Topology optimisation of a 2D cantilever beam

This demo showcases the mother of all topology optimisation problems: compliance minimisation of a solid isotropic material with penalisation (SIMP), regularised with a Helmholtz filter.

In particular this demo emphasizes

the use of custom optimisation solvers and
multi-step adjoint computations.

import argparse

from mpi4py import MPI
from petsc4py import PETSc
from petsc4py.PETSc import ScalarType  # type: ignore

import dolfinx
import dolfinx.fem.petsc
import ufl

import numpy as np
import pyvista as pv

import dolfiny

output = False
parser = argparse.ArgumentParser(description="Truss sizing demo")
parser.add_argument(
    "-a",
    "--algorithm",
    choices=["conlin", "mma"],
    default="mma",
    help="Choose optimisation algorithm",
)
args, _unknown = parser.parse_known_args()

Computational domain¶

For the topological dimension $d \in \{ 2, 3\}$ the computational domain is given by $\Omega = (0, 2)^{d-1} \times (0,1) \subset \mathbb{R}^d$ and is discretized by $(2n, n)$ quadrilateral or $(2n, n, n)$ hexahedral elements respectively, yielding a tesseltation $\mathcal{T}$ .

tdim = 2
n = 50
comm = MPI.COMM_WORLD
mesh = (
    dolfinx.mesh.create_rectangle(
        comm, [[0, 0], [2, 1]], (2 * n, n), cell_type=dolfinx.mesh.CellType.quadrilateral
    )
    if tdim == 2
    else dolfinx.mesh.create_box(
        comm, [[0, 0, 0], [2, 1, 1]], (2 * n, n, n), cell_type=dolfinx.mesh.CellType.hexahedron
    )
)

Problem formulation¶

We define three function spaces associated with density $\rho$ , filtered-density $\hat{\rho}$ , and displacement $u$ :

$V_\rho = \mathcal{P}_0 (\mathcal{T})$
$V_{\hat{\rho}} = \mathcal{P}_1 (\mathcal{T})$
$V_u = \mathcal{P}_1^3 (\mathcal{T})$ .

V_u = dolfinx.fem.functionspace(mesh, ("Lagrange", 1, (tdim,)))
V_ρ = dolfinx.fem.functionspace(mesh, ("Discontinuous Lagrange", 0))
V_ρ_f = dolfinx.fem.functionspace(mesh, ("Lagrange", 1))

ρ = dolfinx.fem.Function(V_ρ, name="density")
ρ_f = dolfinx.fem.Function(V_ρ_f, name="density-filtered")
u = dolfinx.fem.Function(V_u, name="u")

State problem (elasticity)¶

The next step is to define the elasticity problem. We consider a linear isotropic material model, together with classic SIMP penalisation Bendsøe & Kikuchi (1988), which defines the Young’s modulus E as

E(\hat{\rho}) = (\rho_\text{min} + (1-\rho_\text{min}) \hat{\rho}^p) E_0

(1)

where $E_0$ is Young’s modulus of the solid material (associated with the phase $\rho=1$ ), and $p > 1$ is the penalty factor.

As boundary conditions we fix the $x=0$ plane of the design and apply a constant force $f$ (Neumann boundary condition) on the center of the facet at $x=2$ .

ρ_min = np.float64(1e-9)
penalty = 3

# ASTM A-36 / EN S235J2 steel
E0 = 2.11e11  # Pa
E = (ρ_min + (1 - ρ_min) * ρ_f**penalty) * E0
nu = 0.29  # dimensionless


def ε(u):  # strain
    return ufl.sym(ufl.grad(u))


def σ(u):  # stress
    # Lamé parameters λ and μ
    λ = E * nu / ((1 + nu) * (1 - 2 * nu))
    μ = E / (2 * (1 + nu))
    return λ * ufl.nabla_div(u) * ufl.Identity(len(u)) + 2 * μ * ε(u)


def on_rhs(x):
    mask = np.isclose(x[0], 2.0) & np.greater_equal(x[1], 0.45) & np.less_equal(x[1], 0.55)
    if tdim == 3:
        mask &= np.greater_equal(x[2], 0.45) & np.less_equal(x[2], 0.55)
    return mask


facets_rhs = dolfinx.mesh.locate_entities_boundary(mesh, tdim - 1, on_rhs)
facet_tag = dolfinx.mesh.meshtags(mesh, tdim - 1, np.unique(facets_rhs), 1)
ds = ufl.Measure("ds", domain=mesh, subdomain_data=facet_tag)


def compliance(u):
    load = -8.6e4  # Nw
    f = ufl.as_vector((0, load) if tdim == 2 else (0, load, 0))
    return ufl.inner(f, u) * ds(1)


def elastic_energy(u):
    E = 1 / 2 * ufl.inner(σ(u), ε(u)) * ufl.dx
    E -= compliance(u)
    return E


fixed_entities = dolfinx.mesh.locate_entities_boundary(
    mesh, tdim - 1, lambda x: np.isclose(x[0], 0.0)
)
fixed_dofs = dolfinx.fem.locate_dofs_topological(V_u, tdim - 1, fixed_entities)
bc_u = dolfinx.fem.dirichletbc(np.zeros(tdim, dtype=ScalarType), fixed_dofs, V_u)

# State solver
a = ufl.derivative(ufl.derivative(elastic_energy(u), u), u)
L = -ufl.derivative(elastic_energy(u), u)
L = ufl.replace(L, {u: ufl.as_vector((0, 0) if tdim == 2 else (0, 0, 0))})

elas_prob = dolfinx.fem.petsc.LinearProblem(
    a,
    L,
    bcs=[bc_u],
    u=u,
    petsc_options=(
        {
            "ksp_error_if_not_converged": "True",
            "ksp_type": "preonly",
            "pc_type": "cholesky",
            "pc_factor_mat_solver_type": "mumps",
        }
        if tdim == 2
        else {
            # Combination of https://github.com/FEniCS/performance-test and https://doi.org/10.1007/s00158-020-02618-z
            "ksp_error_if_not_converged": True,
            "ksp_type": "cg",
            "ksp_rtol": 1.0e-8,
            "pc_type": "gamg",
            "pc_gamg_type": "agg",
            "pc_gamg_agg_nsmooths": 1,
            "pc_gamg_threshold": 0.001,
            "mg_levels_esteig_ksp_type": "cg",
            "mg_levels_ksp_type": "chebyshev",
            "mg_levels_ksp_chebyshev_esteig_steps": 50,
            "mg_levels_pc_type": "sor",
            "matptap_via": "scalable",
            "pc_gamg_coarse_eq_limit": 1000,
        }
    ),
    petsc_options_prefix="elasticity_ksp",
)

Filtering¶

We use a Helmholtz filter on the density field, first introduced by Lazarov & Sigmund (2010) in the context of topology optimisation.

In short, this boils down to solving for a given density $\rho$ a Helmholtz equation, yielding the filtered-density $\hat{\rho}$

\int_\Omega r^2 \nabla \hat{\rho} \cdot \nabla \tau + \hat{\rho} \tau \ \text{d}x = \int_\Omega \rho \tau \ \text{d}x \qquad \forall \tau \in V_{\hat{\rho}}.

(2)

$r$ is a parameter that controls the filter radius, we choose $r$ to be dependent on the local cell diameter.

Since the Helmholtz equation is self-adjoint and we need to evaluate the adjoint of it for the gradient computation later on, we set up the solver to allow for handling of generic right hand sides. Thus we only have one linear solver and operator matrix stored for both forward and adjoint problem.

r = 0.45 * ufl.CellDiameter(mesh)  # factor 1-3
u_f, v_f = ufl.TrialFunction(V_ρ_f), ufl.TestFunction(V_ρ_f)
a_filter = dolfinx.fem.form(
    r**2 * ufl.inner(ufl.grad(u_f), ufl.grad(v_f)) * ufl.dx + u_f * v_f * ufl.dx
)
L_filter_ρ = dolfinx.fem.form(ρ * v_f * ufl.dx)
s = dolfinx.fem.Function(V_ρ_f, name="s")
L_filter_s = dolfinx.fem.form(s * v_f * ufl.dx)

A_filter = dolfinx.fem.petsc.create_matrix(a_filter)
dolfinx.fem.petsc.assemble_matrix(A_filter, a_filter)
A_filter.assemble()

b_filter = dolfinx.fem.petsc.create_vector(V_ρ_f)

opts = PETSc.Options("filter")  # type: ignore
opts["ksp_type"] = "cg"
opts["pc_type"] = "jacobi"
opts["ksp_error_if_not_converged"] = True

filter_ksp = PETSc.KSP().create()  # type: ignore
filter_ksp.setOptionsPrefix("filter")
filter_ksp.setFromOptions()
filter_ksp.setOperators(A_filter)


def apply_filter(rhs, f) -> None:
    """Compute filtered f from rhs."""

    with b_filter.localForm() as b_local:
        b_local.set(0.0)

    dolfinx.fem.petsc.assemble_vector(b_filter, rhs)
    b_filter.ghostUpdate(PETSc.InsertMode.ADD, PETSc.ScatterMode.REVERSE)  # type: ignore
    b_filter.ghostUpdate(addv=PETSc.InsertMode.INSERT, mode=PETSc.ScatterMode.FORWARD)  # type: ignore

    filter_ksp.solve(b_filter, f.x.petsc_vec)
    f.x.petsc_vec.ghostUpdate(PETSc.InsertMode.INSERT, PETSc.ScatterMode.FORWARD)  # type: ignore

Optimisation problem¶

With the state and filtering problems defined we can define the objective and gradient of the (reduced) optimisation problem.

The objective, to be minimised, is compliance

\int_\Omega f \cdot u \ \text{d}x.

(3)

We constrain the density to lower and upper bounds

\rho_\text{min} \leq \rho \leq 1,

(4)

and the volume of the design to a volume fraction $V_f \in (0, 1)$

\frac{1}{\text{Vol} (\Omega)} \int_\Omega \rho \ \text{d}x \leq V_f.

(5)

The the optimisation problem is stated in reduced form in $\rho$ . So, $\hat{\rho}$ and $u$ only appear as intermediates. Gradients are then computed through the adjoint formulation.


J_form = dolfinx.fem.form(compliance(u))
DJ_form = dolfinx.fem.form(-ufl.derivative(elastic_energy(u), ρ_f))

mesh_volume = comm.allreduce(
    dolfinx.fem.assemble_scalar(dolfinx.fem.form(dolfinx.fem.Constant(mesh, 1.0) * ufl.dx))
)
volume_fraction = ρ / mesh_volume * ufl.dx
max_volume_fraction = 0.4 if tdim == 2 else 0.1

g = volume_fraction <= max_volume_fraction

ρ.x.array[:] = max_volume_fraction
ρ_f.interpolate(ρ)

apply_filter(L_filter_ρ, ρ_f)
elas_prob.solve()

c0 = comm.allreduce(dolfinx.fem.assemble_scalar(J_form))

J_scale = 1 / c0  # normalize
J_scale *= 10  # target objective range


@dolfiny.taoproblem.sync_functions([ρ])
def J(tao, _):
    apply_filter(L_filter_ρ, ρ_f)

    # Compute displacement from filtered density.
    elas_prob.solve()

    return comm.allreduce(dolfinx.fem.assemble_scalar(J_form)) * J_scale


Dρ = dolfinx.fem.Function(V_ρ_f)
z = dolfinx.fem.Function(V_ρ_f, name="z")
tmpDG0 = dolfinx.fem.Function(V_ρ)


@dolfiny.taoproblem.sync_functions([ρ])
def DJ(tao, _, G):
    # TODO: surely not necessary?

