Multiple Load SIMP
We employ here a multi-app scheme similar to that employed in Thermal and mechanical optimization. In this example, each subapp is solving the same mechanics problem with a different applied load. The sensitivity of the structure to each load is computed separately and used in a global optimization algorithm. Considering the optimization of a problem with the separate effect of the loads, instead of accounting for its simultaneous application, can yield a final optimized system whose material distribution is a significantly different from a simple concomitant approach (see Bendsoe and Sigmund (2003)). In the extreme case, having two loads acting on the same point in opposite directions would yield an ill-defined optimized structure if such loads are considered to be concomitant. However, the independent combination of each of the load's sensitivities would yield a structure that is optimized to reduce the compliance under the desired load sensitivity weighting.
In this example, we consider a bridge-like structure on which two vertical loads are applied. The multiapp approach sends the updated pseudo-densities to the subapps, whereas said subapps send their parent app the new load sensitivities. The subapp input files are almost identical, with different load application points:
Listing 1: MBB Application point one
[UserObjects]
[rad_avg]
type = RadialAverage
radius = 3
weights = linear
prop_name = sensitivity
force_preaux = true
execute_on = 'TIMESTEP_END'
[]
# No SIMP optimization in subapp
[calc_sense]
type = SensitivityFilter
density_sensitivity = Dc
design_density = mat_den
filter_UO = rad_avg
force_postaux = true
execute_on = 'TIMESTEP_END'
[]
[]
Listing 2: MBB Application point two
[UserObjects]
[rad_avg]
type = RadialAverage
radius = 3
weights = linear
prop_name = sensitivity
execute_on = TIMESTEP_END
force_preaux = true
[]
# No SIMP optimization in subapp
[calc_sense]
type = SensitivityFilter
density_sensitivity = Dc
design_density = mat_den
filter_UO = rad_avg
execute_on = TIMESTEP_END
force_postaux = true
[]
[]
Filtering of sensitivities takes place in the subapps with the main application being responsible for performing the optimization process.
Listing 3: MBB Density update in main application
[UserObjects]
# We do filtering in the subapps
[update]
type = DensityUpdate
density_sensitivity = total_sensitivity
design_density = mat_den
volume_fraction = ${vol_frac}
execute_on = MULTIAPP_FIXED_POINT_BEGIN
[]
[]
If considered to act simultaneously, the optimized structure becomes an arch. If, on the other hand, the structural responses to each of loads are combined, a more robust triangular frame resurfaces as an optimized design. See figure below for one solution to multi-load problem:

References
- Martin Philip Bendsoe and Ole Sigmund.
Topology optimization: theory, methods, and applications.
Springer Science & Business Media, 2003.[BibTeX]
@book{bendsoe2003topology, author = "Bendsoe, Martin Philip and Sigmund, Ole", title = "Topology optimization: theory, methods, and applications", year = "2003", publisher = "Springer Science \\& Business Media" }
(modules/combined/examples/optimization/multi-load/single_subapp_one.i)
power = 2
E0 = 1.0
Emin = 1.0e-6
[GlobalParams]
displacements = 'disp_x disp_y'
[]
[Mesh]
# final_generator = 'MoveRight'
[Bottom]
type = GeneratedMeshGenerator
dim = 2
nx = 80
ny = 40
xmin = 0
xmax = 150
ymin = 0
ymax = 75
[]
[left_load]
type = ExtraNodesetGenerator
input = Bottom
new_boundary = left_load
coord = '37.5 75 0'
[]
[right_load]
type = ExtraNodesetGenerator
input = left_load
new_boundary = right_load
