Equilibrium Configurations Research Articles

Exact closed-form solution for buckling and free vibration of pipes conveying fluid with intermediate elastic supports

In this paper, the exact closed-form solution is given to investigate the influence of the intermediate elastic support on the buckling and free vibration of an elastically supported pipe. According to the Euler–Bernoulli beam theory, the mechanical model of the pipe is established. The exact equilibrium configuration is derived using the generalised function method without enforcing continuity conditions. A simple solution to the eigenvalue problem is formulated using the methods of complex mode superposition and Laplace transformation. The comparative study shows the differences in the supercritical vibration characteristics and highlights the limitations of previous studies. Parametric studies are carried out to investigate the influence of elastic support and intermediate support conditions on the equilibrium configuration, critical flow velocity, and natural frequency. The results demonstrate that the proposed closed-form solution can determine the support conditions that lead to the maximum critical flow velocity and natural frequency of a pipe with multiple intermediate supports. The maximum values are required to adjust the support conditions leading to the nodes of higher-order equilibrium configurations and complex modes. Furthermore, the natural frequencies of the pipe conveying supercritical fluid no longer satisfy the monotonicity for the support stiffness, the symmetry for the support position, and the ‘zero-point’ property for the support number.

Determining State Points through the Radial Distribution Function of Yukawa Fluids at Equilibrium

Based on our previous work [Fluid Phase Equilibria, 2023, 567, 113709], we here use the radial distribution function (RDF) to determine the state points (density and temperature) of a fluid under the Yukawa potential at equilibrium. The reduced density and reduced temperature are defined as ρ*=ρσ3 and β*=1/T*=ϵ/kBT, respectively. Through the Molecular Dynamics (MD) simulations, we obtain equilibrium configurations and use these data for building models via two methods. The first method establishes two empirical correlations for each potential considered, one between the heights of the first peaks of the RDFs and state points, as well as the other between the displacements of the first peaks of the RDFs and state points. Through these empirical correlations, we can determine the state points of new Yukawa fluid systems with 100% accuracy. The second method utilizes artificial neural network models to predict state points from the heights and displacements of the first peaks of the RDFs, achieving 100% accuracy when the predicted results are rounded to one decimal place. The success of these methods again demonstrates the feasibility of determining state points solely based on equilibrium configurations, is an extension from the Lennard-Jones fluids to the Yukawa potential related fluids.

Equilibrium configurations of two-dimensional bubbles in a channel: N -bubble case

Two-dimensional liquid foam systems comprised of an odd number N of bubbles arranged in a staircase configuration are studied as they are packed within a confined straight channel. Energy minimization is employed to characterize the permitted topology and geometry of the system at equilibrium. It is known that in an infinite staircase, bubble films are flat. The configuration for a finite number of bubbles, however, deviates from that of an infinite staircase at the edges. It is known that films near the edges are not flat and structures with a few bubbles (up to N = 3 ) are well characterized. To study what happens as more bubbles are added, and how the behaviour approaches that of an infinite staircase, structures comprised of N = [ 5 , 7 , … ] bubbles are studied here. For these N values, minimum and maximum bubble areas are found that can pack in a staircase configuration within a straight channel. However, for an infinite staircase and also for N = 3 , there are maximum areas, but no minima. Staircases with very short films, which make the structure susceptible to break in various ways, are also identified.

Small field chaos in spin glasses: Universal predictions from the ultrametric tree and comparison with numerical simulations

Replica symmetry breaking (RSB) for spin glasses predicts that the equilibrium configuration at two different magnetic fields are maximally decorrelated. We show that this theory presents quantitative predictions for this chaotic behavior under the application of a vanishing external magnetic field, in the crossover region where the field intensity scales proportionally to [Formula: see text], being N the system size. We show that RSB theory provides universal predictions for chaotic behavior: They depend only on the zero-field overlap probability function [Formula: see text] and are independent of other system features. In the infinite volume limit, each spin-glass sample is characterized by an infinite number of states that have a tree-like structure. We generate the corresponding probability distribution through efficient sampling using a representation based on the Bolthausen-Sznitman coalescent. Using solely [Formula: see text] as input we can analytically compute the statistics of the states in the region of vanishing magnetic field. In this way, we can compute the overlap probability distribution in the presence of a small vanishing field and the increase of chaoticity when increasing the field. To test our computations, we have simulated the Bethe lattice spin glass and the 4D Edwards-Anderson model, finding in both cases excellent agreement with the universal predictions.

