Computational aspects of vector‐like parametrization of three‐dimensional finite rotations

Abstract

Theoretical and computational aspects of vector-like parametrization of three-dimensional finite rotations, which uses only three rotation parameters, are examined in detail in this work. The relationship of the proposed parametrization with the intrinsic representation of finite rotations (via an orthogonal matrix) is clearly identified. Careful considerations of the consistent linearization procedure pertinent to the proposed parametrization of finite rotations are presented for the chosen model problem of Reissner's non-linear beam theory. Pertaining details of numerical implementation are discussed for the simplest choice of the finite element interpolations for a 2-node three-dimensional beam element. A number of numerical simulations in three-dimensional finite rotation analysis are presented in order to illustrate the proposed approach.KEY WORDS: three-dimensional finite rotations; parametrization; computational procedure *One of the popular choices for three rotation parameters are the Euler angles, which have the well-known nonuniqueness problem (e.g. see Reference 2, p. 144)

Isogeometric collocation for three-dimensional geometrically exact shear-deformable beams

“…It consists of a cantilever initially straight beam of length L = 10 (placed along the x 2 -axis) with a concentrated moment m = me 1 applied at its free end. We assign the same material properties as in [47]. The elasticity tensors are C N = diag(5 × 10 3 , 5 × 10 3 , 10 4 ) and C M = diag(10 2 , 10 2 , 10 4 ) , respectively, which correspond to a beam with circular cross section with radius 0.2 and Young's modulus E = 79 577 and zero Poisson's ratio.…”

“…Different update procedures named Eulerian, total Lagrangian and updated Lagrangian were defined in [46]. In the total Lagrangian formulation, adopted also in [47][48][49], the rotation vector is used to represent the total rotation between initial and current configurations resulting, as highlighted in [48], in a full vector-like parametrization of the rotation operator. The total Lagrangian formulation has two main disadvantages: (i) it suffers singularity problems at rotation angle 2π and its multiples and (ii) it presents difficulties to obtain full linearized equations, see for example appendix of [47].…”

“…In the total Lagrangian formulation, adopted also in [47][48][49], the rotation vector is used to represent the total rotation between initial and current configurations resulting, as highlighted in [48], in a full vector-like parametrization of the rotation operator. The total Lagrangian formulation has two main disadvantages: (i) it suffers singularity problems at rotation angle 2π and its multiples and (ii) it presents difficulties to obtain full linearized equations, see for example appendix of [47]. The latter drawback is addressed by Ritto-Corrêa and Camotim in [48], where second-order directional derivatives of the kinematic operators are provided and a simpler expression for the tangent operator is formulated.…”

A computational framework for polyconvex large strain elasticity for geometrically exact beam theory

“…As reported in reference [43], for a linearised Saint Venant-Kirchhoff model (100) with Poisson ratio ν = 0 and an applied moment of value M = M max , the beam closes on itself forming a closed loop which can be described with an available analytical solution. In general, for a nonlinear constitutive model (or when the Poisson ratio ν is not equal to zero), the exact closure of the beam configuration cannot be a priori guaranteed.…”

Dual extremum principles for geometrically exact finite strain beams

“…Since these pioneering works, considerable progress has been made on the geometrically exact analysis of three-dimensional framed structures, from both theoretical and numerical points of view, see e.g. Cardona and Géradin (1988) [6], Ibrahimbegovic and co-workers (1995,2000,2003) [16,19,18,17], Petrov and Géradin (1998) [31], Saje et al (1998) [36], [7,22], Gruttmann et al (2000) [15], Atluri and co-workers (1988,1989,1996,1998,2001) [20,21,33,2], Betsch and Steinmann (2002) [3], Pimenta and Campello (2003) [32], Zupan and Saje (2003) [50], [24,23], Mata et al (2007) [27], Makinen (2007) [26], Santos et al (2009) [38,37] and many others.…”