 267 1.7e+04 1.8e-03 0.0e+00 1.3e-01 [ Info: 268 1.7e+04 1.7e-03 0.0e+00 1.3e-01 [ Info: 269 1.7e+04 1.7e-03 0.0e+00 1.3e-01 [ Info: 270 1.7e+04 1.8e-03 0.0e+00 1.3e-01 [ Info: 271 1.7e+04 1.6e-03 0.0e+00 1.2e-01 [ Info: 272 1.7e+04 1.5e-03 0.0e+00 1.1e-01 [ Info: 273 1.7e+04 2.4e-03 0.0e+00 1.1e-01 [ Info: 274 1.7e+04 8.8e-04 0.0e+00 9.6e-02 [ Info: 275 1.7e+04 1.8e-03 0.0e+00 1.0e-01 [ Info: 276 1.7e+04 1.9e-03 0.0e+00 8.1e-02 [ Info: 277 1.7e+04 2.0e-03 0.0e+00 1.1e-01 [ Info: 278 1.7e+04 2.0e-03 0.0e+00 9.5e-02 [ Info: 279 1.7e+04 1.8e-03 0.0e+00 1.6e-01 [ Info: 280 1.7e+04 6.6e-04 0.0e+00 2.3e-01 [ Info: 281 1.7e+04 4.4e-04 0.0e+00 3.0e-01 [ Info: 282 1.7e+04 1.9e-03 0.0e+00 2.7e-01 [ Info: 283 1.7e+04 4.4e-03 0.0e+00 1.3e-01 [ Info: 284 1.7e+04 3.1e-03 0.0e+00 7.6e-02 [ Info: 285 1.7e+04 5.9e-04 0.0e+00 8.1e-02 [ Info: 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1.2e+00 [ Info: 309 1.7e+04 2.8e-01 0.0e+00 1.4e+00 [ Info: 310 1.7e+04 2.5e-01 0.0e+00 1.4e+00 [ Info: 311 1.7e+04 1.5e-01 0.0e+00 1.2e+00 [ Info: 312 1.7e+04 7.2e-02 0.0e+00 1.1e+00 [ Info: 313 1.7e+04 8.6e-02 0.0e+00 1.1e+00 [ Info: 314 1.7e+04 1.1e-01 0.0e+00 1.1e+00 [ Info: 315 1.7e+04 1.5e-01 0.0e+00 1.2e+00 [ Info: 316 1.7e+04 1.9e-01 0.0e+00 1.2e+00 [ Info: 317 1.7e+04 2.4e-01 0.0e+00 1.3e+00 [ Info: 318 1.7e+04 2.9e-01 0.0e+00 1.4e+00 [ Info: 319 1.7e+04 3.4e-01 0.0e+00 1.4e+00 [ Info: 320 1.7e+04 3.9e-01 0.0e+00 1.5e+00 [ Info: 321 1.7e+04 3.1e-01 0.0e+00 1.5e+00 [ Info: 322 1.7e+04 2.2e-01 0.0e+00 1.4e+00 [ Info: 323 1.7e+04 7.0e-02 0.0e+00 1.4e-01 [ Info: 324 1.7e+04 3.4e-03 0.0e+00 1.3e-01 [ Info: 325 1.7e+04 2.1e-03 0.0e+00 1.2e-01 [ Info: 326 1.7e+04 1.9e-03 0.0e+00 1.1e-01 [ Info: 327 1.7e+04 1.8e-03 0.0e+00 1.2e-01 [ Info: 328 1.7e+04 1.7e-03 0.0e+00 1.5e-01 [ Info: 329 1.7e+04 2.1e-03 0.0e+00 1.9e-01 [ Info: 330 1.7e+04 9.7e-05 0.0e+00 2.4e-01 [ Info: 331 1.7e+04 1.7e-03 0.0e+00 1.9e-01 [ Info: 332 1.7e+04 1.7e-03 0.0e+00 1.4e-01 [ Info: 333 1.7e+04 2.0e-03 0.0e+00 6.7e-02 [ Info: 334 1.7e+04 4.2e-04 0.0e+00 9.1e-02 [ Info: 335 1.7e+04 