    # Compute filtered density from density.
    # apply_filter()

    # Compute displacement from filtered denstity.
    # elas_prob.solve()

    # Assemble variation (w.r.t. filtered density).
    with s.x.petsc_vec.localForm() as local:
        local.set(0.0)
    dolfinx.fem.petsc.assemble_vector(s.x.petsc_vec, DJ_form)
    s.x.petsc_vec.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
    s.x.petsc_vec.ghostUpdate(addv=PETSc.InsertMode.INSERT, mode=PETSc.ScatterMode.FORWARD)

    # Apply adjoint to DJ/s -> z.
    apply_filter(L_filter_s, z)

    # Interpolate/project z into DG0.
    tmpDG0.interpolate(z)

    # Copy to G.
    tmpDG0.x.petsc_vec.copy(G)
    G.ghostUpdate(addv=PETSc.InsertMode.INSERT, mode=PETSc.ScatterMode.FORWARD)
    G.scale(J_scale)

Custom optimisation routines¶

For the optimisation we rely on our custom implementations of the Method of Moving Asymptotes (MMA) Svanberg (1987) or Convex Linearisation (CONLIN) Fleury (1989).

opts = PETSc.Options()  # type: ignore
opts["tao_type"] = "python"
opts["tao_monitor"] = ""
opts["tao_max_it"] = (max_it := 100)
if args.algorithm == "conlin":
    opts["tao_python_type"] = "dolfiny.conlin.CONLIN"
    opts["tao_conlin_subsolver_tao_monitor"] = ""
else:  # mma
    opts["tao_python_type"] = "dolfiny.mma.MMA"
    opts["tao_mma_move_limit"] = 0.01
    opts["tao_mma_subsolver_tao_monitor"] = ""

problem = dolfiny.taoproblem.TAOProblem(
    J, [ρ], J=(DJ, ρ.x.petsc_vec.copy()), h=[g], lb=ρ_min, ub=np.float64(1)
)

if comm.size == 1:
    plotter = pv.Plotter(off_screen=False, window_size=(1024, int(512 * 1.4)))
    plotter.open_gif("topopt_simp.gif", fps=5)
    pv_grid = pv.UnstructuredGrid(*dolfinx.plot.vtk_mesh(mesh))
    pv_grid.cell_data[ρ.name] = ρ.x.array
    plotter.add_mesh(pv_grid, scalars=ρ.name, clim=[ρ_min, 1], cmap="coolwarm")
    text = plotter.add_text("")
    plotter.view_xy()
    plotter.camera.zoom(1.5)

    plotter_f = pv.Plotter(off_screen=True, window_size=(1024, int(512 * 1.4)))
    plotter_f.open_gif("topopt_simp_filtered.gif", fps=5)
    pv_grid.point_data[ρ_f.name] = ρ_f.x.array
    plotter_f.add_mesh(pv_grid, scalars=ρ_f.name, clim=[ρ_min, 1], cmap="coolwarm")
    text_f = plotter_f.add_text("")
    plotter_f.view_xy()
    plotter_f.camera.zoom(1.5)


def monitor(tao):
    it = tao.getIterationNumber()
    if comm.size == 1:
        text.SetText(0, f"Iteration {it}")
        pv_grid.cell_data[ρ.name] = ρ.x.array
        plotter.render()
        plotter.write_frame()

        text_f.SetText(0, f"Iteration {it}")
        pv_grid.point_data[ρ_f.name] = ρ_f.x.array
        plotter_f.render()
        plotter_f.write_frame()

    if not output:
        return

    with dolfinx.io.XDMFFile(comm, f"topopt_simp/data_{it}.xdmf", "w") as file:
        file.write_mesh(mesh)
        for f in (ρ, ρ_f, u):
            file.write_function(f, it)

problem.tao.setMonitor(monitor)
problem.solve()

  0 TAO,  Function value: 10.,  Residual: 0. 
# TAO   0 (ITERATING)

# sub   0 [  5k] |x|=2.828e+01 |J|=7.809e-01 (density)

# all            |x|=2.828e+01 |J|=7.809e-01 |h|=0.000e+00 |Jh|=0.000e+00 f=1.000e+01

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -9.64549,  Residual: 0.020409 
  1 TAO,  Function value: -9.69352,  Residual: 0.00890096 
  2 TAO,  Function value: -9.70214,  Residual: 0.00429938 
  3 TAO,  Function value: -9.74205,  Residual: 0.00265104 
  4 TAO,  Function value: -9.75701,  Residual: 0.000447574 
  5 TAO,  Function value: -9.75721,  Residual: 0.000244393 
  6 TAO,  Function value: -9.75729,  Residual: 1.48553e-05 
  7 TAO,  Function value: -9.75729,  Residual: 2.78965e-07 
  8 TAO,  Function value: -9.75729,  Residual: 1.85381e-09 
  1 TAO,  Function value: 9.50582,  Residual: 0.71721 
# TAO   1 (ITERATING)

# sub   0 [  5k] |x|=2.828e+01 |J|=7.172e-01 (density)

# all            |x|=2.828e+01 |J|=7.172e-01 |h|=1.232e-13 |Jh|=1.414e-02 f=9.506e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -9.28884,  Residual: 0.000487889 
  1 TAO,  Function value: -9.28894,  Residual: 0.00033167 
  2 TAO,  Function value: -9.28902,  Residual: 3.25708e-05 
  3 TAO,  Function value: -9.28902,  Residual: 1.26597e-06 
  4 TAO,  Function value: -9.28902,  Residual: 4.50746e-09 
  2 TAO,  Function value: 9.06364,  Residual: 0.661569 
# TAO   2 (ITERATING)

# sub   0 [  5k] |x|=2.830e+01 |J|=6.616e-01 (density)

# all            |x|=2.830e+01 |J|=6.616e-01 |h|=1.929e-04 |Jh|=1.414e-02 f=9.064e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -8.87166,  Residual: 0.000404603 
  1 TAO,  Function value: -8.87167,  Residual: 0.000354057 
  2 TAO,  Function value: -8.87172,  Residual: 1.59839e-05 
  3 TAO,  Function value: -8.87172,  Residual: 6.20495e-07 
  4 TAO,  Function value: -8.87172,  Residual: 1.97056e-09 
  3 TAO,  Function value: 8.67278,  Residual: 0.613391 
# TAO   3 (ITERATING)

# sub   0 [  5k] |x|=2.835e+01 |J|=6.134e-01 (density)

# all            |x|=2.835e+01 |J|=6.134e-01 |h|=1.863e-04 |Jh|=1.414e-02 f=8.673e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -8.50316,  Residual: 0.000123495 
  1 TAO,  Function value: -8.50316,  Residual: 9.33418e-05 
  2 TAO,  Function value: -8.50316,  Residual: 1.08584e-06 
  3 TAO,  Function value: -8.50316,  Residual: 1.14491e-08 
  4 TAO,  Function value: 8.32629,  Residual: 0.57163 
# TAO   4 (ITERATING)

# sub   0 [  5k] |x|=2.839e+01 |J|=5.716e-01 (density)

# all            |x|=2.839e+01 |J|=5.716e-01 |h|=1.471e-04 |Jh|=1.414e-02 f=8.326e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -8.17479,  Residual: 0.000843267 
  1 TAO,  Function value: -8.17485,  Residual: 0.000579633 
  2 TAO,  Function value: -8.1749,  Residual: 2.51575e-05 
  3 TAO,  Function value: -8.1749,  Residual: 7.97235e-07 
  4 TAO,  Function value: -8.1749,  Residual: 1.45008e-09 
  5 TAO,  Function value: 8.01499,  Residual: 0.535599 
# TAO   5 (ITERATING)

# sub   0 [  5k] |x|=2.845e+01 |J|=5.356e-01 (density)

# all            |x|=2.845e+01 |J|=5.356e-01 |h|=1.146e-04 |Jh|=1.414e-02 f=8.015e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -7.87799,  Residual: 0.000518069 
  1 TAO,  Function value: -7.87802,  Residual: 0.000349301 
  2 TAO,  Function value: -7.87804,  Residual: 1.35111e-05 
  3 TAO,  Function value: -7.87804,  Residual: 2.51633e-07 
  4 TAO,  Function value: -7.87804,  Residual: 3.21158e-10 
  6 TAO,  Function value: 7.73209,  Residual: 0.504173 
# TAO   6 (ITERATING)

# sub   0 [  5k] |x|=2.851e+01 |J|=5.042e-01 (density)

# all            |x|=2.851e+01 |J|=5.042e-01 |h|=9.846e-05 |Jh|=1.414e-02 f=7.732e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -7.60686,  Residual: 0.000263801 
  1 TAO,  Function value: -7.60686,  Residual: 0.000179228 
  2 TAO,  Function value: -7.60687,  Residual: 3.48587e-06 
  3 TAO,  Function value: -7.60687,  Residual: 4.45298e-08 
  7 TAO,  Function value: 7.47237,  Residual: 0.476522 
# TAO   7 (ITERATING)

# sub   0 [  5k] |x|=2.858e+01 |J|=4.765e-01 (density)

# all            |x|=2.858e+01 |J|=4.765e-01 |h|=8.498e-05 |Jh|=1.414e-02 f=7.472e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -7.35677,  Residual: 6.01834e-05 
  1 TAO,  Function value: -7.35677,  Residual: 4.05623e-05 
  2 TAO,  Function value: -7.35677,  Residual: 2.16564e-07 
  3 TAO,  Function value: -7.35677,  Residual: 1.90934e-09 
  8 TAO,  Function value: 7.23159,  Residual: 0.452101 
# TAO   8 (ITERATING)

# sub   0 [  5k] |x|=2.865e+01 |J|=4.521e-01 (density)

# all            |x|=2.865e+01 |J|=4.521e-01 |h|=7.468e-05 |Jh|=1.414e-02 f=7.232e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -7.12382,  Residual: 0.000323876 
  1 TAO,  Function value: -7.12383,  Residual: 0.000209923 
  2 TAO,  Function value: -7.12384,  Residual: 5.84998e-06 
  3 TAO,  Function value: -7.12384,  Residual: 9.70659e-08 
  4 TAO,  Function value: -7.12384,  Residual: 1.39175e-10 
  9 TAO,  Function value: 7.00649,  Residual: 0.430238 
# TAO   9 (ITERATING)

# sub   0 [  5k] |x|=2.874e+01 |J|=4.302e-01 (density)

# all            |x|=2.874e+01 |J|=4.302e-01 |h|=6.684e-05 |Jh|=1.414e-02 f=7.006e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -6.90525,  Residual: 0.000555924 
  1 TAO,  Function value: -6.90528,  Residual: 0.000331994 
  2 TAO,  Function value: -6.90529,  Residual: 1.45491e-05 
  3 TAO,  Function value: -6.90529,  Residual: 2.85734e-07 
  4 TAO,  Function value: -6.90529,  Residual: 4.15457e-11 
 10 TAO,  Function value: 6.79454,  Residual: 0.410569 
# TAO  10 (ITERATING)

# sub   0 [  5k] |x|=2.883e+01 |J|=4.106e-01 (density)

# all            |x|=2.883e+01 |J|=4.106e-01 |h|=5.927e-05 |Jh|=1.414e-02 f=6.795e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -6.69879,  Residual: 0.000911968 
  1 TAO,  Function value: -6.69886,  Residual: 0.000510445 
  2 TAO,  Function value: -6.69888,  Residual: 6.82803e-05 
  3 TAO,  Function value: -6.69888,  Residual: 2.81017e-06 
  4 TAO,  Function value: -6.69888,  Residual: 9.48415e-09 
 11 TAO,  Function value: 6.59391,  Residual: 0.3927 
# TAO  11 (ITERATING)

