Numerical methods for second‐order shape sensitivity analysis with applications to heat conduction problems

Abstract

Abstract 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.