Topology optimisation of a 3D jet engine bracket
This demo solves the classic industrial optimisation problem of a General Electric (GE) jet engine bracket under multiple load cases, using the Solid Isotropic Material with Penalisation (SIMP), regularised with a Helmholtz filter.
In particular this demo emphasises
working with a
STEPgeometry,multiple load cases,
multi-step adjoint computations, and
the use of custom optimisation solvers.
The GE jet engine bracket challenge was a competition hosted by GrabCAD in 2013. The goal was to design a bracket that connects an engine to the frame of an aircraft, minimising its weight while remaining in the elastic regime under a set of load cases.
Geometry in the STEP file format is provided on the GrabCAD challenge page. We’ve rotated the
geometry to align the faces with coordinate axes and fragmented the geometry into multiple volumes
to disable the optimisation in the bolt areas. This preprocessing was done in
FreeCAD and is available at the dolfiny github repository
here.
The load cases are defined on the GrabCAD challenge page referenced above, but we include a summary here.
Figure 1:GE jet engine bracket challenge geometry and load cases
(source: https://
We start by importing the necessary modules, reading in the geometry, and generating a mesh with
gmsh. Unfortunately, boolean operations in FreeCAD do not always yield valid geometries, so we
need to call removeAllDuplicates before meshing. Otherwise, this results in disconnected
volumes. Removing duplicates renumbers the faces, so we colour the faces in FreeCAD and extract
the face tags based on the colour.
Figure 2:Coloured STEP file pre-processed in FreeCAD, used to identify physical groups.
There are five physical groups:
pin_left(green faces)pin_right(blue faces)bolt_faces(red faces)volume(the main volume for the optimisation)bolts(the bolt volumes where the density is fixed to 1).
The maximum mesh size is set to 3 mm, resulting in approximately 113k tetrahedral elements and 20k
vertices. To resolve the curved geometry, we’ve set Mesh.MeshSizeFromCurvature = 1, which
ensures that the mesh is refined based on the local curvature. More precisely, we have at least 8
elements per . We also use Netgen mesh optimisation to improve the mesh quality.
Additionally, STEP files usually have dimensions in mm, so we ask gmsh to interpret the geometry
in metres by setting Geometry.OCCTargetUnit = M.
from mpi4py import MPI
from petsc4py import PETSc
from petsc4py.PETSc import ScalarType # type: ignore
import dolfinx
import ufl
import gmsh
import numpy as np
import pyvista
import sympy.physics.units as syu
import dolfiny
from dolfiny.units import Quantity
comm = MPI.COMM_WORLD
if comm.rank == 0:
if not gmsh.isInitialized():
gmsh.initialize()
# Convert length units of `CAD` geometry to meters.
gmsh.option.set_string("Geometry.OCCTargetUnit", "M")
gmsh.open("ge_bracket_rotated_nonds.step")
gmsh.model.occ.removeAllDuplicates()
gmsh.model.occ.synchronize()
_, _, surfaces, volumes = (gmsh.model.occ.get_entities(d) for d in range(4))
colors = {
"red": (255, 0, 0, 255),
"green": (0, 255, 0, 255),
"blue": (0, 0, 255, 255),
"yellow": (255, 255, 0, 255),
}
def extract_tags_by_color(a, color):
return list(ai[1] for ai in a if gmsh.model.get_color(*ai) == color)
gmsh.model.addPhysicalGroup(3, [1], name="volume")
gmsh.model.addPhysicalGroup(3, [2, 3, 4, 5], name="bolts")
gmsh.model.addPhysicalGroup(
2, extract_tags_by_color(surfaces, colors["green"]), name="pin_left"
)
gmsh.model.addPhysicalGroup(
2, extract_tags_by_color(surfaces, colors["blue"]), name="pin_right"
)
gmsh.model.addPhysicalGroup(
2, extract_tags_by_color(surfaces, colors["red"]), name="bolt_faces"
)
gmsh.option.setNumber("Mesh.MeshSizeMax", 3e-3) # in meters
gmsh.option.setNumber("Mesh.MeshSizeFromCurvature", 1)
gmsh.option.setNumber("Mesh.MinimumElementsPerTwoPi", 8)
gmsh.option.setNumber("Mesh.OptimizeNetgen", 1)
gmsh.model.occ.synchronize()
gmsh.model.mesh.generate(3)
model = gmsh.model if comm.rank == 0 else None
mesh_data = dolfinx.io.gmsh.model_to_mesh(gmsh.model, comm, rank=0)
mesh = mesh_data.mesh
pin_left_tag = mesh_data.physical_groups["pin_left"].tag
pin_right_tag = mesh_data.physical_groups["pin_right"].tag
bolt_faces_tags = mesh_data.physical_groups["bolt_faces"].tag
volume_tag = mesh_data.physical_groups["volume"].tag
bolts_tag = mesh_data.physical_groups["bolts"].tagOutput
Info : Reading 'ge_bracket_rotated_nonds.step'...
Info : - Label 'Shapes/BooleanFragments' (3D)
Info : - Color (0.447059, 0.47451, 0.501961) (Surface)
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Info : - Color (0.447059, 0.47451, 0.501961) (Surface)
Info : - Color (0.447059, 0.47451, 0.501961) (Surface)
Info : - Color (0, 1, 0) (Surface)
Info : - Color (0, 0, 1) (Surface)
Info : - Color (1, 0, 0) (Surface)
Info : - Color (1, 0, 0) (Surface)
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Info : - Color (1, 0, 0) (Surface)
Info : - Color (1, 0, 0) (Surface)
Info : - Color (1, 0, 0) (Surface)
Info : Done reading 'ge_bracket_rotated_nonds.step'
Info : Meshing 1D...
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Info : Done meshing 2D (Wall 0.535637s, CPU 0.541627s)
Info : Meshing 3D...
Info : 3D Meshing 5 volumes with 1 connected component
Info : Tetrahedrizing 9120 nodes...
Info : Done tetrahedrizing 9128 nodes (Wall 0.110946s, CPU 0.107689s)
Info : Reconstructing mesh...
Info : - Creating surface mesh
Info : - Identifying boundary edges
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Info : Done reconstructing mesh (Wall 0.256788s, CPU 0.237178s)
Info : Found volume 1
Info : Found volume 5
Info : Found volume 3
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Info : Found volume 2
Info : It. 0 - 0 nodes created - worst tet radius 9.33399 (nodes removed 0 0)
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Info : It. 8000 - 7992 nodes created - worst tet radius 1.10477 (nodes removed 0 8)
Info : It. 8500 - 8492 nodes created - worst tet radius 1.08499 (nodes removed 0 8)
Info : It. 9000 - 8992 nodes created - worst tet radius 1.06665 (nodes removed 0 8)
Info : It. 9500 - 9492 nodes created - worst tet radius 1.04942 (nodes removed 0 8)
Info : It. 10000 - 9992 nodes created - worst tet radius 1.03263 (nodes removed 0 8)
Info : It. 10500 - 10492 nodes created - worst tet radius 1.01804 (nodes removed 0 8)
Info : It. 11000 - 10992 nodes created - worst tet radius 1.00323 (nodes removed 0 8)
Info : 3D refinement terminated (20238 nodes total):
Info : - 0 Delaunay cavities modified for star shapeness
Info : - 8 nodes could not be inserted
Info : - 99825 tetrahedra created in 0.559969 sec. (178268 tets/s)
Info : 0 node relocations
Info : Done meshing 3D (Wall 1.38857s, CPU 1.36883s)
Info : Optimizing mesh...
Info : Optimizing volume 1
Info : Optimization starts (volume = 0.00046011) with worst = 0.00518654 / average = 0.768788:
Info : 0.00 < quality < 0.10 : 232 elements
Info : 0.10 < quality < 0.20 : 659 elements
Info : 0.20 < quality < 0.30 : 1184 elements
Info : 0.30 < quality < 0.40 : 1884 elements
Info : 0.40 < quality < 0.50 : 3008 elements
Info : 0.50 < quality < 0.60 : 5135 elements
Info : 0.60 < quality < 0.70 : 11307 elements
Info : 0.70 < quality < 0.80 : 23342 elements
Info : 0.80 < quality < 0.90 : 34205 elements
Info : 0.90 < quality < 1.00 : 16964 elements
Info : 2004 edge swaps, 54 node relocations (volume = 0.00046011): worst = 0.125977 / average = 0.78143 (Wall 0.0470872s, CPU 0.047253s)
Info : 2024 edge swaps, 56 node relocations (volume = 0.00046011): worst = 0.125977 / average = 0.781527 (Wall 0.0591205s, CPU 0.059465s)
Info : No ill-shaped tets in the mesh :-)
Info : 0.00 < quality < 0.10 : 0 elements
Info : 0.10 < quality < 0.20 : 17 elements
Info : 0.20 < quality < 0.30 : 36 elements
Info : 0.30 < quality < 0.40 : 1864 elements
Info : 0.40 < quality < 0.50 : 2900 elements
Info : 0.50 < quality < 0.60 : 5050 elements
Info : 0.60 < quality < 0.70 : 11254 elements
Info : 0.70 < quality < 0.80 : 23547 elements
Info : 0.80 < quality < 0.90 : 34481 elements
Info : 0.90 < quality < 1.00 : 16957 elements
Info : Optimizing volume 2
Info : Optimization starts (volume = 8.77935e-07) with worst = 0.0313673 / average = 0.76375:
Info : 0.00 < quality < 0.10 : 4 elements
Info : 0.10 < quality < 0.20 : 5 elements
Info : 0.20 < quality < 0.30 : 3 elements
Info : 0.30 < quality < 0.40 : 1 elements
Info : 0.40 < quality < 0.50 : 12 elements
Info : 0.50 < quality < 0.60 : 28 elements
Info : 0.60 < quality < 0.70 : 53 elements
Info : 0.70 < quality < 0.80 : 123 elements
Info : 0.80 < quality < 0.90 : 202 elements
Info : 0.90 < quality < 1.00 : 52 elements
Info : 12 edge swaps, 0 node relocations (volume = 8.77935e-07): worst = 0.396093 / average = 0.780308 (Wall 0.000158051s, CPU 0.000271s)
Info : No ill-shaped tets in the mesh :-)
Info : 0.00 < quality < 0.10 : 0 elements
Info : 0.10 < quality < 0.20 : 0 elements
Info : 0.20 < quality < 0.30 : 0 elements
Info : 0.30 < quality < 0.40 : 1 elements
Info : 0.40 < quality < 0.50 : 10 elements
Info : 0.50 < quality < 0.60 : 29 elements
Info : 0.60 < quality < 0.70 : 53 elements
Info : 0.70 < quality < 0.80 : 127 elements
Info : 0.80 < quality < 0.90 : 201 elements
Info : 0.90 < quality < 1.00 : 51 elements
Info : Optimizing volume 3
Info : Optimization starts (volume = 9.30777e-07) with worst = 0.0313673 / average = 0.771053:
Info : 0.00 < quality < 0.10 : 2 elements
Info : 0.10 < quality < 0.20 : 1 elements
Info : 0.20 < quality < 0.30 : 0 elements
Info : 0.30 < quality < 0.40 : 4 elements
Info : 0.40 < quality < 0.50 : 7 elements
Info : 0.50 < quality < 0.60 : 25 elements
Info : 0.60 < quality < 0.70 : 60 elements
