A finite element/quaternion/asymptotic numerical method for the 3D simulation of flexible cables

Abstract

is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. Summary A flexible cable is modeled by a geometrically exact beam model with 3D rotations described using quaternion parameters. The boundary value problem is then discretized by the finite element method. The use of an asymptotic numerical method to solve the problem, requiring quadratic equations, is well suited to the quaternion parametrization. This combination of methods leads to a fast, robust and accurate algorithm very well-adapted for the simulation of the assembly process of cables. This is proved by running many examples involving complicated solutions.

On the frequency response computation of geometrically nonlinear flat structures using reduced-order finite element models

“…N . Depending on the formulation of the geometrical nonlinearities in the finite element discretization, this expansion can be exact (it is the case for 3D elements or for shell/plate/beam elements with a von Kármán strain/displacement nonlinear law [22,34]) or truncated, for instance in the case of large rotation elements (such as geometrically exact beam elements for which the internal force vector include sine and cosine functions of the rotation degrees of freedom of the cross section [56,58]). Notice that Eq.…”

Continuation of periodic solutions of various types of delay differential equations using asymptotic numerical method and harmonic balance method

“…The Asymptotic Numerical Method (ANM) that has first been described in [4,5,6], relies on a high-order Taylor series representation of the solution-branch. This technique has already proven its efficiency for a lot of applications in engineering [10], mechanics [24] or acoustics for example. While some implementations relying on automatic differentiation do exist, see [3] for example, the choice is made here to work with a quadratic framework.…”

“…In order to minimize the number of auxiliary variables, the variable F nl is the sum of the delayed variable ku(t − τ ) (with k constant) and the product v × r. Hence the delayed variable is not isolated like in (10) and 11, but as it appears linearly, this allows to use the approach explained in the previous section. The following quadratic recast is used:…”

“…The following example is a simplified model of clarinet-like instrument, taken from [21] [Eq. (4)-(5)-(6)-(7)- (10)]. This system has been chosen to highlight that our approach copes even with a degenerate form of the general system (9).…”

On conservation of energy and kinematic compatibility in dynamics of nonlinear velocity-based three-dimensional beams

“…Modern techniques in numerical analysis allow many possible approaches in handling the problem of such complexity. It is thus not a surprise that after more than three decades of intensive research this field is still a subject of interest for many researchers, as reflected by recent publications, see, e.g., [1][2][3][4][5][6][7][8][9][10][11][12].…”