Numerical solution of linear elastic problems by spectral collocation methods

Abstract Numerical approximations to the linear elastic system are traditionally based on the finite element method. Here we propose a new formulation based on the spectral collocation ethod. A rigorous theoretical analysis is developed in order to prove the stability and convergance properties of the collocation scheme. We also consider domain decomposition methods in order to handle complex geometries and non-smooth data. Finally we present several numerical results for some examples concerning benchmark problems in geomechanics.

[1]  Cristina Jommi,et al.  On the use of infinite elements in the analysis of two-dimensional layered elastic systems via discretized integral equations , 1989 .

[2]  J. Z. Zhu,et al.  The finite element method , 1977 .

[3]  Claes Johnson.  Numerical solution of partial differential equations by the finite element method , 1988 .

[4]  H. Saunders,et al.  Finite element procedures in engineering analysis , 1982 .

[5]  J. R. Booker,et al.  Finite layer analysis of layered elastic materials using a flexibility approach. Part 1—Strip loadings , 1984 .

[6]  Alfio Quarteroni,et al.  Domain decomposition preconditioners for the spectral collocation method , 1988, J. Sci. Comput..

[7]  C. Canuto.  Spectral methods in fluid dynamics , 1991 .

[8]  Henk A. van der Vorst,et al.  Bi-CGSTAB: A Fast and Smoothly Converging Variant of Bi-CG for the Solution of Nonsymmetric Linear Systems , 1992, SIAM J. Sci. Comput..

[9]  Philip Rabinowitz,et al.  Methods of Numerical Integration , 1985 .

[10]  Alfio Quarteroni,et al.  Finite element preconditioning for legendre spectral collocation approximations to elliptic equations and systems , 1992 .

[11]  G. Sih,et al.  Mathematical theories of brittle fracture. , 1968 .

[12]  L. E. Malvern.  Introduction to the mechanics of a continuous medium , 1969 .

[13]  J. M. Thomas,et al.  Introduction à l'analyse numérique des équations aux dérivées partielles , 1983 .

[14]  Louis A. Hageman,et al.  Iterative Solution of Large Linear Systems. , 1971 .

[15]  I. Babuska,et al.  Finite Element Analysis , 2021 .