Info : 0.70 < quality < 0.80 : 125 elements
Info : 0.80 < quality < 0.90 : 172 elements
Info : 0.90 < quality < 1.00 : 51 elements
Info : 3 edge swaps, 0 node relocations (volume = 9.30777e-07): worst = 0.301081 / average = 0.775766 (Wall 6.48131e-05s, CPU 6.7e-05s)
Info : No ill-shaped tets in the mesh :-)
Info : 0.00 < quality < 0.10 : 0 elements
Info : 0.10 < quality < 0.20 : 0 elements
Info : 0.20 < quality < 0.30 : 0 elements
Info : 0.30 < quality < 0.40 : 4 elements
Info : 0.40 < quality < 0.50 : 6 elements
Info : 0.50 < quality < 0.60 : 27 elements
Info : 0.60 < quality < 0.70 : 60 elements
Info : 0.70 < quality < 0.80 : 123 elements
Info : 0.80 < quality < 0.90 : 173 elements
Info : 0.90 < quality < 1.00 : 51 elements
Info : Optimizing volume 4
Info : Optimization starts (volume = 9.30618e-07) with worst = 0.0313673 / average = 0.772855:
Info : 0.00 < quality < 0.10 : 2 elements
Info : 0.10 < quality < 0.20 : 1 elements
Info : 0.20 < quality < 0.30 : 0 elements
Info : 0.30 < quality < 0.40 : 4 elements
Info : 0.40 < quality < 0.50 : 7 elements
Info : 0.50 < quality < 0.60 : 21 elements
Info : 0.60 < quality < 0.70 : 57 elements
Info : 0.70 < quality < 0.80 : 134 elements
Info : 0.80 < quality < 0.90 : 165 elements
Info : 0.90 < quality < 1.00 : 52 elements
Info : 3 edge swaps, 0 node relocations (volume = 9.30618e-07): worst = 0.300253 / average = 0.777513 (Wall 8.66032e-05s, CPU 8.4e-05s)
Info : No ill-shaped tets in the mesh :-)
Info : 0.00 < quality < 0.10 : 0 elements
Info : 0.10 < quality < 0.20 : 0 elements
Info : 0.20 < quality < 0.30 : 0 elements
Info : 0.30 < quality < 0.40 : 4 elements
Info : 0.40 < quality < 0.50 : 6 elements
Info : 0.50 < quality < 0.60 : 23 elements
Info : 0.60 < quality < 0.70 : 57 elements
Info : 0.70 < quality < 0.80 : 133 elements
Info : 0.80 < quality < 0.90 : 165 elements
Info : 0.90 < quality < 1.00 : 52 elements
Info : Optimizing volume 5
Info : Optimization starts (volume = 9.24493e-07) with worst = 0.012926 / average = 0.743819:
Info : 0.00 < quality < 0.10 : 6 elements
Info : 0.10 < quality < 0.20 : 7 elements
Info : 0.20 < quality < 0.30 : 7 elements
Info : 0.30 < quality < 0.40 : 2 elements
Info : 0.40 < quality < 0.50 : 18 elements
Info : 0.50 < quality < 0.60 : 13 elements
Info : 0.60 < quality < 0.70 : 98 elements
Info : 0.70 < quality < 0.80 : 153 elements
Info : 0.80 < quality < 0.90 : 180 elements
Info : 0.90 < quality < 1.00 : 48 elements
Info : 20 edge swaps, 0 node relocations (volume = 9.24493e-07): worst = 0.31961 / average = 0.769153 (Wall 0.000231315s, CPU 0.000287s)
Info : No ill-shaped tets in the mesh :-)
Info : 0.00 < quality < 0.10 : 0 elements
Info : 0.10 < quality < 0.20 : 0 elements
Info : 0.20 < quality < 0.30 : 0 elements
Info : 0.30 < quality < 0.40 : 3 elements
Info : 0.40 < quality < 0.50 : 14 elements
Info : 0.50 < quality < 0.60 : 15 elements
Info : 0.60 < quality < 0.70 : 88 elements
Info : 0.70 < quality < 0.80 : 165 elements
Info : 0.80 < quality < 0.90 : 180 elements
Info : 0.90 < quality < 1.00 : 48 elements
Info : Done optimizing mesh (Wall 0.197028s, CPU 0.198819s)
Info : Optimizing mesh (Netgen)...
Info : Optimizing volume 1
Info : CalcLocalH: 19935 Points 96122 Elements 17654 Surface Elements
Info : Remove Illegal Elements
Info : 2938 illegal tets
Info : SplitImprove
Info : badmax = 66.5369
Info : 541 splits performed
Info : SwapImprove
Info : 484 swaps performed
Info : SwapImprove2
Info : 20 swaps performed
Info : 1443 illegal tets
Info : SplitImprove
Info : badmax = 133.544
Info : 366 splits performed
Info : SwapImprove
Info : 97 swaps performed
Info : SwapImprove2
Info : 16 swaps performed
Info : 457 illegal tets
Info : SplitImprove
Info : badmax = 9336.96
Info : 125 splits performed
Info : SwapImprove
Info : 24 swaps performed
Info : SwapImprove2
Info : 5 swaps performed
Info : 129 illegal tets
Info : SplitImprove
Info : badmax = 913.977
Info : 35 splits performed
Info : SwapImprove
Info : 5 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : 42 illegal tets
Info : SplitImprove
Info : badmax = 913.977
Info : 12 splits performed
Info : SwapImprove
Info : 1 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : 13 illegal tets
Info : SplitImprove
Info : badmax = 913.977
Info : 5 splits performed
Info : SwapImprove
Info : 0 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : 0 illegal tets
Info : Volume Optimization
Info : CombineImprove
Info : 1067 elements combined
Info : ImproveMesh
Info : Total badness = 138325
Info : Total badness = 131493
Info : SplitImprove
Info : badmax = 126.983
Info : 1 splits performed
Info : ImproveMesh
Info : Total badness = 131496
Info : Total badness = 130643
Info : SwapImprove
Info : 5732 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : ImproveMesh
Info : Total badness = 120595
Info : Total badness = 117516
Info : CombineImprove
Info : 200 elements combined
Info : ImproveMesh
Info : Total badness = 115735
Info : Total badness = 115350
Info : SplitImprove
Info : badmax = 52.6004
Info : 0 splits performed
Info : ImproveMesh
Info : Total badness = 115350
Info : Total badness = 115290
Info : SwapImprove
Info : 1013 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : ImproveMesh
Info : Total badness = 114425
Info : Total badness = 113671
Info : CombineImprove
Info : 55 elements combined
Info : ImproveMesh
Info : Total badness = 113191
Info : Total badness = 113090
Info : SplitImprove
Info : badmax = 34.0974
Info : 1 splits performed
Info : ImproveMesh
Info : Total badness = 113095
Info : Total badness = 113076
Info : SwapImprove
Info : 362 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : ImproveMesh
Info : Total badness = 112836
Info : Total badness = 112585
Info : Optimizing volume 2
Info : CalcLocalH: 188 Points 472 Elements 376 Surface Elements
Info : Remove Illegal Elements
Info : 180 illegal tets
Info : SplitImprove
Info : badmax = 8.90249
Info : 25 splits performed
Info : SwapImprove
Info : 31 swaps performed
Info : SwapImprove2
Info : 2 swaps performed
Info : 114 illegal tets
Info : SplitImprove
Info : badmax = 164.379
Info : 24 splits performed
Info : SwapImprove
Info : 10 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : 39 illegal tets
Info : SplitImprove
Info : badmax = 164.379
Info : 10 splits performed
Info : SwapImprove
Info : 2 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : 7 illegal tets
Info : SplitImprove
Info : badmax = 164.379
Info : 2 splits performed
Info : SwapImprove
Info : 0 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : 3 illegal tets
Info : SplitImprove
Info : badmax = 164.379
Info : 1 splits performed
Info : SwapImprove
Info : 0 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : 0 illegal tets
Info : Volume Optimization
Info : CombineImprove
Info : 18 elements combined
Info : ImproveMesh
Info : Total badness = 1117.7
Info : Total badness = 1084.49
Info : SplitImprove
Info : badmax = 15.5562
Info : 0 splits performed
Info : ImproveMesh
Info : Total badness = 1084.49
Info : Total badness = 1083.6
Info : SwapImprove
Info : 38 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : ImproveMesh
Info : Total badness = 1022.89
Info : Total badness = 1006.75
Info : CombineImprove
Info : 1 elements combined
Info : ImproveMesh
Info : Total badness = 999.392
Info : Total badness = 999.09
Info : SplitImprove
Info : badmax = 6.94992
Info : 0 splits performed
Info : ImproveMesh
Info : Total badness = 999.09
Info : Total badness = 999.092
Info : SwapImprove
Info : 5 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : ImproveMesh
Info : Total badness = 991.706
Info : Total badness = 988.91
Info : CombineImprove
Info : 0 elements combined
Info : ImproveMesh
Info : Total badness = 988.91
Info : Total badness = 988.911
Info : SplitImprove
Info : badmax = 6.94992
Info : 0 splits performed
Info : ImproveMesh
Info : Total badness = 988.911
Info : Total badness = 988.912
Info : SwapImprove
Info : 1 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : ImproveMesh
Info : Total badness = 988.912
Info : Total badness = 988.209
Info : Optimizing volume 3
Info : CalcLocalH: 181 Points 444 Elements 362 Surface Elements
Info : Remove Illegal Elements
Info : 175 illegal tets
Info : SplitImprove
Info : badmax = 14.6469
Info : 22 splits performed
Info : SwapImprove
Info : 23 swaps performed
Info : SwapImprove2
Info : 1 swaps performed
Info : 124 illegal tets
Info : SplitImprove
Info : badmax = 37.585
Info : 22 splits performed
Info : SwapImprove
Info : 12 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : 71 illegal tets
Info : SplitImprove
Info : badmax = 69.7701
Info : 21 splits performed
Info : SwapImprove
Info : 2 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : 12 illegal tets
Info : SplitImprove
Info : badmax = 69.7701
Info : 3 splits performed
Info : SwapImprove
Info : 0 swaps performed
Info : SwapImprove2
Info : 1 swaps performed
Info : 3 illegal tets
Info : SplitImprove
Info : badmax = 69.7701
Info : 1 splits performed
Info : SwapImprove
Info : 0 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : 0 illegal tets
Info : Volume Optimization
Info : CombineImprove
Info : 21 elements combined
Info : ImproveMesh
Info : Total badness = 1167.62
Info : Total badness = 1129.75
Info : SplitImprove
Info : badmax = 28.2773
Info : 0 splits performed
Info : ImproveMesh
Info : Total badness = 1129.75
Info : Total badness = 1128.43
Info : SwapImprove
Info : 49 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : ImproveMesh
Info : Total badness = 1060.25
Info : Total badness = 1022.25
Info : CombineImprove
Info : 5 elements combined
Info : ImproveMesh
Info : Total badness = 980.653
Info : Total badness = 979.16
