Accurate Numerical Approximations of Eigenfrequencies and Eigenfunctions of Elliptic Membranes
Identifieur interne : 000A16 ( Istex/Corpus ); précédent : 000A15; suivant : 000A17Accurate Numerical Approximations of Eigenfrequencies and Eigenfunctions of Elliptic Membranes
Auteurs : Jörg Hettich ; E. Haaren ; M. Ries ; G. StillSource :
- ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik [ 0044-2267 ] ; 1987.
Abstract
In earlier papers [3, 4] a defect‐minimization method was proposed to compute approximate eigenfunctions of membranes by means of a parametric semi‐infinite optimization problem. The method gives approximations and error bounds of eigenvalues and eigenfunctions. An algorithm based on this method has been implemented for the special case of elliptic membranes with variable eccentricity. In this paper we show that by this method we obtain very accurate results even for eccentricities near one. In addition the method is very appropriate to study the behavior of eigenfunctions.
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DOI: 10.1002/zamm.19870671201
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Haaren</name><affiliation><mods:affiliation>Universität Trier, FB IV – Mathematik, Postfach 3825, D‐5500 Trier, F.R.G.</mods:affiliation></affiliation></author><author><name sortKey="Ries, M" sort="Ries, M" uniqKey="Ries M" first="M." last="Ries">M. Ries</name><affiliation><mods:affiliation>Universität Trier, FB IV – Mathematik, Postfach 3825, D‐5500 Trier, F.R.G.</mods:affiliation></affiliation></author><author><name sortKey="Still, G" sort="Still, G" uniqKey="Still G" first="G." last="Still">G. Still</name><affiliation><mods:affiliation>Universität Trier, FB IV – Mathematik, Postfach 3825, D‐5500 Trier, F.R.G.</mods:affiliation></affiliation></author></analytic><monogr></monogr><series><title level="j">ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik</title><title level="j" type="abbrev">Z. angew. Math. 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The method gives approximations and error bounds of eigenvalues and eigenfunctions. An algorithm based on this method has been implemented for the special case of elliptic membranes with variable eccentricity. In this paper we show that by this method we obtain very accurate results even for eccentricities near one. In addition the method is very appropriate to study the behavior of eigenfunctions.</div></front></TEI><istex><corpusName>wiley</corpusName><author><json:item><name>Professor Dr. Hettich</name><affiliations><json:string>Universität Trier, FB IV – Mathematik, Postfach 3825, D‐5500 Trier, F.R.G.</json:string></affiliations></json:item><json:item><name>E. Haaren</name><affiliations><json:string>Universität Trier, FB IV – Mathematik, Postfach 3825, D‐5500 Trier, F.R.G.</json:string></affiliations></json:item><json:item><name>M. Ries</name><affiliations><json:string>Universität Trier, FB IV – Mathematik, Postfach 3825, D‐5500 Trier, F.R.G.</json:string></affiliations></json:item><json:item><name>G. Still</name><affiliations><json:string>Universität Trier, FB IV – Mathematik, Postfach 3825, D‐5500 Trier, F.R.G.</json:string></affiliations></json:item></author><articleId><json:string>ZAMM19870671201</json:string></articleId><language><json:string>eng</json:string></language><originalGenre><json:string>article</json:string></originalGenre><abstract>In earlier papers [3, 4] a defect‐minimization method was proposed to compute approximate eigenfunctions of membranes by means of a parametric semi‐infinite optimization problem. The method gives approximations and error bounds of eigenvalues and eigenfunctions. An algorithm based on this method has been implemented for the special case of elliptic membranes with variable eccentricity. In this paper we show that by this method we obtain very accurate results even for eccentricities near one. In addition the method is very appropriate to study the behavior of eigenfunctions.