{"title":"在波浪有限元框架内进行部分频散分析的高效计算 k(ω)-spectral 形式","authors":"","doi":"10.1016/j.jsv.2024.118652","DOIUrl":null,"url":null,"abstract":"<div><p>This paper addresses the computation of frequency-dependent dispersion curves (i.e., <span><math><mrow><mi>k</mi><mrow><mo>(</mo><mi>ω</mi><mo>)</mo></mrow></mrow></math></span>) and wave modes within the framework of the Wave Finite Element Method (WFEM) and in the context of high-dimensional periodic unit cell models. Numerous applications, ranging from phononics to vibroacoustics, now rely on dispersion analyses or wave expansion over a subset of eigensolutions – complex wavenumbers and Bloch waves – resulting from the resolution of an eigenvalue problem with a <span><math><mi>T</mi></math></span>-palindromic quadratic structure (<span><math><mi>T</mi></math></span>-PQEP). To exploit the structure of finite element models, various structure-preserving linearizations such as the Zhong-Williams and the <span><math><mrow><mo>(</mo><mi>S</mi><mo>+</mo><msup><mrow><mi>S</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></math></span>-transform have already been developed to achieve partial wave resolution of large <span><math><mi>T</mi></math></span>-PQEP, primarily targeting the dominating (least decaying) waves. In this paper we derive an alternative linearization of the <span><math><mi>T</mi></math></span>-PQEP for the <span><math><mrow><mi>k</mi><mrow><mo>(</mo><mi>ω</mi><mo>)</mo></mrow></mrow></math></span> problem, which leads to enhanced targeting of the eigenvalues around the unit circle and reduces the inaccuracies induced by root multiplicity. A specific form of the problem is then proposed as an optimal compromise between ease of implementation, numerical stability, convergence and accuracy enhancement. The performance of our proposed linearization is compared against existing ones across various iterative eigensolvers, since the generalized eigenvalue problems involve complex non-hermitian matrices, which are not extensively included in eigensolvers. Results indicate that the proposed linearization should be favored for the WFEM, as it provides numerical enhancements in dispersion and wave vectors computation for large eigenvalue problems, as well as for further wave expansion applications.</p></div>","PeriodicalId":17233,"journal":{"name":"Journal of Sound and Vibration","volume":null,"pages":null},"PeriodicalIF":4.3000,"publicationDate":"2024-08-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"A computationally efficient k(ω)-spectral form for partial dispersion analyses within the wave finite element framework\",\"authors\":\"\",\"doi\":\"10.1016/j.jsv.2024.118652\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>This paper addresses the computation of frequency-dependent dispersion curves (i.e., <span><math><mrow><mi>k</mi><mrow><mo>(</mo><mi>ω</mi><mo>)</mo></mrow></mrow></math></span>) and wave modes within the framework of the Wave Finite Element Method (WFEM) and in the context of high-dimensional periodic unit cell models. Numerous applications, ranging from phononics to vibroacoustics, now rely on dispersion analyses or wave expansion over a subset of eigensolutions – complex wavenumbers and Bloch waves – resulting from the resolution of an eigenvalue problem with a <span><math><mi>T</mi></math></span>-palindromic quadratic structure (<span><math><mi>T</mi></math></span>-PQEP). To exploit the structure of finite element models, various structure-preserving linearizations such as the Zhong-Williams and the <span><math><mrow><mo>(</mo><mi>S</mi><mo>+</mo><msup><mrow><mi>S</mi></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></mrow></math></span>-transform have already been developed to achieve partial wave resolution of large <span><math><mi>T</mi></math></span>-PQEP, primarily targeting the dominating (least decaying) waves. In this paper we derive an alternative linearization of the <span><math><mi>T</mi></math></span>-PQEP for the <span><math><mrow><mi>k</mi><mrow><mo>(</mo><mi>ω</mi><mo>)</mo></mrow></mrow></math></span> problem, which leads to enhanced targeting of the eigenvalues around the unit circle and reduces the inaccuracies induced by root multiplicity. A specific form of the problem is then proposed as an optimal compromise between ease of implementation, numerical stability, convergence and accuracy enhancement. The performance of our proposed linearization is compared against existing ones across various iterative eigensolvers, since the generalized eigenvalue problems involve complex non-hermitian matrices, which are not extensively included in eigensolvers. Results indicate that the proposed linearization should be favored for the WFEM, as it provides numerical enhancements in dispersion and wave vectors computation for large eigenvalue problems, as well as for further wave expansion applications.</p></div>\",\"PeriodicalId\":17233,\"journal\":{\"name\":\"Journal of Sound and Vibration\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":4.3000,\"publicationDate\":\"2024-08-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Sound and Vibration\",\"FirstCategoryId\":\"5\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022460X24004140\",\"RegionNum\":2,\"RegionCategory\":\"工程技术\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"ACOUSTICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Sound and Vibration","FirstCategoryId":"5","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022460X24004140","RegionNum":2,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"ACOUSTICS","Score":null,"Total":0}
A computationally efficient k(ω)-spectral form for partial dispersion analyses within the wave finite element framework
This paper addresses the computation of frequency-dependent dispersion curves (i.e., ) and wave modes within the framework of the Wave Finite Element Method (WFEM) and in the context of high-dimensional periodic unit cell models. Numerous applications, ranging from phononics to vibroacoustics, now rely on dispersion analyses or wave expansion over a subset of eigensolutions – complex wavenumbers and Bloch waves – resulting from the resolution of an eigenvalue problem with a -palindromic quadratic structure (-PQEP). To exploit the structure of finite element models, various structure-preserving linearizations such as the Zhong-Williams and the -transform have already been developed to achieve partial wave resolution of large -PQEP, primarily targeting the dominating (least decaying) waves. In this paper we derive an alternative linearization of the -PQEP for the problem, which leads to enhanced targeting of the eigenvalues around the unit circle and reduces the inaccuracies induced by root multiplicity. A specific form of the problem is then proposed as an optimal compromise between ease of implementation, numerical stability, convergence and accuracy enhancement. The performance of our proposed linearization is compared against existing ones across various iterative eigensolvers, since the generalized eigenvalue problems involve complex non-hermitian matrices, which are not extensively included in eigensolvers. Results indicate that the proposed linearization should be favored for the WFEM, as it provides numerical enhancements in dispersion and wave vectors computation for large eigenvalue problems, as well as for further wave expansion applications.
期刊介绍:
The Journal of Sound and Vibration (JSV) is an independent journal devoted to the prompt publication of original papers, both theoretical and experimental, that provide new information on any aspect of sound or vibration. There is an emphasis on fundamental work that has potential for practical application.
JSV was founded and operates on the premise that the subject of sound and vibration requires a journal that publishes papers of a high technical standard across the various subdisciplines, thus facilitating awareness of techniques and discoveries in one area that may be applicable in others.