Info : SplitImprove
Info : badmax = 6.09611
Info : 0 splits performed
Info : ImproveMesh
Info : Total badness = 979.16
Info : Total badness = 979.086
Info : SwapImprove
Info : 12 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : ImproveMesh
Info : Total badness = 972.064
Info : Total badness = 965.482
Info : CombineImprove
Info : 0 elements combined
Info : ImproveMesh
Info : Total badness = 965.482
Info : Total badness = 965.445
Info : SplitImprove
Info : badmax = 6.85922
Info : 0 splits performed
Info : ImproveMesh
Info : Total badness = 965.445
Info : Total badness = 965.443
Info : SwapImprove
Info : 8 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : ImproveMesh
Info : Total badness = 961.682
Info : Total badness = 957.183
Info : Optimizing volume 4
Info : CalcLocalH: 180 Points 440 Elements 360 Surface Elements
Info : Remove Illegal Elements
Info : 176 illegal tets
Info : SplitImprove
Info : badmax = 15.533
Info : 20 splits performed
Info : SwapImprove
Info : 15 swaps performed
Info : SwapImprove2
Info : 1 swaps performed
Info : 134 illegal tets
Info : SplitImprove
Info : badmax = 123.935
Info : 22 splits performed
Info : SwapImprove
Info : 8 swaps performed
Info : SwapImprove2
Info : 2 swaps performed
Info : 86 illegal tets
Info : SplitImprove
Info : badmax = 123.935
Info : 22 splits performed
Info : SwapImprove
Info : 7 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : 31 illegal tets
Info : SplitImprove
Info : badmax = 123.935
Info : 8 splits performed
Info : SwapImprove
Info : 2 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : 9 illegal tets
Info : SplitImprove
Info : badmax = 123.935
Info : 3 splits performed
Info : SwapImprove
Info : 0 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : 0 illegal tets
Info : Volume Optimization
Info : CombineImprove
Info : 28 elements combined
Info : ImproveMesh
Info : Total badness = 1172.13
Info : Total badness = 1129.58
Info : SplitImprove
Info : badmax = 38.3281
Info : 1 splits performed
Info : ImproveMesh
Info : Total badness = 1133.94
Info : Total badness = 1126.29
Info : SwapImprove
Info : 42 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : ImproveMesh
Info : Total badness = 1077.07
Info : Total badness = 1052.3
Info : CombineImprove
Info : 6 elements combined
Info : ImproveMesh
Info : Total badness = 986.301
Info : Total badness = 982.417
Info : SplitImprove
Info : badmax = 16.0098
Info : 0 splits performed
Info : ImproveMesh
Info : Total badness = 982.417
Info : Total badness = 982.056
Info : SwapImprove
Info : 11 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : ImproveMesh
Info : Total badness = 964.32
Info : Total badness = 953.466
Info : CombineImprove
Info : 1 elements combined
Info : ImproveMesh
Info : Total badness = 946.49
Info : Total badness = 946.027
Info : SplitImprove
Info : badmax = 8.0708
Info : 0 splits performed
Info : ImproveMesh
Info : Total badness = 946.027
Info : Total badness = 946.018
Info : SwapImprove
Info : 4 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : ImproveMesh
Info : Total badness = 938.657
Info : Total badness = 936.104
Info : Optimizing volume 5
Info : CalcLocalH: 202 Points 513 Elements 404 Surface Elements
Info : Remove Illegal Elements
Info : 194 illegal tets
Info : SplitImprove
Info : badmax = 13.6287
Info : 23 splits performed
Info : SwapImprove
Info : 42 swaps performed
Info : SwapImprove2
Info : 2 swaps performed
Info : 136 illegal tets
Info : SplitImprove
Info : badmax = 708.422
Info : 25 splits performed
Info : SwapImprove
Info : 9 swaps performed
Info : SwapImprove2
Info : 3 swaps performed
Info : 68 illegal tets
Info : SplitImprove
Info : badmax = 708.422
Info : 14 splits performed
Info : SwapImprove
Info : 5 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : 31 illegal tets
Info : SplitImprove
Info : badmax = 708.422
Info : 6 splits performed
Info : SwapImprove
Info : 1 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : 22 illegal tets
Info : SplitImprove
Info : badmax = 708.422
Info : 5 splits performed
Info : SwapImprove
Info : 3 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : 6 illegal tets
Info : SplitImprove
Info : badmax = 708.422
Info : 2 splits performed
Info : SwapImprove
Info : 0 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : 0 illegal tets
Info : Volume Optimization
Info : CombineImprove
Info : 19 elements combined
Info : ImproveMesh
Info : Total badness = 1410.27
Info : Total badness = 1335.21
Info : SplitImprove
Info : badmax = 18.5964
Info : 0 splits performed
Info : ImproveMesh
Info : Total badness = 1335.21
Info : Total badness = 1325.79
Info : SwapImprove
Info : 50 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : ImproveMesh
Info : Total badness = 1252.57
Info : Total badness = 1219.42
Info : CombineImprove
Info : 4 elements combined
Info : ImproveMesh
Info : Total badness = 1166.01
Info : Total badness = 1162.49
Info : SplitImprove
Info : badmax = 6.57123
Info : 0 splits performed
Info : ImproveMesh
Info : Total badness = 1162.49
Info : Total badness = 1162.37
Info : SwapImprove
Info : 19 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : ImproveMesh
Info : Total badness = 1151.01
Info : Total badness = 1140.9
Info : CombineImprove
Info : 2 elements combined
Info : ImproveMesh
Info : Total badness = 1122.58
Info : Total badness = 1121.86
Info : SplitImprove
Info : badmax = 6.57123
Info : 0 splits performed
Info : ImproveMesh
Info : Total badness = 1121.86
Info : Total badness = 1121.86
Info : SwapImprove
Info : 8 swaps performed
Info : SwapImprove2
Info : 0 swaps performed
Info : ImproveMesh
Info : Total badness = 1116.18
Info : Total badness = 1112.33
Info : Done optimizing mesh (Wall 6.03324s, CPU 6.05532s)
Info : 20179 nodes 113212 elements
if comm.size == 1:
grid = pyvista.UnstructuredGrid(*dolfinx.plot.vtk_mesh(mesh))
plotter = pyvista.Plotter(off_screen=True, theme=dolfiny.pyvista.theme)
plotter.add_mesh(
grid, show_edges=True, color="white", line_width=dolfiny.pyvista.pixels // 1000
)
plotter.show_axes()
plotter.camera.elevation = 30
plotter.show()
plotter.close()
plotter.deep_clean()
State problem (linear elasticity)¶
The next step is to define the elasticity problem. We consider a linear isotropic material model, together with the classic SIMP penalisation Bendsøe (1989), which interpolates the Young’s modulus as
where is Young’s modulus of the solid material (associated with the phase ), and is the penalty factor. In this demo we set .
Lower bound on the density, i.e. the density of the void phase is enforced during the optimisation (see below). The lower bound guarantees well-posedness of the elasticity problem. Young’s modulus of the solid phase follows the material specification of Ti-6Al-4V, which is a common aerospace alloy. We’ve taken the approximate value from Ti-6Al-4V. The Poisson’s ratio is set to .
We derive the -th state problem also as a minimisation problem, where the total potential energy is minimised. The total potential energy is defined as
where is the stress tensor, is the small strain tensor, are the applied forces on the boundary parts , and is the displacement field for the -th load condition. The first term is the elastic energy stored in the deformed body, and the second term is the work done by the external forces, which at equilibrium coincides with the compliance.
tdim = mesh.topology.dim
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")
ρ_f.x.array[:] = 1.0
ρ_min = np.float64(1e-3)
penalty = 3
E0 = Quantity(mesh, 110, syu.giga * syu.pascal, "E0") # Ti-6Al-4V
E = ρ_f**penalty * E0
nu = 0.31
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)
ds = ufl.Measure("ds", domain=mesh, subdomain_data=mesh_data.facet_tags)
dx = ufl.Measure("dx", domain=mesh, subdomain_data=mesh_data.cell_tags)
def compliance(load_condition):
u = load_condition["u"]
return sum(
[ufl.inner(f, u) * m for f, m in zip(load_condition["force"], load_condition["measure"])]
)
def elastic_energy(load_condition):
u = load_condition["u"]
E = 1 / 2 * ufl.inner(σ(u), ε(u)) * dx
E -= compliance(load_condition)
return E
assert isinstance(mesh_data.facet_tags, dolfinx.mesh.MeshTags)
assert isinstance(mesh_data.cell_tags, dolfinx.mesh.MeshTags)
fixed_entities = mesh_data.facet_tags.find(bolt_faces_tags)
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)
V_ρ_f_bolt_dofs = dolfinx.fem.locate_dofs_topological(
V_ρ_f, tdim, mesh_data.cell_tags.find(mesh_data.physical_groups["bolts"].tag)
)In the challenge specification there are four load cases defined. We need to prepare the
elasticity problem for each load case and store the associated linear problem. This is achieved by
defining a dictionary with the load case name as the key, and the load vectors, measure, the
function to store the solution, and later the LinearProblem.
The first load case is a vertical upward force of on the pin faces. The Neumann boundary force in the weak form is the force per area, so we need to compute the area of the pin faces. In addition, we convert pounds-force to newtons.
The second and third load cases are horizontal and 42 degree angled forces of approximately and , respectively.
The fourth load case is a torsional load, which we model as a couple of forces in opposite directions of on the pin faces. We assume a lever arm of , which we measured as the approximate distance from the centre of the pin holes to the axis around which the torsion acts (midpoint between the pin holes). This couple of forces is the reason why we need to store the force and measure as lists, to allow for multiple Neumann conditions per load case.
Dirichlet boundary conditions are the same for all load cases and correspond to fixing the bolt faces.