# sub   0 [  5k] |x|=2.894e+01 |J|=3.927e-01 (density)

# all            |x|=2.894e+01 |J|=3.927e-01 |h|=5.074e-05 |Jh|=1.414e-02 f=6.594e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -6.50284,  Residual: 0.00110954 
  1 TAO,  Function value: -6.50293,  Residual: 0.00058612 
  2 TAO,  Function value: -6.50296,  Residual: 0.000140064 
  3 TAO,  Function value: -6.50296,  Residual: 1.209e-05 
  4 TAO,  Function value: -6.50296,  Residual: 2.61481e-07 
  5 TAO,  Function value: -6.50296,  Residual: 2.68517e-10 
 12 TAO,  Function value: 6.40286,  Residual: 0.376336 
# TAO  12 (ITERATING)

# sub   0 [  5k] |x|=2.905e+01 |J|=3.763e-01 (density)

# all            |x|=2.905e+01 |J|=3.763e-01 |h|=4.286e-05 |Jh|=1.414e-02 f=6.403e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -6.31562,  Residual: 0.00104357 
  1 TAO,  Function value: -6.3157,  Residual: 0.000527842 
  2 TAO,  Function value: -6.31573,  Residual: 2.71195e-05 
  3 TAO,  Function value: -6.31573,  Residual: 6.5681e-07 
  4 TAO,  Function value: -6.31573,  Residual: 3.48528e-09 
 13 TAO,  Function value: 6.22022,  Residual: 0.361227 
# TAO  13 (ITERATING)

# sub   0 [  5k] |x|=2.916e+01 |J|=3.612e-01 (density)

# all            |x|=2.916e+01 |J|=3.612e-01 |h|=3.662e-05 |Jh|=1.414e-02 f=6.220e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -6.13614,  Residual: 0.000742456 
  1 TAO,  Function value: -6.13618,  Residual: 0.000468291 
  2 TAO,  Function value: -6.13621,  Residual: 2.70642e-05 
  3 TAO,  Function value: -6.13621,  Residual: 3.44617e-07 
  4 TAO,  Function value: -6.13621,  Residual: 1.96914e-09 
 14 TAO,  Function value: 6.04435,  Residual: 0.347106 
# TAO  14 (ITERATING)

# sub   0 [  5k] |x|=2.928e+01 |J|=3.471e-01 (density)

# all            |x|=2.928e+01 |J|=3.471e-01 |h|=3.235e-05 |Jh|=1.414e-02 f=6.044e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -5.96312,  Residual: 0.00145277 
  1 TAO,  Function value: -5.9633,  Residual: 0.000956439 
  2 TAO,  Function value: -5.96341,  Residual: 0.000185721 
  3 TAO,  Function value: -5.96342,  Residual: 1.37905e-05 
  4 TAO,  Function value: -5.96342,  Residual: 2.61697e-07 
  5 TAO,  Function value: -5.96342,  Residual: 5.09513e-10 
 15 TAO,  Function value: 5.87469,  Residual: 0.333745 
# TAO  15 (ITERATING)

# sub   0 [  5k] |x|=2.941e+01 |J|=3.337e-01 (density)

# all            |x|=2.941e+01 |J|=3.337e-01 |h|=3.182e-05 |Jh|=1.414e-02 f=5.875e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -5.79651,  Residual: 0.000725048 
  1 TAO,  Function value: -5.79656,  Residual: 0.000470705 
  2 TAO,  Function value: -5.79659,  Residual: 3.79913e-05 
  3 TAO,  Function value: -5.79659,  Residual: 6.76119e-07 
  4 TAO,  Function value: -5.79659,  Residual: 5.80795e-09 
 16 TAO,  Function value: 5.71037,  Residual: 0.321231 
# TAO  16 (ITERATING)

# sub   0 [  5k] |x|=2.955e+01 |J|=3.212e-01 (density)

# all            |x|=2.955e+01 |J|=3.212e-01 |h|=2.634e-05 |Jh|=1.414e-02 f=5.710e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -5.63452,  Residual: 0.0013467 
  1 TAO,  Function value: -5.63468,  Residual: 0.000966763 
  2 TAO,  Function value: -5.63482,  Residual: 0.000205991 
  3 TAO,  Function value: -5.63483,  Residual: 2.51127e-05 
  4 TAO,  Function value: -5.63483,  Residual: 6.57993e-07 
  5 TAO,  Function value: -5.63483,  Residual: 1.33146e-09 
 17 TAO,  Function value: 5.55128,  Residual: 0.309285 
# TAO  17 (ITERATING)

# sub   0 [  5k] |x|=2.971e+01 |J|=3.093e-01 (density)

# all            |x|=2.971e+01 |J|=3.093e-01 |h|=2.417e-05 |Jh|=1.414e-02 f=5.551e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -5.47788,  Residual: 0.000947894 
  1 TAO,  Function value: -5.47795,  Residual: 0.000599971 
  2 TAO,  Function value: -5.47801,  Residual: 4.01508e-05 
  3 TAO,  Function value: -5.47801,  Residual: 1.81423e-06 
  4 TAO,  Function value: -5.47801,  Residual: 1.00724e-08 
 18 TAO,  Function value: 5.39677,  Residual: 0.297936 
# TAO  18 (ITERATING)

# sub   0 [  5k] |x|=2.986e+01 |J|=2.979e-01 (density)

# all            |x|=2.986e+01 |J|=2.979e-01 |h|=2.025e-05 |Jh|=1.414e-02 f=5.397e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -5.32528,  Residual: 0.0014213 
  1 TAO,  Function value: -5.32546,  Residual: 0.00105906 
  2 TAO,  Function value: -5.32558,  Residual: 0.000605489 
  3 TAO,  Function value: -5.32562,  Residual: 8.72802e-05 
  4 TAO,  Function value: -5.32562,  Residual: 3.66048e-07 
  5 TAO,  Function value: -5.32562,  Residual: 4.48046e-09 
 19 TAO,  Function value: 5.2467,  Residual: 0.287054 
# TAO  19 (ITERATING)

# sub   0 [  5k] |x|=3.003e+01 |J|=2.871e-01 (density)

# all            |x|=3.003e+01 |J|=2.871e-01 |h|=1.765e-05 |Jh|=1.414e-02 f=5.247e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -5.17723,  Residual: 0.00113815 
  1 TAO,  Function value: -5.17733,  Residual: 0.000712745 
  2 TAO,  Function value: -5.17741,  Residual: 7.15005e-05 
  3 TAO,  Function value: -5.17741,  Residual: 7.67163e-06 
  4 TAO,  Function value: -5.17741,  Residual: 5.52405e-08 
  5 TAO,  Function value: -5.17741,  Residual: 3.45043e-10 
 20 TAO,  Function value: 5.10047,  Residual: 0.276606 
# TAO  20 (ITERATING)

# sub   0 [  5k] |x|=3.020e+01 |J|=2.766e-01 (density)

# all            |x|=3.020e+01 |J|=2.766e-01 |h|=1.529e-05 |Jh|=1.414e-02 f=5.100e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -5.03263,  Residual: 0.00143543 
  1 TAO,  Function value: -5.03281,  Residual: 0.00107219 
  2 TAO,  Function value: -5.03296,  Residual: 0.000474029 
  3 TAO,  Function value: -5.03298,  Residual: 4.96267e-05 
  4 TAO,  Function value: -5.03299,  Residual: 1.76385e-07 
  5 TAO,  Function value: -5.03299,  Residual: 2.591e-10 
 21 TAO,  Function value: 4.95795,  Residual: 0.266525 
# TAO  21 (ITERATING)

# sub   0 [  5k] |x|=3.038e+01 |J|=2.665e-01 (density)

# all            |x|=3.038e+01 |J|=2.665e-01 |h|=1.432e-05 |Jh|=1.414e-02 f=4.958e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -4.89185,  Residual: 0.0013155 
  1 TAO,  Function value: -4.892,  Residual: 0.000886015 
  2 TAO,  Function value: -4.89209,  Residual: 0.000208756 
  3 TAO,  Function value: -4.89209,  Residual: 2.20308e-05 
  4 TAO,  Function value: -4.89209,  Residual: 2.2461e-06 
  5 TAO,  Function value: -4.89209,  Residual: 3.48075e-08 
 22 TAO,  Function value: 4.81878,  Residual: 0.256795 
# TAO  22 (ITERATING)

# sub   0 [  5k] |x|=3.058e+01 |J|=2.568e-01 (density)

# all            |x|=3.058e+01 |J|=2.568e-01 |h|=1.420e-05 |Jh|=1.414e-02 f=4.819e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -4.7541,  Residual: 0.00142696 
  1 TAO,  Function value: -4.75428,  Residual: 0.00104315 
  2 TAO,  Function value: -4.75443,  Residual: 0.000326085 
  3 TAO,  Function value: -4.75445,  Residual: 1.42984e-05 
  4 TAO,  Function value: -4.75445,  Residual: 3.57834e-06 
  5 TAO,  Function value: -4.75445,  Residual: 6.92314e-08 
  6 TAO,  Function value: -4.75445,  Residual: 1.4907e-10 
 23 TAO,  Function value: 4.68266,  Residual: 0.24735 
# TAO  23 (ITERATING)

# sub   0 [  5k] |x|=3.078e+01 |J|=2.473e-01 (density)

# all            |x|=3.078e+01 |J|=2.473e-01 |h|=1.322e-05 |Jh|=1.414e-02 f=4.683e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -4.61934,  Residual: 0.00141466 
  1 TAO,  Function value: -4.61952,  Residual: 0.000977327 
  2 TAO,  Function value: -4.61962,  Residual: 0.000282855 
  3 TAO,  Function value: -4.61963,  Residual: 2.48573e-05 
  4 TAO,  Function value: -4.61963,  Residual: 2.17757e-06 
  5 TAO,  Function value: -4.61963,  Residual: 3.26809e-08 
 24 TAO,  Function value: 4.54919,  Residual: 0.238197 
# TAO  24 (ITERATING)

# sub   0 [  5k] |x|=3.099e+01 |J|=2.382e-01 (density)

# all            |x|=3.099e+01 |J|=2.382e-01 |h|=1.470e-05 |Jh|=1.414e-02 f=4.549e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -4.48698,  Residual: 0.00132561 
  1 TAO,  Function value: -4.48713,  Residual: 0.000963505 
  2 TAO,  Function value: -4.4873,  Residual: 0.000101533 
  3 TAO,  Function value: -4.4873,  Residual: 1.82857e-05 
  4 TAO,  Function value: -4.4873,  Residual: 1.10558e-06 
  5 TAO,  Function value: -4.4873,  Residual: 4.54283e-08 
  6 TAO,  Function value: -4.4873,  Residual: 8.27293e-13 
 25 TAO,  Function value: 4.41805,  Residual: 0.229234 
# TAO  25 (ITERATING)

# sub   0 [  5k] |x|=3.121e+01 |J|=2.292e-01 (density)

# all            |x|=3.121e+01 |J|=2.292e-01 |h|=1.487e-05 |Jh|=1.414e-02 f=4.418e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -4.35686,  Residual: 0.00130815 
  1 TAO,  Function value: -4.35701,  Residual: 0.000961079 
  2 TAO,  Function value: -4.3571,  Residual: 0.000413534 
  3 TAO,  Function value: -4.35713,  Residual: 6.5083e-05 
  4 TAO,  Function value: -4.35713,  Residual: 8.58175e-06 
  5 TAO,  Function value: -4.35713,  Residual: 5.74809e-08 
  6 TAO,  Function value: -4.35713,  Residual: 9.53692e-10 
 26 TAO,  Function value: 4.28898,  Residual: 0.220484 
# TAO  26 (ITERATING)

# sub   0 [  5k] |x|=3.144e+01 |J|=2.205e-01 (density)

# all            |x|=3.144e+01 |J|=2.205e-01 |h|=1.645e-05 |Jh|=1.414e-02 f=4.289e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -4.22862,  Residual: 0.00139381 
  1 TAO,  Function value: -4.22879,  Residual: 0.00103989 
  2 TAO,  Function value: -4.22896,  Residual: 0.000170226 
  3 TAO,  Function value: -4.22897,  Residual: 8.47705e-06 
  4 TAO,  Function value: -4.22897,  Residual: 3.23689e-09 
 27 TAO,  Function value: 4.16186,  Residual: 0.211909 
# TAO  27 (ITERATING)

