SUMMARYWe present in this paper a new ÿnite element formulation of geometrically exact rod models in the threedimensional dynamic elastic range. The proposed formulation leads to an objective (or frame-indi erent under superposed rigid body motions) approximation of the strain measures of the rod involving ÿnite rotations of the director frame, in contrast with some existing formulations. This goal is accomplished through a direct ÿnite element interpolation of the director ÿelds deÿning the motion of the rod's crosssection. Furthermore, the proposed framework allows the development of time-stepping algorithms that preserve the conservation laws of the underlying continuum Hamiltonian system. The conservation laws of linear and angular momenta are inherited by construction, leading to an improved approximation of the rod's dynamics. Several numerical simulations are presented illustrating these properties.
SUMMARYWe present the basic theory for developing novel monolithic and staggered time-stepping algorithms for general non-linear, coupled, thermomechanical problems. The proposed methods are thermodynamically consistent in the sense that their solutions rigorously comply with the two laws of thermodynamics: for isolated systems they preserve the total energy and the entropy never decreases. Furthermore, if the governing equations of the problem have symmetries, the proposed integrators preserve them too. The formulation of such methods is based on two ideas: expressing the evolution equation in the socalled General Equations for Non-Equilibrium Reversible Irreversible Coupling format and enforcing from their inception certain directionality and degeneracy conditions on the discrete vector fields. The new methods can be considered as an extension of the energy-momentum integration algorithms to coupled thermomechanical problems, to which they reduce in the purely Hamiltonian case. In the article, the new ideas are applied to a simple coupled problem: a double thermoelastic pendulum with symmetry. Numerical simulations verify the qualitative features of the proposed methods and illustrate their excellent numerical stability, which stems precisely from their ability to preserve the structure of the evolution equations they discretize.
The finite element formulation of geometrically exact rod models depends crucially on the interpolation of the rotation field from the nodes to the integration points where the internal forces and tangent stiffness are evaluated. Since the rotational group is a nonlinear space, standard (isoparametric) interpolation of these degrees of freedom does not guarantee the orthogonality of the interpolated field hence, more sophisticated interpolation strategies have to be devised. We review and classify the rotation interpolation techniques most commonly used in the context of nonlinear rod models and suggest new ones. All of them are compared and their advantages and disadvantages discussed. In particular, their effect on the frame invariance of the resulting discrete models is analyzed.
IntroductionExisting nonlinear rod theories allow to formulate problems involving arbitrarily large displacements, rotations and strains (see e.g. for a review). By deriving the rod equations from the three-dimensional nonlinear theory of deformable bodies with a projection onto a restricted class of rod-like geometries and motions, the former inherit much of the mathematical structure of the fully dimensional problem. This includes the conservation laws of the dynamic problem and the objectivity of the equations under superposed rigid body motions.In the Computational Mechanics literature, the first attempts to formulate finite elements for these rod theories go back to  and  where a plane rod model due to Reissner  was implemented. A three dimensional extension of this theory was later presented in  by Simo, and its finite element formulation in . Many works followed this pioneering one: [29,9,30,15,18,5,31], to name a few, contributing to a rapid development of the topic.A distinctive feature of nonlinear rod theories of the type mentioned above is that the kinematic variables at every point of the model include both displacements and rotations, and as such they can be classified as Cosserat theories. Apart from its geometric implications, this aspect has important consequences in the formulation of numerical models and in particular finite element ones. The rotation variables belong to a nonlinear manifold and thus their treatment requires special attention which has prompted several authors study this subject in many publications (e.g. [2,6,8]). An issue which is particularly important in the context of the finite element method is the interpolation of rotation variables, which is needed to obtain the values of the rotation field at integration points of the rod, given its values only at the nodes of the finite element mesh. Since a standard interpolation of the nodal rotations does not give, in general, an orthogonal field of rotations, several different solutions have been proposed in the literature. See, the articles mentioned in the previous paragraph and [10, 23].As mentioned above, a fundamental property of nonlinear rod models is the frame invariance of the equations. It is thus desirable that nume...
We formulate a theory of non-equilibrium statistical thermodynamics for ensembles of atoms or molecules. The theory is an application of Jayne's maximum entropy principle, which allows the statistical treatment of systems away from equilibrium. In particular, neither temperature nor atomic fractions are required to be uniform but instead are allowed to take different values from particle to particle. In addition, following the Coleman-Noll method of continuum thermodynamics we derive a dissipation inequality expressed in terms of discrete thermodynamic fluxes and forces. This discrete dissipation inequality effectively sets the structure for discrete kinetic potentials that couple the microscopic field rates to the corresponding driving forces, thus resulting in a closed set of equations governing the evolution of the system. We complement the general theory with a variational meanfield theory that provides a basis for the formulation of computationally tractable approximations. We present several validation cases, concerned with equilibrium properties of alloys, heat conduction in silicon nanowires and hydrogen desorption from palladium thin films, that demonstrate the range and scope of the method and assess its fidelity and predictiveness. These validation cases are characterized by the need or desirability to account for atomiclevel properties while simultaneously entailing time scales much longer than * Corresponding author E-mail address: email@example.com (M. Ortiz). September 27, 2014 those accessible to direct molecular dynamics. The ability of simple meanfield models and discrete kinetic laws to reproduce equilibrium properties and long-term behavior of complex systems is remarkable.
Preprint submitted to Journal of the Mechanics and Physics of Solids
Two of the most popular finite element formulations for solving nonlinear beams are the absolute nodal coordinate and the geometrically exact approaches. Both can be applied to problems with very large deformations and strains, but they differ substantially at the continuous and the discrete levels. In addition, implementation and run-time computational costs also vary significantly. In the current work, we summarize the main features of the two formulations, highlighting their differences and similarities, and perform numerical benchmarks to assess their accuracy and robustness. The article concludes with recommendations for the choice of one formulation over the other.
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