From shape variation to topological changes in constrained minimization: a velocity method-based concept
The ability of velocity methods to describe changes in topology by creating defects such as holes is investigated. For shape optimization, energy-type objective functions are considered, which depend on the geometry by means of the state variables. The state system is represented by abstract, quadratic and constrained minimization problems stated over domains with defects. The velocity method provides the shape derivative of the objective function due to finite variations of a defect. Sufficient conditions are derived, which allow us to pass the shape derivative to the limit with respect to the diminishing defect and, thus, to obtain the 'topological derivative' of the objective function due to a topological change. An illustrative example is presented for a circular hole bored at the tip of a crack.
constrained minimization velocity method structure optimization shape and topological derivative variational methods domain with hole