We can use convenient unit handling provided by the dolfiny.units.Quantity class to convert the
imperial units to SI units. This will happen automatically, since when we create a Quantity, the
value gets internally converted to base SI units.
pin_area = comm.allreduce(
dolfinx.fem.assemble_scalar(
dolfinx.fem.form(1.0 * ds((pin_left_tag, pin_right_tag), domain=mesh))
)
)
g = Quantity(mesh, 9.81, syu.meter / syu.second**2, "g")
torsion_lever_arm = Quantity(mesh, 0.01, syu.meter, "torsion_lever_arm")
pin_area = Quantity(mesh, pin_area, syu.meter**2, "pin_area")
F1 = Quantity(mesh, 8000, syu.pound, "F1")
F2 = Quantity(mesh, 8500, syu.pound, "F2")
F3 = Quantity(mesh, 9500, syu.pound, "F3")
F4 = Quantity(mesh, 5000, syu.pound * syu.inch, "F4")
load_conditions = {
"vertical_up": {
"force": [F1 * g / pin_area * ufl.as_vector((0, 0, 1))],
"measure": [ds((pin_left_tag, pin_right_tag))],
"u": dolfinx.fem.Function(V_u),
},
"horizontal_out": {
"force": [F2 * g / pin_area * ufl.as_vector((0, -1, 0))],
"measure": [ds((pin_left_tag, pin_right_tag))],
"u": dolfinx.fem.Function(V_u),
},
"42deg_vertical_out": {
"force": [
F3 * g / pin_area * ufl.as_vector((0, -np.sin(np.deg2rad(42)), np.cos(np.deg2rad(42))))
],
"measure": [ds((pin_left_tag, pin_right_tag))],
"u": dolfinx.fem.Function(V_u),
},
"torsion": {
"force": [
F4 / torsion_lever_arm * g / pin_area / 2 * ufl.as_vector((0, -1, 0)),
F4 / torsion_lever_arm * g / pin_area / 2 * ufl.as_vector((0, 1, 0)),
],
"measure": [ds(pin_left_tag), ds(pin_right_tag)],
"u": dolfinx.fem.Function(V_u),
},
}
def build_nullspace(V):
"""Build PETSc nullspace for 3D elasticity
Copied from https://github.com/FEniCS/dolfinx/blob/main/python/demo/demo_elasticity.py
"""
# Create vectors that will span the nullspace
bs = V.dofmap.index_map_bs
length0 = V.dofmap.index_map.size_local
basis = [dolfinx.la.vector(V.dofmap.index_map, bs=bs, dtype=PETSc.ScalarType) for i in range(6)]
b = [b.array for b in basis]
# Get dof indices for each subspace (x, y and z dofs)
dofs = [V.sub(i).dofmap.list.flatten() for i in range(3)]
# Set the three translational rigid body modes
for i in range(3):
b[i][dofs[i]] = 1.0
# Set the three rotational rigid body modes
x = V.tabulate_dof_coordinates()
dofs_block = V.dofmap.list.flatten()
x0, x1, x2 = x[dofs_block, 0], x[dofs_block, 1], x[dofs_block, 2]
b[3][dofs[0]] = -x1
b[3][dofs[1]] = x0
b[4][dofs[0]] = x2
b[4][dofs[2]] = -x0
b[5][dofs[2]] = x1
b[5][dofs[1]] = -x2
dolfinx.la.orthonormalize(basis)
basis_petsc = [
PETSc.Vec().createWithArray(x[: bs * length0], bsize=3, comm=V.mesh.comm) for x in b
]
return PETSc.NullSpace().create(vectors=basis_petsc)
ns = build_nullspace(V_u)
for lc in load_conditions.values():
u_lc = lc["u"]
a = ufl.derivative(
ufl.derivative(elastic_energy(lc), u_lc),
u_lc,
)
L = -ufl.derivative(elastic_energy(lc), u_lc)
L = ufl.replace(L, {u_lc: ufl.as_vector((0, 0, 0))})
elas_prob = dolfinx.fem.petsc.LinearProblem(
a,
L,
bcs=[bc_u],
u=u_lc, # type: ignore
petsc_options=(
{
# 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-6,
"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",
"pc_gamg_coarse_eq_limit": 1000,
}
),
petsc_options_prefix="elasticity_ksp",
)
elas_prob._A.setNearNullSpace(ns)
lc["linear_problem"] = elas_prob # type: ignore
J_form = dolfinx.fem.form(compliance(lc))
DJ_form = dolfinx.fem.form(-ufl.derivative(elastic_energy(lc), ρ_f))
lc["J_form"] = J_form
lc["DJ_form"] = DJ_formWe can solve one of the load cases to visualise the displacement field. Below is the displacement field for the torsional load case, converted to mm and scaled by a factor of 0.2.
# Solve all load cases
for name in load_conditions.keys():
load_conditions[name]["linear_problem"].solve() # type: ignore
# Plot subplots for all load cases
if comm.size == 1:
plotter = pyvista.Plotter(theme=dolfiny.pyvista.theme, shape=(2, 2))
for i, name in enumerate(load_conditions.keys()):
plotter.subplot(i // 2, i % 2)
u_lc = load_conditions[name]["u"]
grid = pyvista.UnstructuredGrid(*dolfinx.plot.vtk_mesh(u_lc.function_space.mesh)) # type: ignore
assert isinstance(u_lc, dolfinx.fem.Function)
grid.point_data["u"] = u_lc.x.array.reshape((-1, 3)) * 1000 # displacement in mm
grid_warped = grid.warp_by_vector("u", factor=0.2)
plotter.add_mesh(
grid_warped,
show_edges=True,
n_colors=10,
scalar_bar_args={"title": "Displacement [mm]"},
)
plotter.add_text(
name,
font_size=dolfiny.pyvista.pixels // 100,
font="courier",
position="lower_edge",
)
plotter.camera.elevation = -15
plotter.show_axes()
plotter.show()
plotter.close()
plotter.deep_clean()
Filtering¶
We use a Helmholtz filter on the density field, first introduced by Lazarov & Sigmund (2010) in the context of topology optimisation.
In short, for a given density , we solve a (positive definite) Helmholtz equation, yielding the filtered density :
Here is a parameter that controls the filter radius, we choose to be dependent on the maximum cell diameter. In addition to the volumetric term, we also include a boundary term, which acts like a penalisation towards zero Neumann conditions, i.e., it prevents the density from sticking to the boundary, see Wallin et al. (2020). The boundary is here defined as all boundary facets except those with Dirichlet and Neumann conditions applied.
Since the Helmholtz equation is self-adjoint and we need to evaluate its adjoint 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 one operator matrix stored for both the forward and adjoint problems.
In addition to the filter, we need to enforce that the density equals 1 in the bolt volumes. This is achieved by setting the corresponding degrees of freedom in the filtered density function to 1 after applying the filter. We also set the unfiltered density to 1 in the bolt volumes when assembling the right-hand side of the filter problem. Setting the density to 1 in the bolt volumes is important from a practical point of view, as the bolts need to be in contact with solid material.
num_cells = mesh.topology.index_map(tdim).size_local
h = mesh.h(tdim, np.arange(0, num_cells))
hmax = comm.allreduce(h.max(), MPI.MAX)
r = 0.5 * hmax
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)) * dx
+ u_f * v_f * dx
+ r * ufl.inner(u_f, v_f) * ds
- r * ufl.inner(u_f, v_f) * ds((pin_left_tag, pin_right_tag, bolt_faces_tags))
)
L_filter_ρ = dolfinx.fem.form(ρ * v_f * dx(volume_tag) + 1.0 * v_f * dx(bolts_tag))
s = dolfinx.fem.Function(V_ρ_f, name="s")
L_filter_s = dolfinx.fem.form(s * v_f * 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: ignoreOptimisation 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 the total compliance for all load cases:
which is the sum of the compliance for each load case. This choice (i.e., to sum the individual compliances) is one of many possible options for how to account for multiple load cases. Other options include minimising the maximum compliance or minimising a weighted sum of compliances. For simplicity, we follow the sum approach, as it is linear (wrt. load cases), and thus the adjoint problems can be solved independently.
We constrain the density to lower and upper bounds:
and constrain the volume of the design to a volume fraction :
The optimisation problem is stated in reduced form in . So and only appear as intermediates. Gradients are then computed through the adjoint formulation. Since the total compliance is the sum of the compliance for each load case, the gradient is also the sum of the gradients for each load case.
The volume fraction in this demo was chosen as 0.3 (30% of the active part of the mesh). Note that the GrabCAD challenge does not specify any value for the volume fraction. The goal is to produce shapes as light as possible under stress limit constraints. This would require a different optimisation setup and is out of the scope of this demo.
mesh_volume = comm.allreduce(
dolfinx.fem.assemble_scalar(dolfinx.fem.form(dolfinx.fem.Constant(mesh, 1.0) * dx(volume_tag)))
)
volume_fraction = ρ / mesh_volume * dx(volume_tag)
max_volume_fraction = 0.3
g = volume_fraction <= max_volume_fraction
ρ.x.array[:] = max_volume_fraction
ρ_f.interpolate(ρ)
apply_filter(L_filter_ρ, ρ_f)
ρ_f.x.array[V_ρ_f_bolt_dofs] = 1.0
@dolfiny.taoproblem.sync_functions([ρ])
def J(_tao, _):
apply_filter(L_filter_ρ, ρ_f)
ρ_f.x.array[V_ρ_f_bolt_dofs] = 1.0
ρ_f.x.array[:] = np.clip(ρ_f.x.array, ρ_min, 1.0)
total = 0.0
for lc_name, lc in load_conditions.items():
lc["linear_problem"].solve()
comp = comm.allreduce(dolfinx.fem.assemble_scalar(lc["J_form"]))
if comm.rank == 0:
print(f"Objective ({lc_name}): {comp:.4g}")
total += comp
return total
Dρ = dolfinx.fem.Function(V_ρ_f)
z = dolfinx.fem.Function(V_ρ_f, name="z")
tmpDG0 = dolfinx.fem.Function(V_ρ)
s_lc_vec = s.x.petsc_vec.copy() # vector to store the gradient contrib. from each load case
@dolfiny.taoproblem.sync_functions([ρ])
def DJ(_tao, _, G):
with s.x.petsc_vec.localForm() as s_local, s_lc_vec.localForm() as s_lc_local:
s_local.set(0.0)
s_lc_local.set(0.0)
for _lc_name, lc in load_conditions.items():
dolfinx.fem.petsc.assemble_vector(s_lc_vec, lc["DJ_form"])
s_lc_vec.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
s_lc_vec.ghostUpdate(addv=PETSc.InsertMode.INSERT, mode=PETSc.ScatterMode.FORWARD)
with s.x.petsc_vec.localForm() as s_local, s_lc_vec.localForm() as s_lc_local:
s_local += s_lc_local
s_lc_local.set(0.0)
# Apply adjoint to DJ/s -> z.
apply_filter(L_filter_s, z)
z.x.array[V_ρ_f_bolt_dofs] = 0.0
# 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)
opts = PETSc.Options() # type: ignore
opts["tao_type"] = "python"
opts["tao_monitor"] = ""
opts["tao_max_it"] = (max_it := 50)
opts["tao_python_type"] = "dolfiny.mma.MMA"
opts["tao_mma_move_limit"] = 0.05
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)
)
problem.solve()
with dolfinx.io.XDMFFile(comm, "ge_bracket/result.xdmf", "w") as file:
file.write_mesh(mesh)
for f in (ρ, ρ_f, load_conditions["vertical_up"]["u"]):