# sub   0 [  5k] |x|=3.168e+01 |J|=2.119e-01 (density)

# all            |x|=3.168e+01 |J|=2.119e-01 |h|=1.594e-05 |Jh|=1.414e-02 f=4.162e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -4.10243,  Residual: 0.00131612 
  1 TAO,  Function value: -4.10258,  Residual: 0.000931379 
  2 TAO,  Function value: -4.10268,  Residual: 0.000291273 
  3 TAO,  Function value: -4.10269,  Residual: 2.96072e-05 
  4 TAO,  Function value: -4.10269,  Residual: 5.5777e-06 
  5 TAO,  Function value: -4.10269,  Residual: 9.45211e-09 
 28 TAO,  Function value: 4.03666,  Residual: 0.203541 
# TAO  28 (ITERATING)

# sub   0 [  5k] |x|=3.193e+01 |J|=2.035e-01 (density)

# all            |x|=3.193e+01 |J|=2.035e-01 |h|=1.717e-05 |Jh|=1.414e-02 f=4.037e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -3.97801,  Residual: 0.00142575 
  1 TAO,  Function value: -3.97819,  Residual: 0.00106433 
  2 TAO,  Function value: -3.97835,  Residual: 0.000235346 
  3 TAO,  Function value: -3.97836,  Residual: 1.29387e-05 
  4 TAO,  Function value: -3.97836,  Residual: 8.03736e-07 
  5 TAO,  Function value: -3.97836,  Residual: 1.45162e-08 
 29 TAO,  Function value: 3.9134,  Residual: 0.195367 
# TAO  29 (ITERATING)

# sub   0 [  5k] |x|=3.219e+01 |J|=1.954e-01 (density)

# all            |x|=3.219e+01 |J|=1.954e-01 |h|=1.570e-05 |Jh|=1.414e-02 f=3.913e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -3.85572,  Residual: 0.00137142 
  1 TAO,  Function value: -3.85588,  Residual: 0.000944271 
  2 TAO,  Function value: -3.85599,  Residual: 0.000176175 
  3 TAO,  Function value: -3.85599,  Residual: 2.16821e-05 
  4 TAO,  Function value: -3.85599,  Residual: 1.93311e-06 
  5 TAO,  Function value: -3.85599,  Residual: 6.86106e-09 
 30 TAO,  Function value: 3.79221,  Residual: 0.187397 
# TAO  30 (ITERATING)

# sub   0 [  5k] |x|=3.247e+01 |J|=1.874e-01 (density)

# all            |x|=3.247e+01 |J|=1.874e-01 |h|=1.634e-05 |Jh|=1.414e-02 f=3.792e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -3.73543,  Residual: 0.00144662 
  1 TAO,  Function value: -3.73562,  Residual: 0.00106612 
  2 TAO,  Function value: -3.73576,  Residual: 0.000283569 
  3 TAO,  Function value: -3.73577,  Residual: 4.3391e-05 
  4 TAO,  Function value: -3.73577,  Residual: 9.98057e-07 
  5 TAO,  Function value: -3.73577,  Residual: 3.44081e-08 
 31 TAO,  Function value: 3.67323,  Residual: 0.17966 
# TAO  31 (ITERATING)

# sub   0 [  5k] |x|=3.275e+01 |J|=1.797e-01 (density)

# all            |x|=3.275e+01 |J|=1.797e-01 |h|=1.532e-05 |Jh|=1.414e-02 f=3.673e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -3.61751,  Residual: 0.00135395 
  1 TAO,  Function value: -3.61767,  Residual: 0.00102071 
  2 TAO,  Function value: -3.6178,  Residual: 0.000318326 
  3 TAO,  Function value: -3.61782,  Residual: 7.21785e-05 
  4 TAO,  Function value: -3.61782,  Residual: 1.74871e-05 
  5 TAO,  Function value: -3.61782,  Residual: 1.48281e-06 
  6 TAO,  Function value: -3.61782,  Residual: 1.24024e-09 
 32 TAO,  Function value: 3.5566,  Residual: 0.172158 
# TAO  32 (ITERATING)

# sub   0 [  5k] |x|=3.304e+01 |J|=1.722e-01 (density)

# all            |x|=3.304e+01 |J|=1.722e-01 |h|=1.534e-05 |Jh|=1.414e-02 f=3.557e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -3.50199,  Residual: 0.00135595 
  1 TAO,  Function value: -3.50215,  Residual: 0.00100736 
  2 TAO,  Function value: -3.50229,  Residual: 0.000241431 
  3 TAO,  Function value: -3.5023,  Residual: 6.09699e-05 
  4 TAO,  Function value: -3.5023,  Residual: 1.69754e-05 
  5 TAO,  Function value: -3.5023,  Residual: 8.44271e-07 
  6 TAO,  Function value: -3.5023,  Residual: 1.86429e-08 
 33 TAO,  Function value: 3.44248,  Residual: 0.164889 
# TAO  33 (ITERATING)

# sub   0 [  5k] |x|=3.334e+01 |J|=1.649e-01 (density)

# all            |x|=3.334e+01 |J|=1.649e-01 |h|=1.488e-05 |Jh|=1.414e-02 f=3.442e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -3.3891,  Residual: 0.00128554 
  1 TAO,  Function value: -3.38924,  Residual: 0.000966226 
  2 TAO,  Function value: -3.38937,  Residual: 0.000298897 
  3 TAO,  Function value: -3.38939,  Residual: 4.45146e-05 
  4 TAO,  Function value: -3.38939,  Residual: 1.45036e-05 
  5 TAO,  Function value: -3.38939,  Residual: 7.14474e-07 
  6 TAO,  Function value: -3.38939,  Residual: 1.27103e-08 
 34 TAO,  Function value: 3.33108,  Residual: 0.157862 
# TAO  34 (ITERATING)

# sub   0 [  5k] |x|=3.365e+01 |J|=1.579e-01 (density)

# all            |x|=3.365e+01 |J|=1.579e-01 |h|=1.447e-05 |Jh|=1.414e-02 f=3.331e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -3.279,  Residual: 0.00122682 
  1 TAO,  Function value: -3.27913,  Residual: 0.000958082 
  2 TAO,  Function value: -3.27924,  Residual: 0.000358677 
  3 TAO,  Function value: -3.27927,  Residual: 6.88343e-05 
  4 TAO,  Function value: -3.27927,  Residual: 1.37281e-05 
  5 TAO,  Function value: -3.27927,  Residual: 1.01186e-07 
  6 TAO,  Function value: -3.27927,  Residual: 4.1811e-09 
 35 TAO,  Function value: 3.22254,  Residual: 0.151083 
# TAO  35 (ITERATING)

# sub   0 [  5k] |x|=3.398e+01 |J|=1.511e-01 (density)

# all            |x|=3.398e+01 |J|=1.511e-01 |h|=1.401e-05 |Jh|=1.414e-02 f=3.223e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -3.17186,  Residual: 0.00113554 
  1 TAO,  Function value: -3.17197,  Residual: 0.000880167 
  2 TAO,  Function value: -3.17207,  Residual: 0.000324222 
  3 TAO,  Function value: -3.17209,  Residual: 3.73576e-05 
  4 TAO,  Function value: -3.17209,  Residual: 9.15009e-06 
  5 TAO,  Function value: -3.17209,  Residual: 2.04615e-07 
  6 TAO,  Function value: -3.17209,  Residual: 1.38828e-09 
 36 TAO,  Function value: 3.11701,  Residual: 0.144548 
# TAO  36 (ITERATING)

# sub   0 [  5k] |x|=3.431e+01 |J|=1.445e-01 (density)

# all            |x|=3.431e+01 |J|=1.445e-01 |h|=1.372e-05 |Jh|=1.414e-02 f=3.117e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -3.06779,  Residual: 0.00108721 
  1 TAO,  Function value: -3.06789,  Residual: 0.000794559 
  2 TAO,  Function value: -3.06801,  Residual: 8.14414e-05 
  3 TAO,  Function value: -3.06801,  Residual: 1.50644e-05 
  4 TAO,  Function value: -3.06801,  Residual: 5.94201e-07 
  5 TAO,  Function value: -3.06801,  Residual: 1.49755e-08 
 37 TAO,  Function value: 3.01464,  Residual: 0.138263 
# TAO  37 (ITERATING)

# sub   0 [  5k] |x|=3.466e+01 |J|=1.383e-01 (density)

# all            |x|=3.466e+01 |J|=1.383e-01 |h|=1.333e-05 |Jh|=1.414e-02 f=3.015e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -2.96693,  Residual: 0.00103504 
  1 TAO,  Function value: -2.96702,  Residual: 0.000764533 
  2 TAO,  Function value: -2.96712,  Residual: 0.000154888 
  3 TAO,  Function value: -2.96713,  Residual: 3.91056e-06 
  4 TAO,  Function value: -2.96713,  Residual: 1.1125e-06 
  5 TAO,  Function value: -2.96713,  Residual: 2.37045e-09 
 38 TAO,  Function value: 2.91551,  Residual: 0.13223 
# TAO  38 (ITERATING)

# sub   0 [  5k] |x|=3.501e+01 |J|=1.322e-01 (density)

# all            |x|=3.501e+01 |J|=1.322e-01 |h|=1.300e-05 |Jh|=1.414e-02 f=2.916e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -2.86932,  Residual: 0.00102124 
  1 TAO,  Function value: -2.86942,  Residual: 0.000784201 
  2 TAO,  Function value: -2.86952,  Residual: 0.000242796 
  3 TAO,  Function value: -2.86953,  Residual: 2.46726e-05 
  4 TAO,  Function value: -2.86953,  Residual: 8.58645e-06 
  5 TAO,  Function value: -2.86953,  Residual: 3.76103e-07 
  6 TAO,  Function value: -2.86953,  Residual: 2.07237e-08 
 39 TAO,  Function value: 2.81966,  Residual: 0.126454 
# TAO  39 (ITERATING)

# sub   0 [  5k] |x|=3.538e+01 |J|=1.265e-01 (density)

# all            |x|=3.538e+01 |J|=1.265e-01 |h|=1.253e-05 |Jh|=1.414e-02 f=2.820e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -2.77503,  Residual: 0.000963057 
  1 TAO,  Function value: -2.77511,  Residual: 0.000757009 
  2 TAO,  Function value: -2.7752,  Residual: 0.000310607 
  3 TAO,  Function value: -2.77522,  Residual: 1.78628e-05 
  4 TAO,  Function value: -2.77522,  Residual: 3.99496e-06 
  5 TAO,  Function value: -2.77522,  Residual: 3.88155e-08 
  6 TAO,  Function value: -2.77522,  Residual: 5.05642e-12 
 40 TAO,  Function value: 2.72709,  Residual: 0.12093 
# TAO  40 (ITERATING)

# sub   0 [  5k] |x|=3.575e+01 |J|=1.209e-01 (density)

# all            |x|=3.575e+01 |J|=1.209e-01 |h|=1.200e-05 |Jh|=1.414e-02 f=2.727e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -2.69591,  Residual: 0.000941255 
  1 TAO,  Function value: -2.69599,  Residual: 0.000745009 
  2 TAO,  Function value: -2.69612,  Residual: 9.09717e-06 
  3 TAO,  Function value: -2.69612,  Residual: 1.29867e-06 
  4 TAO,  Function value: -2.69612,  Residual: 1.33231e-08 
 41 TAO,  Function value: 2.66119,  Residual: 0.116919 
# TAO  41 (ITERATING)