</abstract><qualityIndicators><score>4.073</score><pdfVersion>1.3</pdfVersion><pdfPageSize>576 x 828 pts</pdfPageSize><refBibsNative>true</refBibsNative><abstractCharCount>580</abstractCharCount><pdfWordCount>3041</pdfWordCount><pdfCharCount>14816</pdfCharCount><pdfPageCount>9</pdfPageCount><abstractWordCount>86</abstractWordCount></qualityIndicators><title>Accurate Numerical Approximations of Eigenfrequencies and Eigenfunctions of Elliptic Membranes</title><refBibs><json:item><host><author></author><title>Abramowitz, M.;Stegun, I. A.,Handbook of Mathematical Functions,Nat. Bur. Standards,Washington, DC1964.</title></host></json:item><json:item><author><json:item><name>R. Hettich</name></json:item></author><host><pages><last>89</last><first>79</first></pages><author></author><title>Infinite Programming, Springer Lecture Notes in Economics and Mathem</title></host><title>On the computation of membrane‐eigenvalues by semi‐infinite programming</title></json:item><json:item><host><author></author><title>Local aspects of a method for solving membrane‐eigenvalue problems by parametric semi‐infinite programming</title></host></json:item><json:item><author><json:item><name>R. Hettich</name></json:item><json:item><name>P. Zencke</name></json:item></author><host><pages><first>132</first></pages><author></author><title>Systems and Optimization, Springer Lecture Notes in Control and Information</title></host><title>Two case‐studies in parametric semi‐infinite programming</title></json:item><json:item><author><json:item><name>R. Mifflin</name></json:item></author><host><volume>28</volume><pages><last>71</last><first>50</first></pages><author></author><title>Math. Programming</title></host><title>Stationarity and superlinear convergence of an algorithm for univariate locally Lipschitz constrained minimization</title></json:item><json:item><author><json:item><name>C. Moler</name></json:item><json:item><name>L. F. Payne</name></json:item></author><host><volume>5</volume><pages><last>70</last><first>64</first></pages><author></author><title>SIAM J. Numer. Anal.</title></host><title>Bounds for eigenvalues and eigenvectors of symmetric operators</title></json:item><json:item><host><author></author><title>Tables Relating to Mathieu Functions, Nat. Bur. Standards</title></host></json:item><json:item><author><json:item><name> </name></json:item></author><host><volume>7</volume><pages><last>61</last><first>39</first></pages><author></author><title>J. Engineering Math.</title></host><title>“Numerical Analysis” at Delft University of Technology, On the computation of Mathieu functions</title></json:item><json:item><author><json:item><name>B. A. Troesch</name></json:item><json:item><name>H. R. Troesch</name></json:item></author><host><volume>27</volume><pages><last>765</last><first>755</first></pages><author></author><title>Math. 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KGaA, Weinheim</p></availability><date>1987</date></publicationStmt><sourceDesc><biblStruct type="inbook"><analytic><title level="a" type="main" xml:lang="en">Accurate Numerical Approximations of Eigenfrequencies and Eigenfunctions of Elliptic Membranes</title><author xml:id="author-1"><persName><surname>Hettich</surname></persName><roleName type="degree">Professor Dr.</roleName><affiliation>Universität Trier, FB IV – Mathematik, Postfach 3825, D‐5500 Trier, F.R.G.</affiliation></author><author xml:id="author-2"><persName><forename type="first">E.</forename><surname>Haaren</surname></persName><affiliation>Universität Trier, FB IV – Mathematik, Postfach 3825, D‐5500 Trier, F.R.G.</affiliation></author><author xml:id="author-3"><persName><forename type="first">M.</forename><surname>Ries</surname></persName><affiliation>Universität Trier, FB IV – Mathematik, Postfach 3825, D‐5500 Trier, F.R.G.</affiliation></author><author xml:id="author-4"><persName><forename type="first">G.</forename><surname>Still</surname></persName><affiliation>Universität Trier, FB IV – Mathematik, Postfach 3825, D‐5500 Trier, F.R.G.</affiliation></author></analytic><monogr><title level="j">ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik</title><title level="j" type="abbrev">Z. angew. Math. Mech.