NUMERICAL-METHODS FOR 2ND-ORDER SHAPE SENSITIVITY ANALYSIS WITH APPLICATIONS TO HEAT-CONDUCTION PROBLEMS

In this paper, two finite-element-based schemes for second-order shape sensitivity analysis are presented. In the first formulation, the AV-DD method, the first-order shape sensitivity equation is derived and expressed in terms of state and adjoint variables. The resultant equation is then directly differentiated to obtain the second-order shape sensitivity equation. In the second formulation, the DD-AV method, the functional of concern is differentiated twice to yield the second-order sensitivity equation in which the second-order shape derivatives can be eliminated by introducing a proper adjoint equation. A thermal fin problem and a thermal insulation layer problem have been studied to validate the proposed schemes. It is shown that both methods yield identical results, though the DD-AV method is computationally more efficient.