coord = '112.5 75 0'
[]
[left_support]
type = ExtraNodesetGenerator
input = right_load
new_boundary = left_support
coord = '0 0 0'
[]
[right_support]
type = ExtraNodesetGenerator
input = left_support
new_boundary = right_support
coord = '150 0 0'
[]
[]
[Variables]
[disp_x]
[]
[disp_y]
[]
[]
[AuxVariables]
[Dc]
family = MONOMIAL
order = SECOND
initial_condition = -1.0
[]
[Cc]
family = MONOMIAL
order = CONSTANT
initial_condition = -1.0
[]
[mat_den]
family = MONOMIAL
order = CONSTANT
initial_condition = 0.1
[]
[sensitivity_var]
family = MONOMIAL
order = SECOND
initial_condition = -1.0
[]
[]
[AuxKernels]
[sensitivity_kernel]
type = MaterialRealAux
property = sensitivity
variable = sensitivity_var
check_boundary_restricted = false
execute_on = 'TIMESTEP_END'
[]
[]
[Physics/SolidMechanics/QuasiStatic]
[all]
strain = SMALL
add_variables = true
incremental = false
[]
[]
[BCs]
[no_y]
type = DirichletBC
variable = disp_y
boundary = left_support
value = 0.0
[]
[no_x]
type = DirichletBC
variable = disp_x
boundary = left_support
value = 0.0
[]
[no_y_right]
type = DirichletBC
variable = disp_y
boundary = right_support
value = 0.0
[]
[]
[NodalKernels]
[push_left]
type = NodalGravity
variable = disp_y
boundary = left_load
gravity_value = -1e-3
mass = 1
[]
[]
[Materials]
[elasticity_tensor]
type = ComputeVariableIsotropicElasticityTensor
youngs_modulus = E_phys
poissons_ratio = poissons_ratio
args = 'mat_den'
[]
[E_phys]
type = DerivativeParsedMaterial
# Emin + (density^penal) * (E0 - Emin)
expression = '${Emin} + (mat_den ^ ${power}) * (${E0}-${Emin})'
coupled_variables = 'mat_den'
property_name = E_phys
[]
[poissons_ratio]
type = GenericConstantMaterial
prop_names = poissons_ratio
prop_values = 0.0
[]
[stress]
type = ComputeLinearElasticStress
[]
[dc]
type = ComplianceSensitivity
design_density = mat_den
youngs_modulus = E_phys
[]
[]
[Preconditioning]
[smp]
type = SMP
full = true
[]
[]
[UserObjects]
[rad_avg]
type = RadialAverage
radius = 3
weights = linear
prop_name = sensitivity
force_preaux = true
execute_on = 'TIMESTEP_END'
[]
# No SIMP optimization in subapp
[calc_sense]
type = SensitivityFilter
density_sensitivity = Dc
design_density = mat_den
filter_UO = rad_avg
force_postaux = true
execute_on = 'TIMESTEP_END'
[]
[]
[Executioner]
type = Transient
solve_type = NEWTON
petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
petsc_options_value = 'lu superlu_dist'
nl_abs_tol = 1e-10
dt = 1.0
num_steps = 25
[]
[Outputs]
exodus = true
[out]
type = CSV
execute_on = 'TIMESTEP_END'
[]
print_linear_residuals = false
[]
[Postprocessors]
[mesh_volume]
type = VolumePostprocessor
execute_on = 'initial timestep_end'
[]
[total_vol]
type = ElementIntegralVariablePostprocessor
variable = mat_den
execute_on = 'INITIAL TIMESTEP_END'
[]
[vol_frac]
type = ParsedPostprocessor
expression = 'total_vol / mesh_volume'
pp_names = 'total_vol mesh_volume'
[]
[sensitivity]
type = ElementIntegralMaterialProperty
mat_prop = sensitivity
execute_on = 'TIMESTEP_BEGIN TIMESTEP_END NONLINEAR'
[]
[objective]
type = ElementIntegralMaterialProperty
mat_prop = strain_energy_density
execute_on = 'INITIAL TIMESTEP_END'
[]
[]
(modules/combined/examples/optimization/multi-load/single_subapp_two.i)
power = 2
E0 = 1.0
Emin = 1.0e-6
[GlobalParams]
displacements = 'disp_x disp_y'
[]
[Mesh]
[Bottom]
type = GeneratedMeshGenerator
dim = 2
nx = 80
ny = 40
xmin = 0
xmax = 150
ymin = 0
ymax = 75
[]
[left_load]
type = ExtraNodesetGenerator
input = Bottom
new_boundary = left_load
coord = '37.5 75 0'
[]
[right_load]
type = ExtraNodesetGenerator
input = left_load
new_boundary = right_load
coord = '112.5 75 0'
