Design sensitivity analysis for transient eddy current problems using finite element discretization and adjoint variable method

For shape design problems subjected to the transient eddy current equation, a shape design sensitivity expressed explicitly in terms of design variables is derived using a discrete system equation of finite elements and an adjoint equation method. The original state equation is an initial-value problem which is to be solved by time-stepping finite element method. On the other hand, the adjoint equation is obtained as a terminal-value problem which is also to be solved by time-stepping finite element method. With the state and adjoint variables solved, the sensitivity is evaluated, which is employed in a gradient-based optimization algorithm. As a numerical example, an eddy current distribution control on a metal surface is treated using the proposed method in an induction heating system