Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T21:20:17.251Z Has data issue: false hasContentIssue false
  • English
  • Français

Computation of 3D vertex singularities for linear elasticity: Error estimates for a finite element methodon graded meshes

Published online by Cambridge University Press:  15 January 2003

Thomas Apel ,
Anna-Margarete Sändig  and
Sergey I. Solov'ev
Thomas Apel
Affiliation:
TU Chemnitz, Fakultät für Mathematik, 09107 Chemnitz, Germany. apel@mathematik.tu-chemnitz.de.
Anna-Margarete Sändig
Affiliation:
Universität Stuttgart, Mathematisches Institut A, Pfaffenwaldring 57, 70511 Stuttgart, Germany. saendig@mathematik.uni-stuttgart.de.
Sergey I. Solov'ev
Affiliation:
Kazan State University, Faculty of computer science and cybernetics, Kremlevskaya 18, 420008 Kazan, Russia. sergei.solovyev@ksu.ru.
Get access

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.

Keywords

Quadratic eigenvalue problemslinear elasticity3D vertex singularitiesfinite element methodserror estimates.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 36 , Issue 6 , November 2002 , pp. 1043 - 1070
Copyright
© EDP Sciences, SMAI, 2002

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

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).
T. Apel, Anisotropic finite elements: Local estimates and applications. Teubner, Stuttgart, Adv. Numer. Math. (1999). Habilitationsschrift.
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).
Barnhill, R.E. and Gregory, J.A., Interpolation remainder theory from Taylor expansions on triangles. Numer. Math. 25 (1976) 401-408. CrossRef
Bazant, Z.P. and Keer, L.M., Singularities of elastic stresses and of harmonic functions at conical notches or inclusions. Internat. J. Solids Structures 10 (1974) 957-964. CrossRef
Beagles, A.E. and Sändig, A.-M., Singularities of rotationally symmetric solutions of boundary value problems for the Lamé equations. ZAMM 71 (1990) 423-431. CrossRef
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).
Costabel, M. and Dauge, M., 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. CrossRef
M. Dauge, Elliptic boundary value problems on corner domains - smoothness and asymptotics of solutions. Lecture Notes in Math. 1341 , Springer, Berlin (1988).
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.
M. Dauge, ``Simple'' corner-edge asymptotics. Internet publication,http://www.maths.univ-rennes1.fr/~dauge/publis/corneredge.pdf (2000).
J.W. Demmel, J.R. Gilbert and X.S. Li, SuperLU Users' Guide. Technical Report LBNL-44289, Lawrence Berkeley National Laboratory (1999).
Dimitrov, A., Andrä, H. and Schnack, E., Efficient computation of order and mode of corner singularities in 3d-elasticity. Internat. J. Numer. Methods Engrg. 52 (2001) 805-827. CrossRef
A. Dimitrov and E. Schnack, Asymptotical expansion in non-Lipschitzian domains: a numerical approach using h-fem. Numer. Linear Algebra Appl. (to appear).
P. Grisvard, Elliptic problems in nonsmooth domains. Pitman, Boston-London-Melbourne, Monographs and Studies in Mathematics 21 (1985).
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.
Jeggle, H. and Wendland, E., On the discrete approximation of eigenvalue problems with holomorphic parameter dependence. Proc. Roy. Soc. Edinburgh Sect. A 78 (1977) 1-29. CrossRef
Karma, O.O., Approximation of operator functions and convergence of approximate eigenvalues. Tr. Vychisl. Tsentra Tartu. Gosudarst. Univ. 24 (1971) 3-143. In Russian.
Karma, O.O., Asymptotic error estimates for approximate characteristic value of holomorphic Fredholm operator functions. Zh. Vychisl. Mat. Mat. Fiz. 11 (1971) 559-568. In Russian.
Karma, O.O., Approximation in eigenvalue problems for holomorphic Fredholm operator functions. I. Numer. Funct. Anal. Optim. 17 (1996) 365-387. CrossRef
Karma, O.O., Approximation in eigenvalue problems for holomorphic Fredholm operator functions. II: Convergence rate. Numer. Funct. Anal. Optim. 17 (1996) 389-408. CrossRef
Kondrat'ev, V.A., Boundary value problems for elliptic equations on domains with conical or angular points. Tr. Mosk. Mat. Obs. 16 (1967) 209-292. In Russian.
V.A. Kozlov, V.G. Maz'ya and J. Roßmann, Elliptic Boundary Value Problems in Domains with Point Singularities. American Mathematical Society (1997).
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).
Krejn, S.G. and Trofimov, V.P., 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.
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.
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).
D. Leguillon and E. Sanchez-Palencia, Computation of singular solutions in elliptic problems and elasticity. Masson, Paris (1987).
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).
Markus, A.S., On holomorphic operator functions. Dokl. Akad. Nauk 119 (1958) 1099-1102. In Russian.
A.S. Markus, Introduction to spectral theory of polynomial operator pencils. American Mathematical Society, Providence (1988).
Markus, A.S. and Sigal, E.I., The multiplicity of the characteristic number of an analytic operator function. Mat. Issled. 5 (1970) 129-147. In Russian.
Maz'ya, V.G. and Plamenevskiĭ, B., 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.
Maz'ya, V.G. and Plamenevskiĭ, B., 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. CrossRef
Maz'ya, V.G. and Roßmann, J., Über die Asymptotik der Lösung elliptischer Randwertaufgaben in der Umgebung von Kanten. Math. Nachr. 138 (1988) 27-53. CrossRef
Maz'ya, V.G. and Roßmann, J., On the Agmon-Miranda maximum principle for solutions of elliptic equations in polyhedral and polygonal domains. Ann. Global Anal. Geom. 9 (1991) 253-303. CrossRef
Maz'ya, V.G. and Roßmann, J., 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. CrossRef
Mehrmann, V. and Watkins, D., Structure-preserving methods for computing eigenpairs of large sparse skew-Hamiltonian/ Hamiltonian pencils. SIAM J. Sci. Comput. 22 (2001) 1905-1925. CrossRef
Mercier, B. and Raugel, G., 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. CrossRef
S.A. Nazarov and B.A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundary. Walter de Gruyter, Berlin, Exposition. Math. 13 (1994).
Nicaise, S., Regularity of the solutions of elliptic systems in polyhedral domains. Bull. Belg. Math. Soc. Simon Stevin 4 (1997) 411-429.
M. Pester, Grafik-Ausgabe vom Parallelrechner für 2D-Gebiete. Preprint SPC94_24, TU Chemnitz-Zwickau (1994).
G. Raugel, Résolution numérique de problèmes elliptiques dans des domaines avec coins. Ph.D. thesis, Université de Rennes, France (1978).
Raugel, G., 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.
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.
Schmitz, H., Volk, K. and Wendland, W.L., On three-dimensional singularities of elastic fields near vertices. Numer. Methods Partial Differential Equations 9 (1993) 323-337. CrossRef
V. Staroverov, G. Kobelkov, E. Schnack and A. Dimitrov, On numerical methods for flat crack propagation. IMF-Preprint 99-2, Universität Karlsruhe (1999).
Trofimov, V.P., The root subspaces of operators that depend analytically on a parameter. Mat. Issled. 3 (1968) 117-125. In Russian.
Vainikko, G.M. and Karma, O.O., 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.