### Thickness control

Some of the most important manufacturing constraints are related to the thickness of the structure. Defining a notion of thickness for continuous structures and setting limits on its value, shape and topology optimization can result in very different topologies and thus different loading paths.

#### Maximum Thickness

This type of constraint is frequently met in cast parts. Avoiding large features in the shape usually requires a chaper feeding system in order to drive the shrinkage porosity during solidification out of the structure. Moreover, it also appears in a great variety of industrial applications, because of tooling limitations, mechanical specifications such as structural redundancy, etc...

2d bridge: volume minimization under a compliance constraint.
Boundary conditions. Optimized shape with maximum thickness 0.16. Initialization. Optimized shape with maximum thickness 0.14. Optimized shape without thickness constraint. Optimized shape with maximum thickness 0.12.

2d cantilever: volume minimization under a compliance constraint.
 Boundary conditions. Optimized shape with maximum thickness 0.50. Initialization. Optimized shape with maximum thickness 0.40. Optimized shape without thickness constraint. Optimized shape with maximum thickness 0.35.

2d MBB beam: volume minimization under a compliance constraint.
 Boundary conditions. Optimized shape with maximum thickness 0.30. Initialization. Optimized shape with maximum thickness 0.25. Optimized shape without thickness constraint. Optimized shape with maximum thickness 0.20.

3d cantilever: volume minimization under a compliance constraint.
 Boundary conditions. Optimized shape without thickness constraint (2). Optimized shape with maximum thickness 0.40 (2). Optimized shape with maximum thickness 0.40 (5). Optimized shape with maximum thickness 0.35 (1). Optimized shape with maximum thickness 0.35 (4). Initialization. Optimized shape without thickness constraint (3). Optimized shape with maximum thickness 0.40 (3). Optimized shape with maximum thickness 0.40 (6). Optimized shape with maximum thickness 0.35 (2). Optimized shape with maximum thickness 0.35 (5). Optimized shape without thickness constraint (1). Optimized shape with maximum thickness 0.40 (1). Optimized shape with maximum thickness 0.40 (4). Optimized shape with maximum thickness 0.40 (7). Optimized shape with maximum thickness 0.35 (3). Optimized shape with maximum thickness 0.35 (6).

3d MBB beam: volume minimization under a compliance constraint.
 Boundary conditions. Optimized shape without thickness constraint (2). Optimized shape with maximum thickness 0.60 (1). Optimized shape with maximum thickness 0.60 (4). Optimized shape with maximum thickness 0.50 (2). Optimized shape with maximum thickness 0.50 (5). Initialization. Optimized shape without thickness constraint (3). Optimized shape with maximum thickness 0.60 (2). Optimized shape with maximum thickness 0.60 (5). Optimized shape with maximum thickness 0.50 (3). Optimized shape with maximum thickness 0.50 (6). Optimized shape without thickness constraint (1). Optimized shape without thickness constraint (4). Optimized shape with maximum thickness 0.60 (3). Optimized shape with maximum thickness 0.50 (1). Optimized shape with maximum thickness 0.50 (4). Optimized shape with maximum thickness 0.50 (7).

3d box: volume minimization under a compliance constraint.
 Boundary conditions. Optimized shape without thickness constraint (2). Optimized shape with maximum thickness 0.60 (1). Optimized shape with maximum thickness 0.60 (4). Optimized shape with maximum thickness 0.40 (1). Optimized shape with maximum thickness 0.40 (4). Initialization. Optimized shape without thickness constraint (3). Optimized shape with maximum thickness 0.60 (2). Optimized shape with maximum thickness 0.60 (5). Optimized shape with maximum thickness 0.40 (2). Optimized shape with maximum thickness 0.40 (5). Optimized shape without thickness constraint (1). Optimized shape without thickness constraint (4). Optimized shape with maximum thickness 0.60 (3). Optimized shape with maximum thickness 0.60 (6). Optimized shape with maximum thickness 0.40 (3). Optimized shape with maximum thickness 0.40 (6).

#### Minimum Thickness

Controling the minimum feature size is very useful in industrial applications for a great variety of reasons. In cast parts, thin parts are avoided due to problems in the filling process of the mould. In micromechanisms (MEMS), tooling imprecission and geometric uncertainty exclude design with tiny characteristics. This constraint can also be seen as a way to avoid mesh-dependency in shape and topology optimization, or even to treat implicitly complicated criteria, like buckling.

2d cantilever: volume minimization under a compliance constraint.
Boundary conditions. Optimized shape with minimum thickness 0.15. Initialization. Optimized shape with minimum thickness 0.20. Optimized shape without thickness constraint. Optimized shape with minimum thickness 0.30.

2d MBB beam: volume minimization under a compliance constraint.
 Boundary conditions. Optimized shape with minimum thickness 0.10. Initialization. Optimized shape with minimum thickness 0.20. Optimized shape without thickness constraint. Optimized shape with minimum thickness 0.25.

2d displacement inverter: displacement maximization under a volume constraint.
 Boundary conditions. Optimized shape with minimum thickness 0.06. Optimized shape without thickness constraint. Optimized shape with minimum thickness 0.07. Optimized shape with minimum thickness 0.05. Optimized shape with minimum thickness 0.08.