2.6e-05 0.0e+00 8.7e-02 [ Info: 336 1.7e+04 7.6e-04 0.0e+00 8.4e-02 [ Info: 337 1.7e+04 7.6e-04 0.0e+00 8.2e-02 [ Info: 338 1.7e+04 8.3e-04 0.0e+00 7.4e-02 [ Info: 339 1.7e+04 8.9e-04 0.0e+00 7.1e-02 [ Info: 340 1.7e+04 1.0e-03 0.0e+00 6.2e-02 [ Info: 341 1.7e+04 1.0e-03 0.0e+00 5.9e-02 [ Info: 342 1.7e+04 1.1e-03 0.0e+00 6.3e-02 [ Info: 343 1.7e+04 1.1e-03 0.0e+00 5.9e-02 [ Info: 344 1.7e+04 1.1e-03 0.0e+00 7.1e-02 [ Info: 345 1.7e+04 1.0e-03 0.0e+00 6.3e-02 [ Info: 346 1.7e+04 1.1e-03 0.0e+00 8.5e-02 [ Info: 347 1.7e+04 9.8e-04 0.0e+00 8.6e-02 [ Info: 348 1.7e+04 1.0e-03 0.0e+00 1.2e-01 [ Info: 349 1.7e+04 6.3e-04 0.0e+00 1.5e-01 [ Info: 350 1.7e+04 8.0e-04 0.0e+00 1.9e-01 [ Info: 351 1.7e+04 2.9e-04 0.0e+00 2.1e-01 [ Info: 352 1.7e+04 1.2e-03 0.0e+00 1.7e-01 [ Info: 353 1.7e+04 1.2e-03 0.0e+00 1.0e-01 [ Info: 354 1.7e+04 6.8e-04 0.0e+00 8.0e-02 [ Info: 355 1.7e+04 5.8e-04 0.0e+00 7.0e-02 [ Info: 356 1.7e+04 6.8e-04 0.0e+00 7.0e-02 [ Info: 357 1.7e+04 7.9e-04 0.0e+00 6.9e-02 [ Info: 358 1.7e+04 8.9e-04 0.0e+00 6.0e-02 [ Info: 359 1.7e+04 9.0e-04 0.0e+00 6.7e-02 [ Info: 360 1.7e+04 9.0e-04 0.0e+00 6.2e-02 [ Info: 361 1.7e+04 9.0e-04 0.0e+00 6.6e-02 [ Info: 362 1.7e+04 9.0e-04 0.0e+00 6.2e-02 [ Info: 363 1.7e+04 9.1e-04 0.0e+00 6.6e-02 [ Info: 364 1.7e+04 8.9e-04 0.0e+00 6.1e-02 [ Info: 365 1.7e+04 9.1e-04 0.0e+00 6.8e-02 [ Info: 366 1.7e+04 8.8e-04 0.0e+00 5.8e-02 [ Info: 367 1.7e+04 8.9e-04 0.0e+00 7.2e-02 [ Info: 368 1.7e+04 8.4e-04 0.0e+00 6.5e-02 [ Info: 369 1.7e+04 8.9e-04 0.0e+00 8.3e-02 [ Info: 370 1.7e+04 7.9e-04 0.0e+00 8.2e-02 [ Info: 371 1.7e+04 8.7e-04 0.0e+00 1.0e-01 [ Info: 372 1.7e+04 7.3e-04 0.0e+00 9.7e-02 [ Info: 373 1.7e+04 8.6e-04 0.0e+00 1.0e-01 [ Info: 374 1.7e+04 8.3e-04 0.0e+00 6.6e-02 [ Info: 375 1.7e+04 9.5e-04 0.0e+00 3.0e-02 [ Info: 376 1.7e+04 5.1e-04 0.0e+00 2.5e-02 [ Info: 377 1.7e+04 5.3e-05 0.0e+00 2.6e-02 [ Info: 378 1.7e+04 4.5e-05 0.0e+00 2.5e-02 [ Info: 379 1.7e+04 4.9e-04 0.0e+00 2.2e-02 [ Info: 380 1.7e+04 4.1e-04 0.0e+00 1.9e-02 [ Info: 381 1.7e+04 6.2e-04 0.0e+00 1.6e-02 [ Info: 