Ideally, free-shape design regions should be selected where it can be assumed that the shape of the structure is most sensitive to the concerned responses. For example, it would be appropriate to select grids in a high stress region when the objective is to reduce stress.

Free-shape design regions should be defined at different locations on the structure where it is desired for the shape to change independently. For solid structures, feature lines often define natural boundaries for free-shape design regions. Containing any feature lines inside a free-shape design region should be avoided unless the intention is to smooth the feature lines during an optimization. Likewise for a shell structure, sharp corners should not be contained inside a free-shape design region unless the intention is to smooth out such corners.

The DSHAPE card identifies the design region through the GRID continuation card, shown here:

The five parameters that affect the way in which the free-shape design region deforms are the direction type, the move factor, the number of layers for mesh smoothing, the maximum shrinkage, and maximum growth.

Direction Type

This provides a general constraint on the direction of the movement of the free-shape design region. It is defined on the PERT continuation line of the DSHAPE entry in the DTYPE field, shown below:

DTYPE has three distinct options:

Move Factor

The maximum allowable movement in one iteration of the grids defining a free-shape design region is specified as:

MVFACTOR * mesh_size

where, "mesh_size" is the average mesh size of the design region defined in the same DSHAPE card.

MVFACTOR is defined on the PERT continuation line of the DSHAPE entry.

The default value of MVFACTOR is 0.5. A smaller MVFACTOR will make free-shape optimization run slower but with more stability. Conversely, a larger MVFACTOR will make free-shape optimization run faster but with less stability.

MVFACTOR affects the maximum movement in one iteration.

Number of Layers for Mesh Smoothing

With free-shape optimization, internal grids adjacent to those grids defining the design region are moved to avoid mesh distortion. The number of layers of grids to be included in the mesh smoothing buffer may be defined by the NSMOOTH field on the PERT continuation line of the DSHAPE entry.

The default value of NSMOOTH is 10. A larger NSMOOTH will give a larger smoothing buffer, and consequently will work better in avoiding mesh distortion; however, it will result in a slower optimization.

Maximum Shrinkage and Growth

The maximum shrinkage and growth provide a simple way to limit the total amount of deformation of the free-shape design region. These parameters are defined on the PERT continuation line of the DSHAPE entry.

The design region is offset to form two barriers; MXSHRK is the offset in the shrinkage direction and MXGROW is the offset in the growth direction. The design region is then constrained to deform between these two barriers.

Deformation space defined by the maximum growing/shrinking distance

For more details and an example, refer to the section on the Mesh Barrier Constraint below.

Additional treatment to grids in the Transition Zone

When the entire surface or edge of a system is not a design zone and both design and non-design regions exist adjacent to one another, a transition zone can be defined using NTRANS which helps to smooth out the transition. Sharp changes can occur in the design region during optimization and the sections of the design region closest to the non-design region are designated as a transition zone where the corresponding location of the adjacent non-design region is taken into consideration allowing for a smoother transition from the design to non-design region.

NTRANS defines the number of design grid layers in the transition zone to non-design area, where additional treatment will be applied to produce smooth transition.

Defining the Transition Zone grid points for a smooth transition between Design and Non-Design regions (NTRANS=3)

The resulting optimized design will incorporate the effect of non-design regions while moving the transition zone grid points to achieve a smoother final design. The three regions illustrated in the figure above consist of the following highlighted nodes:

The design nodes are separated into two groups:
 
- Design nodes in transition zone (highlighted nodes enclosed by red circles, defined by NTRANS=3)
 
- Design nodes that are NOT in the transition zone (highlighted nodes enclosed by a black circle)

The design nodes in the transition zone will be adjusted during Free-shape optimization to build a smooth transition between “(1) non-design nodes” and “(3) Design nodes that are NOT in the transition zone”. Otherwise, discontinuous or sharp sections may occur, which is explained in the illustration below.

Defining the Transition Zone grid points for a smooth transition between Design and Non-Design regions (NTRANS=3)

It is possible to identify additional constraints on certain grids in free-shape design regions. Three types of constraints are available for specified grids as defined by CTYPE# on the GRIDCON continuation line of the DSHAPE entry:

Constraints are defined on the GRIDCON continuation line as follows:

Example Showing CTYPE = VECTOR

This example demonstrates a simple case where it is necessary to use the "DIR" constraint type to force grids to move in a predefined direction.

A free-shape optimization is performed on a quarter model of a rectangular plate with a hole, shown here:

The curved edge is the free-shape design region. Without any constraints on the free-shape design region, the grids at the ends of the curved edge do not move exactly along the line of the straight edge, but move slightly outward, as shown here:

In order to prevent this phenomenon, the grids at the ends of the curved edge (shown in yellow below) are both constrained to move along the vector indicated by the red arrows.