# sub   0 [  5k] |x|=3.600e+01 |J|=1.169e-01 (density)

# all            |x|=3.600e+01 |J|=1.169e-01 |h|=1.184e-05 |Jh|=1.414e-02 f=2.661e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -2.63209,  Residual: 0.000139256 
  1 TAO,  Function value: -2.63209,  Residual: 0.000102669 
  2 TAO,  Function value: -2.63209,  Residual: 3.66118e-06 
  3 TAO,  Function value: -2.63209,  Residual: 1.96081e-07 
  4 TAO,  Function value: -2.63209,  Residual: 2.16384e-09 
 42 TAO,  Function value: 2.59889,  Residual: 0.113193 
# TAO  42 (ITERATING)

# sub   0 [  5k] |x|=3.623e+01 |J|=1.132e-01 (density)

# all            |x|=3.623e+01 |J|=1.132e-01 |h|=1.050e-05 |Jh|=1.414e-02 f=2.599e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -2.57157,  Residual: 4.13987e-05 
  1 TAO,  Function value: -2.57157,  Residual: 2.62598e-05 
  2 TAO,  Function value: -2.57157,  Residual: 9.85449e-07 
  3 TAO,  Function value: -2.57157,  Residual: 2.35266e-08 
 43 TAO,  Function value: 2.54015,  Residual: 0.109727 
# TAO  43 (ITERATING)

# sub   0 [  5k] |x|=3.647e+01 |J|=1.097e-01 (density)

# all            |x|=3.647e+01 |J|=1.097e-01 |h|=1.045e-05 |Jh|=1.414e-02 f=2.540e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -2.51404,  Residual: 0.000108448 
  1 TAO,  Function value: -2.51404,  Residual: 7.18645e-05 
  2 TAO,  Function value: -2.51405,  Residual: 4.1655e-06 
  3 TAO,  Function value: -2.51405,  Residual: 6.05648e-07 
  4 TAO,  Function value: -2.51405,  Residual: 9.90651e-11 
 44 TAO,  Function value: 2.48378,  Residual: 0.106424 
# TAO  44 (ITERATING)

# sub   0 [  5k] |x|=3.669e+01 |J|=1.064e-01 (density)

# all            |x|=3.669e+01 |J|=1.064e-01 |h|=9.894e-06 |Jh|=1.414e-02 f=2.484e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -2.45873,  Residual: 0.000278125 
  1 TAO,  Function value: -2.45874,  Residual: 0.000197419 
  2 TAO,  Function value: -2.45875,  Residual: 3.56925e-05 
  3 TAO,  Function value: -2.45875,  Residual: 1.1809e-07 
  4 TAO,  Function value: -2.45875,  Residual: 9.39959e-09 
 45 TAO,  Function value: 2.42959,  Residual: 0.103271 
# TAO  45 (ITERATING)

# sub   0 [  5k] |x|=3.692e+01 |J|=1.033e-01 (density)

# all            |x|=3.692e+01 |J|=1.033e-01 |h|=9.410e-06 |Jh|=1.414e-02 f=2.430e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -2.40561,  Residual: 0.000223961 
  1 TAO,  Function value: -2.40562,  Residual: 0.000154008 
  2 TAO,  Function value: -2.40562,  Residual: 4.772e-05 
  3 TAO,  Function value: -2.40562,  Residual: 5.3439e-06 
  4 TAO,  Function value: -2.40562,  Residual: 1.85683e-07 
  5 TAO,  Function value: -2.40562,  Residual: 4.6265e-09 
 46 TAO,  Function value: 2.37756,  Residual: 0.100265 
# TAO  46 (ITERATING)

# sub   0 [  5k] |x|=3.714e+01 |J|=1.003e-01 (density)

# all            |x|=3.714e+01 |J|=1.003e-01 |h|=9.052e-06 |Jh|=1.414e-02 f=2.378e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -2.3547,  Residual: 0.000232126 
  1 TAO,  Function value: -2.35471,  Residual: 0.000163065 
  2 TAO,  Function value: -2.35471,  Residual: 3.10448e-05 
  3 TAO,  Function value: -2.35471,  Residual: 2.54616e-06 
  4 TAO,  Function value: -2.35471,  Residual: 2.30816e-07 
  5 TAO,  Function value: -2.35471,  Residual: 4.13931e-09 
 47 TAO,  Function value: 2.32781,  Residual: 0.0974215 
# TAO  47 (ITERATING)

# sub   0 [  5k] |x|=3.736e+01 |J|=9.742e-02 (density)

# all            |x|=3.736e+01 |J|=9.742e-02 |h|=8.702e-06 |Jh|=1.414e-02 f=2.328e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -2.30567,  Residual: 0.00024435 
  1 TAO,  Function value: -2.30568,  Residual: 0.00017603 
  2 TAO,  Function value: -2.30568,  Residual: 3.9128e-05 
  3 TAO,  Function value: -2.30568,  Residual: 9.38036e-06 
  4 TAO,  Function value: -2.30568,  Residual: 4.56232e-07 
  5 TAO,  Function value: -2.30568,  Residual: 3.37334e-10 
 48 TAO,  Function value: 2.27955,  Residual: 0.0946866 
# TAO  48 (ITERATING)

# sub   0 [  5k] |x|=3.758e+01 |J|=9.469e-02 (density)

# all            |x|=3.758e+01 |J|=9.469e-02 |h|=8.355e-06 |Jh|=1.414e-02 f=2.280e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -2.25829,  Residual: 0.000202926 
  1 TAO,  Function value: -2.25829,  Residual: 0.000155211 
  2 TAO,  Function value: -2.2583,  Residual: 2.27169e-05 
  3 TAO,  Function value: -2.2583,  Residual: 1.38637e-05 
  4 TAO,  Function value: -2.2583,  Residual: 8.56163e-08 
  5 TAO,  Function value: -2.2583,  Residual: 5.56174e-09 
 49 TAO,  Function value: 2.23312,  Residual: 0.092079 
# TAO  49 (ITERATING)

# sub   0 [  5k] |x|=3.780e+01 |J|=9.208e-02 (density)

# all            |x|=3.780e+01 |J|=9.208e-02 |h|=8.124e-06 |Jh|=1.414e-02 f=2.233e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -2.21248,  Residual: 0.000185934 
  1 TAO,  Function value: -2.21248,  Residual: 0.000140728 
  2 TAO,  Function value: -2.21249,  Residual: 4.7911e-05 
  3 TAO,  Function value: -2.21249,  Residual: 1.24381e-05 
  4 TAO,  Function value: -2.21249,  Residual: 3.08549e-06 
  5 TAO,  Function value: -2.21249,  Residual: 1.25633e-08 
 50 TAO,  Function value: 2.18799,  Residual: 0.0895622 
# TAO  50 (ITERATING)

# sub   0 [  5k] |x|=3.802e+01 |J|=8.956e-02 (density)

# all            |x|=3.802e+01 |J|=8.956e-02 |h|=7.903e-06 |Jh|=1.414e-02 f=2.188e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -2.16795,  Residual: 0.000212636 
  1 TAO,  Function value: -2.16795,  Residual: 0.000175456 
  2 TAO,  Function value: -2.16796,  Residual: 6.84999e-05 
  3 TAO,  Function value: -2.16796,  Residual: 5.26689e-06 
  4 TAO,  Function value: -2.16796,  Residual: 7.24963e-07 
  5 TAO,  Function value: -2.16796,  Residual: 2.28157e-08 
  6 TAO,  Function value: -2.16796,  Residual: 6.33044e-13 
 51 TAO,  Function value: 2.14411,  Residual: 0.0871345 
# TAO  51 (ITERATING)

# sub   0 [  5k] |x|=3.823e+01 |J|=8.713e-02 (density)

# all            |x|=3.823e+01 |J|=8.713e-02 |h|=7.733e-06 |Jh|=1.414e-02 f=2.144e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -2.12454,  Residual: 0.00029393 
  1 TAO,  Function value: -2.12455,  Residual: 0.000245375 
  2 TAO,  Function value: -2.12456,  Residual: 8.43624e-05 
  3 TAO,  Function value: -2.12456,  Residual: 2.05084e-05 
  4 TAO,  Function value: -2.12456,  Residual: 1.67865e-05 
  5 TAO,  Function value: -2.12456,  Residual: 6.33139e-07 
  6 TAO,  Function value: -2.12456,  Residual: 2.71139e-08 
  7 TAO,  Function value: -2.12456,  Residual: 6.8095e-13 
 52 TAO,  Function value: 2.10124,  Residual: 0.0847674 
# TAO  52 (ITERATING)

# sub   0 [  5k] |x|=3.845e+01 |J|=8.477e-02 (density)

# all            |x|=3.845e+01 |J|=8.477e-02 |h|=7.580e-06 |Jh|=1.414e-02 f=2.101e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -2.08226,  Residual: 0.000264461 
  1 TAO,  Function value: -2.08227,  Residual: 0.000218394 
  2 TAO,  Function value: -2.08228,  Residual: 3.9721e-05 
  3 TAO,  Function value: -2.08228,  Residual: 9.50075e-06 
  4 TAO,  Function value: -2.08228,  Residual: 7.8253e-07 
  5 TAO,  Function value: -2.08228,  Residual: 4.36975e-08 
  6 TAO,  Function value: -2.08228,  Residual: 1.43733e-09 
 53 TAO,  Function value: 2.0596,  Residual: 0.0824842 
# TAO  53 (ITERATING)

# sub   0 [  5k] |x|=3.867e+01 |J|=8.248e-02 (density)

# all            |x|=3.867e+01 |J|=8.248e-02 |h|=7.495e-06 |Jh|=1.414e-02 f=2.060e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -2.04105,  Residual: 0.000227409 
  1 TAO,  Function value: -2.04106,  Residual: 0.000186368 
  2 TAO,  Function value: -2.04107,  Residual: 2.81134e-05 
  3 TAO,  Function value: -2.04107,  Residual: 1.02789e-05 
  4 TAO,  Function value: -2.04107,  Residual: 8.02402e-07 
  5 TAO,  Function value: -2.04107,  Residual: 4.47044e-08 
  6 TAO,  Function value: -2.04107,  Residual: 2.11867e-12 
 54 TAO,  Function value: 2.01888,  Residual: 0.0802661 
# TAO  54 (ITERATING)

# sub   0 [  5k] |x|=3.888e+01 |J|=8.027e-02 (density)

# all            |x|=3.888e+01 |J|=8.027e-02 |h|=7.427e-06 |Jh|=1.414e-02 f=2.019e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -2.00089,  Residual: 0.000162574 
  1 TAO,  Function value: -2.00089,  Residual: 0.00013477 
  2 TAO,  Function value: -2.0009,  Residual: 1.50817e-05 
  3 TAO,  Function value: -2.0009,  Residual: 2.40523e-06 
  4 TAO,  Function value: -2.0009,  Residual: 1.21162e-06 
  5 TAO,  Function value: -2.0009,  Residual: 1.9447e-10 
 55 TAO,  Function value: 1.97932,  Residual: 0.0781224 
# TAO  55 (ITERATING)

# sub   0 [  5k] |x|=3.909e+01 |J|=7.812e-02 (density)

# all            |x|=3.909e+01 |J|=7.812e-02 |h|=7.249e-06 |Jh|=1.414e-02 f=1.979e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.96166,  Residual: 0.00021868 
  1 TAO,  Function value: -1.96166,  Residual: 0.000180966 
  2 TAO,  Function value: -1.96167,  Residual: 3.14614e-05 
  3 TAO,  Function value: -1.96167,  Residual: 6.62405e-07 
  4 TAO,  Function value: -1.96167,  Residual: 4.66119e-08 
  5 TAO,  Function value: -1.96167,  Residual: 2.09814e-12 
 56 TAO,  Function value: 1.94048,  Residual: 0.0760314 
# TAO  56 (ITERATING)

