3D MBB Beam with a PDE Filter
In this example is included to show that TROUT works in 3D too. Only new material not covered in the previous example will be covered here 2D Topology Optimization with Radial Average Filter, 2D Topology Optimization with PDE Filter and Boundary Penalty, 3D Topology Optimization with PDE Filter.
In this example, the mesh block is 3D.
Listing 1: MBB Mesh
block
[Mesh]
[MeshGenerator]
type = GeneratedMeshGenerator
dim = 3
nx = 60
ny = 20
nz = 20
xmin = 0
xmax = 30
ymin = 0
ymax = 10
zmin = 0
zmax = 10
[]
[node]
type = ExtraNodesetGenerator
input = MeshGenerator
new_boundary = hold_y
coord = '0 0 0; 0 0 10'
[]
[push]
type = ExtraNodesetGenerator
input = node
new_boundary = push
coord = '30 10 5'
[]
[]
The results below show the final design for the 3D MBB problem.
(modules/combined/examples/optimization/3d_mbb.i)
vol_frac = 0.5
E0 = 1
Emin = 1e-8
power = 2
[GlobalParams]
displacements = 'disp_x disp_y disp_z'
[]
[Mesh]
[MeshGenerator]
type = GeneratedMeshGenerator
dim = 3
nx = 60
ny = 20
nz = 20
xmin = 0
xmax = 30
ymin = 0
ymax = 10
zmin = 0
zmax = 10
[]
[node]
type = ExtraNodesetGenerator
input = MeshGenerator
new_boundary = hold_y
coord = '0 0 0; 0 0 10'
[]
[push]
type = ExtraNodesetGenerator
input = node
new_boundary = push
coord = '30 10 5'
[]
[]
[Variables]
[disp_z]
[]
[Dc]
initial_condition = -1.0
[]
[]
[AuxVariables]
[Emin]
family = MONOMIAL
order = CONSTANT
initial_condition = ${Emin}
[]
[power]
family = MONOMIAL
order = CONSTANT
initial_condition = ${power}
[]
[E0]
family = MONOMIAL
order = CONSTANT
initial_condition = ${E0}
[]
[sensitivity]
family = MONOMIAL
order = FIRST
initial_condition = -1.0
[AuxKernel]
type = MaterialRealAux
variable = sensitivity
property = sensitivity
execute_on = LINEAR
[]
[]
[mat_den]
family = MONOMIAL
order = CONSTANT
initial_condition = ${vol_frac}
[]
[Dc_elem]
family = MONOMIAL
order = CONSTANT
initial_condition = -1.0
[AuxKernel]
type = SelfAux
variable = Dc_elem
v = Dc
execute_on = 'TIMESTEP_END'
[]
[]
[mat_den_nodal]
family = L2_LAGRANGE
order = FIRST
initial_condition = ${vol_frac}
[AuxKernel]
type = SelfAux
execute_on = TIMESTEP_END
variable = mat_den_nodal
v = mat_den
[]
[]
[]
[Physics/SolidMechanics/QuasiStatic]
[all]
strain = SMALL
add_variables = true
incremental = false
[]
[]
[Kernels]
[diffusion]
type = FunctionDiffusion
variable = Dc
function = 0.15 # radius coeff
[]
[potential]
type = Reaction
variable = Dc
[]
[source]
type = CoupledForce
variable = Dc
v = sensitivity
[]
[]
[BCs]
[no_x]
type = DirichletBC
variable = disp_y
boundary = hold_y
value = 0.0
[]
[no_y]
type = DirichletBC
variable = disp_x
boundary = right
value = 0.0
[]
[boundary_penalty]
type = ADRobinBC
variable = Dc
boundary = 'left top front back'
coefficient = 10
[]
[boundary_penalty_right]
type = ADRobinBC
variable = Dc
boundary = 'right'
coefficient = 10
[]
[]
[NodalKernels]
[push]
type = NodalGravity
variable = disp_y
boundary = push
gravity_value = -1
mass = 1
[]
[]
[Materials]
[elasticity_tensor]
type = ComputeVariableIsotropicElasticityTensor
youngs_modulus = E_phys
poissons_ratio = poissons_ratio
args = 'Emin mat_den power E0'
[]
[E_phys]
type = DerivativeParsedMaterial
# Emin + (density^penal) * (E0 - Emin)
expression = '${Emin} + (mat_den ^ ${power}) * (${E0}-${Emin})'
coupled_variables = 'mat_den'
property_name = E_phys
[]
[poissons_ratio]
type = GenericConstantMaterial
prop_names = poissons_ratio
prop_values = 0.3
[]
[stress]
type = ComputeLinearElasticStress
[]
[dc]
type = ComplianceSensitivity
design_density = mat_den
youngs_modulus = E_phys
incremental = false
[]
[]
[Preconditioning]
[smp]
type = SMP
full = true
[]
[]
[UserObjects]
[update]
type = DensityUpdate
density_sensitivity = Dc_elem
design_density = mat_den
volume_fraction = ${vol_frac}
execute_on = TIMESTEP_BEGIN
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'
line_search = none
nl_abs_tol = 1e-4
l_max_its = 200
start_time = 0.0
dt = 1.0
num_steps = 70
[]
[Outputs]
[out]
type = Exodus
execute_on = 'INITIAL TIMESTEP_END'
[]
print_linear_residuals = false
[]
[Postprocessors]
[total_vol]
type = ElementIntegralVariablePostprocessor
variable = mat_den
execute_on = 'INITIAL TIMESTEP_END'
[]
[]
[Controls]
[first_period]
type = TimePeriod
start_time = 0.0
end_time = 10
enable_objects = 'BCs::boundary_penalty_right'
execute_on = 'initial timestep_begin'
[]
[]