论文信息 - Computational aspects of the symmetric eigenvalue problem of second order tensors

Computational aspects of the symmetric eigenvalue problem of second order tensors

The symmetric eigenvalue problem of second order tensors or 3 x 3 matrices, respectively, is a frequently treated topic in the field of Computational Mechanics because there exist analytical solutions. However, there are several known difficulties in the numerical computation of the analytical formulae which are recapitulated in this article. In order to show this, a sensitivity analysis is applied and a comparison of different procedures to calculate the eigenvalues, eigenvectors and eigendyades is carried out.

Stefan Hartmann | S. Hartmann

[1] V. Smirnov. Lehrgang der höheren mathematik , 1963 .

[2] W. Schmeidler,et al. W. I. Smirnow, Lehrgang der Höheren Mathematik, Teil V. XIII + 569 S. m. 3 Abb. Berlin 1962. VEB Deutscher Verlag der Wissenschaften. Preis geb. DM 36,— , 1963 .

[3] I N Bronstein,et al. Taschenbuch der Mathematik , 1966 .

[4] H. Schwarz,et al. Numerik symmetrischer Matrizen , 1970 .

[5] Donald E. Carlson,et al. Determination of the stretch and rotation in the polar decomposition of the deformation gradient , 1984 .

[6] T. C. T. Ting,et al. Determination of C1/2, C-1/2 and more general isotropic tensor functions of C , 1985 .

[7] A. Bunse-Gerstner,et al. Numerische lineare Algebra , 1985 .

[8] K. N. Morman. The Generalized Strain Measure With Application to Nonhomogeneous Deformations in Rubber-Like Solids , 1986 .

[9] L. Franca,et al. An algorithm to compute the square root of a 3 × 3 positive definite matrix , 1989 .

[10] William H. Press,et al. Numerical Recipes: FORTRAN , 1988 .

[11] J. C. Simo,et al. Quasi-incompressible finite elasticity in principal stretches. Continuum basis and numerical algorithms , 1991 .

[12] Christian Miehe,et al. Computation of isotropic tensor functions , 1993 .

[13] Christian Miehe,et al. Comparison of two algorithms for the computation of fourth-order isotropic tensor functions , 1998 .

[14] Gilbert Strang,et al. Numerische lineare Algebra , 2003 .