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1. Convergence Characteristics of Geometrically Accurate Spatial Finite Elements

   https://doi.org/10.1115/1.4048731  

   Tinsley, Brian ; Shabana, Ahmed A. ( January 2021 , Journal of Computational and Nonlinear Dynamics)

   Abstract The convergence characteristics of three geometrically accurate spatial finite elements (FEs) are examined in this study using an eigenvalue analysis. The spatial beam, plate, and solid elements considered in this investigation are suited for both structural and multibody system (MBS) applications. These spatial elements are based on geometry derived from the kinematic description of the absolute nodal coordinate formulation (ANCF). In order to allow for an accurate reference-configuration geometry description, the element shape functions are formulated using constant geometry coefficients defined using the position-vector gradients in the reference configuration. The change in the position-vector gradients is used to define a velocity transformation matrix that leads to constant element inertia and stiffness matrices in the case of infinitesimal rotations. In contrast to conventional structural finite elements, the elements considered in this study can be used to describe the initial geometry with the same degree of accuracy as B-spline and nonuniform rational B-spline (NURBS) representations, widely used in the computer-aided design (CAD). An eigenvalue analysis is performed to evaluate the element convergence characteristics in the case of different geometries, including straight, tapered, and curved configurations. The frequencies obtained are compared with those obtained using a commercial FE software and analytical solutions.more »The stiffness matrix is obtained using both the general continuum mechanics (GCM) approach and the newly proposed strain split method (SSM) in order to investigate its effectiveness as a locking alleviation technique.« less
2. Visible and near-infrared spectro-interferometric analysis of the edge-on Be star o Aquarii

   https://doi.org/10.1051/0004-6361/201936039  

   de Almeida, E. S. ; Meilland, A. ; Domiciano de Souza, A. ; Stee, P. ; Mourard, D. ; Nardetto, N. ; Ligi, R. ; Tallon-Bosc, I. ; Faes, D. M. ; Carciofi, A. C. ; et al ( April 2020 , Astronomy & Astrophysics)

   Aims. We present a detailed visible and near-infrared spectro-interferometric analysis of the Be-shell star o Aquarii from quasi-contemporaneous CHARA/VEGA and VLTI/AMBER observations. Methods. We analyzed spectro-interferometric data in the H α (VEGA) and Br γ (AMBER) lines using models of increasing complexity: simple geometric models, kinematic models, and radiative transfer models computed with the 3D non-LTE code HDUST. Results. We measured the stellar radius of o Aquarii in the visible with a precision of 8%: 4.0 ± 0.3 R ⊙ . We constrained the circumstellar disk geometry and kinematics using a kinematic model and a MCMC fitting procedure. The emitting disk sizes in the H α and Br γ lines were found to be similar, at ~10–12 stellar diameters, which is uncommon since most results for Be stars indicate a larger extension in H α than in Br γ . We found that the inclination angle i derived from H α is significantly lower (~15°) than the one derived from Br γ : i ~ 61.2° and 75.9°, respectively. While the two lines originate from a similar region of the disk, the disk kinematics were found to be near to the Keplerian rotation (i.e., β = −0.5) in Br γmore »( β ~ −0.43), but not in H α ( β ~ −0.30). After analyzing all our data using a grid of HDUST models (BeAtlas), we found a common physical description for the circumstellar disk in both lines: a base disk surface density Σ 0 = 0.12 g cm −2 and a radial density law exponent m = 3.0. The same kind of discrepancy, as with the kinematic model, is found in the determination of i using the BeAtlas grid. The stellar rotational rate was found to be very close (~96%) to the critical value. Despite being derived purely from the fit to interferometric data, our best-fit HDUST model provides a very reasonable match to non-interferometric observables of o Aquarii: the observed spectral energy distribution, H α and Br γ line profiles, and polarimetric quantities. Finally, our analysis of multi-epoch H α profiles and imaging polarimetry indicates that the disk structure has been (globally) stable for at least 20 yr. Conclusions. Looking at the visible continuum and Br γ emission line only, o Aquarii fits in the global scheme of Be stars and their circumstellar disk: a (nearly) Keplerian rotating disk well described by the viscous decretion disk (VDD) model. However, the data in the H α line shows a substantially different picture that cannot fully be understood using the current generation of physical models of Be star disks. The Be star o Aquarii presents a stable disk (close to the steady-state), but, as in previous analyses, the measured m is lower than the standard value in the VDD model for the steady-state regime ( m = 3.5). This suggests that some assumptions of this model should be reconsidered. Also, such long-term disk stability could be understood in terms of the high rotational rate that we measured for this star, the rate being a main source for the mass injection in the disk. Our results on the stellar rotation and disk stability are consistent with results in the literature showing that late-type Be stars are more likely to be fast rotators and have stable disks.« less
3. ANCF curvature continuity: application to soft and fluid materials

   https://doi.org/10.1007/s11071-020-05550-5  

   Shabana, Ahmed A. ; Zhang, Dayu ( April 2020 , Nonlinear Dynamics)