file.write_function(f)Output
Objective (vertical_up): 411.2
Objective (horizontal_out): 350.4
Objective (42deg_vertical_out): 236.5
Objective (torsion): 147.9
0 TAO, Function value: 1145.95, Residual: 0.
# TAO 0 (ITERATING)
# sub 0 [ 92k] |x|=9.145e+01 |J|=3.404e+02 (density)
# all |x|=9.145e+01 |J|=3.404e+02 |h|=0.000e+00 |Jh|=0.000e+00 f=1.146e+03
Objective (vertical_up): 411.2
Objective (horizontal_out): 350.4
Objective (42deg_vertical_out): 236.5
Objective (torsion): 147.9
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: 553.745, Residual: 0.11102
1 TAO, Function value: 458.34, Residual: 0.0441879
2 TAO, Function value: 279.593, Residual: 0.00741164
3 TAO, Function value: 273.012, Residual: 0.00151835
4 TAO, Function value: 272.704, Residual: 9.4577e-05
5 TAO, Function value: 272.702, Residual: 1.30908e-06
Objective (vertical_up): 288.4
Objective (horizontal_out): 241.6
Objective (42deg_vertical_out): 167.1
Objective (torsion): 95.39
1 TAO, Function value: 792.448, Residual: 186.605
# TAO 1 (ITERATING)
# sub 0 [ 92k] |x|=9.223e+01 |J|=1.866e+02 (density)
# all |x|=9.223e+01 |J|=1.866e+02 |h|=2.554e-15 |Jh|=3.664e-03 f=7.924e+02
Objective (vertical_up): 288.4
Objective (horizontal_out): 241.6
Objective (42deg_vertical_out): 167.1
Objective (torsion): 95.39
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: 71.2955, Residual: 0.0152474
1 TAO, Function value: 63.9108, Residual: 0.0131327
2 TAO, Function value: 46.6231, Residual: 0.00335417
3 TAO, Function value: 45.6722, Residual: 0.000431405
4 TAO, Function value: 45.6557, Residual: 9.20281e-06
5 TAO, Function value: 45.6557, Residual: 3.05897e-08
Objective (vertical_up): 209.6
Objective (horizontal_out): 175.9
Objective (42deg_vertical_out): 123.4
Objective (torsion): 65.75
2 TAO, Function value: 574.619, Residual: 111.375
# TAO 2 (ITERATING)
# sub 0 [ 92k] |x|=9.580e+01 |J|=1.114e+02 (density)
# all |x|=9.580e+01 |J|=1.114e+02 |h|=3.883e-03 |Jh|=3.664e-03 f=5.746e+02
Objective (vertical_up): 209.6
Objective (horizontal_out): 175.9
Objective (42deg_vertical_out): 123.4
Objective (torsion): 65.75
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -29.5256, Residual: 0.0128871
1 TAO, Function value: -34.5999, Residual: 0.010174
2 TAO, Function value: -42.0436, Residual: 0.0015038
3 TAO, Function value: -42.1903, Residual: 0.000117422
4 TAO, Function value: -42.1912, Residual: 8.53503e-07
5 TAO, Function value: -42.1912, Residual: 5.84959e-10
Objective (vertical_up): 157.7
Objective (horizontal_out): 133.7
Objective (42deg_vertical_out): 94.51
Objective (torsion): 47.87
3 TAO, Function value: 433.765, Residual: 71.0753
# TAO 3 (ITERATING)
# sub 0 [ 92k] |x|=1.011e+02 |J|=7.108e+01 (density)
# all |x|=1.011e+02 |J|=7.108e+01 |h|=3.811e-03 |Jh|=3.664e-03 f=4.338e+02
Objective (vertical_up): 157.7
Objective (horizontal_out): 133.7
Objective (42deg_vertical_out): 94.51
Objective (torsion): 47.87
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -65.8425, Residual: 0.0118753
1 TAO, Function value: -69.9593, Residual: 0.00841921
2 TAO, Function value: -73.8114, Residual: 0.000776926
3 TAO, Function value: -73.8422, Residual: 3.80438e-05
4 TAO, Function value: -73.8422, Residual: 1.43041e-07
Objective (vertical_up): 121.9
Objective (horizontal_out): 104.6
Objective (42deg_vertical_out): 74
Objective (torsion): 36.37
4 TAO, Function value: 336.877, Residual: 47.788
# TAO 4 (ITERATING)
# sub 0 [ 92k] |x|=1.077e+02 |J|=4.779e+01 (density)
# all |x|=1.077e+02 |J|=4.779e+01 |h|=3.266e-03 |Jh|=3.664e-03 f=3.369e+02
Objective (vertical_up): 121.9
Objective (horizontal_out): 104.6
Objective (42deg_vertical_out): 74
Objective (torsion): 36.37
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -71.8576, Residual: 0.011884
1 TAO, Function value: -75.7628, Residual: 0.00735368
2 TAO, Function value: -78.1022, Residual: 0.000282702
3 TAO, Function value: -78.1056, Residual: 4.26605e-06
4 TAO, Function value: -78.1056, Residual: 2.42867e-09
Objective (vertical_up): 96.16
Objective (horizontal_out): 83.01
Objective (42deg_vertical_out): 58.56
Objective (torsion): 28.37
5 TAO, Function value: 266.088, Residual: 33.38
# TAO 5 (ITERATING)
# sub 0 [ 92k] |x|=1.155e+02 |J|=3.338e+01 (density)
# all |x|=1.155e+02 |J|=3.338e+01 |h|=2.847e-03 |Jh|=3.664e-03 f=2.661e+02
Objective (vertical_up): 96.16
Objective (horizontal_out): 83.01
Objective (42deg_vertical_out): 58.56
Objective (torsion): 28.37
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -66.2217, Residual: 0.0122544
1 TAO, Function value: -67.4219, Residual: 0.0107871
2 TAO, Function value: -71.4902, Residual: 8.61848e-05
3 TAO, Function value: -71.4904, Residual: 3.24222e-06
4 TAO, Function value: -71.4904, Residual: 2.1953e-09
Objective (vertical_up): 76.94
Objective (horizontal_out): 66.49
Objective (42deg_vertical_out): 46.71
Objective (torsion): 22.41
6 TAO, Function value: 212.56, Residual: 23.9794
# TAO 6 (ITERATING)
# sub 0 [ 92k] |x|=1.242e+02 |J|=2.398e+01 (density)
# all |x|=1.242e+02 |J|=2.398e+01 |h|=2.510e-03 |Jh|=3.664e-03 f=2.126e+02
Objective (vertical_up): 76.94
Objective (horizontal_out): 66.49
Objective (42deg_vertical_out): 46.71
Objective (torsion): 22.41
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -101.433, Residual: 0.00343852
1 TAO, Function value: -101.526, Residual: 0.0029152
2 TAO, Function value: -101.767, Residual: 6.56005e-05
3 TAO, Function value: -101.767, Residual: 1.74992e-06
4 TAO, Function value: -101.767, Residual: 2.36812e-09
Objective (vertical_up): 66.9
Objective (horizontal_out): 57.15
Objective (42deg_vertical_out): 40.21
Objective (torsion): 18.39
7 TAO, Function value: 182.656, Residual: 18.3428
# TAO 7 (ITERATING)
# sub 0 [ 92k] |x|=1.293e+02 |J|=1.834e+01 (density)
# all |x|=1.293e+02 |J|=1.834e+01 |h|=2.187e-03 |Jh|=3.664e-03 f=1.827e+02
Objective (vertical_up): 66.9
Objective (horizontal_out): 57.15
Objective (42deg_vertical_out): 40.21
Objective (torsion): 18.39
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -101.094, Residual: 0.00143029
1 TAO, Function value: -101.11, Residual: 0.00121698
2 TAO, Function value: -101.152, Residual: 5.27433e-06
3 TAO, Function value: -101.152, Residual: 4.53779e-09
Objective (vertical_up): 59.35
Objective (horizontal_out): 50.31
Objective (42deg_vertical_out): 35.37
Objective (torsion): 15.51
8 TAO, Function value: 160.54, Residual: 14.5952
# TAO 8 (ITERATING)
# sub 0 [ 92k] |x|=1.339e+02 |J|=1.460e+01 (density)
# all |x|=1.339e+02 |J|=1.460e+01 |h|=1.222e-03 |Jh|=3.664e-03 f=1.605e+02
Objective (vertical_up): 59.35
Objective (horizontal_out): 50.31
Objective (42deg_vertical_out): 35.37
Objective (torsion): 15.51
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -95.6418, Residual: 0.00325832
1 TAO, Function value: -95.7255, Residual: 0.00277963
2 TAO, Function value: -95.9481, Residual: 4.63337e-06
3 TAO, Function value: -95.9481, Residual: 4.61301e-08
Objective (vertical_up): 53.2
Objective (horizontal_out): 44.89
Objective (42deg_vertical_out): 31.49
Objective (torsion): 13.33
9 TAO, Function value: 142.916, Residual: 11.9738
# TAO 9 (ITERATING)
# sub 0 [ 92k] |x|=1.388e+02 |J|=1.197e+01 (density)
# all |x|=1.388e+02 |J|=1.197e+01 |h|=1.104e-03 |Jh|=3.664e-03 f=1.429e+02
Objective (vertical_up): 53.2
Objective (horizontal_out): 44.89
Objective (42deg_vertical_out): 31.49
Objective (torsion): 13.33
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -89.774, Residual: 0.00350541
1 TAO, Function value: -89.8707, Residual: 0.00298595
2 TAO, Function value: -90.1262, Residual: 1.31798e-05
3 TAO, Function value: -90.1262, Residual: 2.14326e-07
Objective (vertical_up): 48.05
Objective (horizontal_out): 40.46
Objective (42deg_vertical_out): 28.27
Objective (torsion): 11.66
10 TAO, Function value: 128.437, Residual: 10.0648
# TAO 10 (ITERATING)
# sub 0 [ 92k] |x|=1.438e+02 |J|=1.006e+01 (density)
# all |x|=1.438e+02 |J|=1.006e+01 |h|=9.739e-04 |Jh|=3.664e-03 f=1.284e+02
Objective (vertical_up): 48.05
Objective (horizontal_out): 40.46
Objective (42deg_vertical_out): 28.27
Objective (torsion): 11.66
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -84.1098, Residual: 0.00334509
1 TAO, Function value: -84.1978, Residual: 0.00284549
2 TAO, Function value: -84.4274, Residual: 5.10285e-06
3 TAO, Function value: -84.4274, Residual: 5.50262e-08
Objective (vertical_up): 43.67
Objective (horizontal_out): 36.76
Objective (42deg_vertical_out): 25.54
Objective (torsion): 10.34
11 TAO, Function value: 116.307, Residual: 8.62376
# TAO 11 (ITERATING)
# sub 0 [ 92k] |x|=1.491e+02 |J|=8.624e+00 (density)
# all |x|=1.491e+02 |J|=8.624e+00 |h|=8.380e-04 |Jh|=3.664e-03 f=1.163e+02
Objective (vertical_up): 43.67
Objective (horizontal_out): 36.76
Objective (42deg_vertical_out): 25.54
Objective (torsion): 10.34
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -79.2091, Residual: 0.00287415
1 TAO, Function value: -79.2741, Residual: 0.00244618
2 TAO, Function value: -79.4397, Residual: 8.24293e-05
3 TAO, Function value: -79.4399, Residual: 6.79877e-08
