An energy decaying scheme for nonlinear dynamics of shells

Bottasso, Carlo L.; Bauchau, Olivier A.; Choi, Jou-Young

doi:10.1016/s0045-7825(02)00243-8

Cited by 39 publications

(28 citation statements)

References 27 publications

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“…6. In summary, the present energy-momentum scheme essentially leads to the same results as reported in [2,9] for the energy-decaying scheme. In contrast to the conserving scheme in [2,9] our energy-momentum method does not lead to spurious oscillations in the stress resultants.…”

Section: Beam With Concentrated Massessupporting

confidence: 85%

“…As in the standard case, the equations of motion can be calculated from the augmented Hamiltonian, cf. (9). In this connection both primary and secondary constraints have to be added.…”

Section: The Degenerate Rigid Bodymentioning

confidence: 99%

“…This example has been taken from [2], see also [9]. The initial configuration of the beam with concentrated masses is depicted in Fig.…”

Section: Beam With Concentrated Massesmentioning

confidence: 99%

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Constrained dynamics of geometrically exact beams

Betsch¹,

Steinmann²

2003

Computational Mechanics

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Geometrically exact beams are regarded from the outset as constrained mechanical systems. This viewpoint facilitates the discretization in space and time of the underlying continuous beam formulation without using rotational variables. The present semi-discrete beam equations assume the form of differential-algebraic equations which are discretized in time. The resulting energymomentum scheme satisfies the algebraic constraint equations on both configuration and momentum level. IntroductionThe present work deals with the dynamics of nonlinear beams in three-dimensional space. Modern finite element formulations for beams are inherently related to the approximation of finite rotations in space and time, see, for example, Crisfield [11, Chapter 17]. However, finite elements based on the space interpolation of rotational variables may be afflicted with problems such as nonobjective and path-dependent solutions, see Crisfield and Jelenić [12,20]. Moreover the use of rotational variables may further burden the discretization in time, see Jelenić and Crisfield [19] for an investigation of alternative timestepping procedures.The present work is based on the so-called 'geometrically exact beam theory' which has been at the heart of many previously developed beam finite elements, see, for example, [10,12,[15][16][17][18][19][20][21][22][25][26][27][28]. All of these works make use of rotational variables in the discretization process. In contrast to that, the present discretization approach does not rest on rotational variables. This is achieved by regarding nonlinear beams from the outset as constrained mechanical systems. In particular, we extend the constrained semi-discrete beam formulation in [8, Section 3.1] to the dynamic range.An outline of the remaining part of the paper is as follows. Section 2 contains a short summary of the computational treatment of constrained mechanical systems needed subsequently. Section 3 provides a first illustration of the present approach within the context of rigid body dynamics. Then in Sect. 4 the present approach is extended to the semi-discrete beam formulation. Numerical examples are given in Sect. 5 and conclusions are drawn in Sect. 6. Numerical treatment of constrained mechanical systemsIn this section we consider finite-dimensional mechanical systems subject to holonomic constraints. For the present purposes it suffices to restrict our attention to autonomous systems. Within the Lagrangian framework of classical mechanics, the motion of a constrained mechanical system can be described byThe configuration of the mechanical system is characterized by the vector of coordinates qðtÞ 2 R n . The coordinates have to satisfy m holonomic constraint conditions of the formThese constraints restrict possible motions to a ðn À mÞ-dimensional constraint manifold Q ¼ fqðtÞ 2 R n jU b ðqÞ ¼ 0; b ¼ 1; . . . ; mg. Accordingly, the constraint manifold Q is regarded as embedded in a real, n-dimensional vector space. The constraint functions U 1 ; . . . ; U m : R n ! R are assumed to be irreduc...

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Section: Beam With Concentrated Massessupporting

confidence: 85%

“…As in the standard case, the equations of motion can be calculated from the augmented Hamiltonian, cf. (9). In this connection both primary and secondary constraints have to be added.…”

Section: The Degenerate Rigid Bodymentioning

confidence: 99%

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Constrained dynamics of geometrically exact beams

Betsch¹,

Steinmann²

2003

Computational Mechanics

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“…Other authors, instead, have preserved the basic structure of the original EM method and extended it to ensure positive dissipation (see [2,8]), so that situations involving frictional contact and impact could be considered. Additional innovations as discussed in [6] and [28] are also worth noting.…”

Section: Introductionmentioning

confidence: 99%

An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Part 1: Rods

2008

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A fully conserving algorithm is developed in this paper for the integration of the equations of motion in nonlinear rod dynamics. The starting point is a re-parameterization of the rotation field in terms of the so-called Rodrigues rotation vector, which results in an extremely simple update of the rotational variables. The weak form is constructed with a non-orthogonal projection corresponding to the application of the virtual power theorem. Together with an appropriate time-collocation, it ensures exact conservation of momentum and total energy in the absence of external forces. Appealing is the fact that nonlinear hyperelastic materials (and not only materials with quadratic potentials) are permitted without any prejudice on the conservation properties. Spatial discretization is performed via the finite element method and the performance of the scheme is assessed by means of several numerical simulations.

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“…A number of approaches for integration of differential-algebraic equations have been presented, see, e.g., [4,5,7,12,16]. Similarly, Betsch and Steinmann [6] treated the mechanical system as constrained from the onset, resulting in a formulation with a greater number of multipliers than in other formulations.…”

Section: Introductionmentioning

confidence: 99%

A heuristic viscosity-type dissipation for high frequency oscillation damping in time integration algorithms

2007

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A heuristic viscosity-type dissipation of strains is applied for controlling high frequency oscillation damping in a one-step time integration. The basic idea of the approach is to apply damping in locations, where it is needed, at times when it is needed. To this end a computationally efficient algorithm for identifying the need for damping is developed. The identification algorithm keeps track of the number of recent changes of signs of strain increments, which helps in decision whether damping should be engaged or not. A special scaling function is introduced to smoothen the transition between the damped and undamped phases. The approach is verified in the analysis of the geometrically exact plane elastic beam.

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An energy decaying scheme for nonlinear dynamics of shells

Cited by 39 publications

References 27 publications

Constrained dynamics of geometrically exact beams

Constrained dynamics of geometrically exact beams

An exact conserving algorithm for nonlinear dynamics with rotational DOFs and general hyperelasticity. Part 1: Rods

A heuristic viscosity-type dissipation for high frequency oscillation damping in time integration algorithms

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