Final Equilibrium Configuration Research Articles

Tokamak plasma equilibria with n=1 toroidal asymmetry

A general approach of how to construct plasma equilibrium in a tokamak with n=1 violation of toroidal symmetry is proposed. For an arbitrary axisymmetric tokamak plasma equilibrium, there exists the small n=1 deformation of the initial magnetic configuration that keeps the nesting of the magnetic surfaces (as in the initial configuration) and provides plasma equilibrium; such deformation and final equilibrium configuration are calculated analytically. The asymmetric analogue of the Solov'ev's equilibrium with non-degenerated plasma pressure and current density profiles is presented as an example of the application of the developed algorithm.

Novel nodal relaxation algorithm with its application on nonlinear analysis of tensegrity-based structures

This study proposes a robust technique for static and dynamic nonlinear analysis of tensegrity-based structures using a novel method, entitled the Node Relaxation algorithm (NR). Regular numerical algorithms are usually invalid in dealing with flexible tensegrities due to singularities arising from large displacements and/or rotations following any alteration of the internal actuated components or externally applied loads. Especially in tackling dynamic problems, it becomes even more challenging to predict structural deformation when its member forces are varied constantly. In this paper, a novel iterative algorithm, consisting of consecutive releasing and restraining of each node, is raised to oblige the structural form to gradually evolve to its final equilibrium configuration. The tangential unbalance force on each node, selected as the sensitivity index, is employed to accelerate the structure searching for its stable state. The Newmark-β method is also adopted to update the position of free nodes in each iterative step, which assists the NR algorithm in finding updated equilibrium structural configuration during the nonlinear dynamic analysis. Two kinds of tensegrity-based structures, namely the spring-rod tensegrity and the cable-rod tensegrity, are given as the illustrative examples to confirm the effectiveness and robustness of the presented method.

An entropy current and the second law in higher derivative theories of gravity

We construct a proof of the second law of thermodynamics in an arbitrary diffeomorphism invariant theory of gravity working within the approximation of linearized dynamical fluctuations around stationary black holes. We achieve this by establishing the existence of an entropy current defined on the horizon of the dynamically perturbed black hole in such theories. By construction, this entropy current has non-negative divergence, suggestive of a mechanism for the dynamical black hole to approach a final equilibrium configuration via entropy production as well as the spatial flow of it on the null horizon. This enables us to argue for the second law in its strongest possible form, which has a manifest locality at each space-time point. We explicitly check that the form of the entropy current that we construct in this paper exactly matches with previously reported expressions computed considering specific four derivative theories of higher curvature gravity. Using the same set up we also provide an alternative proof of the physical process version of the first law applicable to arbitrary higher derivative theories of gravity.

Investigation of curing deformation behavior of curved fiber metal laminates

To keep the size and shape accuracy of fiber metal laminates (FMLs) with an initial curvature during manufacturing, curing deformation behavior is investigated by using analytical, numerical and experimental methods. Based on Eckstein’s theory, an analytical model considering slippage effect between metal layer and fiber layer is developed to study the warping deformation modes of curved FMLs after curing process. The warping displacements of the curved FMLs predicted by the analytical method are in good agreement with the finite element results and experimental data. The final equilibrium configurations of the curved FMLs can be captured by using the analytical method, which can guide the size and shape control for FMLs. The effect of the side length, initial curvature radius and stacking sequence on the curing deformation characteristics of curved FMLs is studied. The curing deformation mechanisms of the curved FMLs with [Al/0/90/Al] and [Al/90/0/Al] stacking sequences are also illustrated from the perspectives of force and total strain energy.

Isogeometric configuration design sensitivity analysis of geometrically exact shear-deformable beam structures

In this paper, using an isogeometric approach, a continuum-based adjoint configuration design sensitivity analysis (DSA) method is presented for three-dimensional finite deformation shear-deformable beam structures. A geometrically exact beam model together with a multiplicative update of finite rotation by an exponential map of a skew-symmetric matrix is utilized. The material derivative of the orthogonal transformation matrix can be evaluated at final equilibrium configuration, which enables to compute design sensitivity using the tangent stiffness at the equilibrium without further iterations. We also present a procedure of explicit parameterization of initial orthonormal frame using the smallest rotation (SR) method within the isogeometric analysis framework. Furthermore, it is shown that for curve entities embedded to a smooth surface, the convected basis of the surface can be effectively utilized for reference orthonormal frames in the SR method. Various numerical examples including pressure loads and nonhomogeneous kinematic boundary conditions in built-up structures demonstrate the effectiveness of the developed DSA method.