382 1.7e+04 8.0e-05 0.0e+00 1.3e-02 [ Info: 383 1.7e+04 6.0e-04 0.0e+00 9.6e-03 [ Info: 384 1.7e+04 2.1e-05 0.0e+00 7.8e-03 [ Info: 385 1.7e+04 3.6e-05 0.0e+00 7.9e-03 [ Info: 386 1.7e+04 3.2e-05 0.0e+00 7.9e-03 [ Info: 387 1.7e+04 2.2e-05 0.0e+00 7.8e-03 [ Info: 388 1.7e+04 1.2e-05 0.0e+00 7.5e-03 [ Info: 389 1.7e+04 5.1e-06 0.0e+00 7.3e-03 [ Info: 390 1.7e+04 4.4e-06 0.0e+00 7.0e-03 [ Info: 391 1.7e+04 4.9e-06 0.0e+00 6.8e-03 [ Info: 392 1.7e+04 2.6e-06 0.0e+00 6.5e-03 [ Info: 393 1.7e+04 6.3e-07 0.0e+00 6.2e-03 [ Info: 394 1.7e+04 3.1e-07 0.0e+00 5.8e-03 [ Info: 395 1.7e+04 3.6e-08 0.0e+00 5.3e-03 [ Info: 396 1.7e+04 1.6e-06 0.0e+00 4.7e-03 [ Info: 397 1.7e+04 1.4e-06 0.0e+00 4.7e-03 [ Info: 398 1.7e+04 1.1e-06 0.0e+00 4.8e-03 [ Info: 399 1.7e+04 1.1e-06 0.0e+00 4.6e-03 [ Info: 400 1.7e+04 3.5e-06 0.0e+00 4.3e-03 [ Info: 401 1.7e+04 2.6e-06 0.0e+00 3.7e-03 [ Info: 402 1.7e+04 3.2e-06 0.0e+00 3.2e-03 [ Info: 403 1.7e+04 1.8e-06 0.0e+00 2.4e-03 [ Info: 404 1.7e+04 5.6e-07 0.0e+00 2.0e-03 [ Info: 405 1.7e+04 2.8e-06 0.0e+00 1.5e-03 [ Info: 406 1.7e+04 2.6e-06 0.0e+00 1.5e-03 [ Info: 407 1.7e+04 3.9e-06 0.0e+00 1.8e-03 [ Info: 408 1.7e+04 3.2e-06 0.0e+00 1.4e-03 [ Info: 409 1.7e+04 2.5e-06 0.0e+00 9.2e-04 [ Info: 410 1.7e+04 2.2e-06 0.0e+00 7.4e-04 [ Info: 411 1.7e+04 7.3e-07 0.0e+00 7.4e-04 [ Info: 412 1.7e+04 1.3e-06 0.0e+00 7.4e-04 [ Info: 413 1.7e+04 2.3e-06 0.0e+00 7.4e-04 [ Info: 414 1.7e+04 3.0e-06 0.0e+00 7.4e-04 [ Info: 415 1.7e+04 2.5e-06 0.0e+00 7.4e-04 [ Info: 416 1.7e+04 1.2e-06 0.0e+00 7.5e-04 [ Info: 417 1.7e+04 2.7e-07 0.0e+00 7.5e-04 [ Info: 418 1.7e+04 2.3e-07 0.0e+00 7.5e-04 [ Info: 419 1.7e+04 7.5e-07 0.0e+00 7.5e-04 [ Info: 420 1.7e+04 2.7e-07 0.0e+00 7.5e-04 [ Info: 421 1.7e+04 9.2e-07 0.0e+00 7.5e-04 [ Info: 422 1.7e+04 8.3e-07 0.0e+00 1.1e-03 [ Info: 423 1.7e+04 1.4e-06 0.0e+00 1.4e-03 [ Info: 424 1.7e+04 9.0e-06 0.0e+00 8.8e-04 [ Info: 425 1.7e+04 8.0e-06 0.0e+00 7.3e-04 [ Info: 426 1.7e+04 1.0e-05 0.0e+00 7.2e-04 [ Info: 427 1.7e+04 8.1e-06 0.0e+00 7.3e-04 [ Info: 428 1.7e+04 1.8e-05 0.0e+00 7.4e-04 [ Info: 429 1.7e+04 3.8e-06 0.0e+00 7.4e-04 [ Info: 430 1.7e+04 5.7e-06 0.0e+00 7.3e-04 [ Info: 431 1.7e+04 1.6e-06 0.0e+00 7.3e-04 [ Info: 432 1.7e+04 3.2e-06 0.0e+00 7.3e-04 [ Info: 433 1.7e+04 3.4e-06 0.0e+00 7.3e-04 [ Info: 434 1.7e+04 6.7e-06 0.0e+00 7.4e-04 [ Info: 435 1.7e+04 5.3e-06 0.0e+00 7.4e-04 [ Info: 436 1.7e+04 1.2e-06 0.0e+00 7.5e-04 [ Info: 437 1.7e+04 8.2e-07 0.0e+00 7.5e-04 [ Info: 438 1.7e+04 8.0e-07 0.0e+00 7.5e-04 [ Info: 439 1.7e+04 4.9e-07 0.0e+00 7.5e-04 [ Info: 440 1.7e+04 4.7e-07 0.0e+00 7.5e-04 [ Info: 441 1.7e+04 1.5e-06 0.0e+00 7.5e-04 [ Info: 442 1.7e+04 2.8e-06 0.0e+00 7.4e-04 [ Info: 443 1.7e+04 1.3e-05 0.0e+00 7.3e-04 [ Info: 444 1.7e+04 5.8e-06 0.0e+00 7.3e-04 [ Info: 445 1.7e+04 1.4e-05 0.0e+00 7.4e-04 [ Info: 446 1.7e+04 1.1e-05 0.0e+00 7.5e-04 [ Info: 447 1.7e+04 1.3e-05 0.0e+00 7.4e-04 [ Info: 448 1.7e+04 1.4e-05 0.0e+00 7.2e-04 [ Info: 449 1.7e+04 8.4e-06 0.0e+00 7.2e-04 [ Info: 450 1.7e+04 5.7e-06 0.0e+00 7.1e-04 [ Info: 451 1.7e+04 4.7e-06 0.0e+00 7.1e-04 [ Info: 452 1.7e+04 8.3e-06 0.0e+00 7.2e-04 [ Info: 453 1.7e+04 9.0e-06 0.0e+00 7.3e-04 [ Info: 454 1.7e+04 8.0e-06 0.0e+00 7.4e-04 [ Info: 455 1.7e+04 2.6e-06 0.0e+00 7.4e-04 [ Info: 456 1.7e+04 1.2e-06 0.0e+00 7.4e-04 [ Info: 457 1.7e+04 1.4e-06 0.0e+00 7.4e-04 [ Info: 458 1.7e+04 1.2e-06 0.0e+00 7.5e-04 [ Info: 459 1.7e+04 2.7e-06 0.0e+00 7.5e-04 [ Info: 460 1.7e+04 2.1e-06 0.0e+00 7.5e-04 [ Info: 461 1.7e+04 8.8e-07 0.0e+00 7.5e-04 [ Info: 462 1.7e+04 6.2e-07 0.0e+00 7.5e-04 [ Info: 463 1.7e+04 1.7e-05 0.0e+00 7.3e-04 [ Info: 464 1.7e+04 1.8e-05 0.0e+00 7.2e-04 [ Info: 465 1.7e+04 8.8e-06 0.0e+00 7.2e-04 [ Info: 466 1.7e+04 2.0e-05 0.0e+00 7.5e-04 [ Info: 467 1.7e+04 5.4e-06 0.0e+00 7.4e-04 [ Info: 468 1.7e+04 1.6e-06 0.0e+00 7.4e-04 [ Info: 469 1.7e+04 1.5e-05 0.0e+00 7.2e-04 [ Info: 470 1.7e+04 1.4e-05 0.0e+00 7.1e-04 [ Info: 471 1.7e+04 1.8e-06 0.0e+00 7.1e-04 [ Info: 472 1.7e+04 2.8e-06 0.0e+00 7.1e-04 [ Info: 473 1.7e+04 6.6e-06 0.0e+00 7.2e-04 [ Info: 474 1.7e+04 5.9e-06 0.0e+00 7.3e-04 [ Info: 475 1.7e+04 3.3e-06 0.0e+00 