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

Andrew N. Norris, Adam J. Nagy, Feruza A. Amirkulova

Research output: Contribution to journalArticle

9 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)2520-2531
Number of pages12
JournalJournal of Sound and Vibration
Volume332
Issue number10
DOIs
StatePublished - Jan 28 2013

Fingerprint

impedance
matrices
Riccati equation
cylindrical coordinates
Taylor series
state vectors
traction
isotropy
Riccati equations
polynomials
differential equations
harmonics
Differential equations
expansion
Polynomials
Scattering
approximation
scattering

All Science Journal Classification (ASJC) codes

  • Condensed Matter Physics
  • Mechanics of Materials
  • Acoustics and Ultrasonics
  • Mechanical Engineering

Cite this

@article{fb0aa3f7650f488cb39cd965eb04b1f8,
title = "Stable methods to solve the impedance matrix for radially inhomogeneous cylindrically anisotropic structures",
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.",
author = "Norris, {Andrew N.} and Nagy, {Adam J.} and Amirkulova, {Feruza A.}",
year = "2013",
month = "1",
day = "28",
doi = "10.1016/j.jsv.2012.12.016",
language = "English (US)",
volume = "332",
pages = "2520--2531",
journal = "Journal of Sound and Vibration",
issn = "0022-460X",
publisher = "Academic Press Inc.",
number = "10",

}

Stable methods to solve the impedance matrix for radially inhomogeneous cylindrically anisotropic structures. / Norris, Andrew N.; Nagy, Adam J.; Amirkulova, Feruza A.

In: Journal of Sound and Vibration, Vol. 332, No. 10, 28.01.2013, p. 2520-2531.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Stable methods to solve the impedance matrix for radially inhomogeneous cylindrically anisotropic structures

AU - Norris, Andrew N.

AU - Nagy, Adam J.

AU - Amirkulova, Feruza A.

PY - 2013/1/28

Y1 - 2013/1/28

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84875430909&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84875430909&partnerID=8YFLogxK

U2 - 10.1016/j.jsv.2012.12.016

DO - 10.1016/j.jsv.2012.12.016

M3 - Article

AN - SCOPUS:84875430909

VL - 332

SP - 2520

EP - 2531

JO - Journal of Sound and Vibration

JF - Journal of Sound and Vibration

SN - 0022-460X

IS - 10

ER -