Using these constraints - corner grids moving along the constrained direction - the grids at the ends of the curved edge now move as desired, along the line of the straight edge, as shown here:

Example showing CTYPE = PLANAR

In this example, the total volume of a cantilever beam is to be minimized subject to a displacement constraint in the loading direction at the free-end of the beam. The model is shown here:

Two free-shape design regions are defined in this example. Both of the vertical sides of the beam are selected as design regions and a free-shape optimization is performed.

Without any constraints on the free-shape design region, the top and bottom surfaces of the beam do not remain strictly on the X-Z plane.

To ensure that the top and bottom surfaces remain on the X-Z plane, the grids along the edges of the design regions DSHAPE1 and DSHAPE2 are constrained to move only on the X-Z plane.

Using these constraints – constrained grids moving only on the X-Z plane – the top and bottom surfaces of the beam remain on the X-Z plane as desired.

It is often desirable to produce a symmetric design. Even if the loads and boundary conditions are perfectly symmetric, there is no guarantee that the resulting design will be perfectly symmetric. In order to ensure a symmetric design, a symmetry constraint must be defined. An additional advantage of this constraint is that it will produce symmetric designs regardless of the initial mesh, loads or boundary conditions.

The 1-plane symmetry constraint is defined on the PATRN continuation line:

Example Showing 1-plane Symmetry Constraints in 2-dimensions

In this example, the objective is to minimize the total volume subject to a stress constraint using free-shape optimization. Results are shown with and without symmetry constraints.

Example Showing 1-plane Symmetry Constraints in 3-dimensions

In this example, the objective is to minimize the compliance subject to a volume constraint using free-shape optimization. Results are shown with and without symmetry constraints.

3D Model showing free-shape design grids

It is often desirable to produce a design with a constant cross-section along a given path, particularly in the case of parts manufactured by an extrusion process. By using extrusion manufacturing constraints with free-shape optimization, constant cross-section designs can be attained for solid models (regardless of the initial mesh, loads or boundary conditions).

The extrusion constraint is defined on the EXTR continuation line:

Two types of extrusion path are available for free-shape optimization – straight line and circular.

Example Showing Extrusion Constraint Along a Straight Line

The FE model, optimization problem and design variables definition are the same as in the previous example, so the result without the extrusion constraint is the same as shown above. The result with the extrusion constraint (straight line) is shown here.

Result with extrusion path (along x-axis)

Example Showing Extrusion Constraint Along a Circular Path

In this example, the objective is to minimize the von Mises stress subject to a volume constraint using free-shape optimization. A circular extrusion path is defined using a cylindrical coordinate system (θ direction). Results are shown with and without extrusion constraints (circular).

In the casting process, cavities that are not open and lined up with the sliding direction of the die are not feasible. Draw direction constraints may be defined for the design region so that the optimized shape will allow the die to slide in a specified direction. Only a single die is considered for each design region (defined in each DSHAPE card), and non-design regions will not be considered for this constraint.

The draw direction constraint is defined on the DRAW continuation line:

Example Showing Draw Direction Constraint

The FE model, optimization problem and design grids definition are the same as those in the example showing 1-plane symmetry constraints in 3 dimensions. Results with draw direction constraint are shown here.

Result with draw direction constraint (along Y-axis)

Example Showing Combination of 1-plane Symmetry and Draw Direction Constraints

The FE model, optimization problem and design grids definition are the same as those in the example showing 1-plane symmetry constraints in 3 dimensions. Results with 1-plane symmetry and draw direction constraints are shown here.

Result with both draw direction constraint (Y-axis) and 1-plane symmetry constraint (XY-plane)

Similar to the maximum shrinkage and growth parameters as defined on the PERT continuation line, it is possible to limit the extent of the total deformation of the design region by way of side constraints. Side constraints allow the deformation space to be defined as a coordinate range; i.e. between (x1, y1, z1) and (x2, y2, z2). These ranges may be with reference to rectangular, cylindrical or spherical systems.

Example Showing Side Constraints

In this example, the objective is to minimize the von Mises stress subject to a volume constraint using free-shape optimization. Results are shown with and without side constraints.

Aside from shrinkage and growth parameters and side constraints, a more general capability to limit the extent of the total deformation of the design region is available by way of defining a mesh barrier constraint. The mesh barrier is composed of special shell elements (BMFACE), and in order to keep computational effort to a minimum, as few elements as possible should be used in its definition.

The mesh barrier is defined on the BMESH continuation line.

Example Showing Mesh Barrier Constraint

The FE model, optimization problem and design grids definition are the same as those in the example showing 1-plane symmetry constraints in 3 dimensions. A mesh barrier is added (large red tria elements).

From the results, you can see how the mesh barrier constrains the model deformation, but if the mesh barrier is not big enough, the design region deformation is unconstrained beyond its limits.

Example Showing Maximum Shrinkage and Growth Parameters

The FE model, optimization problem and design grids definition are the same as those in the example showing 1-plane symmetry constraints in 3 dimensions. In addition, maximum shrinkage and growth parameters (2.0) and a 1-plane symmetry constraint (XZ-plane), are defined.

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