Behavior of a phase field model for wetting on structured surfaces

AbstractTechnical surfaces are generally not perfectly smooth, but usually exhibit roughness. Additionally, as micro production techniques continue to evolve, geometrical surface structures at the micro scale can be manufactured, which presents a challenge for wetting models that are commonly formulated for smooth surfaces. A phase field model for surface wetting, proposed by Diewald et al., serves as a basis for this investigation. On smooth surfaces, the width of the gas‐liquid interface can be widened for scale bridging purposes, as this allows for accurate computations on coarse grids. However, it was observed that this interface scaling impacts the results when the model is applied to rough surfaces, and therefore a free choice of the interface width is not always sensible. In this work, the ability of the model to deal with sinusoidally shaped surfaces is investigated. An Allen–Cahn evolution equation is used to determine static equilibrium configurations for droplets on such surfaces.

Direct computations of viscoelastic moduli of biomolecular condensates.

Biomolecular condensates are viscoelastic materials defined by time-dependent, sequence-specific complex shear moduli. Here, we show that viscoelastic moduli can be computed directly using a generalization of the Rouse model that leverages information regarding intra- and inter-chain contacts, which we extract from equilibrium configurations of lattice-based Metropolis Monte Carlo (MMC) simulations of phase separation. The key ingredient of the generalized Rouse model is a graph Laplacian that we compute from equilibrium MMC simulations. We compute two flavors of graph Laplacians, one based on a single-chain graph that accounts only for intra-chain contacts, and the other referred to as a collective graph that accounts for inter-chain interactions. Calculations based on the single-chain graph systematically overestimate the storage and loss moduli, whereas calculations based on the collective graph reproduce the measured moduli with greater fidelity. However, in the long time, low-frequency domain, a mixture of the two graphs proves to be most accurate. In line with the theory of Rouse and contrary to recent assertions, we find that a continuous distribution of relaxation times exists in condensates. The single crossover frequency between dominantly elastic vs dominantly viscous behaviors does not imply a single relaxation time. Instead, it is influenced by the totality of the relaxation modes. Hence, our analysis affirms that viscoelastic fluid-like condensates are best described as generalized Maxwell fluids. Finally, we show that the complex shear moduli can be used to solve an inverse problem to obtain the relaxation time spectra that underlie the dynamics within condensates. This is of practical importance given advancements in passive and active microrheology measurements of condensate viscoelasticity.

Elastic ribbons in bubble columns: When elasticity, capillarity, and gravity govern equilibrium configurations.

Taking advantage of the competition between elasticity and capillarity has proven to be an efficient way to design structures by folding, bending, or assembling elastic objects in contact with liquid interfaces. Elastocapillary effects often occur at scales where gravity does not play an important role, such as in microfabrication processes. However, the influence of gravity can become significant at the desktop scale, which is relevant for numerous situations including model experiments used to provide a fundamental physics understanding, working at easily accessible scales. We focus here on the case of elastic ribbons placed in two-dimensional bubble columns: by introducing an elastic ribbon inside the central soap films of a staircase bubble structure in a square cross-sectioncolumn, the deviation from Plateau's laws (capillarity-dominated case dictating the shape of usual foams) can be quantified as a function of the rigidity of the ribbon. For long ribbons, gravity cannot be neglected. We provide a detailed theoretical analysis of the ribbon profile, taking into account capillarity, elasticity, and gravity. We compute the total energy of the system and perform energy minimization under constraints, using Lagrangian mechanics. The model is then validated via a comparison with experiments with three different ribbon thicknesses.

Design and analysis of a flexible struts V-expander tensegrity robot for navigating pipes

Tensegrity robots have increasingly attracted attention in recent years. Traditionally, these robots rely on rigid struts and cables to maintain equilibrium configurations. However, the inflexibility inherent in these rigid struts curtails the robot’s capacity for deformation, thereby amplifying structural intricacy and imposing limitations on potential applications, particularly in the realm of pipe inspection. Drawing inspiration from the V-expander tensegrity structure, this paper presents a design, analysis, and validation of a flexible struts tensegrity robot. The integration of flexible struts enables the robot to exhibit a compact structure, passive compliance, and excellent adaptability. Through the actuation of three active cables, the robot exhibits inchworm-like motion capabilities for pipes ranging from 50 mm to 110 mm in diameters. A kinetostatics modeling approach is presented to predict the shapes of flexible struts and control the motion behaviors of the robot. To validate the capabilities of the proposed robot and assess the effectiveness of the kinetostatics model, a prototype was constructed and subjected to a series of experiments. The results demonstrate that the prototype exhibits remarkable shape changeability, mobility, and adaptability, while precisely controlling the contact force.