Objective (vertical_up): 39.95
Objective (horizontal_out): 33.67
Objective (42deg_vertical_out): 23.23
Objective (torsion): 9.276
12 TAO, Function value: 106.123, Residual: 7.51262
# TAO 12 (ITERATING)
# sub 0 [ 92k] |x|=1.544e+02 |J|=7.513e+00 (density)
# all |x|=1.544e+02 |J|=7.513e+00 |h|=7.029e-04 |Jh|=3.664e-03 f=1.061e+02
Objective (vertical_up): 39.95
Objective (horizontal_out): 33.67
Objective (42deg_vertical_out): 23.23
Objective (torsion): 9.276
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -74.9827, Residual: 0.00235873
1 TAO, Function value: -75.0264, Residual: 0.00200215
2 TAO, Function value: -75.1348, Residual: 9.92008e-05
3 TAO, Function value: -75.1351, Residual: 2.15147e-06
4 TAO, Function value: -75.1351, Residual: 5.16475e-10
Objective (vertical_up): 36.79
Objective (horizontal_out): 31.08
Objective (42deg_vertical_out): 21.27
Objective (torsion): 8.429
13 TAO, Function value: 97.5786, Residual: 6.64329
# TAO 13 (ITERATING)
# sub 0 [ 92k] |x|=1.597e+02 |J|=6.643e+00 (density)
# all |x|=1.597e+02 |J|=6.643e+00 |h|=5.688e-04 |Jh|=3.664e-03 f=9.758e+01
Objective (vertical_up): 36.79
Objective (horizontal_out): 31.08
Objective (42deg_vertical_out): 21.27
Objective (torsion): 8.429
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -71.1313, Residual: 0.00211294
1 TAO, Function value: -71.1664, Residual: 0.00179304
2 TAO, Function value: -71.2524, Residual: 0.000117338
3 TAO, Function value: -71.2527, Residual: 3.84009e-06
4 TAO, Function value: -71.2528, Residual: 1.26159e-09
Objective (vertical_up): 34.09
Objective (horizontal_out): 28.91
Objective (42deg_vertical_out): 19.61
Objective (torsion): 7.743
14 TAO, Function value: 90.3499, Residual: 5.9505
# TAO 14 (ITERATING)
# sub 0 [ 92k] |x|=1.649e+02 |J|=5.951e+00 (density)
# all |x|=1.649e+02 |J|=5.951e+00 |h|=4.514e-04 |Jh|=3.664e-03 f=9.035e+01
Objective (vertical_up): 34.09
Objective (horizontal_out): 28.91
Objective (42deg_vertical_out): 19.61
Objective (torsion): 7.743
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -80.9441, Residual: 0.0101286
1 TAO, Function value: -81.7655, Residual: 0.00891468
2 TAO, Function value: -83.7289, Residual: 0.00283132
3 TAO, Function value: -83.9505, Residual: 6.49474e-05
4 TAO, Function value: -83.9506, Residual: 1.66643e-06
5 TAO, Function value: -83.9506, Residual: 3.31796e-09
Objective (vertical_up): 33.29
Objective (horizontal_out): 27.89
Objective (42deg_vertical_out): 19.07
Objective (torsion): 7.652
15 TAO, Function value: 87.9008, Residual: 5.70653
# TAO 15 (ITERATING)
# sub 0 [ 92k] |x|=1.662e+02 |J|=5.707e+00 (density)
# all |x|=1.662e+02 |J|=5.707e+00 |h|=3.591e-04 |Jh|=3.664e-03 f=8.790e+01
Objective (vertical_up): 33.29
Objective (horizontal_out): 27.89
Objective (42deg_vertical_out): 19.07
Objective (torsion): 7.652
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -83.6306, Residual: 0.00228325
1 TAO, Function value: -83.6714, Residual: 0.00192038
2 TAO, Function value: -83.7667, Residual: 0.000103544
3 TAO, Function value: -83.7669, Residual: 5.98804e-06
4 TAO, Function value: -83.7669, Residual: 2.00735e-08
Objective (vertical_up): 32.7
Objective (horizontal_out): 27.18
Objective (42deg_vertical_out): 18.66
Objective (torsion): 7.566
16 TAO, Function value: 86.1087, Residual: 5.54081
# TAO 16 (ITERATING)
# sub 0 [ 92k] |x|=1.672e+02 |J|=5.541e+00 (density)
# all |x|=1.672e+02 |J|=5.541e+00 |h|=2.099e-04 |Jh|=3.664e-03 f=8.611e+01
Objective (vertical_up): 32.7
Objective (horizontal_out): 27.18
Objective (42deg_vertical_out): 18.66
Objective (torsion): 7.566
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -82.9332, Residual: 0.00112043
1 TAO, Function value: -82.943, Residual: 0.000934455
2 TAO, Function value: -82.9654, Residual: 2.27937e-07
Objective (vertical_up): 32.21
Objective (horizontal_out): 26.64
Objective (42deg_vertical_out): 18.34
Objective (torsion): 7.486
17 TAO, Function value: 84.6628, Residual: 5.41332
# TAO 17 (ITERATING)
# sub 0 [ 92k] |x|=1.680e+02 |J|=5.413e+00 (density)
# all |x|=1.680e+02 |J|=5.413e+00 |h|=1.795e-04 |Jh|=3.664e-03 f=8.466e+01
Objective (vertical_up): 32.21
Objective (horizontal_out): 26.64
Objective (42deg_vertical_out): 18.34
Objective (torsion): 7.486
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -82.2727, Residual: 0.000679793
1 TAO, Function value: -82.2763, Residual: 0.000561086
2 TAO, Function value: -82.2839, Residual: 4.24188e-07
Objective (vertical_up): 31.79
Objective (horizontal_out): 26.23
Objective (42deg_vertical_out): 18.07
Objective (torsion): 7.417
18 TAO, Function value: 83.5062, Residual: 5.31422
# TAO 18 (ITERATING)
# sub 0 [ 92k] |x|=1.688e+02 |J|=5.314e+00 (density)
# all |x|=1.688e+02 |J|=5.314e+00 |h|=1.328e-04 |Jh|=3.664e-03 f=8.351e+01
Objective (vertical_up): 31.79
Objective (horizontal_out): 26.23
Objective (42deg_vertical_out): 18.07
Objective (torsion): 7.417
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -81.7499, Residual: 0.000315934
1 TAO, Function value: -81.7507, Residual: 0.000257612
2 TAO, Function value: -81.7522, Residual: 3.56011e-06
3 TAO, Function value: -81.7522, Residual: 3.60164e-08
Objective (vertical_up): 31.46
Objective (horizontal_out): 25.93
Objective (42deg_vertical_out): 17.86
Objective (torsion): 7.357
19 TAO, Function value: 82.6032, Residual: 5.23818
# TAO 19 (ITERATING)
# sub 0 [ 92k] |x|=1.694e+02 |J|=5.238e+00 (density)
# all |x|=1.694e+02 |J|=5.238e+00 |h|=9.890e-05 |Jh|=3.664e-03 f=8.260e+01
Objective (vertical_up): 31.46
Objective (horizontal_out): 25.93
Objective (42deg_vertical_out): 17.86
Objective (torsion): 7.357
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -81.3315, Residual: 0.000142467
1 TAO, Function value: -81.3317, Residual: 0.000115919
2 TAO, Function value: -81.332, Residual: 1.31033e-06
3 TAO, Function value: -81.332, Residual: 6.2522e-09
Objective (vertical_up): 31.19
Objective (horizontal_out): 25.72
Objective (42deg_vertical_out): 17.69
Objective (torsion): 7.305
20 TAO, Function value: 81.9085, Residual: 5.18052
# TAO 20 (ITERATING)
# sub 0 [ 92k] |x|=1.699e+02 |J|=5.181e+00 (density)
# all |x|=1.699e+02 |J|=5.181e+00 |h|=7.296e-05 |Jh|=3.664e-03 f=8.191e+01
Objective (vertical_up): 31.19
Objective (horizontal_out): 25.72
Objective (42deg_vertical_out): 17.69
Objective (torsion): 7.305
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -80.9494, Residual: 0.0002216
1 TAO, Function value: -80.9497, Residual: 0.000179123
2 TAO, Function value: -80.9505, Residual: 4.57567e-06
3 TAO, Function value: -80.9505, Residual: 6.67314e-08
Objective (vertical_up): 30.98
Objective (horizontal_out): 25.56
Objective (42deg_vertical_out): 17.56
Objective (torsion): 7.261
21 TAO, Function value: 81.3558, Residual: 5.13512
# TAO 21 (ITERATING)
# sub 0 [ 92k] |x|=1.702e+02 |J|=5.135e+00 (density)
# all |x|=1.702e+02 |J|=5.135e+00 |h|=5.512e-05 |Jh|=3.664e-03 f=8.136e+01
Objective (vertical_up): 30.98
Objective (horizontal_out): 25.56
Objective (42deg_vertical_out): 17.56
Objective (torsion): 7.261
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -80.6172, Residual: 0.000284727
1 TAO, Function value: -80.6178, Residual: 0.000230323
2 TAO, Function value: -80.6191, Residual: 8.21191e-06
3 TAO, Function value: -80.6191, Residual: 2.86494e-07
Objective (vertical_up): 30.81
Objective (horizontal_out): 25.42
Objective (42deg_vertical_out): 17.45
Objective (torsion): 7.224
22 TAO, Function value: 80.9092, Residual: 5.09868
# TAO 22 (ITERATING)
# sub 0 [ 92k] |x|=1.705e+02 |J|=5.099e+00 (density)
# all |x|=1.705e+02 |J|=5.099e+00 |h|=4.347e-05 |Jh|=3.664e-03 f=8.091e+01
Objective (vertical_up): 30.81
Objective (horizontal_out): 25.42
Objective (42deg_vertical_out): 17.45
Objective (torsion): 7.224
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -80.3173, Residual: 0.000325873
1 TAO, Function value: -80.3181, Residual: 0.000266543
2 TAO, Function value: -80.3198, Residual: 7.23329e-06
3 TAO, Function value: -80.3198, Residual: 2.42464e-07
Objective (vertical_up): 30.68
Objective (horizontal_out): 25.31
Objective (42deg_vertical_out): 17.35
Objective (torsion): 7.191
23 TAO, Function value: 80.5345, Residual: 5.06824
# TAO 23 (ITERATING)
# sub 0 [ 92k] |x|=1.708e+02 |J|=5.068e+00 (density)
# all |x|=1.708e+02 |J|=5.068e+00 |h|=3.464e-05 |Jh|=3.664e-03 f=8.053e+01
Objective (vertical_up): 30.68
Objective (horizontal_out): 25.31
Objective (42deg_vertical_out): 17.35
Objective (torsion): 7.191
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -80.0595, Residual: 0.000260863
1 TAO, Function value: -80.06, Residual: 0.000215014
2 TAO, Function value: -80.0612, Residual: 6.51381e-06
3 TAO, Function value: -80.0612, Residual: 2.7154e-07
Objective (vertical_up): 30.57
Objective (horizontal_out): 25.22
Objective (42deg_vertical_out): 17.27
Objective (torsion): 7.162
24 TAO, Function value: 80.2167, Residual: 5.04247
# TAO 24 (ITERATING)
# sub 0 [ 92k] |x|=1.710e+02 |J|=5.042e+00 (density)
# all |x|=1.710e+02 |J|=5.042e+00 |h|=2.737e-05 |Jh|=3.664e-03 f=8.022e+01
Objective (vertical_up): 30.57
