×

Structured eigenvalue methods for the computation of corner singularities in 3D anisotropic elastic structures. (English) Zbl 1029.74042

Summary: This paper is concerned with the computation of three-dimensional vertex singularities of anisotropic elastic fields. The singularities are described by eigenpairs of a corresponding operator pencil on a subdomain of the sphere. The solution approach is to introduce a modified quadratic variational boundary eigenvalue problem which consists of two self-adjoint, positive definite sesquilinear forms and a skew-Hermitian form. This eigenvalue problem is discretized by finite element method. The resulting quadratic matrix eigenvalue problem is then solved by skew Hamiltonian implicitly restarted Arnoldi method which is specially adapted to the structure of this problem. Some numerical examples are given that show the performance of this approach.

MSC:

74S05 Finite element methods applied to problems in solid mechanics
74G70 Stress concentrations, singularities in solid mechanics
74E10 Anisotropy in solid mechanics
74B05 Classical linear elasticity
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] 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, Preprint, SFB393/01-33, TU Chemnitz 2001; 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, Preprint, SFB393/01-33, TU Chemnitz 2001
[2] Bai, Z.; Sleijpen, G.; van der Vorst, H., Quadratic eigenvalue problems (Section 9.2), (Bai, Z.; Demmel, J.; Dongarra, J.; Ruhe, A.; van der Vorst, H., Templates for the Solution of Algebraic Eigenvalue Problems: A Practical Guide (2000), SIAM: SIAM Philadelphia), 281-289
[3] Bažant, Z. P.; Estenssoro, L. F., Surface singularity and crack propagation, Int. J. Solids Struct., 15, 405-426 (1979) · Zbl 0424.73086
[4] Benner, P.; Byers, R.; Faßbender, H.; Mehrmann, V.; Watkins, D., Cholesky-like factorizations of skew-symmetric matrices, ETNA, 11, 85-93 (2000) · Zbl 0963.65033
[5] P. Benner, R. Byers, V. Mehrmann, H. Xu, Numerical computation of deflating subspaces of embedded Hamiltonian pencils, SIAM J. Matrix Anal. Appl., to appear; P. Benner, R. Byers, V. Mehrmann, H. Xu, Numerical computation of deflating subspaces of embedded Hamiltonian pencils, SIAM J. Matrix Anal. Appl., to appear · Zbl 1035.49022
[6] Dauge, M., Elliptic Boundary Value Problems on Corner Domains-Smoothness and Asymptotics of Solutions, (Lecture Notes in Mathematics, vol. 1341 (1988), Springer: Springer Berlin) · Zbl 0668.35001
[7] T.A. Davis, UMFPACK Version 3.0 User Guide, Dept. of Computer and Information Science and Engineering, Univ. of Florida, Gainesville, FL, 2001; T.A. Davis, UMFPACK Version 3.0 User Guide, Dept. of Computer and Information Science and Engineering, Univ. of Florida, Gainesville, FL, 2001
[8] J.W. Demmel, J.R. Gilbert, X.S. Li, SuperLU Users’ Guide, Technical Report Institution LBNL-44289, Lawrence Berkeley National Laboratory, 1999; J.W. Demmel, J.R. Gilbert, X.S. Li, SuperLU Users’ Guide, Technical Report Institution LBNL-44289, Lawrence Berkeley National Laboratory, 1999
[9] Dimitrov, A.; Andrä, H.; Schnack, E., Efficient computation of order and mode of corner singularities in 3D-elasticity, Int. J. Numer. Meth. Engrg., 52, 805-827 (2001) · Zbl 1043.74042
[10] Glushkov, E. V.; Glushkova, N. V.; Lapina, O. N., 3D elastic stress singularity at polyhedral corner points, Int. J. Solids Struct., 36, 1105-1128 (1999) · Zbl 0927.74027
[11] Grisvard, P., Elliptic Problems in Nonsmooth Domains, (Monographs and Studies in Mathematics, vol. 21 (1985), Pitman: Pitman Boston) · Zbl 0695.35060
[12] Grisvard, P., Singularité en elasticité, Arch. Ration. Mech., 107, 157-180 (1989) · Zbl 0706.73013
[13] G. Haase, Th. Hommel, A. Meyer, M. Pester, Bibliotheken zur Entwicklung paralleler Algorithmen, Preprint, SPC95_20, TU Chemnitz-Zwickau, 1995, updated version of SPC94_4 and SPC93_1; G. Haase, Th. Hommel, A. Meyer, M. Pester, Bibliotheken zur Entwicklung paralleler Algorithmen, Preprint, SPC95_20, TU Chemnitz-Zwickau, 1995, updated version of SPC94_4 and SPC93_1
[14] Kågström, B.; Ruhe, A., An algorithm for numerical computation of the Jordan normal form of a complex matrix, ACM Trans. Math. Software, 6, 398-419 (1980) · Zbl 0434.65020
[15] Kågström, B.; Ruhe, A., Algorithm 560: JNF, an algorithm for numerical computation of the Jordan normal form of a complex matrix, ACM Trans. Math. Software, 6, 398-419 (1980) · Zbl 0434.65020
[16] Karma, O. O., Approximation in eigenvalue problems for holomorphic Fredholm operator functions. II: Convergence rate, Numer. Funct. Anal. Optim., 17, 389-408 (1996) · Zbl 0880.47010
[17] Kondrat’ev, V. A., Boundary value problems for elliptic equations on domains with conical or angular points, Trudy Moskov. Mat. Obshch., 16, 209-292 (1967), (in Russian)