7.3e-04 [ Info: 476 1.7e+04 3.1e-06 0.0e+00 7.3e-04 [ Info: 477 1.7e+04 5.9e-06 0.0e+00 7.4e-04 [ Info: 478 1.7e+04 6.9e-06 0.0e+00 7.5e-04 [ Info: 479 1.7e+04 4.4e-06 0.0e+00 7.5e-04 [ Info: 480 1.7e+04 1.6e-06 0.0e+00 7.5e-04 [ Info: 481 1.7e+04 4.9e-06 0.0e+00 7.4e-04 [ Info: 482 1.7e+04 1.4e-05 0.0e+00 7.3e-04 [ Info: 483 1.7e+04 4.9e-06 0.0e+00 7.2e-04 [ Info: 484 1.7e+04 3.7e-06 0.0e+00 7.3e-04 [ Info: 485 1.7e+04 1.9e-07 0.0e+00 7.3e-04 [ Info: 486 1.7e+04 1.3e-06 0.0e+00 7.3e-04 [ Info: 487 1.7e+04 6.6e-06 0.0e+00 7.4e-04 [ Info: 488 1.7e+04 7.2e-07 0.0e+00 7.4e-04 [ Info: 489 1.7e+04 1.0e-05 0.0e+00 7.3e-04 [ Info: 490 1.7e+04 1.1e-06 0.0e+00 7.3e-04 [ Info: 491 1.7e+04 5.4e-06 0.0e+00 7.2e-04 [ Info: 492 1.7e+04 1.3e-06 0.0e+00 7.2e-04 [ Info: 493 1.7e+04 3.3e-06 0.0e+00 7.3e-04 [ Info: 494 1.7e+04 5.6e-07 0.0e+00 7.3e-04 [ Info: 495 1.7e+04 2.7e-06 0.0e+00 7.3e-04 [ Info: 496 1.7e+04 1.5e-06 0.0e+00 7.3e-04 [ Info: 497 1.7e+04 4.8e-06 0.0e+00 7.4e-04 [ Info: 498 1.7e+04 1.1e-06 0.0e+00 7.3e-04 [ Info: 499 1.7e+04 9.3e-06 0.0e+00 7.2e-04 [ Info: 500 1.7e+04 1.5e-05 0.0e+00 7.4e-04 [ Info: 501 1.7e+04 1.1e-05 0.0e+00 7.5e-04 [ Info: 502 1.7e+04 2.5e-08 0.0e+00 7.5e-04 [ Info: 503 1.7e+04 4.7e-07 0.0e+00 7.5e-04 [ Info: 504 1.7e+04 1.1e-05 0.0e+00 7.4e-04 [ Info: 505 1.7e+04 1.1e-05 0.0e+00 7.5e-04 [ Info: 506 1.7e+04 5.2e-04 0.0e+00 2.3e-04 [ Info: 507 1.7e+04 5.9e-05 0.0e+00 2.9e-04 [ Info: 508 1.7e+04 4.7e-04 0.0e+00 7.6e-04 [ Info: 509 1.7e+04 2.0e-07 0.0e+00 7.6e-04 [ Info: 510 1.7e+04 2.7e-06 0.0e+00 7.5e-04 [ Info: 511 1.7e+04 9.1e-07 0.0e+00 7.5e-04 [ Info: 512 1.7e+04 9.5e-07 0.0e+00 7.5e-04 [ Info: 513 1.7e+04 3.3e-05 0.0e+00 7.8e-04 [ Info: 514 1.7e+04 5.3e-04 0.0e+00 2.5e-04 [ Info: 515 1.7e+04 5.0e-04 0.0e+00 7.6e-04 [ Info: 516 1.7e+04 4.8e-05 0.0e+00 7.1e-04 [ Info: 517 1.7e+04 2.2e-04 0.0e+00 4.9e-04 [ Info: 518 1.7e+04 2.0e-04 0.0e+00 6.8e-04 [ Info: 519 1.7e+04 4.4e-04 0.0e+00 2.5e-04 [ Info: 520 1.7e+04 5.1e-04 0.0e+00 7.6e-04 [ Info: 521 1.7e+04 6.0e-07 0.0e+00 7.6e-04 [ Info: 522 1.7e+04 1.9e-06 