# sub   0 [  5k] |x|=3.931e+01 |J|=7.603e-02 (density)

# all            |x|=3.931e+01 |J|=7.603e-02 |h|=7.112e-06 |Jh|=1.414e-02 f=1.940e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.92328,  Residual: 0.000206618 
  1 TAO,  Function value: -1.92328,  Residual: 0.000176286 
  2 TAO,  Function value: -1.92329,  Residual: 6.49287e-05 
  3 TAO,  Function value: -1.92329,  Residual: 2.81164e-05 
  4 TAO,  Function value: -1.92329,  Residual: 2.04956e-06 
  5 TAO,  Function value: -1.92329,  Residual: 4.49769e-08 
  6 TAO,  Function value: -1.92329,  Residual: 3.89633e-12 
 57 TAO,  Function value: 1.90263,  Residual: 0.0740083 
# TAO  57 (ITERATING)

# sub   0 [  5k] |x|=3.952e+01 |J|=7.401e-02 (density)

# all            |x|=3.952e+01 |J|=7.401e-02 |h|=6.991e-06 |Jh|=1.414e-02 f=1.903e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.88579,  Residual: 0.00020285 
  1 TAO,  Function value: -1.8858,  Residual: 0.000173118 
  2 TAO,  Function value: -1.8858,  Residual: 4.29121e-05 
  3 TAO,  Function value: -1.8858,  Residual: 1.86225e-05 
  4 TAO,  Function value: -1.8858,  Residual: 4.97978e-06 
  5 TAO,  Function value: -1.8858,  Residual: 7.85049e-07 
  6 TAO,  Function value: -1.8858,  Residual: 1.63313e-08 
 58 TAO,  Function value: 1.86558,  Residual: 0.0720474 
# TAO  58 (ITERATING)

# sub   0 [  5k] |x|=3.974e+01 |J|=7.205e-02 (density)

# all            |x|=3.974e+01 |J|=7.205e-02 |h|=6.848e-06 |Jh|=1.414e-02 f=1.866e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.84917,  Residual: 0.000172732 
  1 TAO,  Function value: -1.84918,  Residual: 0.000131619 
  2 TAO,  Function value: -1.84918,  Residual: 3.38562e-05 
  3 TAO,  Function value: -1.84918,  Residual: 6.82873e-06 
  4 TAO,  Function value: -1.84918,  Residual: 2.31843e-07 
  5 TAO,  Function value: -1.84918,  Residual: 1.73291e-08 
 59 TAO,  Function value: 1.82946,  Residual: 0.0701504 
# TAO  59 (ITERATING)

# sub   0 [  5k] |x|=3.996e+01 |J|=7.015e-02 (density)

# all            |x|=3.996e+01 |J|=7.015e-02 |h|=6.782e-06 |Jh|=1.414e-02 f=1.829e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.8134,  Residual: 0.000180053 
  1 TAO,  Function value: -1.8134,  Residual: 0.000158491 
  2 TAO,  Function value: -1.81341,  Residual: 8.24239e-05 
  3 TAO,  Function value: -1.81341,  Residual: 4.19042e-06 
  4 TAO,  Function value: -1.81341,  Residual: 1.70309e-07 
  5 TAO,  Function value: -1.81341,  Residual: 5.61487e-11 
 60 TAO,  Function value: 1.7941,  Residual: 0.0683118 
# TAO  60 (ITERATING)

# sub   0 [  5k] |x|=4.017e+01 |J|=6.831e-02 (density)

# all            |x|=4.017e+01 |J|=6.831e-02 |h|=6.678e-06 |Jh|=1.414e-02 f=1.794e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.78279,  Residual: 0.00154829 
  1 TAO,  Function value: -1.78301,  Residual: 0.00129455 
  2 TAO,  Function value: -1.78357,  Residual: 0.000200232 
  3 TAO,  Function value: -1.78359,  Residual: 5.0233e-05 
  4 TAO,  Function value: -1.78359,  Residual: 1.18022e-05 
  5 TAO,  Function value: -1.78359,  Residual: 6.81815e-07 
  6 TAO,  Function value: -1.78359,  Residual: 2.63259e-07 
  7 TAO,  Function value: -1.78359,  Residual: 1.92657e-11 
 61 TAO,  Function value: 1.77209,  Residual: 0.0671504 
# TAO  61 (ITERATING)

# sub   0 [  5k] |x|=4.032e+01 |J|=6.715e-02 (density)

# all            |x|=4.032e+01 |J|=6.715e-02 |h|=6.562e-06 |Jh|=1.414e-02 f=1.772e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.76197,  Residual: 8.0191e-05 
  1 TAO,  Function value: -1.76197,  Residual: 6.83215e-05 
  2 TAO,  Function value: -1.76198,  Residual: 2.65346e-05 
  3 TAO,  Function value: -1.76198,  Residual: 1.07891e-05 
  4 TAO,  Function value: -1.76198,  Residual: 6.01134e-06 
  5 TAO,  Function value: -1.76198,  Residual: 4.65773e-07 
  6 TAO,  Function value: -1.76198,  Residual: 1.2118e-07 
  7 TAO,  Function value: -1.76198,  Residual: 1.76959e-11 
 62 TAO,  Function value: 1.75088,  Residual: 0.0660633 
# TAO  62 (ITERATING)

# sub   0 [  5k] |x|=4.048e+01 |J|=6.606e-02 (density)

# all            |x|=4.048e+01 |J|=6.606e-02 |h|=5.125e-06 |Jh|=1.414e-02 f=1.751e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.74133,  Residual: 2.81451e-05 
  1 TAO,  Function value: -1.74133,  Residual: 1.67991e-05 
  2 TAO,  Function value: -1.74133,  Residual: 9.6108e-07 
  3 TAO,  Function value: -1.74133,  Residual: 2.36462e-08 
  4 TAO,  Function value: -1.74133,  Residual: 4.85065e-12 
 63 TAO,  Function value: 1.73082,  Residual: 0.0650637 
# TAO  63 (ITERATING)

# sub   0 [  5k] |x|=4.063e+01 |J|=6.506e-02 (density)

# all            |x|=4.063e+01 |J|=6.506e-02 |h|=5.367e-06 |Jh|=1.414e-02 f=1.731e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.72163,  Residual: 9.13654e-05 
  1 TAO,  Function value: -1.72163,  Residual: 7.93256e-05 
  2 TAO,  Function value: -1.72163,  Residual: 1.7178e-05 
  3 TAO,  Function value: -1.72163,  Residual: 2.46672e-06 
  4 TAO,  Function value: -1.72163,  Residual: 3.43308e-07 
  5 TAO,  Function value: -1.72163,  Residual: 2.12068e-10 
 64 TAO,  Function value: 1.7115,  Residual: 0.0641286 
# TAO  64 (ITERATING)

# sub   0 [  5k] |x|=4.079e+01 |J|=6.413e-02 (density)

# all            |x|=4.079e+01 |J|=6.413e-02 |h|=5.079e-06 |Jh|=1.414e-02 f=1.711e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.70273,  Residual: 0.000148575 
  1 TAO,  Function value: -1.70273,  Residual: 0.000131176 
  2 TAO,  Function value: -1.70274,  Residual: 2.33857e-05 
  3 TAO,  Function value: -1.70274,  Residual: 5.00851e-06 
  4 TAO,  Function value: -1.70274,  Residual: 7.9334e-07 
  5 TAO,  Function value: -1.70274,  Residual: 1.34996e-07 
  6 TAO,  Function value: -1.70274,  Residual: 2.01279e-11 
 65 TAO,  Function value: 1.69307,  Residual: 0.063255 
# TAO  65 (ITERATING)

# sub   0 [  5k] |x|=4.095e+01 |J|=6.326e-02 (density)

# all            |x|=4.095e+01 |J|=6.326e-02 |h|=4.893e-06 |Jh|=1.414e-02 f=1.693e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.68466,  Residual: 0.000151791 
  1 TAO,  Function value: -1.68466,  Residual: 0.00013436 
  2 TAO,  Function value: -1.68466,  Residual: 4.61139e-05 
  3 TAO,  Function value: -1.68466,  Residual: 4.00531e-06 
  4 TAO,  Function value: -1.68466,  Residual: 2.28559e-06 
  5 TAO,  Function value: -1.68466,  Residual: 1.75487e-07 
  6 TAO,  Function value: -1.68466,  Residual: 1.22137e-08 
 66 TAO,  Function value: 1.67537,  Residual: 0.0624307 
# TAO  66 (ITERATING)

# sub   0 [  5k] |x|=4.110e+01 |J|=6.243e-02 (density)

# all            |x|=4.110e+01 |J|=6.243e-02 |h|=4.750e-06 |Jh|=1.414e-02 f=1.675e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.66747,  Residual: 0.000107846 
  1 TAO,  Function value: -1.66747,  Residual: 9.60242e-05 
  2 TAO,  Function value: -1.66747,  Residual: 3.21511e-05 
  3 TAO,  Function value: -1.66747,  Residual: 1.02109e-05 
  4 TAO,  Function value: -1.66747,  Residual: 6.96004e-06 
  5 TAO,  Function value: -1.66747,  Residual: 9.5877e-07 
  6 TAO,  Function value: -1.66747,  Residual: 7.16705e-10 
 67 TAO,  Function value: 1.65871,  Residual: 0.0616741 
# TAO  67 (ITERATING)

# sub   0 [  5k] |x|=4.126e+01 |J|=6.167e-02 (density)

# all            |x|=4.126e+01 |J|=6.167e-02 |h|=4.632e-06 |Jh|=1.414e-02 f=1.659e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.65128,  Residual: 0.000198306 
  1 TAO,  Function value: -1.65128,  Residual: 0.000164595 
  2 TAO,  Function value: -1.65129,  Residual: 3.16666e-05 
  3 TAO,  Function value: -1.65129,  Residual: 1.4153e-06 
  4 TAO,  Function value: -1.65129,  Residual: 1.49869e-07 
  5 TAO,  Function value: -1.65129,  Residual: 3.52785e-11 
 68 TAO,  Function value: 1.64305,  Residual: 0.0609744 
# TAO  68 (ITERATING)

# sub   0 [  5k] |x|=4.140e+01 |J|=6.097e-02 (density)

# all            |x|=4.140e+01 |J|=6.097e-02 |h|=4.479e-06 |Jh|=1.414e-02 f=1.643e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.63645,  Residual: 0.000310312 
  1 TAO,  Function value: -1.63646,  Residual: 0.000253454 
  2 TAO,  Function value: -1.63648,  Residual: 2.10652e-05 
  3 TAO,  Function value: -1.63648,  Residual: 1.62295e-05 
  4 TAO,  Function value: -1.63648,  Residual: 9.56629e-07 
  5 TAO,  Function value: -1.63648,  Residual: 1.29932e-08 
 69 TAO,  Function value: 1.62919,  Residual: 0.0603506 
# TAO  69 (ITERATING)

# sub   0 [  5k] |x|=4.154e+01 |J|=6.035e-02 (density)

# all            |x|=4.154e+01 |J|=6.035e-02 |h|=4.309e-06 |Jh|=1.414e-02 f=1.629e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.62309,  Residual: 0.000213406 
  1 TAO,  Function value: -1.62309,  Residual: 0.000182045 
  2 TAO,  Function value: -1.6231,  Residual: 9.59766e-05 
  3 TAO,  Function value: -1.6231,  Residual: 1.91353e-05 
  4 TAO,  Function value: -1.6231,  Residual: 3.17211e-06 
  5 TAO,  Function value: -1.6231,  Residual: 5.78269e-07 
  6 TAO,  Function value: -1.6231,  Residual: 1.60484e-10 
 70 TAO,  Function value: 1.61633,  Residual: 0.0597848 
# TAO  70 (ITERATING)