A generalized geometric mechanics theory for multi-curve-fold origami: Vertex constrained universal configurations

Folding paper along curves leads to spatial structures that have curved surfaces meeting at spatial creases, defined as curve-fold origami. In this work, we provide an Eulerian framework focusing on the mechanics of arbitrary curve-fold origami, especially for multi-curve-fold origami with vertices. We start with single-curve-fold origami that has wide panels. Wide panel leads to different domains of mechanical responses induced by various generator distributions of the curved surface. The theories are then extended to multi-curve-fold origami, involving additional geometric correlations between creases. As an illustrative example, the deformation and equilibrium configuration of origami with annular creases are studied both theoretically and numerically. Afterward, single-vertex curved origami theory is studied as a special type of multi-curve-fold origami. We find that the extra periodicity at the vertex strongly constrains the configuration space, leading to a region near the vertex that has a striking universal equilibrium configuration regardless of the mechanical properties. Both theories and numerics confirm the existence of the universality in the near-field region. In addition, the far-field deformation is obtained via energy minimization and validated by finite element analysis. Our generalized multi-curve-fold origami theory, including the vertex-contained universality, is anticipated to provide a new understanding and framework for the shape programming of the curve-fold origami system.

Complexity Scaling of Liquid Dynamics.

According to excess-entropy scaling, dynamic properties of liquids like viscosity and diffusion coefficient are determined by the entropy. This link between dynamics and thermodynamics is increasingly studied and of interest also for industrial applications, but hampered by the challenge of calculating entropy efficiently. Utilizing the fact that entropy is basically the Kolmogorov complexity, which can be estimated from optimal compression algorithms [Avinery etal., Phys. Rev. Lett. 123, 178102 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.178102; Martiniani etal., Phys. Rev. X 9, 011031 (2019)PRXHAE2160-330810.1103/PhysRevX.9.011031], we here demonstrate that the diffusion coefficients of four simple liquids follow a quasiuniversal exponential function of the optimal compression length of a single equilibrium configuration. We conclude that "complexity scaling" has the potential to become a useful tool for estimating dynamic properties of any liquid from a single configuration.

Analytical Framework for Tension Characterization in Submerged Anchor Cables via Nonlinear In-Plane Free Vibrations

Submerged tensioned anchor cables (STACs) are pivotal components utilized extensively for anchoring and supporting offshore floating structures. Unlike tensioned cables in air, STACs exhibit distinctive nonlinear damping characteristics. Although existing studies on the free vibration response and tension identification of STACs often employ conventional Galerkin and average methods, the effect of the quadratic damping coefficient (QDC) on the vibration frequency remains unquantified. This paper re-examines the effect of bending stiffness on the static equilibrium configuration of STACs, and establishes the in-plane transverse free motion equations considering bending stiffness, sag, and hydrodynamic force. By introducing the bending stiffness influence coefficient and the Irvine parameter, the exact analytical solutions of symmetric and antisymmetric frequencies and modal shapes of STACs are derived. An improved Galerkin method is proposed to discretize the nonlinear free motion equations ensuring the accuracy and applicability of the analytical results. Additionally, this paper presents an analytical solution for the nonlinear free vibration response of the STACs using the improved averaging method, along with improved frequency formulas and tension identification methods considering the QDC. Through a case study, it is demonstrated that the improved methods introduced in this paper offer higher accuracy and wider applicability compared to the conventional approaches. These findings provide theoretical guidance and reference for the precise dynamic analysis, monitoring, and evaluation of marine anchor cable structures.

Joint action of phase mixing and nonlinear effects in MHD waves propagating in coronal loops