0.0e+00 7.5e-04 [ Info: 523 1.7e+04 2.3e-06 0.0e+00 7.6e-04 [ Info: 524 1.7e+04 2.8e-07 0.0e+00 7.6e-04 [ Info: 525 1.7e+04 4.7e-07 0.0e+00 7.6e-04 [ Info: 526 1.7e+04 2.3e-07 0.0e+00 7.6e-04 [ Info: 527 1.7e+04 6.8e-04 0.0e+00 2.2e-04 [ Info: 528 1.7e+04 6.8e-04 0.0e+00 7.6e-04 [ Info: 529 1.7e+04 1.3e-06 0.0e+00 7.6e-04 [ Info: 530 1.7e+04 2.6e-07 0.0e+00 7.6e-04 [ Info: 531 1.7e+04 4.4e-07 0.0e+00 7.6e-04 [ Info: 532 1.7e+04 5.3e-07 0.0e+00 7.6e-04 [ Info: 533 1.7e+04 6.6e-04 0.0e+00 2.1e-04 [ Info: 534 1.7e+04 6.6e-04 0.0e+00 7.6e-04 [ Info: 535 1.7e+04 1.4e-06 0.0e+00 7.6e-04 [ Info: 536 1.7e+04 3.3e-07 0.0e+00 7.6e-04 [ Info: 537 1.7e+04 3.3e-07 0.0e+00 7.6e-04 [ Info: 538 1.7e+04 7.1e-07 0.0e+00 7.6e-04 [ Info: 539 1.7e+04 6.4e-04 0.0e+00 2.0e-04 [ Info: 540 1.7e+04 6.4e-04 0.0e+00 7.6e-04 [ Info: 541 1.7e+04 1.4e-06 0.0e+00 7.6e-04 [ Info: 542 1.7e+04 3.7e-07 0.0e+00 7.6e-04 [ Info: 543 1.7e+04 2.1e-07 0.0e+00 7.6e-04 [ Info: 544 1.7e+04 8.6e-07 0.0e+00 7.6e-04 [ Info: 545 1.7e+04 6.1e-04 0.0e+00 1.8e-04 [ Info: 546 1.7e+04 6.1e-04 0.0e+00 7.6e-04 [ Info: 547 1.7e+04 1.4e-06 0.0e+00 7.6e-04 [ Info: 548 1.7e+04 4.2e-07 0.0e+00 7.6e-04 [ Info: 549 1.7e+04 1.2e-07 0.0e+00 7.6e-04 [ Info: 550 1.7e+04 9.9e-07 0.0e+00 7.6e-04 [ Info: 551 1.7e+04 5.9e-04 0.0e+00 1.8e-04 [ Info: 552 1.7e+04 5.9e-04 0.0e+00 7.6e-04 [ Info: 553 1.7e+04 1.4e-06 0.0e+00 7.6e-04 [ Info: 554 1.7e+04 4.7e-07 0.0e+00 7.6e-04 [ Info: 555 1.7e+04 4.8e-08 0.0e+00 7.6e-04 [ Info: 556 1.7e+04 1.1e-06 0.0e+00 7.6e-04 [ Info: 557 1.7e+04 5.6e-04 0.0e+00 2.0e-04 [ Info: 558 1.7e+04 5.6e-04 0.0e+00 7.6e-04 [ Info: 559 1.7e+04 1.5e-06 0.0e+00 7.6e-04 [ Info: 560 1.7e+04 4.4e-07 0.0e+00 7.6e-04 [ Info: 561 1.7e+04 4.4e-09 0.0e+00 7.6e-04 [ Info: 562 1.7e+04 8.2e-04 7.7e-10 2.9e-04 [ Info: 563 1.7e+04 8.2e-04 0.0e+00 7.6e-04 [ Info: 564 1.7e+04 1.2e-06 0.0e+00 7.6e-04 [ Info: 565 1.7e+04 4.7e-07 0.0e+00 7.6e-04 [ Info: 566 1.7e+04 3.4e-07 0.0e+00 7.6e-04 [ Info: 567 1.7e+04 5.3e-08 0.0e+00 7.6e-04 [ Info: 568 1.7e+04 8.0e-04 4.7e-10 2.7e-04 [ Info: 569 1.7e+04 