Issue 29

A. Castellano et alii, Frattura ed Integrità Strutturale, 29 (2014) 128-138; DOI: 10.3221/IGF-ESIS.29.12 129 azimuthal shear to support axially periodic twist-like bifurcation modes similar to the Taylor-Couette axially periodic cellular patterns observed when the flow of a viscous fluid confined in the gap between two rotating concentric cylinders becomes unstable. Here, the hollow cylinder is made of a Levinson-Burgess isotropic compressible elastic material. By restricting our attention to a class of incremental displacements characterized by three unknown scalar functions of the radial coordinate and having an axial periodic structure, we then study the related incremental boundary-value problem. This leads to a somewhat complex non-autonomous homogeneous system of six first order linear ODE’s with homogeneous boundary conditions, whose analysis is performed by a numerical approach. We show that a procedure based on the Magnus method outlined in A n overview of geometric numerical integrators with special reference to the Magnus expansion reveals very effective either for approximating the matricant of the resulting differential problem or for determining the first singular value of the bifurcating load corresponding to a non-trivial twist-like solution. A N OVERVIEW OF GEOMETRIC NUMERICAL INTEGRATORS WITH SPECIAL REFERENCE TO THE MAGNUS EXPANSION number of problems in scientific and engineering areas, ranging from Quantum Mechanics, Optics, Electromagnetism and Molecular Physics to Control Theory and Bifurcation, require the resolution of systems of differential equations with varying coefficients. Besides some very special cases, for which a fundamental matrix solution is explicitly available, the analysis of the problem usually requires the construction of an approximate representation of the solution of the non-autonomous differential system. In the last decade, this issue has attracted many scientists and has also represented a field of intense research activity; in particular, recent studies have shown that in many cases the so-called geometric numerical integration methods may offer better approximation schemes compared to perturbative procedures or to standard numerical integrators. The Magnus method belongs to the class of geometric numerical integrators, based on Lie-Group methods (see [6, 7] for appropriate references), which, although not yet diffused in continuum mechanics, have recently found many applications in areas of molecular physics [9], electromagnetism [10] and celestial mechanics [11]. In this Section, we first give a brief overview of geometric numerical integrators and then we deal with the Magnus method as a particular case. Geometric numerical integration concerns the development of numerical approximations to the solution of an ODE system with the main goal of preserving at any order of approximation the qualitative properties of the exact (but unknown) solution and thereby of exhibiting an improved accuracy with respect to other numerical methods. Although such methods apply also for nonlinear matrix differential equations, for the purposes of bifurcation problems here we consider geometric numerical integrators based on the Magnus expansion for the special case of homogeneous non- autonomous first order linear ODE systems of the form         6x6 R R R , R 0, 0     Y A Y Y I (1) where the unknown 6x6 matrix   R Y , usually referred as the matricant of the Cauchy initial-value problem (1), is the 6- dimensional identity matrix when the real independent variable R is equal to zero. Generally speaking, these new numerical approaches involve Lie-Group methods, which are based on Lie groups and their associated Lie algebras. We recall that a Lie group corresponds to a differential manifold which is endowed with a group structure; the corresponding Lie algebra, defined as the tangent space to the Lie group at the identity, is a linear space endowed with the Lie bracket commutation operation and a natural mapping. Since the unknown matricant of a first order ODE system like (1) must be an invertible 6x6 matrix with variable entries, the setting for finding solutions of (1) is a 6x6 matrix Lie group. In this case, the above general definition of a Lie algebra implies that the 6x6 matrix   R A governing (1) is an element of a matrix Lie algebra. Moreover, in this setting the associated matrix Lie algebra is endowed with the matrix commutation law   , :=  A B AB BA (2) where   , A B is commonly called the matrix commutator . Finally, for these special matrix Lie groups the matrix exponential function plays the role of the natural mapping. We emphasize that in the theory of Lie groups the role of the exponential mapping is crucial. Indeed, within the context of differential geometry, the matrix exponential maps an element belonging to a matrix Lie algebra into an invertible matrix belonging to its corresponding Lie group; in other words, the exponential mapping allows to recapture the local A