</title><idno type="pISSN">0044-2267</idno><idno type="eISSN">1521-4001</idno><idno type="DOI">10.1002/(ISSN)1521-4001</idno><imprint><publisher>WILEY‐VCH Verlag</publisher><pubPlace>Berlin</pubPlace><date type="published" when="1987"></date><biblScope unit="volume">67</biblScope><biblScope unit="issue">12</biblScope><biblScope unit="page" from="589">589</biblScope><biblScope unit="page" to="597">597</biblScope></imprint></monogr><idno type="istex">666106E33616CBF47958F575A675E2D93AA844B2</idno><idno type="DOI">10.1002/zamm.19870671201</idno><idno type="ArticleID">ZAMM19870671201</idno></biblStruct></sourceDesc></fileDesc><profileDesc><creation><date>1987</date></creation><langUsage><language ident="en">en</language></langUsage><abstract xml:lang="en"><p>In earlier papers [3, 4] a defect‐minimization method was proposed to compute approximate eigenfunctions of membranes by means of a parametric semi‐infinite optimization problem. The method gives approximations and error bounds of eigenvalues and eigenfunctions. An algorithm based on this method has been implemented for the special case of elliptic membranes with variable eccentricity. In this paper we show that by this method we obtain very accurate results even for eccentricities near one. In addition the method is very appropriate to study the behavior of eigenfunctions.</p></abstract><abstract xml:lang="de"><p>In vorangegangenen Arbeiten [3, 4] wurde ein Defekt‐Minimierungsverfahren zur Berechnung genäherter Eigenfunktionen für Membranen mittels parametrischer semi‐infiniter Optimierung vorgeschlagen. Das Verfahren liefert Approximationen und Fehlerschranken für Eigenwerte und Eigenfunktionen. Für den Spezialfall elliptischer Membranen mit variabler Exzentrizität wurde ein auf diesem Verfahren basierender Algorithmus implementiert. In dieser Arbeit wird gezeigt, daß man mit diesem Verfahren selbst für Exzentrizitäten nahe Eins sehr genaue Ergebnisse erhält. Darüber hinaus ist dieses Verfahren sehr gut zur Untersuchung des Verhaltens von Eigenfunktionen geeignet.</p></abstract><textClass><keywords scheme="Journal Subject"><list><head>article-category</head><item><term>Article</term></item></list></keywords></textClass></profileDesc><revisionDesc><change when="1987-03-16">Received</change><change when="1987">Published</change></revisionDesc></teiHeader></istex:fulltextTEI><json:item><extension>txt</extension><original>false</original><mimetype>text/plain</mimetype><uri>https://api.istex.fr/document/666106E33616CBF47958F575A675E2D93AA844B2/fulltext/txt</uri></json:item></fulltext><metadata><istex:metadataXml wicri:clean="Wiley, elements deleted: body"><istex:xmlDeclaration>version="1.0" encoding="UTF-8" standalone="yes"</istex:xmlDeclaration><istex:document><component version="2.0" type="serialArticle" xml:lang="en"><header><publicationMeta level="product"><publisherInfo><publisherName>WILEY‐VCH Verlag</publisherName><publisherLoc>Berlin</publisherLoc></publisherInfo><doi registered="yes">10.1002/(ISSN)1521-4001</doi><issn type="print">0044-2267</issn><issn type="electronic">1521-4001</issn><idGroup><id type="product" value="ZAMM"></id></idGroup><titleGroup><title type="main" xml:lang="en" sort="ZAMM ZEITSCHRIFT FUER ANGEWANDTE MATHEMATIK UND MECHANIK">ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik</title><title type="tocForm">ZAMM ‐ Zeitschrift für Angewandte Mathematik und Mechanik / Journal of Applied Mathematics and Mechanics</title><title type="short">Z. angew. 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The method gives approximations and error bounds of eigenvalues and eigenfunctions. An algorithm based on this method has been implemented for the special case of elliptic membranes with variable eccentricity. In this paper we show that by this method we obtain very accurate results even for eccentricities near one. In addition the method is very appropriate to study the behavior of eigenfunctions.