2D MBB Beam with a Convolution Filter and Adaptive Mesh Refinement
In this example we will go over the SIMP approach to topology optimization using adaptive mesh refinement. 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.
The main addition for this example is showing an effective indicator for adaptive mesh refinement, which is the mat_den_nodal variable below.. In the AuxVariables block there is a new variable.
Listing 1: Nodal material density variable definitions
[AuxVariables]
[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
[]
[]
[]
(modules/combined/examples/optimization/2d_mbb_pde_amr.i)The mat_den_nodal variable works better than using the mat_den variable for the indicator. Using the ValueJumpIndicator on the mat_den_nodal variable and refining the mesh where the indicator is greater than a small threshold (0.1), will result in a mesh that is refined in the areas where the material is not 0 or 1. This helps to create a "sharp" interface between the void and the solid elements, without having to refine the entire mesh.
Listing 2: MBB Adaptivity block
[Adaptivity]
max_h_level = 2
recompute_markers_during_cycles = true
interval = 1
cycles_per_step = 1
marker = density_marker
[Indicators]
[density_jump]
type = ValueJumpIndicator
variable = mat_den_nodal
[]
[]
[Markers]
[density_marker]
type = ErrorToleranceMarker
indicator = density_jump
coarsen = 0.1
refine = 0.1
[]
[]
[]
(modules/combined/examples/optimization/2d_mbb_pde_amr.i)(modules/combined/examples/optimization/2d_mbb_pde_amr.i)
vol_frac = 0.5
E0 = 1
Emin = 1e-8
power = 2
[GlobalParams]
displacements = 'disp_x disp_y'
[]
[Mesh]
[MeshGenerator]
type = GeneratedMeshGenerator
dim = 2
nx = 30
ny = 10
xmin = 0
xmax = 30
ymin = 0
ymax = 10
[]
[node]
type = ExtraNodesetGenerator
input = MeshGenerator
new_boundary = pull
nodes = 0
[]
[push]
type = ExtraNodesetGenerator
input = node
new_boundary = push
coord = '30 10 0'
[]
[]
[Variables]
[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 = ${vol_frac}
[AuxKernel]
type = SelfAux
execute_on = TIMESTEP_END
variable = mat_den_nodal
v = mat_den
[]
[]
[]
[Physics/SolidMechanics/QuasiStatic]
[all]
strain = SMALL
add_variables = true
incremental = false
[]
[]
[Kernels]
[diffusion]
type = FunctionDiffusion
variable = Dc
function = 0.15 # radius coeff
[]
[potential]
type = Reaction
variable = Dc
[]
[source]
type = CoupledForce
variable = Dc
v = sensitivity
[]
[]
[BCs]
[no_x]
type = DirichletBC
variable = disp_y
boundary = pull
value = 0.0
[]
[no_y]
type = DirichletBC
variable = disp_x
boundary = right
value = 0.0
[]
[boundary_penalty]
type = ADRobinBC
variable = Dc
boundary = 'left top'
coefficient = 10
[]
[boundary_penalty_right]
type = ADRobinBC
variable = Dc
boundary = 'right'
coefficient = 10
[]
[]
[NodalKernels]
[pull]
type = NodalGravity
variable = disp_y
boundary = push
gravity_value = -1
mass = 1
[]
[]
[Materials]
[elasticity_tensor]
type = ComputeVariableIsotropicElasticityTensor
youngs_modulus = E_phys
poissons_ratio = poissons_ratio
args = 'Emin mat_den power E0'
[]
[E_phys]
type = DerivativeParsedMaterial
# Emin + (density^penal) * (E0 - Emin)
expression = '${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 = 40
enable_objects = 'BCs::boundary_penalty_right'
execute_on = 'initial timestep_begin'
[]
[]
[Adaptivity]
max_h_level = 2
recompute_markers_during_cycles = true
interval = 1
cycles_per_step = 1
marker = density_marker
[Indicators]
[density_jump]
type = ValueJumpIndicator
variable = mat_den_nodal
[]
[]
[Markers]
[density_marker]
type = ErrorToleranceMarker
indicator = density_jump
coarsen = 0.1
refine = 0.1
[]
[]
[]
(modules/combined/examples/optimization/2d_mbb_pde_amr.i)
vol_frac = 0.5
E0 = 1
Emin = 1e-8
power = 2
[GlobalParams]
displacements = 'disp_x disp_y'
[]
[Mesh]
[MeshGenerator]
type = GeneratedMeshGenerator
dim = 2
nx = 30
ny = 10
xmin = 0
xmax = 30
ymin = 0
ymax = 10
[]
[node]
type = ExtraNodesetGenerator
input = MeshGenerator
new_boundary = pull
nodes = 0
[]
[push]
type = ExtraNodesetGenerator
input = node
new_boundary = push
coord = '30 10 0'
[]
[]
[Variables]
[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 = ${vol_frac}
[AuxKernel]
type = SelfAux
execute_on = TIMESTEP_END
variable = mat_den_nodal
v = mat_den
[]
[]
[]
[Physics/SolidMechanics/QuasiStatic]
[all]
strain = SMALL
add_variables = true
incremental = false
[]
[]
[Kernels]
[diffusion]
type = FunctionDiffusion
variable = Dc
function = 0.15 # radius coeff
[]
[potential]
type = Reaction
variable = Dc
[]
[source]
type = CoupledForce
variable = Dc
v = sensitivity
[]
[]
[BCs]
[no_x]
type = DirichletBC
variable = disp_y
boundary = pull
value = 0.0
[]
[no_y]
type = DirichletBC
variable = disp_x
boundary = right
value = 0.0
[]
[boundary_penalty]
type = ADRobinBC
variable = Dc
boundary = 'left top'
coefficient = 10
[]
[boundary_penalty_right]
type = ADRobinBC
variable = Dc
boundary = 'right'
coefficient = 10
[]
[]
[NodalKernels]
[pull]
type = NodalGravity
variable = disp_y
boundary = push
gravity_value = -1
mass = 1
[]
[]
[Materials]
[elasticity_tensor]
type = ComputeVariableIsotropicElasticityTensor
youngs_modulus = E_phys
poissons_ratio = poissons_ratio
args = 'Emin mat_den power E0'
[]
[E_phys]
type = DerivativeParsedMaterial
# Emin + (density^penal) * (E0 - Emin)
expression = '${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 = 40
enable_objects = 'BCs::boundary_penalty_right'
execute_on = 'initial timestep_begin'
[]
[]
[Adaptivity]
max_h_level = 2
recompute_markers_during_cycles = true
interval = 1
cycles_per_step = 1
marker = density_marker
[Indicators]
[density_jump]
type = ValueJumpIndicator
variable = mat_den_nodal
[]
[]
[Markers]
[density_marker]
type = ErrorToleranceMarker
indicator = density_jump
coarsen = 0.1
refine = 0.1
[]
[]
[]