[18] Konstantinov, M. M.; Mehrmann, V.; Petkov, P. Hr., Perturbation theory for the Hamiltonian Schur form, SIAM J. Matrix Anal. Appl., 23, 387-424 (2002) · Zbl 1016.15007
[19] Kozlov, V. A.; Maz’ya, V. G., Spectral properties of operator pencils, generated through elliptic boundary value problems in a cone, Funkcionalniı̆ analis i ego priloshenija, 2, 38-46 (1998), (in Russian)
[20] Kozlov, V. A.; Maz’ya, V. G.; Roßmann, J., Elliptic Boundary Value Problems in Domains with Point Singularities (1997), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0947.35004
[21] Kozlov, V. A.; Maz’ya, V. G.; Roßmann, J., Spectral properties of operator pencils generated by elliptic boundary value problems for the Lamé system, Rostocker Math. Kollog., 51, 5-24 (1997) · Zbl 0910.35083
[22] Kozlov, V. A.; Maz’ya, V. G.; Roßmann, J., Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations (2001), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0965.35003
[23] Kozlov, V. A.; Maz’ya, V. G.; Schwab, C., On singularities of solutions of the displacement problem of linear elasticity near the vertex of a cone, Arch. Ration. Mech. Anal., 119, 197-227 (1992) · Zbl 0770.73014
[24] Kufner, A.; Sändig, A.-M., Some Applications of Weighted Sobolev Spaces (1987), Teubner: Teubner Leipzig · Zbl 0662.46034
[25] Leguillon, D., Computation of 3D-singularities in elasticity, (Costabel, M.; Dauge, M.; Nicaise, S., Boundary Value Problems and Integral Equations in Nonsmooth Domains. Boundary Value Problems and Integral Equations in Nonsmooth Domains, Lect. Notes Pure Appl. Math., vol. 167 (1995), Marcel Dekker: Marcel Dekker New York), 161-170, Proceedings of a Conference at CIRM, Luminy, France, May 3-7, 1993 · Zbl 0876.35031
[26] Lehoucq, R. B.; Sorensen, D. C.; Yang, C., ARPACK User’s Guide. Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, (Software-Environments-Tools, vol. 6 (1998), SIAM: SIAM Philadelphia, PA) · Zbl 0901.65021
[27] Meerbergen, K., Locking and restarting quadratic eigenvalue solvers, SIAM J. Sci. Comput., 22, 1814-1839 (2001) · Zbl 0985.65027
[28] Meerbergen, K.; Tisseur, F., The quadratic eigenvalue problem, SIAM Rev., 43, 235-286 (2001) · Zbl 0985.65028
[29] C. Mehl, Compatible Lie and Jordan algebras and applications to structured matrices and pencils, Ph.D. thesis, TU Chemnitz, 1998; C. Mehl, Compatible Lie and Jordan algebras and applications to structured matrices and pencils, Ph.D. thesis, TU Chemnitz, 1998
[30] Mehl, C., Condensed forms for skew-Hamiltonian/Hamiltonian pencils, SIAM J. Matrix Anal. Appl., 21, 454-476 (1999) · Zbl 0947.15003
[31] Mehrmann, V.; Watkins, D., Structure-preserving methods for computing eigenpairs of large sparse skew-Hamiltonian/Hamiltonian pencils, SIAM J. Sci. Comput., 22, 1905-1925 (2001) · Zbl 0986.65033
[32] M. Pester, Grafik-Ausgabe vom Parallelrechner für 2D-Gebiete, Preprint, SPC94_24 TU Chemnitz-Zwickau, 1994; M. Pester, Grafik-Ausgabe vom Parallelrechner für 2D-Gebiete, Preprint, SPC94_24 TU Chemnitz-Zwickau, 1994
[33] U. Reichel, NetMake-C-Subroutinensammlung zur Netzgenerierung, TU Chemnitz, Fak. f. Mathematik, 1999; U. Reichel, NetMake-C-Subroutinensammlung zur Netzgenerierung, TU Chemnitz, Fak. f. Mathematik, 1999
[34] Rössle, A.; Sändig, A.-M., Stress singularities in bonded dissimilar materials under mechanical and thermal loading, Comput. Mater. Sci., 7, 48-55 (1996)
[35] Schmitz, H.; Volk, K.; Wendland, W. L., On three-dimensional singularities of elastic fields near vertices, Numer. Meth. Partial Diff. Equat., 9, 323-337 (1993) · Zbl 0771.73014
[36] Sleijpen, G. L.G.; van der Vorst, H. A.; van Gijzen, M. B., Quadratic eigenproblems are no problem, SIAM News, 29, 8-9 (1996)
[37] Sorensen, D. C., Implicit application of polynomial filters in a \(k\)-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13, 357-385 (1992) · Zbl 0763.65025
[38] V. Staroverov, G. Kobelkov, E. Schnack, A. Dimitrov, On numerical methods for flat crack propagation, IMF-Oreorubnt 99-2, Universität Karlsruhe, 1999; V. Staroverov, G. Kobelkov, E. Schnack, A. Dimitrov, On numerical methods for flat crack propagation, IMF-Oreorubnt 99-2, Universität Karlsruhe, 1999
[39] Stewart, G. W.; Sun, J.-G., Matrix Perturbation Theory (1990), Academic Press: Academic Press Boston, MA
[40] Tisseur, F., Stability of structured Hamiltonian eigensolvers, SIAM J. Matrix Anal. Appl., 23, 103-125 (2001) · Zbl 0996.65041
[41] K. Volk, Zur Berechnung von Singulärfunktionen dreidimensionaler elastischer Felder, Ph.D. thesis, Universität Stuttgart, 1989; K. Volk, Zur Berechnung von Singulärfunktionen dreidimensionaler elastischer Felder, Ph.D. thesis, Universität Stuttgart, 1989 · Zbl 0704.73013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.