Shape Optimization

An optimization method wherein the outer boundary of the structure is modified to solve the optimization problem.

Using finite element models, the shape is defined by the grid point locations and shape optimization modifies these locations to update the shape.

Shape variables are required to implement shape optimization. Each shape variable is defined by using a DESVAR Bulk Data Entry. If a discrete design variable is desired, a DDVAL Bulk Data Entry needs to be referenced for the design variable values. DVGRID Bulk Data Entries define how much a particular grid point location is changed by the design variable. Any number of DVGRID Bulk Data Entries can be added to the model. Each DVGRID Bulk Data Entry must reference an existing DESVAR Bulk Data Entry if it is to be a part of the optimization. The DVGRID data in OptiStruct contains grid location perturbations, not basis shapes.

The OUTPUT, DVGRID option creates shape variable definitions for displacement or eigenvector results of linear static, normal modes, or linear buckling analyses. These shape variable definitions can then be used in subsequent optimizations. This process facilitates the use of "natural" shape functions.

The generation of the design variables and of the DVGRID Bulk Data Entries is facilitated by the HyperMorph utility, which is part of the HyperMesh software.

Large shape changes in the contact interface during shape optimization are supported. The shape changes of the slave and master surfaces in contact are updated for each iteration during shape optimization. Currently, the large shape change with contact is supported only for node-to-surface type of contact. Large shape change will not be turned on, if surface-to-surface type of contact is present in input file regardless of whether such contact is on design region or not.

The following responses (see Responses for a description) are currently available as the objective or as constraint functions:

Mass	Volume	Center of Gravity
Moment of Inertia	Static Compliance	Static Displacement
Natural Frequency	Buckling Factor	Static Stress, Strain, Forces
Static Composite Stress, Strain, Failure Index	Frequency Response Displacement, Velocity, Acceleration	Frequency Response Stress, Strain, Forces
Weighted Compliance	Weighted Frequency	Combined Compliance Index
Function	Temperature

Design Variables

In finite elements, the shape of a structure is defined by the vector of nodal coordinates $x$ .

In order to avoid mesh distortions due to shape changes, changes of the shape of the structural boundary must be translated into changes of the interior of the mesh.

The two most commonly used approaches to account for mesh changes during a shape optimization are the basis vector approach and the perturbation vector approach. Both approaches refer to the definition of the structural shape as a linear combination of vectors.

With the basis vector approach, the structural shape is defined as a linear combination of basis vectors. The basis vectors define nodal locations.(1)

x = \sum_{i} d_{i} b_{i}

Where,

$x$: Vector of nodal coordinates
$b_{i}$: Basis vector associated with the design variable $d_{i}$

With the perturbation vector approach, the structural shape change is defined as a linear combination of perturbation vectors. The perturbation vectors define changes of nodal locations with respect to the original finite element mesh.(2)

x = x_{0} + \sum_{i} d_{i} p_{i}

Where,

$x$: Vector of nodal coordinates
$x_{0}$: Vector of nodal coordinates of the initial design
$p_{i}$: Perturbation vector associated to the design variable $d_{i}$

The initial nodal coordinates are those defined with the GRID entity. The perturbation vectors are defined on the DVGRID statement, which is referenced by the design variable entity DESVAR.

If a discrete design variable is desired, a DDVAL Bulk Data Entry needs to be referenced on the DESVAR Bulk Data Entry for the design variable values.

Note: In OptiStruct, only the perturbation vector approach is available. The DVGRID cards must contain perturbation vectors.