8.0e-04 0.0e+00 7.6e-04 [ Info: 570 1.7e+04 1.2e-06 0.0e+00 7.6e-04 [ Info: 571 1.7e+04 2.6e-07 0.0e+00 7.6e-04 [ Info: 572 1.7e+04 5.9e-07 0.0e+00 7.6e-04 [ Info: 573 1.7e+04 1.1e-07 0.0e+00 7.6e-04 [ Info: 574 1.7e+04 7.7e-04 1.6e-10 2.6e-04 [ Info: 575 1.7e+04 7.7e-04 0.0e+00 7.6e-04 [ Info: 576 1.7e+04 1.3e-06 0.0e+00 7.6e-04 [ Info: 577 1.7e+04 9.5e-08 0.0e+00 7.6e-04 [ Info: 578 1.7e+04 8.0e-07 0.0e+00 7.6e-04 [ Info: 579 1.7e+04 1.6e-07 0.0e+00 7.6e-04 [ Info: 580 1.7e+04 7.4e-04 0.0e+00 2.5e-04 [ Info: 581 1.7e+04 7.5e-04 0.0e+00 7.6e-04 [ Info: 582 1.7e+04 1.3e-06 0.0e+00 7.6e-04 [ Info: 583 1.7e+04 3.6e-08 0.0e+00 7.6e-04 [ Info: 584 1.7e+04 8.8e-07 0.0e+00 7.6e-04 [ Info: 585 1.7e+04 1.1e-07 0.0e+00 7.6e-04 [ Info: 586 1.7e+04 7.2e-04 0.0e+00 2.4e-04 [ Info: 587 1.7e+04 7.2e-04 0.0e+00 7.6e-04 [ Info: 588 1.7e+04 1.3e-06 0.0e+00 7.6e-04 [ Info: 589 1.7e+04 1.4e-07 0.0e+00 7.6e-04 [ Info: 590 1.7e+04 6.7e-07 0.0e+00 7.6e-04 [ Info: 591 1.7e+04 2.0e-07 0.0e+00 7.6e-04 [ Info: 592 1.7e+04 6.9e-04 0.0e+00 2.3e-04 [ Info: 593 1.7e+04 6.9e-04 0.0e+00 7.6e-04 [ Info: 594 1.7e+04 1.3e-06 0.0e+00 7.6e-04 [ Info: 595 1.7e+04 2.3e-07 0.0e+00 7.6e-04 [ Info: 596 1.7e+04 5.0e-07 0.0e+00 7.6e-04 [ Info: 597 1.7e+04 4.4e-07 0.0e+00 7.6e-04 [ Info: 598 1.7e+04 6.7e-04 0.0e+00 2.2e-04 [ Info: 599 1.7e+04 6.7e-04 0.0e+00 7.6e-04 [ Info: 600 1.7e+04 1.3e-06 0.0e+00 7.6e-04 [ Info: 601 1.7e+04 3.0e-07 0.0e+00 7.6e-04 [ Info: 602 1.7e+04 3.8e-07 0.0e+00 7.6e-04 [ Info: 603 1.7e+04 6.3e-07 0.0e+00 7.6e-04 [ Info: 604 1.7e+04 6.5e-04 0.0e+00 2.0e-04 [ Info: 605 1.7e+04 6.5e-04 0.0e+00 7.6e-04 [ Info: 606 1.7e+04 1.4e-06 0.0e+00 7.6e-04 [ Info: 607 1.7e+04 3.7e-07 0.0e+00 7.6e-04 [ Info: 608 1.7e+04 2.7e-07 0.0e+00 7.6e-04 [ Info: 609 1.7e+04 7.9e-07 0.0e+00 7.6e-04 [ Info: 610 1.7e+04 6.2e-04 0.0e+00 1.9e-04 [ Info: 611 1.7e+04 6.2e-04 0.0e+00 7.6e-04 [ Info: 612 1.7e+04 1.4e-06 0.0e+00 7.6e-04 [ Info: 613 1.7e+04 4.2e-07 0.0e+00 7.6e-04 [ Info: 614 1.7e+04 1.8e-07 0.0e+00 7.6e-04 [ Info: 615 1.7e+04 9.3e-07 0.0e+00 7.6e-04 [ Info: 616 