# sub   0 [  5k] |x|=4.167e+01 |J|=5.978e-02 (density)

# all            |x|=4.167e+01 |J|=5.978e-02 |h|=4.021e-06 |Jh|=1.414e-02 f=1.616e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.6107,  Residual: 0.000145893 
  1 TAO,  Function value: -1.6107,  Residual: 0.000113282 
  2 TAO,  Function value: -1.6107,  Residual: 1.03425e-05 
  3 TAO,  Function value: -1.6107,  Residual: 5.80652e-07 
  4 TAO,  Function value: -1.6107,  Residual: 9.46201e-08 
  5 TAO,  Function value: -1.6107,  Residual: 7.26692e-12 
 71 TAO,  Function value: 1.60443,  Residual: 0.0592759 
# TAO  71 (ITERATING)

# sub   0 [  5k] |x|=4.180e+01 |J|=5.928e-02 (density)

# all            |x|=4.180e+01 |J|=5.928e-02 |h|=3.915e-06 |Jh|=1.414e-02 f=1.604e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.5991,  Residual: 0.000147204 
  1 TAO,  Function value: -1.5991,  Residual: 0.000122581 
  2 TAO,  Function value: -1.59911,  Residual: 2.46326e-05 
  3 TAO,  Function value: -1.59911,  Residual: 3.12504e-06 
  4 TAO,  Function value: -1.59911,  Residual: 2.11324e-07 
  5 TAO,  Function value: -1.59911,  Residual: 6.06208e-11 
 72 TAO,  Function value: 1.59317,  Residual: 0.0588018 
# TAO  72 (ITERATING)

# sub   0 [  5k] |x|=4.193e+01 |J|=5.880e-02 (density)

# all            |x|=4.193e+01 |J|=5.880e-02 |h|=3.795e-06 |Jh|=1.414e-02 f=1.593e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.58833,  Residual: 0.00010658 
  1 TAO,  Function value: -1.58833,  Residual: 8.45224e-05 
  2 TAO,  Function value: -1.58833,  Residual: 4.43781e-05 
  3 TAO,  Function value: -1.58833,  Residual: 8.88382e-06 
  4 TAO,  Function value: -1.58833,  Residual: 8.61767e-07 
  5 TAO,  Function value: -1.58833,  Residual: 4.11199e-07 
  6 TAO,  Function value: -1.58833,  Residual: 3.68288e-11 
 73 TAO,  Function value: 1.5829,  Residual: 0.0583813 
# TAO  73 (ITERATING)

# sub   0 [  5k] |x|=4.205e+01 |J|=5.838e-02 (density)

# all            |x|=4.205e+01 |J|=5.838e-02 |h|=3.685e-06 |Jh|=1.414e-02 f=1.583e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.57839,  Residual: 0.000114014 
  1 TAO,  Function value: -1.57839,  Residual: 9.31034e-05 
  2 TAO,  Function value: -1.5784,  Residual: 3.80451e-05 
  3 TAO,  Function value: -1.5784,  Residual: 2.83424e-06 
  4 TAO,  Function value: -1.5784,  Residual: 1.92275e-07 
  5 TAO,  Function value: -1.5784,  Residual: 5.90327e-11 
 74 TAO,  Function value: 1.57333,  Residual: 0.057997 
# TAO  74 (ITERATING)

# sub   0 [  5k] |x|=4.217e+01 |J|=5.800e-02 (density)

# all            |x|=4.217e+01 |J|=5.800e-02 |h|=3.520e-06 |Jh|=1.414e-02 f=1.573e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.56918,  Residual: 0.000128094 
  1 TAO,  Function value: -1.56918,  Residual: 8.78271e-05 
  2 TAO,  Function value: -1.56918,  Residual: 5.57798e-05 
  3 TAO,  Function value: -1.56918,  Residual: 4.06517e-05 
  4 TAO,  Function value: -1.56918,  Residual: 2.74815e-05 
  5 TAO,  Function value: -1.56918,  Residual: 6.02594e-06 
  6 TAO,  Function value: -1.56918,  Residual: 1.99797e-06 
  7 TAO,  Function value: -1.56918,  Residual: 8.14505e-08 
  8 TAO,  Function value: -1.56918,  Residual: 1.30489e-11 
 75 TAO,  Function value: 1.5645,  Residual: 0.0576432 
# TAO  75 (ITERATING)

# sub   0 [  5k] |x|=4.229e+01 |J|=5.764e-02 (density)

# all            |x|=4.229e+01 |J|=5.764e-02 |h|=3.398e-06 |Jh|=1.414e-02 f=1.564e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.56068,  Residual: 0.000139511 
  1 TAO,  Function value: -1.56068,  Residual: 0.00011243 
  2 TAO,  Function value: -1.56069,  Residual: 1.07268e-05 
  3 TAO,  Function value: -1.56069,  Residual: 3.20793e-08 
  4 TAO,  Function value: -1.56069,  Residual: 8.05056e-10 
 76 TAO,  Function value: 1.55636,  Residual: 0.0573193 
# TAO  76 (ITERATING)

# sub   0 [  5k] |x|=4.239e+01 |J|=5.732e-02 (density)

# all            |x|=4.239e+01 |J|=5.732e-02 |h|=3.223e-06 |Jh|=1.414e-02 f=1.556e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.55298,  Residual: 9.08157e-05 
  1 TAO,  Function value: -1.55298,  Residual: 7.186e-05 
  2 TAO,  Function value: -1.55299,  Residual: 2.56527e-05 
  3 TAO,  Function value: -1.55299,  Residual: 9.74075e-06 
  4 TAO,  Function value: -1.55299,  Residual: 4.08585e-06 
  5 TAO,  Function value: -1.55299,  Residual: 1.02554e-06 
  6 TAO,  Function value: -1.55299,  Residual: 1.22511e-07 
  7 TAO,  Function value: -1.55299,  Residual: 1.13016e-08 
 77 TAO,  Function value: 1.54913,  Residual: 0.0570306 
# TAO  77 (ITERATING)

# sub   0 [  5k] |x|=4.249e+01 |J|=5.703e-02 (density)

# all            |x|=4.249e+01 |J|=5.703e-02 |h|=3.078e-06 |Jh|=1.414e-02 f=1.549e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.54596,  Residual: 5.53681e-05 
  1 TAO,  Function value: -1.54597,  Residual: 4.29629e-05 
  2 TAO,  Function value: -1.54597,  Residual: 1.24775e-05 
  3 TAO,  Function value: -1.54597,  Residual: 4.63027e-06 
  4 TAO,  Function value: -1.54597,  Residual: 3.21052e-07 
  5 TAO,  Function value: -1.54597,  Residual: 6.2712e-08 
  6 TAO,  Function value: -1.54597,  Residual: 4.45728e-12 
 78 TAO,  Function value: 1.54234,  Residual: 0.0567591 
# TAO  78 (ITERATING)

# sub   0 [  5k] |x|=4.259e+01 |J|=5.676e-02 (density)

# all            |x|=4.259e+01 |J|=5.676e-02 |h|=2.954e-06 |Jh|=1.414e-02 f=1.542e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.5395,  Residual: 6.28024e-05 
  1 TAO,  Function value: -1.5395,  Residual: 5.43179e-05 
  2 TAO,  Function value: -1.5395,  Residual: 1.65992e-05 
  3 TAO,  Function value: -1.5395,  Residual: 5.19067e-06 
  4 TAO,  Function value: -1.5395,  Residual: 1.92683e-07 
  5 TAO,  Function value: -1.5395,  Residual: 1.76299e-10 
 79 TAO,  Function value: 1.53624,  Residual: 0.0565185 
# TAO  79 (ITERATING)

# sub   0 [  5k] |x|=4.268e+01 |J|=5.652e-02 (density)

# all            |x|=4.268e+01 |J|=5.652e-02 |h|=2.867e-06 |Jh|=1.414e-02 f=1.536e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.53359,  Residual: 7.08367e-05 
  1 TAO,  Function value: -1.53359,  Residual: 5.64992e-05 
  2 TAO,  Function value: -1.53359,  Residual: 8.23658e-07 
  3 TAO,  Function value: -1.53359,  Residual: 3.54519e-07 
  4 TAO,  Function value: -1.53359,  Residual: 7.31309e-11 
 80 TAO,  Function value: 1.53053,  Residual: 0.0562962 
# TAO  80 (ITERATING)

# sub   0 [  5k] |x|=4.277e+01 |J|=5.630e-02 (density)

# all            |x|=4.277e+01 |J|=5.630e-02 |h|=2.693e-06 |Jh|=1.414e-02 f=1.531e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.52817,  Residual: 4.54423e-05 
  1 TAO,  Function value: -1.52817,  Residual: 3.81677e-05 
  2 TAO,  Function value: -1.52817,  Residual: 2.3473e-05 
  3 TAO,  Function value: -1.52817,  Residual: 3.55327e-07 
  4 TAO,  Function value: -1.52817,  Residual: 1.04586e-08 
 81 TAO,  Function value: 1.52542,  Residual: 0.0560986 
# TAO  81 (ITERATING)

# sub   0 [  5k] |x|=4.286e+01 |J|=5.610e-02 (density)

# all            |x|=4.286e+01 |J|=5.610e-02 |h|=2.626e-06 |Jh|=1.414e-02 f=1.525e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.52326,  Residual: 3.14282e-05 
  1 TAO,  Function value: -1.52326,  Residual: 2.82715e-05 
  2 TAO,  Function value: -1.52326,  Residual: 1.27956e-05 
  3 TAO,  Function value: -1.52326,  Residual: 2.74854e-06 
  4 TAO,  Function value: -1.52326,  Residual: 7.82618e-07 
  5 TAO,  Function value: -1.52326,  Residual: 1.94099e-10 
 82 TAO,  Function value: 1.52073,  Residual: 0.0559175 
# TAO  82 (ITERATING)

# sub   0 [  5k] |x|=4.293e+01 |J|=5.592e-02 (density)

# all            |x|=4.293e+01 |J|=5.592e-02 |h|=2.486e-06 |Jh|=1.414e-02 f=1.521e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.51878,  Residual: 3.73516e-05 
  1 TAO,  Function value: -1.51878,  Residual: 1.80721e-05 
  2 TAO,  Function value: -1.51878,  Residual: 6.29112e-06 
  3 TAO,  Function value: -1.51878,  Residual: 9.93973e-07 
  4 TAO,  Function value: -1.51878,  Residual: 6.11673e-08 
  5 TAO,  Function value: -1.51878,  Residual: 8.45084e-09 
 83 TAO,  Function value: 1.51647,  Residual: 0.0557516 
# TAO  83 (ITERATING)

# sub   0 [  5k] |x|=4.300e+01 |J|=5.575e-02 (density)

# all            |x|=4.300e+01 |J|=5.575e-02 |h|=2.328e-06 |Jh|=1.414e-02 f=1.516e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.51461,  Residual: 4.09704e-05 
  1 TAO,  Function value: -1.51461,  Residual: 3.56817e-05 
  2 TAO,  Function value: -1.51461,  Residual: 3.65574e-06 
  3 TAO,  Function value: -1.51461,  Residual: 2.08507e-06 
  4 TAO,  Function value: -1.51461,  Residual: 9.02968e-09 
 84 TAO,  Function value: 1.51242,  Residual: 0.0555917 
# TAO  84 (ITERATING)

# sub   0 [  5k] |x|=4.307e+01 |J|=5.559e-02 (density)

# all            |x|=4.307e+01 |J|=5.559e-02 |h|=2.212e-06 |Jh|=1.414e-02 f=1.512e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.51068,  Residual: 4.06456e-06 
  1 TAO,  Function value: -1.51068,  Residual: 3.42231e-06 
  2 TAO,  Function value: -1.51068,  Residual: 1.17102e-07 
  3 TAO,  Function value: -1.51068,  Residual: 2.60806e-08 
  4 TAO,  Function value: -1.51068,  Residual: 4.76324e-13 
 85 TAO,  Function value: 1.50862,  Residual: 0.0554424 
# TAO  85 (ITERATING)