Capture into first-order resonances and long-term stability of pairs of equal-mass planets

Massive planets form within the lifetime of protoplanetary disks and undergo orbital migration due to planet-disk interactions. When the first planet reaches the inner edge of the disk its migration stops and the second planet is locked in resonance. We detail the resonant trapping comparing semi-analytical formulae and numerical simulations in the case of two equal-mass coplanar planets trapped in first order resonances. We describe the family of resonant stable equilibrium points (zero-amplitude libration orbits) using expanded and non-expanded Hamiltonians. We show that during convergent migration the planets evolve along these families of equilibria. Eccentricity damping from the disk leads to a final equilibrium configuration that we predict analytically. The fact that observed multi-exoplanetary systems are rarely seen in resonances suggests that the resonant configurations achieved by migration can become unstable after the removal of the disk. We probe the stability of the resonances as a function of planetary mass. To do this, we fictitiously increase the masses of the planets, adiabatically maintaining the low-amplitude libration regime until instability. We discuss two hypotheses for instability: low-order secondary resonance of the libration frequency with a fast synodic frequency, and minimal approach distance between planets. We show that secondary resonances do not seem to impact resonant systems at low-amplitude of libration. Resonant systems are more stable than non-resonant ones for a given minimal distance at close encounters, but we show that the latter still play the decisive role in the destabilisation of resonant pairs. We show that, as the planetary mass increases and the minimal distance between planets gets smaller in terms of mutual Hill radius, the region of stability around the resonance center shrinks, until the equilibrium point itself becomes unstable.

Spreading of a pendant liquid drop underneath a textured substrate

A pendant drop spreading underneath a partially wetting surface from an initial shape to its final equilibrium configuration and contact angle is studied. A mathematical formulation that quantifies spreading behavior of liquid drops over textured surfaces is discussed. The drop volume and the equilibrium contact angle are treated as parameters in the study. The unbalanced force at the three-phase contact line is modeled as being proportional to the degree of departure from the equilibrium state. Model predictions are verified against the available experimental data in the literature. Results show that the flow dynamics is strongly influenced by the fluid properties, drop volume, and contact angle of the liquid with the partially wetting surface. The drop exhibits rich dynamical behavior including inertial oscillations and gravitational instability, given that gravity tries to detach the drop against wetting contributions. Flow characteristics of drop motion, namely, the radius of the footprint, slip length, and dynamic contact angle in the pendant configuration are presented. Given the interplay among the competing time-dependent forces, a spreading drop can momentarily be destabilized and not achieve a stable equilibrium shape. Instability is then controlled by the initial drop shape as well. The spreading model is used to delineate stable and unstable regimes in the parameter space. Predictions of the drop volume based on the Young-Laplace equation are seen to be conservative relative to the estimates of the dynamical model discussed in the present study.

Skyrmion states in thin confined polygonal nanostructures

Recent studies have demonstrated that skyrmionic states can be the ground state in thin-film FeGe disk nanostructures in the absence of a stabilising applied magnetic field. In this work, we advance this understanding by investigating to what extent this stabilisation of skyrmionic structures through confinement exists in geometries that do not match the cylindrical symmetry of the skyrmion—such as squares and triangles. Using simulation, we show that skyrmionic states can form the ground state for a range of system sizes in both triangular and square-shaped FeGe nanostructures of 10 nm thickness in the absence of an applied field. We further provide data to assist in the experimental verification of our prediction; to imitate an experiment where the system is saturated with a strong applied field before the field is removed, we compute the time evolution and show the final equilibrium configuration of magnetization fields, starting from a uniform alignment.

