From shape variation to topological changes in constrained minimization: a velocity method-based concept

Abstract

The ability of velocity methods to describe changes of topology by creating defects like holes is investigated. For the shape optimization energy-type objective functions are considered, which depend on the geometry by means of state variables. The state system is represented by abstract, quadratic, 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 diminishing defect, thus, to obtain the "topological derivative" of the objective function due to a topology change. An illustrative example is presented for a circular hole bored at the tip of a crack.

Optimal control of the thickness of a rigid inclusion in equilibrium problems for inhomogeneous two-dimensional bodies with a crack

“…Differentiability of the energy functionals for the boundary value problems with unilateral constraints on the boundary was studied in many works, e.g. [6,10,20,23,30,32]. Using the velocity method, the general result of shape differentiability in the abstract form of the quadratic functional is formulated in [6], and it is proven for bijective feasible sets.…”

Characterization of the Fréchet derivative of the elasto-acoustic field with respect to Lipschitz domains

“…The determination of the shape of an obstacle from its effects on known acoustic or electromagnetic waves is an important problem in many technologies such as sonar, radar, geophysical exploration, medical imaging and non destructive testing (see, for example, [3,5,[8][9][10] and the references cited therein). This inverse scattering problem is difficult to solve, especially from a numerical point of view because it is ill-posed and nonlinear.…”

Optimal location of a rigid inclusion in equilibrium problems for inhomogeneous two‐dimensional bodies with a crack

“…For a heterogeneous two dimensional body with a micro‐object (defect) and a macro‐object (crack), the antiplane strain energy release rate is expressed by means of the mode‐III stress intensity factor that is examined with respect to small defects such as microcracks, holes, and inclusions . The ability of velocity methods to describe changes of topology by creating defects like holes is investigated in []. Using the shape‐topological sensitivity analysis, the existence of a solution of an optimal control problem concerning the best choice of the location and shape of elastic inclusions was proved in [].…”