Adjoint Topology Optimization Model Reference

The adjoint topology optimization model determines the optimal distribution of material in a specified domain with respect to the optimization objective and constraints, which are defined as adjoint cost functions.

Topology Optimization Model Reference


Provided By	[physics continuum] > Models > Optional Adjoint Models
Example Node Path	Continua > Physics 1 > Models > Topology Optimization
Requires	Time: Steady Flow: Coupled Flow Optional Models: Adjoint
Properties	See Topology Optimization Solver Properties.
Compatible Meshers	Trimmed Cell Mesher Polyhedral Mesher Tetrahedral Mesher Thin Mesh Mesher Advanced Layer Mesher
Activates	Physics Models	Topology Physics
	Solvers	See Topology Optimization Solver Properties.
	Field Functions	See Topology Optimization Field Functions.

Topology Optimization Model Properties

Allow Hole Formation

Specifies the source term

ω

for the level set equation Eqn. (5124).

When activated, pockets of solid can appear anywhere in the optimization domain.

Deactive
No source term is defined.
Active
By activating Allow Hole Formation you define a Source Strength for the level set equation. It governs how quickly the holes are formed. For large values (>1), holes are introduced within one or two optimization iterations bases on the sign of the gradient. For small values (~0.1), more optimization iterations are required before forming the holes in the domain. As the optimization proceeds, the sign of the gradient may change from iteration to iteration, inhibiting the formation of holes. See also: Source Strength.

Topology Optimization Boundary Settings

The following boundary Physics Conditions are activated when Topology Optimization Model is selected.

Wall

Topology Option

Specifies the material indicator on this boundary.

Specified (default): Activates the Physics Values > Material Indicator node, where you specify the value of the material indicator ∈ [0, 1].
- 0 — only solid material is allowed
- 1 — only primary (fluid) material is allowed
Fixed: Specifies that on this boundary only solid material is allowed. It is equivalent to Specified with a value of 0.
Free: Indicates that the material indicator on this boundary is extrapolated from the near wall cell. The material is free to evolve as the optimization progresses.

Topology Optimization Solver Properties

The default settings of several solver properties reflect the best practices in most application cases. Start the first simulation with default settings before modifying their values.

Mode: Specifies whether the optimizer acts to Minimize or Maximize the defined objective.

Objective: Specifies the single objective for optimization from the list of available cost functions.

Iteration: Displays the current iteration of the optimization loop.

Step Size: Specifies the normalized step size for optimization.
The optimizer should lead to predictable treads efficiently, decreasing the objective and/or constraint violation consistently over multiple iterations. If the objective and constraint violation fluctuate from iteration to iteration, the step size should be decreased.

Optimization Settings

Penalty
Governs how aggressively the optimizer attempts to satisfy the specified constraints. The default value is 1000.
If the objective fails to improve but constraints are rapidly satisfied, reduce the penalty value. If the optimizer fails to satisfy the constraints after many iterations, the penalty should be increased.
Use Sensitivity Normalization
When activated, the optimization problem is normalized by the max sensitivity values in Eqn. (5135).
The optimizer weights the constraint violations with the objective based on normalizing by the max absolute sensitivity value for each.
In case of multiple constraints, if the optimizer seems to consistently target only one constraint over all others, it may be a sign that this normalization is not working. A manual rescaling by introducing a scale factor into this constraint could help. The ideal scaling for an optimization problem puts the objective value near unity and normalizes all constraint violations such that departures from constraint bounds carry equal weight (for example, % departure from constraint bound can be used instead of absolute value of the constraint departure).
Re-normalization Frequency
Specifies the frequency with which the optimization problem is re-normalized.
If the sensitivity values for the objective and constraints vary significantly as the optimization proceeds, the scaling described above may no longer apply. To address this issue, the normalization can be re-applied at a fixed frequency (defined as number of iterations per renormalization). If the scaling of the problems seems to degrade as the optimization progresses, the frequency should be increased by specifying a smaller number of iterations. The default value is 0, which means that the normalization is performed only once at the first iteration.
ADAM:Beta 1
Specifies the parameter $β_{1}$ for the ADAM (Adaptive Moment Update Estimation) update rule. See also: Eqn. (5126).
This parameter determines the under-relaxation performed on the gradient. Increasing it toward 1 should give smoother optimization performance with the objective and constraints varying more slowly from iteration to iteration. This slower variation can help limit oscillations in the optimum as the optimizer approaches constraint boundaries. It can also prevent the optimizer from terminating prematurely in a local minimum.
ADAM:Beta 2
Specifies the parameter $β_{2}$ for the ADAM (Adaptive Moment Update Estimation) update rule. See also: Eqn. (5127).
This parameter determines the under-relaxation of the variance in the gradient, which is used to scale the optimization search direction. Increasing this value toward 1 should limit oscillations as the optimizer approaches a minima.