[]
[left_support]
type = ExtraNodesetGenerator
input = right_load
new_boundary = left_support
coord = '0 0 0'
[]
[right_support]
type = ExtraNodesetGenerator
input = left_support
new_boundary = right_support
coord = '150 0 0'
[]
[]
[Variables]
[disp_x]
[]
[disp_y]
[]
[]
[AuxVariables]
[Dc]
family = MONOMIAL
order = SECOND
initial_condition = -1.0
[]
[Cc]
family = MONOMIAL
order = CONSTANT
initial_condition = -1.0
[]
[mat_den]
family = MONOMIAL
order = CONSTANT
initial_condition = 0.1
[]
[sensitivity_var]
family = MONOMIAL
order = SECOND
initial_condition = -1.0
[]
[]
[AuxKernels]
[sensitivity_kernel]
type = MaterialRealAux
check_boundary_restricted = false
property = sensitivity
variable = sensitivity_var
execute_on = 'TIMESTEP_END'
[]
[]
[Physics/SolidMechanics/QuasiStatic]
[all]
strain = SMALL
add_variables = true
incremental = false
[]
[]
[BCs]
[no_y]
type = DirichletBC
variable = disp_y
boundary = left_support
value = 0.0
[]
[no_x]
type = DirichletBC
variable = disp_x
boundary = left_support
value = 0.0
[]
[no_y_right]
type = DirichletBC
variable = disp_y
boundary = right_support
value = 0.0
[]
[]
[NodalKernels]
[push_right]
type = NodalGravity
variable = disp_y
boundary = right_load
gravity_value = -1e-3
mass = 1
[]
[]
[Materials]
[elasticity_tensor]
type = ComputeVariableIsotropicElasticityTensor
youngs_modulus = E_phys
poissons_ratio = poissons_ratio
args = 'mat_den'
[]
[E_phys]
type = DerivativeParsedMaterial
# Emin + (density^penal) * (E0 - Emin)
expression = '${Emin} + (mat_den ^ ${power}) * (${E0}-${Emin})'
coupled_variables = 'mat_den'
property_name = E_phys
[]
[poissons_ratio]
type = GenericConstantMaterial
prop_names = poissons_ratio
prop_values = 0.0
[]
[stress]
type = ComputeLinearElasticStress
[]
[dc]
type = ComplianceSensitivity
design_density = mat_den
youngs_modulus = E_phys
[]
[]
[Preconditioning]
[smp]
type = SMP
full = true
[]
[]
[UserObjects]
[rad_avg]
type = RadialAverage
radius = 3
weights = linear
prop_name = sensitivity
execute_on = TIMESTEP_END
force_preaux = true
[]
# No SIMP optimization in subapp
[calc_sense]
type = SensitivityFilter
density_sensitivity = Dc
design_density = mat_den
filter_UO = rad_avg
execute_on = TIMESTEP_END
force_postaux = true
[]
[]
[Executioner]
type = Transient
solve_type = NEWTON
petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
petsc_options_value = 'lu superlu_dist'
nl_abs_tol = 1e-10
dt = 1.0
num_steps = 25
[]
[Outputs]
exodus = true
[out]
type = CSV
execute_on = 'TIMESTEP_END'
[]
print_linear_residuals = false
[]
[Postprocessors]
[mesh_volume]
type = VolumePostprocessor
execute_on = 'initial timestep_end'
[]
[total_vol]
type = ElementIntegralVariablePostprocessor
variable = mat_den
execute_on = 'INITIAL TIMESTEP_END'
[]
[vol_frac]
type = ParsedPostprocessor
expression = 'total_vol / mesh_volume'
pp_names = 'total_vol mesh_volume'
[]
[sensitivity]
type = ElementIntegralMaterialProperty
mat_prop = sensitivity
execute_on = 'TIMESTEP_BEGIN TIMESTEP_END NONLINEAR'
[]
[objective]
type = ElementIntegralMaterialProperty
mat_prop = strain_energy_density
execute_on = 'INITIAL TIMESTEP_END'
[]
[]
(modules/combined/examples/optimization/multi-load/single_main.i)
vol_frac = 0.3
power = 1.1
E0 = 1.0
Emin = 1.0e-6
[GlobalParams]
displacements = 'disp_x disp_y'
[]
[Mesh]
# final_generator = 'MoveRight'
[Bottom]
type = GeneratedMeshGenerator
dim = 2
nx = 80
ny = 40
xmin = 0
xmax = 150
ymin = 0
ymax = 75
[]
[left_load]
type = ExtraNodesetGenerator
input = Bottom
new_boundary = left_load
coord = '37.5 75 0'
[]
[right_load]
type = ExtraNodesetGenerator
input = left_load
new_boundary = right_load