</p></abstract><abstract type="main" xml:lang="de"><p>In vorangegangenen Arbeiten [3, 4] wurde ein Defekt‐Minimierungsverfahren zur Berechnung genäherter Eigenfunktionen für Membranen mittels parametrischer semi‐infiniter Optimierung vorgeschlagen. Das Verfahren liefert Approximationen und Fehlerschranken für Eigenwerte und Eigenfunktionen. Für den Spezialfall elliptischer Membranen mit variabler Exzentrizität wurde ein auf diesem Verfahren basierender Algorithmus implementiert. In dieser Arbeit wird gezeigt, daß man mit diesem Verfahren selbst für Exzentrizitäten nahe Eins sehr genaue Ergebnisse erhält. Darüber hinaus ist dieses Verfahren sehr gut zur Untersuchung des Verhaltens von Eigenfunktionen geeignet.</p></abstract></abstractGroup></contentMeta></header></component></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Accurate Numerical Approximations of Eigenfrequencies and Eigenfunctions of Elliptic Membranes</title></titleInfo><titleInfo type="alternative" contentType="CDATA" lang="en"><title>Accurate Numerical Approximations of Eigenfrequencies and Eigenfunctions of Elliptic Membranes</title></titleInfo><name type="personal"><namePart type="termsOfAddress">Professor Dr.</namePart><namePart type="family">Hettich</namePart><affiliation>Universität Trier, FB IV – Mathematik, Postfach 3825, D‐5500 Trier, F.R.G.</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">E.</namePart><namePart type="family">Haaren</namePart><affiliation>Universität Trier, FB IV – Mathematik, Postfach 3825, D‐5500 Trier, F.R.G.</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">M.</namePart><namePart type="family">Ries</namePart><affiliation>Universität Trier, FB IV – Mathematik, Postfach 3825, D‐5500 Trier, F.R.G.</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">G.</namePart><namePart type="family">Still</namePart><affiliation>Universität Trier, FB IV – Mathematik, Postfach 3825, D‐5500 Trier, F.R.G.</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="article" displayLabel="article"></genre><originInfo><publisher>WILEY‐VCH Verlag</publisher><place><placeTerm type="text">Berlin</placeTerm></place><dateIssued encoding="w3cdtf">1987</dateIssued><dateCaptured encoding="w3cdtf">1987-03-16</dateCaptured><copyrightDate encoding="w3cdtf">1987</copyrightDate></originInfo><language><languageTerm type="code" authority="rfc3066">en</languageTerm><languageTerm type="code" authority="iso639-2b">eng</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType><extent unit="figures">8</extent><extent unit="tables">1</extent><extent unit="references">9</extent></physicalDescription><abstract lang="en">In earlier papers [3, 4] a defect‐minimization method was proposed to compute approximate eigenfunctions of membranes by means of a parametric semi‐infinite optimization problem. The method gives approximations and error bounds of eigenvalues and eigenfunctions. An algorithm based on this method has been implemented for the special case of elliptic membranes with variable eccentricity. In this paper we show that by this method we obtain very accurate results even for eccentricities near one. In addition the method is very appropriate to study the behavior of eigenfunctions.</abstract><abstract lang="de">In vorangegangenen Arbeiten [3, 4] wurde ein Defekt‐Minimierungsverfahren zur Berechnung genäherter Eigenfunktionen für Membranen mittels parametrischer semi‐infiniter Optimierung vorgeschlagen. Das Verfahren liefert Approximationen und Fehlerschranken für Eigenwerte und Eigenfunktionen. Für den Spezialfall elliptischer Membranen mit variabler Exzentrizität wurde ein auf diesem Verfahren basierender Algorithmus implementiert. In dieser Arbeit wird gezeigt, daß man mit diesem Verfahren selbst für Exzentrizitäten nahe Eins sehr genaue Ergebnisse erhält. 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|area= UnivTrevesV1
|flux= Istex
|étape= Corpus
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|clé= ISTEX:666106E33616CBF47958F575A675E2D93AA844B2
|texte= Accurate Numerical Approximations of Eigenfrequencies and Eigenfunctions of Elliptic Membranes
}}
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