Dependence of equilibrium Griffith surface energy on crack speed in phase-field models for fracture coupled to elastodynamics

Phase-field models for crack propagation enable the simulation of complex crack patterns without complex and expensive tracking and remeshing as cracks grow. In the setting without inertia, the crack evolution is obtained from a variational energetic starting point, and leads to an equation for the order parameter coupled to elastostatics. Careful mathematical analysis has shown that this is consistent with the Griffith model for fracture. Recent efforts to include inertia in this formulation have replaced elastostatics by elastodynamics. In this brief note, we examine the elastodynamic augmentation, and find that it effectively causes the Griffith surface energy to depend on the velocity of the crack. That is, considering two identical specimens that are each fractured by a single crack that grows at different velocities in the two specimens, it is expected that the final equilibrium configurations are nominally identical; however, the phase-field fracture models augmented with elastodynamics achieve final configurations—in particular, the Griffiths surface energy contributions—that depend on the crack velocity. The physical reason is that the finite relaxation time for the stresses in the elastodynamic setting enables the cracked region to widen, beyond the value observed in the quasistatic setting. Once the crack widens, the “no-healing” condition prevents it from relaxing even after the specimen reaches equilibrium. In phase-field models, crack width in the reference configuration is unrelated to the physical opening of the crack but is instead a measure of Griffiths surface energy. This observation suggests that elastodynamic phase-field fracture models should not be used in settings where the crack velocity is large.

Combining Precipitation and Vitrification to Control the Number of Surface Patches on Polymer Nanocolloids.

In an effort to incorporate increasingly higher levels of functionality into soft nanoparticles, heterogeneously structured particles stand out as a simple means to enhance functionality by tailoring only particle architecture. Various means exist for the fabrication of particles with specific structural configurations; however, the tunability of particle morphology is still a challenging and often laborious task, especially in self-assembled systems where a single equilibrium configuration dominates. Improved strategies for multipatch particle assembly are therefore needed to allow for the tailoring of particle structure via a single, continuous assembly route. One means of accomplishing this is through kinetic trapping of particle morphologies along the path to the final equilibrium configuration in precipitation-induced, phase-separating polymer blends. Here, we demonstrate this capability by using rapid nanoprecipitation to control the overall size, composition, and patch distribution of soft colloids. In particular, we illustrate that polymer feed concentration, blend ratio, and polymer molecular weight can all serve as functional handles with which to consistently alter particle patch distributions in a self-assembling homopolymer system without redesigning the starting materials. We furthermore delineate the role of polymer vitrification in the determination of particle structure.

Cellular entry of graphene nanosheets: the role of thickness, oxidation and surface adsorption

Coarse grained molecular dynamics simulations are conducted to study the interaction of graphene nanosheets with a lipid bilayer, focusing on the effects of graphene thicknesses (single/multi-layered graphene), oxidation and surface absorption by lipid molecules. The results show that a hydrophobic corner of graphene can pierce into the bilayer, while different oxidations of the nanosheets affect their final equilibrium configurations in the bilayer: lying across or within the hydrophobic core of the bilayer. The underlying mechanism is clarified by calculating the energy barrier for graphene piercing into the bilayer. Our studies provide fundamental guidance towards understanding how graphene enters cells, which is important for biomedical diagnostics and therapies, and for managing health impacts following occupational or environmental exposure.

Analytic and numerical models for the expansion of a compact Hii region

Abstract ‘Compact’ Hii regions (embedded in dense molecular clouds) relax to a pressure equilibrium configuration in shorter time-scales than the main-sequence lifetime of the exciting O stars. This result motivates a rederivation of the analytic model for the expansion of an Hii region into a uniform environment. We have modified the classical model so as to include the breaking of the expansion necessary to reach the correct, final pressure equilibrium configuration. We find that this extended model also has a full analytic solution. A comparison of the analytic solution with a (spherically symmetric) numerical simulation is also presented.

Structure Variation and Evolution in Microphase-Separated Grafted Diblock Copolymer Films

The phase behavior of grafted d-polystyrene-block-poly(methyl methacrylate) diblock copolymer films is examined, with particular focus on the effect of solvent and annealing time. It was observed that the films undergo a two-step transformation from an initially disordered state, through an ordered metastable state, to the final equilibrium configuration. It was also found that altering the solvent used to wash the films, or complete removal of the solvent prior to thermal annealing using supercritical CO2, could influence the structure of the films in the metastable state, though the final equilibrium state was unaffected. To aid in the understanding to these experimental results, a series of self-consistent field theory calculations were done on a model diblock copolymer brush containing solvent. Of the different models examined, those which contained a solvent selective for the grafted polymer block most accurately matched the observed experimental behavior. We hypothesize that the structure of the films in the metastable state results from solvent enrichment of the film near the film/substrate interface in the case of films washed with solvent or faster relaxation of the nongrafted block for supercritical CO2 treated (solvent free) films. The persistence of the metastable structures was attributed to the slow reorganization of the polymer chains in the absence of solvent.

