Computation of 3D vertex singularities for linear elasticity : error estimates for a finite element method on graded meshes

Thomas Apel; Anna-Margarete Sändig; Sergey I. Solov'ev

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2002)

  • Volume: 36, Issue: 6, page 1043-1070
  • ISSN: 0764-583X

Abstract

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This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields with Dirichlet boundary conditions, focusing on the derivation of error estimates for a finite element method on graded meshes. The singularities are described by eigenpairs of a corresponding operator pencil on spherical polygonal domains. The main idea is to introduce a modified quadratic variational boundary eigenvalue problem which consists of two self-adjoint, positive definite sesquilinear forms and a skew-Hermitean form. This eigenvalue problem is discretized by a finite element method on graded meshes. Based on regularity results for the eigensolutions estimates for the finite element error are derived both for the eigenvalues and the eigensolutions. Finally, some numerical results are presented.

How to cite

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Apel, Thomas, Sändig, Anna-Margarete, and Solov'ev, Sergey I.. "Computation of 3D vertex singularities for linear elasticity : error estimates for a finite element method on graded meshes." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 36.6 (2002): 1043-1070. <http://eudml.org/doc/245618>.

@article{Apel2002,
abstract = {This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields with Dirichlet boundary conditions, focusing on the derivation of error estimates for a finite element method on graded meshes. The singularities are described by eigenpairs of a corresponding operator pencil on spherical polygonal domains. The main idea is to introduce a modified quadratic variational boundary eigenvalue problem which consists of two self-adjoint, positive definite sesquilinear forms and a skew-Hermitean form. This eigenvalue problem is discretized by a finite element method on graded meshes. Based on regularity results for the eigensolutions estimates for the finite element error are derived both for the eigenvalues and the eigensolutions. Finally, some numerical results are presented.},
author = {Apel, Thomas, Sändig, Anna-Margarete, Solov'ev, Sergey I.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {quadratic eigenvalue problems; linear elasticity; 3D vertex singularities; finite element methods; error estimates},
language = {eng},
number = {6},
pages = {1043-1070},
publisher = {EDP-Sciences},
title = {Computation of 3D vertex singularities for linear elasticity : error estimates for a finite element method on graded meshes},
url = {http://eudml.org/doc/245618},
volume = {36},
year = {2002},
}

TY - JOUR
AU - Apel, Thomas
AU - Sändig, Anna-Margarete
AU - Solov'ev, Sergey I.
TI - Computation of 3D vertex singularities for linear elasticity : error estimates for a finite element method on graded meshes
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2002
PB - EDP-Sciences
VL - 36
IS - 6
SP - 1043
EP - 1070
AB - This paper is concerned with the computation of 3D vertex singularities of anisotropic elastic fields with Dirichlet boundary conditions, focusing on the derivation of error estimates for a finite element method on graded meshes. The singularities are described by eigenpairs of a corresponding operator pencil on spherical polygonal domains. The main idea is to introduce a modified quadratic variational boundary eigenvalue problem which consists of two self-adjoint, positive definite sesquilinear forms and a skew-Hermitean form. This eigenvalue problem is discretized by a finite element method on graded meshes. Based on regularity results for the eigensolutions estimates for the finite element error are derived both for the eigenvalues and the eigensolutions. Finally, some numerical results are presented.
LA - eng
KW - quadratic eigenvalue problems; linear elasticity; 3D vertex singularities; finite element methods; error estimates
UR - http://eudml.org/doc/245618
ER -