# sub   0 [  5k] |x|=4.314e+01 |J|=5.544e-02 (density)

# all            |x|=4.314e+01 |J|=5.544e-02 |h|=2.155e-06 |Jh|=1.414e-02 f=1.509e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.50694,  Residual: 8.89323e-06 
  1 TAO,  Function value: -1.50694,  Residual: 6.856e-06 
  2 TAO,  Function value: -1.50694,  Residual: 1.01459e-06 
  3 TAO,  Function value: -1.50694,  Residual: 1.86975e-07 
  4 TAO,  Function value: -1.50694,  Residual: 3.73368e-11 
 86 TAO,  Function value: 1.50494,  Residual: 0.0552984 
# TAO  86 (ITERATING)

# sub   0 [  5k] |x|=4.321e+01 |J|=5.530e-02 (density)

# all            |x|=4.321e+01 |J|=5.530e-02 |h|=2.088e-06 |Jh|=1.414e-02 f=1.505e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.50339,  Residual: 3.35311e-06 
  1 TAO,  Function value: -1.50339,  Residual: 2.24394e-06 
  2 TAO,  Function value: -1.50339,  Residual: 7.13013e-10 
 87 TAO,  Function value: 1.50152,  Residual: 0.0551632 
# TAO  87 (ITERATING)

# sub   0 [  5k] |x|=4.327e+01 |J|=5.516e-02 (density)

# all            |x|=4.327e+01 |J|=5.516e-02 |h|=2.040e-06 |Jh|=1.414e-02 f=1.502e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.50004,  Residual: 5.24664e-05 
  1 TAO,  Function value: -1.50004,  Residual: 4.18776e-05 
  2 TAO,  Function value: -1.50004,  Residual: 8.35333e-06 
  3 TAO,  Function value: -1.50004,  Residual: 5.76334e-07 
  4 TAO,  Function value: -1.50004,  Residual: 3.81681e-08 
  5 TAO,  Function value: -1.50004,  Residual: 2.89837e-12 
 88 TAO,  Function value: 1.49828,  Residual: 0.0550337 
# TAO  88 (ITERATING)

# sub   0 [  5k] |x|=4.333e+01 |J|=5.503e-02 (density)

# all            |x|=4.333e+01 |J|=5.503e-02 |h|=1.952e-06 |Jh|=1.414e-02 f=1.498e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.4969,  Residual: 3.77343e-05 
  1 TAO,  Function value: -1.4969,  Residual: 3.34644e-05 
  2 TAO,  Function value: -1.4969,  Residual: 1.305e-06 
  3 TAO,  Function value: -1.4969,  Residual: 3.73371e-07 
  4 TAO,  Function value: -1.4969,  Residual: 1.03385e-10 
 89 TAO,  Function value: 1.49527,  Residual: 0.0549116 
# TAO  89 (ITERATING)

# sub   0 [  5k] |x|=4.339e+01 |J|=5.491e-02 (density)

# all            |x|=4.339e+01 |J|=5.491e-02 |h|=1.887e-06 |Jh|=1.414e-02 f=1.495e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.49397,  Residual: 2.92257e-05 
  1 TAO,  Function value: -1.49397,  Residual: 2.45737e-05 
  2 TAO,  Function value: -1.49397,  Residual: 4.03904e-06 
  3 TAO,  Function value: -1.49397,  Residual: 6.99698e-07 
  4 TAO,  Function value: -1.49397,  Residual: 1.72558e-08 
  5 TAO,  Function value: -1.49397,  Residual: 3.64404e-12 
 90 TAO,  Function value: 1.49244,  Residual: 0.0547961 
# TAO  90 (ITERATING)

# sub   0 [  5k] |x|=4.344e+01 |J|=5.480e-02 (density)

# all            |x|=4.344e+01 |J|=5.480e-02 |h|=1.814e-06 |Jh|=1.414e-02 f=1.492e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.49126,  Residual: 2.77293e-06 
  1 TAO,  Function value: -1.49126,  Residual: 2.17219e-06 
  2 TAO,  Function value: -1.49126,  Residual: 5.98685e-09 
 91 TAO,  Function value: 1.48989,  Residual: 0.0546882 
# TAO  91 (ITERATING)

# sub   0 [  5k] |x|=4.349e+01 |J|=5.469e-02 (density)

# all            |x|=4.349e+01 |J|=5.469e-02 |h|=1.730e-06 |Jh|=1.414e-02 f=1.490e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.48876,  Residual: 3.05926e-05 
  1 TAO,  Function value: -1.48876,  Residual: 2.29509e-05 
  2 TAO,  Function value: -1.48876,  Residual: 6.21022e-07 
  3 TAO,  Function value: -1.48876,  Residual: 1.36519e-08 
 92 TAO,  Function value: 1.48746,  Residual: 0.0545835 
# TAO  92 (ITERATING)

# sub   0 [  5k] |x|=4.354e+01 |J|=5.458e-02 (density)

# all            |x|=4.354e+01 |J|=5.458e-02 |h|=1.623e-06 |Jh|=1.414e-02 f=1.487e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.48642,  Residual: 3.38799e-05 
  1 TAO,  Function value: -1.48642,  Residual: 2.16467e-05 
  2 TAO,  Function value: -1.48642,  Residual: 3.80934e-06 
  3 TAO,  Function value: -1.48642,  Residual: 1.2574e-07 
  4 TAO,  Function value: -1.48642,  Residual: 7.34038e-09 
 93 TAO,  Function value: 1.48522,  Residual: 0.0544848 
# TAO  93 (ITERATING)

# sub   0 [  5k] |x|=4.359e+01 |J|=5.448e-02 (density)

# all            |x|=4.359e+01 |J|=5.448e-02 |h|=1.573e-06 |Jh|=1.414e-02 f=1.485e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.48424,  Residual: 4.27713e-05 
  1 TAO,  Function value: -1.48424,  Residual: 2.95745e-05 
  2 TAO,  Function value: -1.48424,  Residual: 4.87176e-06 
  3 TAO,  Function value: -1.48424,  Residual: 1.50231e-07 
  4 TAO,  Function value: -1.48424,  Residual: 6.90882e-11 
 94 TAO,  Function value: 1.48311,  Residual: 0.0543892 
# TAO  94 (ITERATING)

# sub   0 [  5k] |x|=4.363e+01 |J|=5.439e-02 (density)

# all            |x|=4.363e+01 |J|=5.439e-02 |h|=1.463e-06 |Jh|=1.414e-02 f=1.483e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.48221,  Residual: 4.62298e-05 
  1 TAO,  Function value: -1.48221,  Residual: 2.76077e-05 
  2 TAO,  Function value: -1.48221,  Residual: 2.1366e-07 
  3 TAO,  Function value: -1.48221,  Residual: 4.69512e-09 
 95 TAO,  Function value: 1.48117,  Residual: 0.054303 
# TAO  95 (ITERATING)

# sub   0 [  5k] |x|=4.367e+01 |J|=5.430e-02 (density)

# all            |x|=4.367e+01 |J|=5.430e-02 |h|=1.421e-06 |Jh|=1.414e-02 f=1.481e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.48031,  Residual: 5.74815e-05 
  1 TAO,  Function value: -1.48031,  Residual: 3.12546e-05 
  2 TAO,  Function value: -1.48031,  Residual: 6.8086e-06 
  3 TAO,  Function value: -1.48031,  Residual: 3.21539e-07 
  4 TAO,  Function value: -1.48031,  Residual: 1.24937e-10 
 96 TAO,  Function value: 1.47931,  Residual: 0.054221 
# TAO  96 (ITERATING)

# sub   0 [  5k] |x|=4.371e+01 |J|=5.422e-02 (density)

# all            |x|=4.371e+01 |J|=5.422e-02 |h|=1.353e-06 |Jh|=1.414e-02 f=1.479e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.4785,  Residual: 7.13033e-05 
  1 TAO,  Function value: -1.4785,  Residual: 2.72517e-05 
  2 TAO,  Function value: -1.4785,  Residual: 3.06835e-06 
  3 TAO,  Function value: -1.4785,  Residual: 1.7484e-07 
  4 TAO,  Function value: -1.4785,  Residual: 9.48102e-09 
 97 TAO,  Function value: 1.47757,  Residual: 0.0541412 
# TAO  97 (ITERATING)

# sub   0 [  5k] |x|=4.375e+01 |J|=5.414e-02 (density)

# all            |x|=4.375e+01 |J|=5.414e-02 |h|=1.291e-06 |Jh|=1.414e-02 f=1.478e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.47677,  Residual: 4.80152e-05 
  1 TAO,  Function value: -1.47677,  Residual: 1.93967e-05 
  2 TAO,  Function value: -1.47677,  Residual: 5.97826e-07 
  3 TAO,  Function value: -1.47677,  Residual: 5.9306e-08 
  4 TAO,  Function value: -1.47677,  Residual: 2.14403e-12 
 98 TAO,  Function value: 1.47587,  Residual: 0.0540633 
# TAO  98 (ITERATING)

# sub   0 [  5k] |x|=4.378e+01 |J|=5.406e-02 (density)

# all            |x|=4.378e+01 |J|=5.406e-02 |h|=1.256e-06 |Jh|=1.414e-02 f=1.476e+00

  Iteration information for tao_mma_subsolver_ solve.
  0 TAO,  Function value: -1.47511,  Residual: 3.84967e-05 
  1 TAO,  Function value: -1.47511,  Residual: 1.42594e-05 
  2 TAO,  Function value: -1.47511,  Residual: 2.23998e-07 
  3 TAO,  Function value: -1.47511,  Residual: 4.54775e-09 
 99 TAO,  Function value: 1.47426,  Residual: 0.0539886 
# TAO  99 (ITERATING)

# sub   0 [  5k] |x|=4.382e+01 |J|=5.399e-02 (density)

# all            |x|=4.382e+01 |J|=5.399e-02 |h|=1.225e-06 |Jh|=1.414e-02 f=1.474e+00

if comm.size == 1:
    plotter.close()
    plotter_f.close()

with dolfinx.io.XDMFFile(comm, "topopt_simp/data.xdmf", "w") as file:
    file.write_mesh(mesh)
    for f in (ρ, ρ_f, u):
        file.write_function(f)

Results¶

We plot the density and filtered-density across MMA iterations.

The density field drives for a ‘black/white’ design, i.e. for a discrete split into phase ( $\rho=1$ ) and void ( $\rho=0$ ). This results (due to the nature of discretization) in heavy stair-casing to occur along the phase interface.

In contrast the filtered-density (computed from the corresponding density field in every iteration) smears our the interface and thus will never result in a ‘black/white’ design.

References¶

Bendsøe, M. P., & Kikuchi, N. (1988). Generating optimal topologies in structural design using a homogenization method. Computer Methods in Applied Mechanics and Engineering, 71(2), 197–224. 10.1016/0045-7825(88)90086-2
Lazarov, B. S., & Sigmund, O. (2010). Filters in topology optimization based on Helmholtz‐type differential equations. International Journal for Numerical Methods in Engineering, 86(6), 765–781. 10.1002/nme.3072
Svanberg, K. (1987). The method of moving asymptotes—a new method for structural optimization. International Journal for Numerical Methods in Engineering, 24(2), 359–373. 10.1002/nme.1620240207
Fleury, C. (1989). CONLIN: An efficient dual optimizer based on convex approximation concepts. Structural Optimization, 1(2), 81–89. 10.1007/bf01637664

Structural optimisation

Truss sizing optimisation

Structural optimisation

Topology optimisation of a 3D jet engine bracket