Acidification of three-dimensional emeraldine polymers: Search for minimum energy paths from base to salt

We present a numerical simulation of the HCl acidification process of a three-dimensional semiconducting emeraldine base (EB) polymer leading to the corresponding metallic emeraldine salt form. We have searched minimum energy paths connecting the initial configuration, composed of two EB polymer chains per cell each one attached by two HCl molecules, with the Pc2a polaronic configuration which is the final state of the acidification process. For this aim, the variational nudged elastic band method has been adopted. We provide a pictorial representation of the acidification process at T=0 K, monitoring the EB protonation and the evolution of the polymeric chains and of the positions of the Cl(-) counterions on the lowest potential energy surface. To include also temperature effects, we have explored the potential energy surface around the final equilibrium configuration, heating the system and following its dynamics by the Car-Parrinello procedure.

A one‐dimensional model of bifurcations in gravel bed channels with erodible banks

In a recent paper, Bolla Pittaluga et al. proposed a physically based formulation for the nodal point conditions to be adopted at a channel bifurcation in the context of a one‐dimensional approach. They employed such conditions to study the equilibrium configurations of a simple bifurcation and their stability, assuming erodible bed and fixed banks. With the present work we extend the model proposed by Bolla Pittaluga et al. to the case of channels with erodible banks, i.e., channels which may adapt their width to the actual flow conditions. Such an extension is of a particular interest for river bifurcations in gravel bed braided streams, in which bed evolution and bank erosion processes occur over comparable timescales. Moreover, to study the morphological evolution of a bifurcation, we employ a different approach with respect to Bolla Pittaluga et al. and introduce a “local analysis” which does not account for the influence exerted on channel morphology by downstream conditions on longer timescales. For values of the controlling parameters typical of gravel bed braided streams, the model shows that the stable equilibrium solutions of a bifurcation are invariably characterized by a strongly unbalanced partition of water and sediment discharges in the two branches. Numerical simulations, based on a simplified model of the bifurcation evolution, allow us to investigate the role of the mechanism of generation of a bifurcation on its final equilibrium configurations. It is shown that bifurcations which form through the incision of a new, initially narrow, channel may lead to significantly different equilibrium configurations with respect to bifurcations which generate from a central deposition in a wide channel. In spite of its simplicity the model seems to retain the most relevant effects which govern the behavior of gravel bed channel bifurcations.

Gravitational Waves from Phase‐Transition‐Induced Collapse of Neutron Stars

(Abridged) We study the gravitational radiation from gravitational collapses of rapidly rotating neutron stars induced by a phase-transition from normal nuclear matter to a mixed phase of quark and nuclear matter in the core of the stars. The study is based on self-consistent three dimensional hydrodynamic simulations with Newtonian gravity and a high resolution shock capturing scheme, and the quadrupole formula of gravitational radiation. The quark matter of the mixed phase is described by the MIT bag model and the normal nuclear matter is described by an ideal fluid equation of state (EOS). 1. We determined the magnitudes of the emitted gravitational waves for several collapse scenarios. 2. We determined the types and frequencies of the fluid oscillation modes excited by the process. In particular, we find that the gravitational wave signals produced by the collapses are dominated by the fundamental quadrupole and quasi-radial modes of the final equilibrium configurations. In some collapse scenarios, we find that the oscillations are damped out within a few dynamical timescales due to the growth of differential rotations and the formation of strong shock waves. 3. We showed that the spectrum of the gravitational wave signals is sensitive to the EOS, implying that the detection of such gravitational waves could provide useful constraints on the EOS of newly born quark stars. 4. For the range of rotation periods we have studied, we found no sign of the development of nonaxisymmetric dynamical instabilities in the collapse process.

