Background

ProTOp can handle the most important (geometrical) technological constraints. These can be classified into three categories, as follows:

• the opening,
• the symmetry, and
• the periodic pattern

constraints.

█ Input data

The following obligatory or optional data has to be provided.

Constraint type
Constraint types are described in the subsequent section.

Enforce on design limits
Select this option in order to enforce technological constraints on lower limits of design parameters. Note that this option is only needed if shell/lattice configurators are engaged and the configured design space does not reflect the imposed constraints. See the following example.

Figure. Shell surface of a hydraulic splitter example; two uni-direction openings should be applied: the first one upwards from the red line and the second one downwards from the red line.

Figure. Shell configuration of a hydraulic splitter example: Enforce on design limits not checked.

Figure. Shell configuration of a hydraulic splitter example: Enforce on design limits checked.

Intensity
This is a relative parameter that influences the intensity of technological constraints enforcement. Namely, technological constraints are enforced on a discrete finite element mesh. Therefore, they can not be geometrically exact and some numerical procedures have to be used to do the material fill-in as good as possible. The intensity parameter is related to this numerical procedure and its optimal value depends very much on the actual situation. Therefore, the user must be able to tune it to get the desired result. The figure below illustrates what means low intensity (-1.0) and what means high intensity (+1.0) for an opening technological constraint.

Figure. Influence of the enforcement intensity parameter for an opening constraint.

█ Constraint types description

This section illustrates the meaning of various constraints available in ProTOp. The colors in the illustrations are used to denote various domains/materials, as follows

• Light gray - denotes the design domain (dashed grey line is the domain boundary)
  • the design domain corresponds to the CAD model used to import the data into ProTOp
  • at optimization start, the whole design domain may be either completely (solid mode) or partially (shell/lattice mode) filled with material

• Dark green - denotes the unconstrained optimal design, i.e., the material part of the domain after optimization without any technological constraints

• Light green - denotes the material that would be added to the unconstrained optimal design by running the optimization with the specified technological constraint

The following table illustrates the available options and the codes used to indicate a particular constraint in ProTOp.

Table. Illustration of opening, symmetry, and periodic constraints

Illustration	Imposed constraint and its code
	No constraint (original material distribution)
	Unidirectional opening or UniDirOpen constraint Opening direction: green arrow (up) Draft angle: zero or positive (in degrees)
	Bidirectional opening or BiDirOpen constraint Opening direction: either of the green arrows (up or down)
	Bi-Unidirectional opening or BiUniDirOpen constraint Opening directions: green arrows (up and down from the divider plane plane) Draft angle: zero or positive (in degrees)
	Bidirectional opening with exclusions of non-void nodes or BiDirOpenExFix constraint Opening direction: either of the green arrows (up or down) Difference to BiDirOpen constraint: Nodes that can not become void are excluded from processing even if they belong to a free region; an example of such nodes are those within the minimal-thickness of a shell that was configured by a shell configuration tool.
	Plane symmetry or PlaneSym constraint Symmetry plane: red dashed line Plane normal direction: either of the green arrows (up or down)
	Axisymmetry or AxiSym constraint Axis and its direction: green marker (orthogonal to view plane)
	Angular periodic pattern PeriodAng constraint Angular period: in degrees
	Linear periodic pattern PeriodLin constraint Linear period: in length units

█ Numerical accuracy considerations

In ProTOp constraints are enforced by various numerical projection and mapping procedures that run on a discrete FE mesh of the model. This means that the quality of these procedures strongly depends on FE mesh properties and fineness. In order to achieve the best results, the following guideline should be followed:

• Mesh uniformity and fineness
  The mesh should be as uniform as possible and fine enough for the desired accuracy.
  
  
• Symmetry and periodicity of the mesh
  Eventual symmetry and periodicity constraints should be reflected in the mesh structure if possible. This assures excellent results.
  
  
• Symmetry and periodicity of boundary conditions
  Eventual symmetry and periodicity constraints should be reflected in boundary conditions if possible. In this context additional load cases might prove very helpful in improving the results.
  
  

█ Error and warning conditions

A technological constraint data row is flagged by an error status in the following situations:

• The constraint type is other than UniDirOpen, BiDirOpen, BiUniDirOpen, BiDirOpenExFix, PlaneSym, AxiSym, PeriodicAng, or PeriodicLin.
• The volume region reference of a constraint is not valid.
• The referenced volume region of a constraint is flagged by error status.
• The coordinate system reference of a constraint is not valid.
• The referenced coordinate system of a constraint is flagged by error status.
• The constraint axis is other than X, Y, Z, -X, -Y, or -Z.
• The draft angle of a unidirectional opening constraint is less than zero.
• The period of a periodic constraint is less than or equal to zero.