DFG-Project: Numerical solution of quadratic operator eigenvalue problems from continuum mechanics

• Prof. Dr. Thomas Apel

Researcher:

• Dr.rer.nat. Cornelia Pester

Partners: 

• Sven Beuchler
• Dominique Leguillon
• Arnd Meyer
• Serge Nicaise
• Anna-Margarethe Sändig
• Sergey I. Solov'ev
• David Watkins
• Zohar Yosibash

Student assistants in larger subprojects: 

• Mojca Miklavec
• Jan Rosam

 Content: 

• Study of singular parts of solutions of elliptic boundary value problems in domains with concave (re-entrant) corners
• Application in continuum mechanics in the analysis of stress concentrations in the vicinity of corners and crack tips and thus in the prediction of crack initiation and crack propagation
• Consideration of bodies made of anisotropic materials or composites possible
• Determination of singular functions leads to eigenvalue problems for (mostly quadratic) operator pencils with complex eigenvalues and eigenfunctions
• Special structure of the spectrum (Hamiltonian eigenvalue symmetry)
• Of interest are the eigenvalues with smallest positive real part
• Finite element discretization and suitable linearization of the problem leads to a standard eigenvalue problem for a Hamiltonian matrix
• Further development of a priori and a posteriori finite element failure analysis
  Dissertation (C. Pester):
  • Derivation of a residual-based a posteriori error estimator for the solutions of eigenvalue problems for arbitrary operator pencils
  • Model examples: Laplace equation, linear elasticity problem.
    Derivation of the associated eigenvalue problems when considering corner singularities
  • Construction of an adaptive finite element method
  • Special features:
    • Complex solutions can occur
    • Consideration of vector functions
    • Definition domain is generally a part of the sphere surface
    Error estimator theory is also applicable for arbitrary two-dimensional manifolds and independent of the origin of the eigenvalue problem
• Implementation of fast and stable algorithms for solving the algebraic eigenvalue problem by exploiting the structure of the problem
• Effectiveness improvements: Comparison of different methods, comparison of different packages for solving large weakly populated systems of equations
• Verification of all investigations by appropriate numerical tests

 Results: 

• Program package CoCoS
  • different algorithms for solving the discretized eigenvalue problem
  • graphical representation of the eigenfunctions
  • h- and p-version of the finite element method
  • adaptive mesh refinement
  • highly accurate computation of singularity exponents for different example domains:
    
    • polyhedron corner with variable angle (e.g. Fichera corner)
    • edge with variable angle
    •  acute (notch-shaped) crack
    • non-planar crack
    • circular cone
    • Bazant-Estenssoro crack
    • simple crack
  • Interesting facts about the Fichera corner (singularity exponents)
  • Further results are given in the program documentation
• PhD of Cornelia Pester
• Student Research Projects

 Publications: 

• Th. Apel, V. Mehrmann, and D. Watkins Numerical solution of large scale structured polynomial or rational eigenvalue problems. In: F. Cucker, R. DeVore, P. Olver, and E. Süli, editors, Foundations of Computational Mathematics, Minneapolis 2002, volume 312 of Lecture Note Series, Cambridge, 2004. London Mathematical Society, Cambridge University Press. Download
• Th. Apel, A.-M. Sändig, S.I. Solov'ev Computation of 3D vertex singularities for linear elasticity: Error estimates for a finite element method on graded meshes. Math. Model. Numer. Anal., 36:1043-1070, 2002. Download
• Th. Apel, A. Meyer, C. Pester Corrigendum: Computation of 3D vertex singularities for linear elasticity: Error estimates for a finite element method on graded meshes. Download
• Th. Apel, C. Pester Clement-type interpolation on spherical domains - interpolation error estimates and application to a posteriori error estimation. IMA J. Numer. Anal. 25, No.2, 310-336, 2005. Download
• Th. Apel, C. Pester Quadratic eigenvalue problems in the analysis of cracks in brittle materials. Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS). Jyväskylä, 2004. Download
• C. Pester Hamiltonian eigenvalue symmetry for quadratic operator eigenvalue problems. J. Integral Equations Appl., 17:71-89, 2005. Download
• A. Meyer, C. Pester The Laplace and the linear elasticity problems near polyhedral corners and associated eigenvalue problems. Math. Methods Appl. Sci. 30:751-777, 2007. Download
• C. Pester A residual a posteriori error estimator for the eigenvalue problem for the Laplace-Beltrami operator. Preprint SFB393/05-01, Preprint-Reihe des SFB393 der Technischen Universität Chemnitz, 2005. Download
• C. Pester CoCoS -- Computation of corner singularities (Dokumentation zum Programmpaket CoCoS). Preprint SFB393/05-03, Preprint-Reihe des SFB393 der Technischen Universität Chemnitz, 2005. Download
• C. Pester A posteriori error estimation for non-linear eigenvalue problems for differential operators of second order with focus on 3d vertex singularities. Dissertation. Technische Universität Chemnitz, 2006. Download

 Student Research Projects: 

• Chr. Gay Solving of Poisson equations with singularities. Rapport de stage de fin d'études, TU Chemnitz/ENSTA Paris, 2002.
• C. Pester Fehlerschätzer für lineare Eigenwertprobleme. Diplomarbeit. TU Chemnitz, 2002
• J. Rosam Berechnung der Rissgeometrie bei spröden elastischen Körpern. Diplomarbeit, TU Chemnitz, 2004.
• S. Trebesius A singular function method for elliptic boundary value problems in three dimensional domains. Diplomarbeit, TU Chemnitz, 2004.