Maximize band gaps

For dynamic problems it is often problematic to have an eigenvalue at some frequency. This can be done with the special form SIMULIA Tosca Structure handles MINMAX-problems for eigenvalue. Here, the object function $Φ$ for eigenfrequencies by TARGET= MINMAX is given:

Φ | α \frac{1}{f_{k} - f_{k}^{*}} |

By minimizing this expression the eigenfrequencies f will be moved away from the value of $f_{k} *$ . With $α$ as a constant the SIMULIA Tosca Structure commands are:

DRESP
 ID_NAME = dresp_eig_2
 DEF_TYPE = SYSTEM
 TYPE = DYN_FREQ
 LC_SET = Modal,All,2
END_

OBJ_FUNC
 ID_NAME =  move_dresp_eig_2_from_15Hz
 DRESP = dresp_eig_2, 1.0, 15.
 TARGET = MINMAX
END_

Convergence of eigenvalues of a bandgap optimization (15 Hz)

The second eigenmode will be pushed away from 15.0 Hz. The weighting of the design response is simply 1.0.

Important: Flat structures should first have the eigenvalues maximized before doing the band-gap optimization because the optimizer must also be able to move some eigenvalues down.