Documentation of the Config Files for Topology Optimization Problems

Let us take a detailed look at the config files for topology optimization problems and discusss the corresponding parameters. Note that the settings are mostly the same as for optimal control problems (as we deal with the problems on a fixed mesh), so the reader is referred to Documentation of the Config Files for Optimal Control Problems for most parameters and here we only describe those that are not already treated there.

As in Documentation of the Config Files for Optimal Control Problems, we refer to the documentation of the configparser module for a detailed description of how these config files can be structured. Moreover, we remark that cashocs has a default behavior for almost all of these parameters, which is triggered when they are NOT specified in the config file, and we will discuss this behavior for each parameter in this tutorial.

Again, we start with a summary of all parameters before discussing them detailedly. New users might want to skip the summary and continue directly at the detailed description.

Summary

An overview over all configuration parameters for topology optimization and their default values can be found in the following.

TopologyOptimization

Parameter = Default value	Description and remarks
angle_tol = 1.0	Stopping tolerance for the angle (in degrees)
interpolation_scheme = volume	The approach used to interpolate the topological derivative. Possible options: volume or angle
normalize_topological_derivative = False	Whether to normalize the topological derivative in each iteration.
re_normalize_levelset = False	Whether to re-normalize the levelset function in each iteration.
topological_derivative_is_identical = False	Whether the topological derivative is identical for all considered parts
tol_bisection = 1e-4	Tolerance for the bisection procedure (used for incorporating volume constraints)
max_iter_bisection = 100	Maximum number of iterations of the bisection procedure

Detailed Description

Let us now give a detailed description over all parameters available for configuring cashocs’ behavior.

TopologyOptimization

Topology optimization problems contain additional parameters only in the section “TopologyOptimization”. Let us go over the parameters in that section now.

The first parameter is

angle_tol = 1.0

which is the absolute tolerance (in degrees) of the angle between topological derivative and level-set function and is used as stopping criterion for the topology optimization algorithms. The default value is angle_tol = 1.0.

The next parameter is

interpolation_scheme = volume

which describes which scheme is used to interpolate the topological derivative to the mesh nodes. Possible options are interpolation_scheme = volume and interpolation_scheme = angle, where a weighting by the cell volume or angles is used, respectively. The default is given by interpolation_scheme = volume. Note that the angle-weighted interpolation is not available in parallel.

Next up we have

normalize_topological_derivative = False

which is a boolean flag that indicates whether the topological derivative should be normalized in an \(L^2\) sense before using it in the optimization algorithm or not. The default is normalize_topological_derivative = False.

The next parameter is given by

re_normalize_levelset = False

which is a boolean flag used to indicate, whether the level set function should be re-normalized in each iteration of the optimization algorithm. The default is given by re_normalize_levelset = False

Next, we have the following parameter

topological_derivative_is_identical = False

which is a boolean flag that is used to indicate, whether the topological derivative is equal in both parts of the domain. As this is usually not the case, the default setting is topological_derivative_is_identical = False.

The next parameter is given by

tol_bisection = 1e-4

which determines the absolute tolerance for the bisection approach that is used to incorporate volume constraints into the topology optimization. The default tolerance of tol_bisection = 1e-4 should be sufficient for most applications.

Finally, the parameter

max_iter_bisection = 100

determines how many iterations of the bisection approach are carried out in the worst case. The default value for this is given by max_iter_bisection = 100.