### Abstract

Although boundary element methods have been applied to interior problems for many years, the numerical difficulties that can occur have not been thoroughly explored. Various authors have reported low-frequency breakdowns and artificial damping due to discretization errors. In this paper, it is shown through a simple example problem that the numerical difficulties depend on the solution formulation. When the boundary conditions are imposed directly, the solution suffers from artificial damping, which may potentially lead to erroneous predictions when boundary element methods are used to evaluate the performance of damping materials. This difficulty can be alleviated by first computing an impedance or admittance matrix, and then using its reactive component to derive the solution for the acoustic field. Numerical computations are used to demonstrate that this technique eliminates artificial damping, but does not correct errors in the reactive components of the impedance or admittance matrices, which then causes nonexistence and nonuniqueness difficulties at the interior resonance frequencies for hard-wall and pressure release boundary conditions, respectively. It is shown that the admittance formulation is better suited to boundary element computations for interior problems because the resonance frequencies for pressure release boundary conditions do not begin until the smallest dimension of the boundary surface is at least one half the acoustic wavelength. Aside from producing much more accurate predictions, the admittance matrix is also much easier to interpolate at low frequencies due to the absence of interior resonances. For the example problem considered, only the formulation using the reactive component of the admittance matrix produces accurate solutions as long as the surface element discretization satisfies the standard six-element-per-wavelength rule.

Original language | English (US) |
---|---|

Pages (from-to) | 1083-1096 |

Number of pages | 14 |

Journal | Journal of Sound and Vibration |

Volume | 319 |

Issue number | 3-5 |

DOIs | |

State | Published - Jan 23 2009 |

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### All Science Journal Classification (ASJC) codes

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

### Cite this

}

*Journal of Sound and Vibration*, vol. 319, no. 3-5, pp. 1083-1096. https://doi.org/10.1016/j.jsv.2008.06.040

**Numerical difficulties with boundary element solutions of interior acoustic problems.** / Fahnline, John Brian.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Numerical difficulties with boundary element solutions of interior acoustic problems

AU - Fahnline, John Brian

PY - 2009/1/23

Y1 - 2009/1/23

N2 - Although boundary element methods have been applied to interior problems for many years, the numerical difficulties that can occur have not been thoroughly explored. Various authors have reported low-frequency breakdowns and artificial damping due to discretization errors. In this paper, it is shown through a simple example problem that the numerical difficulties depend on the solution formulation. When the boundary conditions are imposed directly, the solution suffers from artificial damping, which may potentially lead to erroneous predictions when boundary element methods are used to evaluate the performance of damping materials. This difficulty can be alleviated by first computing an impedance or admittance matrix, and then using its reactive component to derive the solution for the acoustic field. Numerical computations are used to demonstrate that this technique eliminates artificial damping, but does not correct errors in the reactive components of the impedance or admittance matrices, which then causes nonexistence and nonuniqueness difficulties at the interior resonance frequencies for hard-wall and pressure release boundary conditions, respectively. It is shown that the admittance formulation is better suited to boundary element computations for interior problems because the resonance frequencies for pressure release boundary conditions do not begin until the smallest dimension of the boundary surface is at least one half the acoustic wavelength. Aside from producing much more accurate predictions, the admittance matrix is also much easier to interpolate at low frequencies due to the absence of interior resonances. For the example problem considered, only the formulation using the reactive component of the admittance matrix produces accurate solutions as long as the surface element discretization satisfies the standard six-element-per-wavelength rule.

AB - Although boundary element methods have been applied to interior problems for many years, the numerical difficulties that can occur have not been thoroughly explored. Various authors have reported low-frequency breakdowns and artificial damping due to discretization errors. In this paper, it is shown through a simple example problem that the numerical difficulties depend on the solution formulation. When the boundary conditions are imposed directly, the solution suffers from artificial damping, which may potentially lead to erroneous predictions when boundary element methods are used to evaluate the performance of damping materials. This difficulty can be alleviated by first computing an impedance or admittance matrix, and then using its reactive component to derive the solution for the acoustic field. Numerical computations are used to demonstrate that this technique eliminates artificial damping, but does not correct errors in the reactive components of the impedance or admittance matrices, which then causes nonexistence and nonuniqueness difficulties at the interior resonance frequencies for hard-wall and pressure release boundary conditions, respectively. It is shown that the admittance formulation is better suited to boundary element computations for interior problems because the resonance frequencies for pressure release boundary conditions do not begin until the smallest dimension of the boundary surface is at least one half the acoustic wavelength. Aside from producing much more accurate predictions, the admittance matrix is also much easier to interpolate at low frequencies due to the absence of interior resonances. For the example problem considered, only the formulation using the reactive component of the admittance matrix produces accurate solutions as long as the surface element discretization satisfies the standard six-element-per-wavelength rule.

UR - http://www.scopus.com/inward/record.url?scp=55349097817&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=55349097817&partnerID=8YFLogxK

U2 - 10.1016/j.jsv.2008.06.040

DO - 10.1016/j.jsv.2008.06.040

M3 - Article

AN - SCOPUS:55349097817

VL - 319

SP - 1083

EP - 1096

JO - Journal of Sound and Vibration

JF - Journal of Sound and Vibration

SN - 0022-460X

IS - 3-5

ER -