1.7e+04 6.0e-04 0.0e+00 1.8e-04 [ Info: 617 1.7e+04 6.0e-04 0.0e+00 7.6e-04 [ Info: 618 1.7e+04 1.4e-06 0.0e+00 7.6e-04 [ Info: 619 1.7e+04 4.7e-07 0.0e+00 7.6e-04 [ Info: 620 1.7e+04 1.0e-07 0.0e+00 7.6e-04 [ Info: 621 1.7e+04 1.1e-06 0.0e+00 7.6e-04 [ Info: 622 1.7e+04 5.7e-04 0.0e+00 1.9e-04 [ Info: 623 1.7e+04 5.7e-04 0.0e+00 7.6e-04 [ Info: 624 1.7e+04 1.4e-06 0.0e+00 7.6e-04 [ Info: 625 1.7e+04 5.1e-07 0.0e+00 7.6e-04 [ Info: 626 1.7e+04 3.3e-08 0.0e+00 7.6e-04 [ Info: 627 1.7e+04 1.1e-06 0.0e+00 7.6e-04 [ Info: 628 1.7e+04 5.5e-04 0.0e+00 2.1e-04 [ Info: 629 1.7e+04 2.8e-04 8.1e-10 1.0e-04 [ Info: 630 1.7e+04 2.8e-04 0.0e+00 2.1e-04 [ Info: 631 1.7e+04 5.4e-04 0.0e+00 7.6e-04 [ Info: 632 1.7e+04 8.1e-04 5.7e-10 2.7e-04 [ Info: 633 1.7e+04 8.1e-04 0.0e+00 7.6e-04 [ Info: 634 1.7e+04 6.3e-04 0.0e+00 4.5e-04 [ Info: 635 1.7e+04 7.5e-05 0.0e+00 8.5e-05 Step 3: penalty=3.0, tol=0.0001, obj=17247.963338550737
The MMA87 optimizer iteratively updates the density field to minimize thermal compliance while satisfying the volume constraint. Ramping the penalty drives the final design toward a crisp 0/1 distribution.
Results
@show obj(res.minimizer)
@show constr(res.minimizer)obj(res.minimizer) = 17247.963338550737
constr(res.minimizer) = -6.303111921290849e-10-6.303111921290849e-10The final design should show the characteristic branching “tree” pattern that efficiently conducts heat from the top to the bottom boundary.
Visualization
using CairoMakie
fig = visualize(problem; topology=res.minimizer)Precompiling packages... 6595.9 ms ✓ QuartoNotebookWorkerMakieExt (serial) 1 dependency successfully precompiled in 7 seconds Precompiling packages... 5452.2 ms ✓ QuartoNotebookWorkerCairoMakieExt (serial) 1 dependency successfully precompiled in 6 seconds
The visualization reveals the tree-like structure — high-conductivity material forms branching paths similar to how structural members carry loads in compliance minimization. This analogy between heat conduction and structural mechanics is a fundamental insight in topology optimization.