Nonlinear dynamics of slender structures: a new object-oriented framework

Romero

2021

Acta Mech

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In this work, we present the mathematical formulation and the numerical implementation of a new model for initially straight, transversely isotropic rods. By adopting a configuration space that intrinsically avoids shear deformations and by systemically neglecting the energetic contribution due to torsion, the proposed model admits an unconstrained variational statement. Moreover, as the natural state of the rod is the trivial one and the mechanical properties are homogeneous on the cross section, the need for pull-back and push-forward operations in the formulation is totally circumvented. These features impose, however, some smoothness requirements on the stored energy function that need to be carefully considered when adopting general constitutive models. In addition to introducing the rod model, we propose a spatial discretization with the finite element method, and a time integration with a hybrid, implicit scheme. To illustrate the favorable features of the new model, we provide results corresponding to numerical simulations for plane and three-dimensional problems that are investigated in the static and dynamic settings. Finally, and to put the presented ideas in a suitable context, we compare solutions obtained with the new model against a rod model that allows for torsion and shear.

Section: Resultsmentioning

confidence: 98%

On a nonlinear rod exhibiting only axial and bending deformations: mathematical modeling and numerical implementation

Romero

2021

Acta Mech

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“…Moreover, even for coarse discretizations, no additional residual stress corrections are necessary. For an extensive treatment of geometrically exact beams in the non‐data‐driven finite element setting, see References 29‐33.…”

Section: Specialization Of the Proposed Approachmentioning

confidence: 99%

A framework for data‐driven structural analysis in general elasticity based on nonlinear optimization: The dynamic case

Numerical Meth Engineering

Steinbach

Schillinger

et al. 2020

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Summary In this article, we present an extension of the formulation recently developed by the authors to the structural dynamics setting. Inspired by a structure‐preserving family of variational integrators, our new formulation relies on a discrete balance equation that establishes the dynamic equilibrium. From this point of departure, we first derive an “exact” discrete‐continuous nonlinear optimization problem that works directly with data sets. We then develop this formulation further into an “approximate” nonlinear optimization problem that relies on a general constitutive model. This underlying model can be identified from a data set in an offline phase. To showcase the advantages of our framework, we specialize our methodology to the case of a geometrically exact beam formulation that makes use of all elements of our approach. We investigate three numerical examples of increasing difficulty that demonstrate the excellent computational behavior of the proposed framework and motivate future research in this direction.

“…This very simple construction satisfies, only for the standard rigid body case, the preservation of linear and angular momenta in combination with the kinetic energy in absence of external loads. The discrete version of the Jacobian matrix of the constraints can be computed with the average vector field (Gebhardt et al 2019a, b) as…”

Section: Basic Energy-momentum Algorithmmentioning

confidence: 99%

The Rotating Rigid Body Model Based on a Non-twisting Frame

Romero

2020

J Nonlinear Sci

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This work proposes and investigates a new model of the rotating rigid body based on the non-twisting frame. Such a frame consists of three mutually orthogonal unit vectors whose rotation rate around one of the three axis remains zero at all times and, thus, is represented by a nonholonomic restriction. Then, the corresponding Lagrange–D’Alembert equations are formulated by employing two descriptions, the first one relying on rotations and a splitting approach, and the second one relying on constrained directors. For vanishing external moments, we prove that the new model possesses conservation laws, i.e., the kinetic energy and two nonholonomic momenta that substantially differ from the holonomic momenta preserved by the standard rigid body model. Additionally, we propose a new specialization of a class of energy–momentum integration schemes that exactly preserves the kinetic energy and the nonholonomic momenta replicating the continuous counterpart. Finally, we present numerical results that show the excellent conservation properties as well as the accuracy for the time-discretized governing equations.