Computing Derivatives of Finite Element Solutions by Automatic Derivative Programs

The derivatives of EM responses with respect to the design variables play a crucial role for gradient-based optimizations and sensitivity analysis. This paper focuses on computing the derivatives of EM responses solved using the finite element method (FEM). The primary challenge in obtaining derivatives of FEM solutions lies in determining the differential of the FEM governing equation's residuals. In this study, we employ the automatic derivative (AD) functions available in the PyTorch library to calculate the differentials. By utilizing the explicit expression of the FEM stiffness matrix and excitation vector, the AD tool efficiently computes the differentials of residuals with minimal computational effort, eliminating the need for manual differential calculations. The efficiency and accuracy of this method is validated through an example involving the calculation of the derivative of scattering matrix for a 2D dielectric cylinder scattering problem.