coord = '112.5 75 0'
[]
[left_support]
type = ExtraNodesetGenerator
input = right_load
new_boundary = left_support
coord = '0 0 0'
[]
[right_support]
type = ExtraNodesetGenerator
input = left_support
new_boundary = right_support
coord = '150 0 0'
[]
[]
[Variables]
[disp_x]
[]
[disp_y]
[]
[]
[AuxVariables]
[mat_den]
family = MONOMIAL
order = CONSTANT
initial_condition = 0.02
[]
[sensitivity_one]
family = MONOMIAL
order = SECOND
initial_condition = -1.0
[]
[sensitivity_two]
family = MONOMIAL
order = SECOND
initial_condition = -1.0
[]
[total_sensitivity]
family = MONOMIAL
order = SECOND
initial_condition = -1.0
[]
[]
[AuxKernels]
[total_sensitivity]
type = ParsedAux
variable = total_sensitivity
expression = '0.5*sensitivity_one + 0.5*sensitivity_two'
coupled_variables = 'sensitivity_one sensitivity_two'
execute_on = 'LINEAR TIMESTEP_END'
[]
[]
[Physics/SolidMechanics/QuasiStatic]
[all]
strain = SMALL
add_variables = true
incremental = false
[]
[]
[BCs]
[no_y]
type = DirichletBC
variable = disp_y
boundary = left_support
value = 0.0
[]
[no_x]
type = DirichletBC
variable = disp_x
boundary = left_support
value = 0.0
[]
[no_y_right]
type = DirichletBC
variable = disp_y
boundary = right_support
value = 0.0
[]
[]
[Materials]
[elasticity_tensor]
type = ComputeVariableIsotropicElasticityTensor
youngs_modulus = E_phys
poissons_ratio = poissons_ratio
args = 'mat_den'
[]
[E_phys]
type = DerivativeParsedMaterial
# Emin + (density^penal) * (E0 - Emin)
expression = '${Emin} + (mat_den ^ ${power}) * (${E0}-${Emin})'
coupled_variables = 'mat_den'
property_name = E_phys
[]
[poissons_ratio]
type = GenericConstantMaterial
prop_names = poissons_ratio
prop_values = 0.0
[]
[stress]
type = ComputeLinearElasticStress
[]
[]
[Preconditioning]
[smp]
type = SMP
full = true
[]
[]
[UserObjects]
# We do filtering in the subapps
[update]
type = DensityUpdate
density_sensitivity = total_sensitivity
design_density = mat_den
volume_fraction = ${vol_frac}
execute_on = MULTIAPP_FIXED_POINT_BEGIN
[]
[]
[Executioner]
type = Transient
solve_type = NEWTON
petsc_options_iname = '-pc_type -pc_factor_mat_solver_package'
petsc_options_value = 'lu superlu_dist'
nl_abs_tol = 1e-10
dt = 1.0
num_steps = 25
[]
[Outputs]
exodus = true
[out]
type = CSV
execute_on = 'TIMESTEP_END'
[]
print_linear_residuals = false
[]
[Postprocessors]
[mesh_volume]
type = VolumePostprocessor
execute_on = 'initial timestep_end'
[]
[total_vol]
type = ElementIntegralVariablePostprocessor
variable = mat_den
execute_on = 'INITIAL TIMESTEP_END'
[]
[vol_frac]
type = ParsedPostprocessor
expression = 'total_vol / mesh_volume'
pp_names = 'total_vol mesh_volume'
[]
[sensitivity]
type = ElementIntegralVariablePostprocessor
variable = total_sensitivity
[]
[]
[MultiApps]
[sub_app_one]
type = TransientMultiApp
input_files = single_subapp_one.i
[]
[sub_app_two]
type = TransientMultiApp
input_files = single_subapp_two.i
[]
[]
[Transfers]
# First SUB-APP
# To subapp densities
[subapp_one_density]
type = MultiAppCopyTransfer
to_multi_app = sub_app_one
source_variable = mat_den # Here
variable = mat_den
[]
# From subapp sensitivity
[subapp_one_sensitivity]
type = MultiAppCopyTransfer
from_multi_app = sub_app_one
source_variable = Dc # sensitivity_var
variable = sensitivity_one # Here
[]
# Second SUB-APP
# To subapp densities
[subapp_two_density]
type = MultiAppCopyTransfer
to_multi_app = sub_app_two
source_variable = mat_den # Here
variable = mat_den
[]
# From subapp sensitivity
[subapp_two_sensitivity]
type = MultiAppCopyTransfer
from_multi_app = sub_app_two
source_variable = Dc # sensitivity_var
variable = sensitivity_two # Here
[]
[]