Nonsingular black hole model as a possible end product of gravitational collapse

In this paper we present a non-singular black hole model as a possible end-product of gravitational collapse. The depicted spacetime which is type [II,(II)], by Petrov classification, is an exact solution of the Einstein equations and contains two horizons. The equation of state in the radial direction, is a well-behaved function of the density and smoothly reproduces vacuum-like behavior near r=0 while tending to a polytrope at larger r, low density, values. The final equilibrium configuration comprises of a de Sitter-like inner core surrounded by a family of 2-surfaces of matter fields with variable equation of state. The fields are all concentrated in the vicinity of the radial center r=0. The solution depicts a spacetime that is asymptotically Schwarzschild at large r, while it becomes de Sitter-like for vanishing r. Possible physical interpretations of the macro-state of the black hole interior in the model are offered. We find that the possible state admits two equally viable interpretations, namely either a quintessential intermediary region or a phase transition in which a two-fluid system is in both dynamic and thermodynamic equilibrium. We estimate the ratio of pure matter present to the total energy and in both (interpretations) cases find it to be virtually the same, being 0.83. Finally, the well-behaved dependence of the density and pressure on the radial coordinate provides some insight on dealing with the information loss paradox.

Static equilibrium analysis of compliant mechanical systems using relative coordinates and loop closure equations

This paper deals with the static equilibrium formulation of mechanical systems, including rigid and compliant elements, and the determination of their final equilibrium configuration under known loads. Statically determinate as well as statically indeterminate systems are analysed. Relative coordinates are used as generalised coordinates. The system is modeled using homogeneous transformation matrices and loop closure equations. In the first phase of analysis (assembly), initial correct configuration of the system is obtained, from approximate input provided by the user. For the solution of the equilibrium problem, a potential function is used. Its minimisation, subject to rigid constraints yields the equilibrium configuration. The approach is implemented in a program and tested on a number of planar and spatial mechanical systems. The software program is intended to be used as a design aid for engineers.

Origin of Core-Halo Structure in One-Dimensional Self-Gavitating System

The relaxation process of self-gravitating systems is examined by using one-dimensional numeri­ cal simulation. We get the asymptotic distribution function which disagrees with that proposed by Lynden-Bell. Our distribution function has two peaks in low and high energy regions and a valley in the medium energy region. This characteristic core-halo structure in phase space has been observed in many simulations. Vir e clarify the dynamical mechanism which generates this 'core-halo' structure. The essence of this mechanism is that the elements of the system are accelerated (or decelerated) very effectively by the evolving gravitational potential in a specific energy region, reflecting the initial conditions. Some classes of stellar systems,such as elliptical galaxies, seem to have already settled down to their final equilibrium configuration. This is strongly suggested from the fact that most elliptical galaxies show the universal luminosity profile which is characterized by R1!4-l aw.*} Moreover, most giant elliptical galaxies have little rotation and a random stellar velocity. The velocity dispersion of the stars in the inner few kiloparsecs is also universally correlated with luminosity according to the Faber-Jackson law. 2 ) Therefore, there must be a common relaxation mechanism which guarantees this universality. The ordinary relaxation mechanism due to the two-body gravitational interaction (binary encounter) has been the only well­ established relaxation process in an elliptical galaxy which has poor gas. However, the relaxation time scale is too long (more than 10 15 yrs) to relax an elliptical galaxy within the age of the Universe. Hence we believe that there must be another relaxa­ tion mechanism which is more rapid and more effective for a self-gravitating system. 25 years ago, Lynden-Bell proposed a new relaxation mechanism due to the variation of the mean potential. 3 ) The relaxation time scale is far shorter than that by binary encounters. This new mechanism was expected to solve the above­ mentioned problem. The evolution of self-gravitating systems is generally described by Boltzmann equation and Poisson equation,

A Three-Dimensional Dynamic Model for Determining the Equilibrium Configurations of Buoyantly Unstable Icebergs

Abstract This paper describes a dynamic model developed for the purpose of determining the final equilibrium configurations of buoyantly unstable icebergs. The model places no restrictions on the size, shape, or dimensionality of the iceberg, or on the variation range of the configuration coordinates. Furthermore, it includes all six degrees of freedom and is based on a Lagrangian formulation of the dynamic equations of motion. It can be used to advantage in those situations in which the iceberg has a complicated potential function and can acquire enough momentum and kinetic energy in the initial phase of its motion to make its final configuration uncertain on the basis of a static potential analysis. The behavior of the model is examined through several model simulations. The sensitivity of the final equilibrium position to the initial orientation and shape of the iceberg is clearly evident in the model simulations. Model simulations also show that when an iceberg is released from a nonequilibrium initial state, the time taken for it to settle down varies from about 40 s for a growler to nearly 400 s for a large iceberg. While these absolute times may change with better parameterization of the forces, the relative variations with iceberg size are likely to be preserved.