Constraints

Alongside the main topology optimization objective, you can define further optimization constraints. These constraints can be based on a target cost-function or volume ratio constraint.

Right-Click Actions

New
Adds optimization constraints.
- User-Defined Constraint
  Stipulates that the chosen cost function must satisfy the specified constraint. Because the optimizer requires derivatives, this type of constraint uses adjoint cost functions to specify the desired report.
- Volume Ratio Constraint
  Enforces a constraint on the volume ratio of the solid phase. This type of constraint is useful for ensuring a manufacturable design. Without a volume ratio constraint, the optimizer can stray into trivial solutions.

Properties

When you define a new topology optimization constraint, the corresponding User-Defined Constraint or the Volume Ratio Constraint sub-nodes are created. Set the following properties for each constraint:

Scale Factor: Specifies a user-defined scaling for each individual constraint. Adjust this property to alter the relative weight of one constraint with respect to the others. If one optimization constraint seems to be either dominating the evolution of the design or is unaccounted for, specify a Scale Factor lower than 1 (which is the default) to reduce its influence or greater than 1 to increase the control of the selected constraint.

Type

Each type allows you to specify the value(s) that the constraint must satisfy. The following options are available:


Type	Corresponding Settings
Minimum	Minimum Specify the minimum value for the constrained quantity. The default setting is 0.
Maximum	Maximum Specify the maximum value for the constrained quantity. The default setting is 1.
Equality	Target Specify the target value for the constrained quantity. The default setting is 0.

Phase: Available with the Volume Ratio Constraint only. Selects the [solid phase] for specifying the volume ratio.

Constraint: Available with the User-Defined Constraint only. Selects the [cost function] bound to the user-defined constraint.

Topology Optimization Field Functions

The following field functions are available when applying the Topology Optimization Model.

Topology Level Set: Outputs the level set variable that controls the material distribution in Eqn. (5138). Positive values correspond to the primary material. Negative values correspond to the solid material. If the optimization stalls, check that both positive and negative values of this field function appear in the optimization domain.
Topology Derivative: Outputs the derivative of the optimization Largrangian function w.r.t the material indicator . Includes the effects of the penalty terms and Lagrange multipliers.
Topology Level Set Gradient: Defines the normal direction for the interface. Magnitude is used for AMR. Only available if temporary storage is enabled.

The following field functions are available when Temporary Storage Retained is activated for the Topology Optimization solver:

Topology Hole Source: Indicates the propensity of the optimizer to introduce a hole in the domain. Positive values of the Hole source indicate that solid material will be introduced. The rate at which solid material is introduced is governed by the Source Strength setting in the Topology Optimization model.
Topology Interface Movement: Indicates the propagation speed of the level set interface. If the optimization stalls, make sure that the values are not too small.
Mean Topology Derivative: Outputs the average topology derivative computed based on the ADAM rule.
Variance of Topology Derivative: Outputs the variance of the topology derivative computed based on the ADAM rule.
Topology Level Set Residual: Represents the residual of the level set equation—a debugging quantity if the level set equation fails to converge.
Topology Level Set Correction: Outputs the corrections of the level set computed every iteration—a debugging quantity if the level set equation fails to converge.