References

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  1. [1] E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney and D. Sorensen, LAPACK Users’ Guide. SIAM, Philadelphia, PA, third edition (1999). Zbl0934.65030
  2. [2] T. Apel, Anisotropic finite elements: Local estimates and applications. Teubner, Stuttgart, Adv. Numer. Math. (1999). Habilitationsschrift. Zbl0934.65121MR1716824
  3. [3] T. Apel, V. Mehrmann and D. Watkins, Structured eigenvalue methods for the computation of corner singularities in 3D anisotropic elastic structures. Comput. Methods Appl. Mech. Engrg. (to appear), Preprint SFB393/01-25, TU Chemnitz (2001). Zbl1029.74042
  4. [4] R.E. Barnhill and J.A. Gregory, Interpolation remainder theory from Taylor expansions on triangles. Numer. Math. 25 (1976) 401–408. Zbl0304.65075
  5. [5] Z.P. Bažant and L.M. Keer, Singularities of elastic stresses and of harmonic functions at conical notches or inclusions. Internat. J. Solids Structures 10 (1974) 957–964. 
  6. [6] A.E. Beagles and A.-M. Sändig, Singularities of rotationally symmetric solutions of boundary value problems for the Lamé equations. ZAMM 71 (1990) 423–431. Zbl0751.73009
  7. [7] P. Benner, R. Byers, V. Mehrmann and H. Xu, Numerical computation of deflating subspaces of embedded Hamiltonian pencils. SIAM J. Matrix Anal. Appl. (to appear), Preprint SFB393/99-15, TU Chemnitz (1999). Zbl1035.49022
  8. [8] M. Costabel and M. Dauge, General edge asymptotics of solutions of second order elliptic boundary value problems I, II. Proc. Roy. Soc. Edinburgh Sect. A 123 (1993) 109–155, 157–184. Zbl0791.35033
  9. [9] M. Dauge, Elliptic boundary value problems on corner domains – smoothness and asymptotics of solutions. Lecture Notes in Math. 1341, Springer, Berlin (1988). Zbl0668.35001
  10. [10] M. Dauge, Singularities of corner problems and problems of corner singularities, in: Actes du 30ème Congrés d’Analyse Numérique: CANum ’98 (Arles, 1998), Soc. Math. Appl. Indust., Paris (1999) 19–40. Zbl0921.35048
  11. [11] M. Dauge, “Simple” corner-edge asymptotics. Internet publication, http://www.maths.univ-rennes1.fr/ dauge/publis/corneredge.pdf (2000). 
  12. [12] J.W. Demmel, J.R. Gilbert and X.S. Li, SuperLU Users’ Guide. Technical Report LBNL-44289, Lawrence Berkeley National Laboratory (1999). 
  13. [13] A. Dimitrov, H. Andrä and E. Schnack, Efficient computation of order and mode of corner singularities in 3d-elasticity. Internat. J. Numer. Methods Engrg. 52 (2001) 805–827. Zbl1043.74042
  14. [14] A. Dimitrov and E. Schnack, Asymptotical expansion in non-Lipschitzian domains: a numerical approach using h -fem. Numer. Linear Algebra Appl. (to appear). Zbl1071.65550MR1934872
  15. [15] P. Grisvard, Elliptic problems in nonsmooth domains. Pitman, Boston–London–Melbourne, Monographs and Studies in Mathematics 21 (1985). Zbl0695.35060
  16. [16] G. Haase, T. Hommel, A. Meyer, and M. Pester, Bibliotheken zur Entwicklung paralleler Algorithmen. Preprint SPC95_20, TU Chemnitz–Zwickau (1995). Updated version of SPC94_4 and SPC93_1. 
  17. [17] H. Jeggle and E. Wendland, On the discrete approximation of eigenvalue problems with holomorphic parameter dependence. Proc. Roy. Soc. Edinburgh Sect. A 78 (1977) 1–29. Zbl0383.65038
  18. [18] O.O. Karma, Approximation of operator functions and convergence of approximate eigenvalues. Tr. Vychisl. Tsentra Tartu. Gosudarst. Univ. 24 (1971) 3–143. In Russian. 
  19. [19] O.O. Karma, Asymptotic error estimates for approximate characteristic value of holomorphic Fredholm operator functions. Zh. Vychisl. Mat. Mat. Fiz. 11 (1971) 559–568. In Russian. Zbl0222.47007
  20. [20] O.O. Karma, Approximation in eigenvalue problems for holomorphic Fredholm operator functions. I. Numer. Funct. Anal. Optim. 17 (1996) 365–387. Zbl0880.47009
  21. [21] O.O. Karma, Approximation in eigenvalue problems for holomorphic Fredholm operator functions. II: Convergence rate. Numer. Funct. Anal. Optim. 17 (1996) 389–408. Zbl0880.47010
  22. [22] V.A. Kondrat’ev, Boundary value problems for elliptic equations on domains with conical or angular points. Tr. Mosk. Mat. Obs. 16 (1967) 209–292. In Russian. Zbl0162.16301
  23. [23] V.A. Kozlov, V.G. Maz’ya and J. Roßmann, Elliptic Boundary Value Problems in Domains with Point Singularities. American Mathematical Society (1997). 
  24. [24] V.A. Kozlov, V.G. Maz’ya and J. Roßmann, Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. American Mathematical Society (2001). Zbl0965.35003
  25. [25] S.G. Krejn and V.P. Trofimov, On holomorphic operator functions of several complex variables. Funct. Anal. Appl. 3 (1969) 85–86. In Russian. English transl. in Funct. Anal. Appl. 3 (1969) 330–331. Zbl0201.16901
