Shape Sensitivity Analysis for Optimal Design of Time-Harmonic Electro-quasistatic System Based on Continuum Approach

Abstract

This study proposes a continuum-based sensitivity analysis for an optimal shape design of a time-harmonic electro-quasistatic (EQS) system. Design variables include all the boundaries of the EQS system: the Dirichlet, Neumann, and interface boundaries. The continuum approach: 1) considers an augmented objective function as a continuous functional to be differentiated, which formulates the EQS system and its performance and 2) sets the design variable as a function of a pseudo time. This indicates that the continuum deformation of the design variable is considered. The material derivative concept, which describes the equation of the motion of the design variable in Lagrangian and Eulerian perspectives, is employed to derive the shape sensitivity formula from the augmented objective function. The Lagrange multiplier method, adjoint variable technique, and variational identity are sequentially applied to the material derivatives of the augmented objective function, for taking computational advantages. Owing to the continuum approach, the shape sensitivity can be accurately and precisely calculated using the sensitivity formula in an analytical form, which indicates that the continuum sensitivity is derived before the discretization process of the finite element method (FEM). In addition, the continuum sensitivity analysis is easily implemented using commercial FEM software because the state- and adjoint variables share the same bilinear form. Finally, the continuum sensitivity provides a physical intuition because the sensitivity formula is expressed as a surface integral. Two examples are presented to demonstrate the theoretical validity of the continuum sensitivity formula and the numerical feasibility of the continuum sensitivity analysis.