Objective (horizontal_out): 25.22
Objective (42deg_vertical_out): 17.27
Objective (torsion): 7.162
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -79.8301, Residual: 0.000242742
1 TAO, Function value: -79.8305, Residual: 0.000201871
2 TAO, Function value: -79.8316, Residual: 5.43093e-06
3 TAO, Function value: -79.8316, Residual: 2.66814e-07
Objective (vertical_up): 30.47
Objective (horizontal_out): 25.14
Objective (42deg_vertical_out): 17.2
Objective (torsion): 7.136
25 TAO, Function value: 79.9446, Residual: 5.02049
# TAO 25 (ITERATING)
# sub 0 [ 92k] |x|=1.712e+02 |J|=5.020e+00 (density)
# all |x|=1.712e+02 |J|=5.020e+00 |h|=2.122e-05 |Jh|=3.664e-03 f=7.994e+01
Objective (vertical_up): 30.47
Objective (horizontal_out): 25.14
Objective (42deg_vertical_out): 17.2
Objective (torsion): 7.136
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -79.6349, Residual: 0.000251965
1 TAO, Function value: -79.6354, Residual: 0.000211222
2 TAO, Function value: -79.6366, Residual: 8.18333e-06
3 TAO, Function value: -79.6366, Residual: 2.98965e-07
Objective (vertical_up): 30.4
Objective (horizontal_out): 25.06
Objective (42deg_vertical_out): 17.14
Objective (torsion): 7.114
26 TAO, Function value: 79.7167, Residual: 5.00208
# TAO 26 (ITERATING)
# sub 0 [ 92k] |x|=1.713e+02 |J|=5.002e+00 (density)
# all |x|=1.713e+02 |J|=5.002e+00 |h|=1.699e-05 |Jh|=3.664e-03 f=7.972e+01
Objective (vertical_up): 30.4
Objective (horizontal_out): 25.06
Objective (42deg_vertical_out): 17.14
Objective (torsion): 7.114
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -79.4676, Residual: 0.000225492
1 TAO, Function value: -79.468, Residual: 0.000191615
2 TAO, Function value: -79.4691, Residual: 1.35423e-06
3 TAO, Function value: -79.4691, Residual: 2.94368e-09
Objective (vertical_up): 30.34
Objective (horizontal_out): 25
Objective (42deg_vertical_out): 17.09
Objective (torsion): 7.095
27 TAO, Function value: 79.5229, Residual: 4.98644
# TAO 27 (ITERATING)
# sub 0 [ 92k] |x|=1.715e+02 |J|=4.986e+00 (density)
# all |x|=1.715e+02 |J|=4.986e+00 |h|=1.336e-05 |Jh|=3.664e-03 f=7.952e+01
Objective (vertical_up): 30.34
Objective (horizontal_out): 25
Objective (42deg_vertical_out): 17.09
Objective (torsion): 7.095
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -79.332, Residual: 0.000170157
1 TAO, Function value: -79.3322, Residual: 0.000145629
2 TAO, Function value: -79.3329, Residual: 1.12946e-06
3 TAO, Function value: -79.3329, Residual: 1.80347e-09
Objective (vertical_up): 30.29
Objective (horizontal_out): 24.95
Objective (42deg_vertical_out): 17.04
Objective (torsion): 7.08
28 TAO, Function value: 79.3615, Residual: 4.97342
# TAO 28 (ITERATING)
# sub 0 [ 92k] |x|=1.716e+02 |J|=4.973e+00 (density)
# all |x|=1.716e+02 |J|=4.973e+00 |h|=1.037e-05 |Jh|=3.664e-03 f=7.936e+01
Objective (vertical_up): 30.29
Objective (horizontal_out): 24.95
Objective (42deg_vertical_out): 17.04
Objective (torsion): 7.08
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -79.2171, Residual: 0.000124022
1 TAO, Function value: -79.2172, Residual: 0.000106833
2 TAO, Function value: -79.2176, Residual: 5.76521e-07
Objective (vertical_up): 30.25
Objective (horizontal_out): 24.91
Objective (42deg_vertical_out): 17.01
Objective (torsion): 7.067
29 TAO, Function value: 79.2282, Residual: 4.96262
# TAO 29 (ITERATING)
# sub 0 [ 92k] |x|=1.716e+02 |J|=4.963e+00 (density)
# all |x|=1.716e+02 |J|=4.963e+00 |h|=8.548e-06 |Jh|=3.664e-03 f=7.923e+01
Objective (vertical_up): 30.25
Objective (horizontal_out): 24.91
Objective (42deg_vertical_out): 17.01
Objective (torsion): 7.067
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -79.117, Residual: 8.35246e-05
1 TAO, Function value: -79.1171, Residual: 7.22154e-05
2 TAO, Function value: -79.1172, Residual: 1.18143e-06
3 TAO, Function value: -79.1172, Residual: 3.74702e-09
Objective (vertical_up): 30.21
Objective (horizontal_out): 24.87
Objective (42deg_vertical_out): 16.98
Objective (torsion): 7.055
30 TAO, Function value: 79.1156, Residual: 4.95352
# TAO 30 (ITERATING)
# sub 0 [ 92k] |x|=1.717e+02 |J|=4.954e+00 (density)
# all |x|=1.717e+02 |J|=4.954e+00 |h|=7.701e-06 |Jh|=3.664e-03 f=7.912e+01
Objective (vertical_up): 30.21
Objective (horizontal_out): 24.87
Objective (42deg_vertical_out): 16.98
Objective (torsion): 7.055
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -79.027, Residual: 6.43596e-05
1 TAO, Function value: -79.027, Residual: 5.56739e-05
2 TAO, Function value: -79.0271, Residual: 2.36128e-06
3 TAO, Function value: -79.0271, Residual: 8.2616e-08
Objective (vertical_up): 30.18
Objective (horizontal_out): 24.84
Objective (42deg_vertical_out): 16.95
Objective (torsion): 7.045
31 TAO, Function value: 79.018, Residual: 4.94565
# TAO 31 (ITERATING)
# sub 0 [ 92k] |x|=1.717e+02 |J|=4.946e+00 (density)
# all |x|=1.717e+02 |J|=4.946e+00 |h|=5.990e-06 |Jh|=3.664e-03 f=7.902e+01
Objective (vertical_up): 30.18
Objective (horizontal_out): 24.84
Objective (42deg_vertical_out): 16.95
Objective (torsion): 7.045
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -78.9422, Residual: 6.04702e-05
1 TAO, Function value: -78.9422, Residual: 5.27945e-05
2 TAO, Function value: -78.9423, Residual: 1.59476e-06
3 TAO, Function value: -78.9423, Residual: 3.51107e-08
Objective (vertical_up): 30.15
Objective (horizontal_out): 24.82
Objective (42deg_vertical_out): 16.92
Objective (torsion): 7.036
32 TAO, Function value: 78.9311, Residual: 4.93868
# TAO 32 (ITERATING)
# sub 0 [ 92k] |x|=1.718e+02 |J|=4.939e+00 (density)
# all |x|=1.718e+02 |J|=4.939e+00 |h|=5.248e-06 |Jh|=3.664e-03 f=7.893e+01
Objective (vertical_up): 30.15
Objective (horizontal_out): 24.82
Objective (42deg_vertical_out): 16.92
Objective (torsion): 7.036
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -78.8668, Residual: 5.21797e-05
1 TAO, Function value: -78.8668, Residual: 4.59909e-05
2 TAO, Function value: -78.8669, Residual: 9.46125e-07
3 TAO, Function value: -78.8669, Residual: 2.53114e-08
Objective (vertical_up): 30.13
Objective (horizontal_out): 24.8
Objective (42deg_vertical_out): 16.9
Objective (torsion): 7.028
33 TAO, Function value: 78.8564, Residual: 4.93273
# TAO 33 (ITERATING)
# sub 0 [ 92k] |x|=1.718e+02 |J|=4.933e+00 (density)
# all |x|=1.718e+02 |J|=4.933e+00 |h|=4.586e-06 |Jh|=3.664e-03 f=7.886e+01
Objective (vertical_up): 30.13
Objective (horizontal_out): 24.8
Objective (42deg_vertical_out): 16.9
Objective (torsion): 7.028
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -78.8006, Residual: 4.02763e-05
1 TAO, Function value: -78.8006, Residual: 3.56814e-05
2 TAO, Function value: -78.8006, Residual: 8.90758e-07
3 TAO, Function value: -78.8006, Residual: 1.08043e-08
Objective (vertical_up): 30.11
Objective (horizontal_out): 24.78
Objective (42deg_vertical_out): 16.88
Objective (torsion): 7.021
34 TAO, Function value: 78.7914, Residual: 4.92756
# TAO 34 (ITERATING)
# sub 0 [ 92k] |x|=1.718e+02 |J|=4.928e+00 (density)
# all |x|=1.718e+02 |J|=4.928e+00 |h|=4.042e-06 |Jh|=3.664e-03 f=7.879e+01
Objective (vertical_up): 30.11
Objective (horizontal_out): 24.78
Objective (42deg_vertical_out): 16.88
Objective (torsion): 7.021
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -78.7435, Residual: 2.21171e-05
1 TAO, Function value: -78.7435, Residual: 1.96989e-05
2 TAO, Function value: -78.7435, Residual: 4.02726e-07
Objective (vertical_up): 30.09
Objective (horizontal_out): 24.76
Objective (42deg_vertical_out): 16.87
Objective (torsion): 7.015
35 TAO, Function value: 78.7348, Residual: 4.92307
# TAO 35 (ITERATING)
# sub 0 [ 92k] |x|=1.718e+02 |J|=4.923e+00 (density)
# all |x|=1.718e+02 |J|=4.923e+00 |h|=3.594e-06 |Jh|=3.664e-03 f=7.873e+01
Objective (vertical_up): 30.09
Objective (horizontal_out): 24.76
Objective (42deg_vertical_out): 16.87
Objective (torsion): 7.015
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -78.6927, Residual: 1.39374e-05
1 TAO, Function value: -78.6927, Residual: 1.24327e-05
2 TAO, Function value: -78.6927, Residual: 1.45913e-07
Objective (vertical_up): 30.08
Objective (horizontal_out): 24.75
Objective (42deg_vertical_out): 16.85
Objective (torsion): 7.009
36 TAO, Function value: 78.684, Residual: 4.91908
# TAO 36 (ITERATING)
# sub 0 [ 92k] |x|=1.718e+02 |J|=4.919e+00 (density)
# all |x|=1.718e+02 |J|=4.919e+00 |h|=3.621e-06 |Jh|=3.664e-03 f=7.868e+01
Objective (vertical_up): 30.08
Objective (horizontal_out): 24.75
Objective (42deg_vertical_out): 16.85
Objective (torsion): 7.009
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -78.6481, Residual: 1.26687e-05
1 TAO, Function value: -78.6481, Residual: 1.13525e-05
2 TAO, Function value: -78.6481, Residual: 1.17559e-07
Objective (vertical_up): 30.06
Objective (horizontal_out): 24.74
Objective (42deg_vertical_out): 16.84
Objective (torsion): 7.003
37 TAO, Function value: 78.6385, Residual: 4.9155
# TAO 37 (ITERATING)
# sub 0 [ 92k] |x|=1.719e+02 |J|=4.916e+00 (density)
# all |x|=1.719e+02 |J|=4.916e+00 |h|=3.070e-06 |Jh|=3.664e-03 f=7.864e+01
Objective (vertical_up): 30.06
Objective (horizontal_out): 24.74
