umu.sePublications
Change search
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf
A Hierarchical POD Reduction Method of Finite Element Models with Application to Simulated Mechanical Systems
Umeå University, Faculty of Science and Technology, Department of Physics.
2012 (English)Independent thesis Advanced level (degree of Master (One Year)), 20 credits / 30 HE creditsStudent thesis
Abstract [en]

When simulating mechanical systems the flexibility of the components often has to be taken into account. This is particularly important for simulations when high detailed information is demanded, e.g. to calculate stresses. To this end the Finite Element Method (FEM) is often used. However the models can become very large, containing millions of degrees of freedom. Solving large linear systems are computationally demanding. Therefore ways of reducing the problem is often sought. These reduction does, however, remove much of the details that was to be investigated. In this thesis this problem is addressed by creating a reduction scheme, using Proper Orthogonal Decomposition (POD), that significantly reduces a problem but still captures much of the details. A novel method for enriching regular POD-based model reduction methods with hierarchically determined enrichment POD-modes is developed. The method is proposed and validated in a FEM application towards dynamical simulation. The enriched method is compared against a regular POD reduction technique. An numerical study is made of a model example of linear elasticity in a gearwheel. The numerical study suggests that the error of displacements is around ten times smaller, on average, when using the enriched basis compared to a reference basis of equal dimensionality consisting of only regular POD modes. Also it is shown that local quantities as the von Mises stress in a gearwheel tooth is preserved much better using the enriched basis. An a posteriori error estimate is proposed and proved for the static case, showing that the error is bound.

Abstract [sv]

När man simulerar mekaniska system så måste man ofta ta hänsyn till de ingående komponenternas flexibilitet. Detta är särskilt viktigt då man gör simuleringar med krav på hög detaljkännedom, såsom mätningar av spänningar i kugghjul etc. Till detta ändamål används ofta en Finit Element Metod (FEM). Dock kan modellerna ofta bli väldigt stora, med över en miljon frihetsgrader. Att lösa linjära system av den storleken är beräkningsmässigt krävande. Därför är det naturligt att försöka reducera problemen. Reduktion innebär dock att information försvinner, i synnerhet de detaljer som skulle beräknas. I detta examensarbete så behandlas problemet genom att skapa en ny metod för reducering av stora finita element modeller. Metoden bygger på tidigare kunskap om Proper Orthogonal Decomposition (POD) som ett sätt att reducera modeller. Den nya metoden reducerar finita ellement modeller samtidigt som den bibehåller hög detalj. En ny metod utvecklas för att berika en vanlig POD-baserad modellreduktion med hjälp av hieraktiskt bestämda berikningsmoder. Metoden beskrivs och testas i en dynamisk FEM-applikation av elasticitet i ett kugghjul i 2 dimensioner. Metoden för berikning jämförs numeriskt med en metod som använder vanlig POD-reduktion. Körningar visar att felet i den berikade metoden är omkring 10 gånger mindre, i genomsnitt, jämfört med en vanlig metod. Det visas också att spänningar bevaras på ett mycket bra sätt med den nya berikningsmetoden. Dessutom så formuleras och bevisas ett a posteriori estimat för statiska lastfall, vilket innebär att felet i metoden är bundet.

Place, publisher, year, edition, pages
2012. , 40 p.
Keyword [en]
Model reduction, FEM, POD, enrichment, hierarchy, global and local behaviour
Keyword [sv]
Modellreduktion, FEM, POD, berikning, hirarki, globalt och lokalt be- teende
National Category
Computational Mathematics Other Physics Topics
Identifiers
URN: urn:nbn:se:umu:diva-55056OAI: oai:DiVA.org:umu-55056DiVA: diva2:525308
Subject / course
Examensarbete i teknisk fysik
Educational program
Master of Science Programme in Engineering Physics
Presentation
2012-03-30, Universitetsklubben, Umeå universitet, Umeå, 15:00 (Swedish)
Uppsok
Physics, Chemistry, Mathematics
Supervisors
Examiners
Available from: 2012-05-07 Created: 2012-05-07 Last updated: 2013-03-22Bibliographically approved

Open Access in DiVA

Bjorklund_MasterThesis_HierPOD(4456 kB)810 downloads
File information
File name FULLTEXT01.pdfFile size 4456 kBChecksum SHA-512
d4428853c748ad8d95d564c3378bbf5a6f31997d7b9be55495c6ac32773dd7d2f4d84c078530ac3d5601070f2b1449c8be56b16666ed6e2449df1115d1818ed3
Type fulltextMimetype application/pdf

Search in DiVA

By author/editor
Björklund, Martin
By organisation
Department of Physics
Computational MathematicsOther Physics Topics

Search outside of DiVA

GoogleGoogle Scholar
Total: 810 downloads
The number of downloads is the sum of all downloads of full texts. It may include eg previous versions that are now no longer available

urn-nbn

Altmetric score

urn-nbn
Total: 251 hits
CiteExportLink to record
Permanent link

Direct link
Cite
Citation style
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Other style
More styles
Language
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Other locale
More languages
Output format
  • html
  • text
  • asciidoc
  • rtf