   The continuity of the position-vector gradients at the nodal points of a finite element mesh does not always ensure the continuity of the gradients at the element interfaces. Discontinuity of the gradients at the interface not only adversely affects the quality of the simulation results, but can also lead to computer models that do not properly represent realistic physical system behaviors, particularly in the case of soft and fluid material applications. In this study, the absolute nodal coordinate formulation (ANCF) finite elements are used to define general curvature-continuity conditions that allow for eliminating or minimizing the discontinuity of the position gradients at the element interface. For the ANCF solid element, with four-node surfaces, it is shownthat continuity of the gradients tangent to an arbitrary point on a surface is ensured as the result of the continuity of the gradients at the nodal points. The general ANCF continuity conditions are applicable to both reference-configuration straight and curved geometries. These conditions are formulated without the need for using the computer-aided-design knot vector and knot multiplicity, which do not account properly for the concept of system degrees of freedom. The ANCF curvature-continuity conditions are written in terms of constant geometric coefficients obtained usingmore »the matrix of position-vector gradients that defines the reference-configuration geometry. The formulation of these conditions is demonstrated using the ANCF fully parameterized three-dimensional solid and tetrahedral elements, which employ a complete set of position gradients as nodal coordinates. Numerical results are presented in order to examine the effect of applying the curvature-continuity conditions on achieving a higher degree of smoothness at the element interfaces in the case of soft and fluid materials.« less
4. On formulations of compressible mantle convection

   https://doi.org/10.1093/gji/ggaa078  

   Gassmöller, Rene ; Dannberg, Juliane ; Bangerth, Wolfgang ; Heister, Timo ; Myhill, Robert ( May 2020 , Geophysical Journal International)

   SUMMARY Mantle convection and long-term lithosphere dynamics in the Earth and other planets can be treated as the slow deformation of a highly viscous fluid, and as such can be described using the compressible Navier–Stokes equations. Since on Earth-sized planets the influence of compressibility is not a dominant effect, density deviations from a reference profile are at most on the order of a few percent and using the full governing equations poses numerical challenges, most modelling studies have simplified the governing equations. Common approximations assume a temporally constant, but depth-dependent reference profile for the density (the anelastic liquid approximation), or drop compressibility altogether and use a constant reference density (the Boussinesq approximation). In most previous studies of mantle convection and crustal dynamics, one can assume that the error introduced by these approximations was small compared to the errors that resulted from poorly constrained material behaviour and limited numerical accuracy. However, as model parametrizations have become more realistic, and model resolution has improved, this may no longer be the case and the error due to using simplified conservation equations might no longer be negligible: while such approximations may be reasonable for models of mantle plumes or slabs traversing the whole mantle,more »they may be unsatisfactory for layered materials experiencing phase transitions or materials undergoing significant heating or cooling. For example, at boundary layers or close to dynamically changing density gradients, the error arising from the use of the aforementioned compressibility approximations can be the dominant error source, and common approximations may fail to capture the physical behaviour of interest. In this paper, we discuss new formulations of the continuity equation that include dynamic density variations due to temperature, pressure and composition without using a reference profile for the density. We quantify the improvement in accuracy relative to existing formulations in a number of benchmark models and evaluate for which practical applications these effects are important. Finally, we consider numerical aspects of the new formulations. We implement and test these formulations in the freely available community software aspect, and use this code for our numerical experiments.« less
5. Multiple-Scale Analysis of a Tunable Bi-Stable Piezoelectric Energy Harvester

   https://doi.org/10.1115/1.4046961  

   Qian, Feng ; Abaid, Nicole ; Zuo, Lei ( April 2021 , ASME Letters in Dynamic Systems and Control)

   Abstract This paper presents the theoretical modeling and multiple-scale analysis of a novel piezoelectric energy harvester composed of a metal cantilever beam, piezoelectric films, and an axial preload spring at the moveable end. The harvester experiences mono- and bi-stable regimes as the stiffness of preload spring increases. The governing equations are derived with two high-order coupling terms induced by the axial motion. The literature shows that these high-order coupling terms lead to tedious calculations in the stability analysis of solutions. This work introduces an analytical strategy and the implementation of the multiple-scale method for the harvester in either the mono- or bi-stable status. Numerical simulations are performed to verify the analytical solutions. The influence of the electrical resistance, excitation level, and the spring pre-deformation on the voltage outputs and dynamics are investigated. The spring pre-deformation has a slight influence on the energy harvesting performance of the mono-stable system, but a large effect on that of the bi-stable system.