{"title":"About the Burton–Miller factor in the low frequency region","authors":"","doi":"10.1016/j.enganabound.2024.105883","DOIUrl":null,"url":null,"abstract":"<div><p>The Burton–Miller method is a widely used approach in acoustics to enhance the stability of the boundary element method for exterior Helmholtz problems at so-called critical frequencies. This method depends on a coupling parameter <span><math><mi>η</mi></math></span> and it can be shown that as long as <span><math><mi>η</mi></math></span> has an imaginary part different from 0, the boundary integral formulation for the Helmholtz equation has a unique solution at all frequencies. A popular choice for this parameter is <span><math><mrow><mi>η</mi><mo>=</mo><mfrac><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></math></span>, where <span><math><mi>k</mi></math></span> is the wavenumber. It can be shown that this choice is quasi optimal, at least in the high frequency limit. However, especially in the low frequency region, where the critical frequencies are still sparsely distributed, different choices for this factor result in a smaller condition number and a smaller error of the solution. In this work, alternative choices for this factor are compared based on numerical experiments. Additionally, a way to enhance the Burton–Miller solution with <span><math><mrow><mi>η</mi><mo>=</mo><mfrac><mrow><mi>i</mi></mrow><mrow><mi>k</mi></mrow></mfrac></mrow></math></span> for a sound hard scatterer in the low frequency region by an additional step of a modified Richardson iteration is introduced.</p></div>","PeriodicalId":51039,"journal":{"name":"Engineering Analysis with Boundary Elements","volume":null,"pages":null},"PeriodicalIF":4.2000,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Engineering Analysis with Boundary Elements","FirstCategoryId":"5","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0955799724003576","RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ENGINEERING, MULTIDISCIPLINARY","Score":null,"Total":0}
引用次数: 0
Abstract
The Burton–Miller method is a widely used approach in acoustics to enhance the stability of the boundary element method for exterior Helmholtz problems at so-called critical frequencies. This method depends on a coupling parameter and it can be shown that as long as has an imaginary part different from 0, the boundary integral formulation for the Helmholtz equation has a unique solution at all frequencies. A popular choice for this parameter is , where is the wavenumber. It can be shown that this choice is quasi optimal, at least in the high frequency limit. However, especially in the low frequency region, where the critical frequencies are still sparsely distributed, different choices for this factor result in a smaller condition number and a smaller error of the solution. In this work, alternative choices for this factor are compared based on numerical experiments. Additionally, a way to enhance the Burton–Miller solution with for a sound hard scatterer in the low frequency region by an additional step of a modified Richardson iteration is introduced.
期刊介绍:
This journal is specifically dedicated to the dissemination of the latest developments of new engineering analysis techniques using boundary elements and other mesh reduction methods.
Boundary element (BEM) and mesh reduction methods (MRM) are very active areas of research with the techniques being applied to solve increasingly complex problems. The journal stresses the importance of these applications as well as their computational aspects, reliability and robustness.
The main criteria for publication will be the originality of the work being reported, its potential usefulness and applications of the methods to new fields.
In addition to regular issues, the journal publishes a series of special issues dealing with specific areas of current research.
The journal has, for many years, provided a channel of communication between academics and industrial researchers working in mesh reduction methods
Fields Covered:
• Boundary Element Methods (BEM)
• Mesh Reduction Methods (MRM)
• Meshless Methods
• Integral Equations
• Applications of BEM/MRM in Engineering
• Numerical Methods related to BEM/MRM
• Computational Techniques
• Combination of Different Methods
• Advanced Formulations.