Stable methods to solve the impedance matrix for radially inhomogeneous cylindrically anisotropic structures

TL;DR

This paper introduces a stable numerical method for computing the impedance matrix in radially inhomogeneous, cylindrically anisotropic structures, overcoming exponential instability issues in previous approaches.

Contribution

A novel stable integration scheme combining matricant and impedance matrices for fully anisotropic materials is developed, enabling accurate solutions where previous methods failed.

Findings

Exponential scheme outperforms other approximation methods.

The new method remains stable for fully anisotropic materials.

Comparison with Buchwald potentials validates the approach.

Abstract

A stable approach for integrating the impedance matrix in cylindrical, radial inhomogeneous structures is developed and studied. A Stroh-like system using the time-harmonic displacement-traction state vector is used to derive the Riccati matrix differential equation involving the impedance matrix. It is found that the resulting equation is stiff and leads to exponential instabilities. A stable scheme for integration is found in which a local expansion is performed by combining the matricant and impedance matrices. This method offers a stable solution for fully anisotropic materials, which was previously unavailable in the literature. Several approximation schemes for integrating the Riccati equation in cylindrical coordinates are considered: exponential, Magnus, Taylor series, Lagrange polynomials, with numerical examples indicating that the exponential scheme performs best. The impedance matrix is compared with solutions involving Buchwald potentials in which the material is limited to piecewise constant transverse isotropy. Lastly a scattering example is considered and compared with the literature.