Objective (42deg_vertical_out): 16.84
Objective (torsion): 7.003
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -78.607, Residual: 1.41456e-05
1 TAO, Function value: -78.607, Residual: 1.26763e-05
2 TAO, Function value: -78.607, Residual: 7.54231e-08
Objective (vertical_up): 30.05
Objective (horizontal_out): 24.73
Objective (42deg_vertical_out): 16.82
Objective (torsion): 6.998
38 TAO, Function value: 78.598, Residual: 4.9123
# TAO 38 (ITERATING)
# sub 0 [ 92k] |x|=1.719e+02 |J|=4.912e+00 (density)
# all |x|=1.719e+02 |J|=4.912e+00 |h|=2.779e-06 |Jh|=3.664e-03 f=7.860e+01
Objective (vertical_up): 30.05
Objective (horizontal_out): 24.73
Objective (42deg_vertical_out): 16.82
Objective (torsion): 6.998
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -78.5715, Residual: 1e-05
1 TAO, Function value: -78.5715, Residual: 8.94544e-06
2 TAO, Function value: -78.5715, Residual: 2.75707e-08
Objective (vertical_up): 30.04
Objective (horizontal_out): 24.72
Objective (42deg_vertical_out): 16.81
Objective (torsion): 6.994
39 TAO, Function value: 78.5627, Residual: 4.9095
# TAO 39 (ITERATING)
# sub 0 [ 92k] |x|=1.719e+02 |J|=4.909e+00 (density)
# all |x|=1.719e+02 |J|=4.909e+00 |h|=2.355e-06 |Jh|=3.664e-03 f=7.856e+01
Objective (vertical_up): 30.04
Objective (horizontal_out): 24.72
Objective (42deg_vertical_out): 16.81
Objective (torsion): 6.994
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -78.5393, Residual: 9.37528e-06
1 TAO, Function value: -78.5393, Residual: 8.42415e-06
2 TAO, Function value: -78.5393, Residual: 1.64106e-08
Objective (vertical_up): 30.03
Objective (horizontal_out): 24.71
Objective (42deg_vertical_out): 16.8
Objective (torsion): 6.99
40 TAO, Function value: 78.531, Residual: 4.90695
# TAO 40 (ITERATING)
# sub 0 [ 92k] |x|=1.719e+02 |J|=4.907e+00 (density)
# all |x|=1.719e+02 |J|=4.907e+00 |h|=2.261e-06 |Jh|=3.664e-03 f=7.853e+01
Objective (vertical_up): 30.03
Objective (horizontal_out): 24.71
Objective (42deg_vertical_out): 16.8
Objective (torsion): 6.99
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -78.5102, Residual: 1.07323e-05
1 TAO, Function value: -78.5102, Residual: 9.65778e-06
2 TAO, Function value: -78.5102, Residual: 7.49858e-09
Objective (vertical_up): 30.02
Objective (horizontal_out): 24.7
Objective (42deg_vertical_out): 16.79
Objective (torsion): 6.986
41 TAO, Function value: 78.5026, Residual: 4.90469
# TAO 41 (ITERATING)
# sub 0 [ 92k] |x|=1.719e+02 |J|=4.905e+00 (density)
# all |x|=1.719e+02 |J|=4.905e+00 |h|=2.074e-06 |Jh|=3.664e-03 f=7.850e+01
Objective (vertical_up): 30.02
Objective (horizontal_out): 24.7
Objective (42deg_vertical_out): 16.79
Objective (torsion): 6.986
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -78.4843, Residual: 1.04952e-05
1 TAO, Function value: -78.4843, Residual: 6.43147e-06
2 TAO, Function value: -78.4843, Residual: 3.28398e-09
Objective (vertical_up): 30.01
Objective (horizontal_out): 24.7
Objective (42deg_vertical_out): 16.78
Objective (torsion): 6.982
42 TAO, Function value: 78.4773, Residual: 4.90266
# TAO 42 (ITERATING)
# sub 0 [ 92k] |x|=1.719e+02 |J|=4.903e+00 (density)
# all |x|=1.719e+02 |J|=4.903e+00 |h|=1.954e-06 |Jh|=3.664e-03 f=7.848e+01
Objective (vertical_up): 30.01
Objective (horizontal_out): 24.7
Objective (42deg_vertical_out): 16.78
Objective (torsion): 6.982
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -78.4612, Residual: 1.21063e-05
1 TAO, Function value: -78.4612, Residual: 7.46409e-06
2 TAO, Function value: -78.4612, Residual: 3.93877e-08
Objective (vertical_up): 30.01
Objective (horizontal_out): 24.7
Objective (42deg_vertical_out): 16.78
Objective (torsion): 6.978
43 TAO, Function value: 78.4546, Residual: 4.90085
# TAO 43 (ITERATING)
# sub 0 [ 92k] |x|=1.719e+02 |J|=4.901e+00 (density)
# all |x|=1.719e+02 |J|=4.901e+00 |h|=1.820e-06 |Jh|=3.664e-03 f=7.845e+01
Objective (vertical_up): 30.01
Objective (horizontal_out): 24.7
Objective (42deg_vertical_out): 16.78
Objective (torsion): 6.978
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -78.4402, Residual: 1.16542e-05
1 TAO, Function value: -78.4402, Residual: 7.32725e-06
2 TAO, Function value: -78.4402, Residual: 9.34037e-08
Objective (vertical_up): 30
Objective (horizontal_out): 24.69
Objective (42deg_vertical_out): 16.77
Objective (torsion): 6.975
44 TAO, Function value: 78.4341, Residual: 4.89922
# TAO 44 (ITERATING)
# sub 0 [ 92k] |x|=1.719e+02 |J|=4.899e+00 (density)
# all |x|=1.719e+02 |J|=4.899e+00 |h|=1.728e-06 |Jh|=3.664e-03 f=7.843e+01
Objective (vertical_up): 30
Objective (horizontal_out): 24.69
Objective (42deg_vertical_out): 16.77
Objective (torsion): 6.975
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -78.4214, Residual: 7.0363e-06
1 TAO, Function value: -78.4214, Residual: 4.43201e-06
2 TAO, Function value: -78.4214, Residual: 5.03333e-08
Objective (vertical_up): 29.99
Objective (horizontal_out): 24.69
Objective (42deg_vertical_out): 16.76
Objective (torsion): 6.972
45 TAO, Function value: 78.4153, Residual: 4.89769
# TAO 45 (ITERATING)
# sub 0 [ 92k] |x|=1.719e+02 |J|=4.898e+00 (density)
# all |x|=1.719e+02 |J|=4.898e+00 |h|=1.489e-06 |Jh|=3.664e-03 f=7.842e+01
Objective (vertical_up): 29.99
Objective (horizontal_out): 24.69
Objective (42deg_vertical_out): 16.76
Objective (torsion): 6.972
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -78.4035, Residual: 2.04174e-06
1 TAO, Function value: -78.4035, Residual: 1.29462e-06
2 TAO, Function value: -78.4035, Residual: 2.47819e-09
Objective (vertical_up): 29.99
Objective (horizontal_out): 24.69
Objective (42deg_vertical_out): 16.75
Objective (torsion): 6.969
46 TAO, Function value: 78.3982, Residual: 4.89631
# TAO 46 (ITERATING)
# sub 0 [ 92k] |x|=1.719e+02 |J|=4.896e+00 (density)
# all |x|=1.719e+02 |J|=4.896e+00 |h|=1.437e-06 |Jh|=3.664e-03 f=7.840e+01
Objective (vertical_up): 29.99
Objective (horizontal_out): 24.69
Objective (42deg_vertical_out): 16.75
Objective (torsion): 6.969
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -78.3873, Residual: 8.42673e-06
1 TAO, Function value: -78.3873, Residual: 5.42327e-06
2 TAO, Function value: -78.3873, Residual: 2.18702e-08
Objective (vertical_up): 29.98
Objective (horizontal_out): 24.69
Objective (42deg_vertical_out): 16.75
Objective (torsion): 6.967
47 TAO, Function value: 78.3823, Residual: 4.89503
# TAO 47 (ITERATING)
# sub 0 [ 92k] |x|=1.719e+02 |J|=4.895e+00 (density)
# all |x|=1.719e+02 |J|=4.895e+00 |h|=1.405e-06 |Jh|=3.664e-03 f=7.838e+01
Objective (vertical_up): 29.98
Objective (horizontal_out): 24.69
Objective (42deg_vertical_out): 16.75
Objective (torsion): 6.967
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -78.3724, Residual: 4.82082e-06
1 TAO, Function value: -78.3724, Residual: 3.08042e-06
2 TAO, Function value: -78.3724, Residual: 9.98996e-09
Objective (vertical_up): 29.98
Objective (horizontal_out): 24.69
Objective (42deg_vertical_out): 16.74
Objective (torsion): 6.964
48 TAO, Function value: 78.3675, Residual: 4.89383
# TAO 48 (ITERATING)
# sub 0 [ 92k] |x|=1.719e+02 |J|=4.894e+00 (density)
# all |x|=1.719e+02 |J|=4.894e+00 |h|=1.305e-06 |Jh|=3.664e-03 f=7.837e+01
Objective (vertical_up): 29.98
Objective (horizontal_out): 24.69
Objective (42deg_vertical_out): 16.74
Objective (torsion): 6.964
Iteration information for tao_mma_subsolver_ solve.
0 TAO, Function value: -78.3583, Residual: 7.36359e-06
1 TAO, Function value: -78.3583, Residual: 4.64608e-06
2 TAO, Function value: -78.3583, Residual: 2.83826e-08
Objective (vertical_up): 29.97
Objective (horizontal_out): 24.68
Objective (42deg_vertical_out): 16.73
Objective (torsion): 6.963
49 TAO, Function value: 78.3534, Residual: 4.89269
# TAO 49 (ITERATING)
# sub 0 [ 92k] |x|=1.719e+02 |J|=4.893e+00 (density)
# all |x|=1.719e+02 |J|=4.893e+00 |h|=1.276e-06 |Jh|=3.664e-03 f=7.835e+01
Post-processing¶
Once the optimisation is done, we can visualise the results. We use pyvista for the
post-processing. Below is the filtered density field, clipped at 0.5 to visualise the solid
structure.
if comm.size == 1:
grid = pyvista.UnstructuredGrid(*dolfinx.plot.vtk_mesh(ρ_f.function_space.mesh))
grid.point_data["density-filtered"] = ρ_f.x.array
grid_clipped = grid.clip_scalar(scalars="density-filtered", invert=False, value=0.5)
plotter = pyvista.Plotter(
off_screen=True,
theme=dolfiny.pyvista.theme,
shape=(1, 2),
window_size=(dolfiny.pyvista.pixels, dolfiny.pyvista.pixels // 2),
)
plotter.add_mesh(grid_clipped, color="white")
plotter.show_axes()
plotter.camera.elevation = 30
plotter.subplot(0, 1)
plotter.add_mesh(grid_clipped, color="white")
plotter.show_axes()
plotter.view_xz()
plotter.camera.azimuth = 180
plotter.show()
plotter.close()
plotter.deep_clean()
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- 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
- Wallin, M., Ivarsson, N., Amir, O., & Tortorelli, D. (2020). Consistent boundary conditions for PDE filter regularization in topology optimization. Structural and Multidisciplinary Optimization, 62(3), 1299–1311. 10.1007/s00158-020-02556-w