  26. [26] S.G. Krejn and V.P. Trofimov, On Fredholm operator depending holomorphically on the parameters. Tr. Seminara po funk. anal. Voronezh univ. (1970) 63–85. Zbl0276.47029
  27. [27] D. Leguillon, Computation of 3D-singularities in elasticity, in: Boundary value problems and integral equations in nonsmooth domains, M. Costabel, M. Dauge and S. Nicaise Eds. New York, Lecture Notes in Pure and Appl. Math. 167 (1995) 161–170. Marcel Dekker. Proceedings of a conference at CIRM, Luminy, France, May 3–7 (1993). Zbl0876.35031
  28. [28] D. Leguillon and E. Sanchez-Palencia, Computation of singular solutions in elliptic problems and elasticity. Masson, Paris (1987). Zbl0647.73010MR995254
  29. [29] R.B. Lehoucq, D.C. Sorensen and C. Yang, ARPACK user’s guide. Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods. SIAM, Philadelphia, PA, Software Environ. Tools 6 (1998). Zbl0901.65021
  30. [30] A.S. Markus, On holomorphic operator functions. Dokl. Akad. Nauk 119 (1958) 1099–1102. In Russian. Zbl0089.32203
  31. [31] A.S. Markus, Introduction to spectral theory of polynomial operator pencils. American Mathematical Society, Providence (1988). Zbl0678.47005MR971506
  32. [32] A.S. Markus and E.I. Sigal, The multiplicity of the characteristic number of an analytic operator function. Mat. Issled. 5 (1970) 129–147. In Russian. Zbl0234.47013
  33. [33] V.G. Maz’ya and B. Plamenevskiĭ, L p -estimates of solutions of elliptic boundary value problems in domains with edges. Tr. Mosk. Mat. Obs. 37 (1978) 49–93. In Russian. English transl. in Trans. Moscow Math. Soc. 1 (1980) 49–97. Zbl0453.35025
  34. [34] V.G. Maz’ya and B. Plamenevskiĭ, The first boundary value problem for classical equations of mathematical physics in domains with piecewise smooth boundaries, part I, II. Z. Anal. Anwendungen 2 (1983) 335–359, 523–551. In Russian. Zbl0554.35099
  35. [35] V.G. Maz’ya and J. Roßmann, Über die Asymptotik der Lösung elliptischer Randwertaufgaben in der Umgebung von Kanten. Math. Nachr. 138 (1988) 27–53. Zbl0672.35020
  36. [36] V.G. Maz’ya and J. Roßmann, On the Agmon-Miranda maximum principle for solutions of elliptic equations in polyhedral and polygonal domains. Ann. Global Anal. Geom. 9 (1991) 253–303. Zbl0753.35013
  37. [37] V.G. Maz’ya and J. Roßmann, On the behaviour of solutions to the dirichlet problem for second order elliptic equations near edges and polyhedral vertices with critical angles. Z. Anal. Anwendungen 13 (1994) 19–47. Zbl0802.35034
  38. [38] V. Mehrmann and D. Watkins, Structure-preserving methods for computing eigenpairs of large sparse skew-Hamiltonian/ Hamiltonian pencils. SIAM J. Sci. Comput. 22 (2001) 1905–1925. Zbl0986.65033
  39. [39] B. Mercier and G. Raugel, Résolution d’un problème aux limites dans un ouvert axisymétrique par éléments finis en r , z et séries de Fourier en θ . RAIRO Anal. Numér. 16 (1982) 405–461. Zbl0531.65054
  40. [40] S.A. Nazarov and B.A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundary. Walter de Gruyter, Berlin, Exposition. Math. 13 (1994). Zbl0806.35001MR1283387
  41. [41] S. Nicaise, Regularity of the solutions of elliptic systems in polyhedral domains. Bull. Belg. Math. Soc. Simon Stevin 4 (1997) 411–429. Zbl0918.35031
  42. [42] M. Pester, Grafik-Ausgabe vom Parallelrechner für 2D-Gebiete. Preprint SPC94_24, TU Chemnitz-Zwickau (1994). 
  43. [43] G. Raugel, Résolution numérique de problèmes elliptiques dans des domaines avec coins. Ph.D. thesis, Université de Rennes, France (1978). 
  44. [44] G. Raugel, Résolution numérique par une méthode d’éléments finis du problème de Dirichlet pour le Laplacien dans un polygone. C. R. Acad. Sci. Paris Sér. I Math. 286 (1978) A791–A794. Zbl0377.65058
  45. [45] A.-M. Sändig and R. Sändig, Singularities of non-rotationally symmetric solutions of boundary value problems for the Lamé equations in a three dimensional domain with conical points. Breitenbrunn, Analysis on manifolds with singularities (1990), Teubner-Texte zur Mathematik, Band 131 (1992) 181–193. Zbl0820.35039
  46. [46] H. Schmitz, K. Volk and W.L. Wendland, On three-dimensional singularities of elastic fields near vertices. Numer. Methods Partial Differential Equations 9 (1993) 323–337. Zbl0771.73014
  47. [47] V. Staroverov, G. Kobelkov, E. Schnack and A. Dimitrov, On numerical methods for flat crack propagation. IMF-Preprint 99-2, Universität Karlsruhe (1999). 
  48. [48] V.P. Trofimov, The root subspaces of operators that depend analytically on a parameter. Mat. Issled. 3 (1968) 117–125. In Russian. Zbl0234.47010
  49. [49] G.M. Vainikko and O.O. Karma, Convergence rate of approximate methods in an eigenvalue problem with a parameter entering nonlinearly. Zh. Vychisl. Mat. Mat. Fiz. 14 (1974) 1393–1408. In Russian. Zbl0339.65030

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