DEH Scheme DGTD-Based Transient Modeling Approach for the Cole–Cole Dispersive Media Using Tustin’s Method

Abstract

The Cole–Cole model can simulate the electromagnetic properties of biological tissues more accurately in a wide frequency band. The current simulation of wave propagation in the Cole–Cole dispersive media usually uses the finite-difference time-domain (FDTD) method, but its requirements in small grid size and time step interval for curved objects limit its application in calculating multiscale complex biological tissues. The discontinuous Galerkin time-domain (DGTD) method is more suitable for simulating multiscale complex targets because it can use unstructured grids and high-order basis functions. Therefore, this article proposes a transient modeling approach for the Cole–Cole dispersive media based on DGTD method, in which the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$z$ </tex-math></inline-formula> transform is performed on the Cole–Cole model, and the fractional power terms are approximated to integer power polynomials according to the continued fraction theory. Moreover, in order to avoid the frequency-dependent permittivity in Maxwell’s equations, the DEH scheme is constructed and a weak form equation for the constitutive relationship between D and E is created. Maxwell’s equations and the constitutive relationship are updated by the leap-frog time-stepping method. The accuracy and ability of the new method are verified by simulating the plane wave propagation in various Cole–Cole media.