Context. The evolution of Alfvén waves in cylindrical magnetic flux tubes, which represent a basic model for loops observed in the solar corona, can be affected by phase mixing and turbulent cascade. Phase mixing results from transverse inhomogeneities in the Alfvén speed, causing different shells of the flux tube to oscillate at different frequencies, thus forming increasingly smaller spatial scales in the direction perpendicular to the guide field. Turbulent cascade also contributes to the dissipation of the bulk energy of the waves through the generation of smaller spatial scales. Both processes present characteristic timescales. Different regimes can be envisaged according to how those timescales are related and to the typical timescale at which dissipation is at work. Aims. We investigate the interplay of phase mixing and the nonlinear turbulent cascade in the evolution and dissipation of Alfvén waves using compressible magnetohydrodynamic numerical simulations. We consider perturbations in the form of torsional waves, both propagating and standing, or turbulent fluctuations, or a combination of the two. The main purpose is to study how phase mixing and nonlinear couplings jointly work to produce small scales in different regimes. Methods. We conducted a numerical campaign to explore the typical parameters, such as the loop length, the amplitude and spatial profile of the perturbations, and the dissipative coefficients. A pseudo-spectral code was employed to solve the three-dimensional compressible magnetohydrodynamic equations, modeling the evolution of perturbations propagating in a flux tube corresponding to an equilibrium configuration with cylindrical symmetry. Results. We find that phase mixing takes place for moderate amplitudes of the turbulent component even in a distorted, nonaxisymmetric configuration, building small scales that are locally transverse to the density gradient. The dissipative time decreases with increasing the percentage of the turbulent component. This behavior is verified both for propagating and standing waves. Even in the fully turbulent case, a mechanism qualitatively similar to phase mixing occurs: it actively generates small scales together with the nonlinear cascade, thus providing the shortest dissipative time. General considerations are given to identify this regime in the parameter space. The turbulent perturbation also distorts the background density, locally increasing the Alfvén velocity gradient and further contributing to accelerating the formation of small scales. Conclusions. Our campaign of simulations is relevant for the coronal plasma where Reynolds and Lundquist numbers are extremely high. For sufficiently low perturbation amplitudes, phase mixing and turbulence work synergically, speeding up the dissipation of the perturbation energy: phase mixing dominates at early times and nonlinear effects at later times. We find that the dissipative time is shorter than those of phase mixing and the nonlinear cascade when individually considered.

Bile acid-gut microbiota imbalance in cholestasis and its long-term effect in mice.

Our pre-clinical study using a mouse model of cholestasis underscores that cholestasis not only disrupts the equilibrium and structural configuration of the gut microbiota but also emphasizes the persistence of these adverse effects even after bile stasis restoration. This suggests the need of monitoring and initiating interventions for gut microbiota structural restoration in patients with cholestasis during and after recovery. We believe that our study contributes to novel and better understanding of the intricate interplay among bile acid homeostasis, gut microbiota, and cholestasis-associated complications. Our pre-clinical findings may provide implications for the clinical management of patients with cholestasis.

Enhanced fast inertial relaxation engine (FIRE) for multiscale simulations

In multiscale modeling methods (MMM), the integration of atomistic to continuum coupling is a common practice, where two regions have their own distinct length scales. The equilibrium configurations of such multiscale systems under given conditions are typically obtained through energy minimization algorithms (EMA). However, traditional EMAs, such as the conjugate gradient (CG) and limited-memory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) algorithms, are unable to discern the diverse scales inherent in such systems. In this work, it is found that the convergence rate of energy minimization in multiscale simulations is significantly slower than that in full atomistic simulations, regardless of using CG, LBFGS algorithms or the latest fast inertial relaxation engine (FIRE). The lower efficiency emerges due to the coexistence of atoms and nodes with distinct length scales within the multiscale framework, yet the current EMAs fail to differentiate between them. It results in disparate convergence rates across different scales, which undermines both computational accuracy and efficiency. To address the issue, a multiscale FIRE algorithm which updates positions of atoms and nodes synchronously by employing appropriate effective mass is proposed. The optimal effective mass is determined by synchronizing the vibration of harmonic oscillators across different scales. By employing the multiscale FIRE algorithm, the computational efficiency increased by 24.4 and 23.7 times compared to the CG and LBFGS algorithms when used for multiscale nanoindentation simulations. These findings and the proposed algorithm provide valuable insights for structural relaxations of multiscale physical problems and are promising to further improve the computational accuracy and efficiency of MMMs.

Optimisation of the poloidal field system for advanced divertor configurations in STEP

The power exhaust proves to be one of the most challenging and concept–defining aspects in the design of a commercial fusion power plant, while the magnetic coil system, capable of supporting advanced exhaust solutions, emerges as one of the main design and cost drivers. Consequently, much effort should be dedicated to the optimisation of a robust global magnetic configuration, which integrates both the plasma and edge scenarios, while ensuring engineering feasibility and compatibility with the available technology. Here we present a multidisciplinary framework employed to analyse, evaluate, and optimise the Spherical Tokamak for Energy Production (STEP) equilibrium configuration, coupled with a viable divertor solution, and a compatible poloidal field coil system. The complexity of this task leads to a multitude of potentially conflicting requirements and competing constraints. We identify interfaces and conflicts between the aspects of the design that were previously considered independently, and highlight the key benefits, trends, and trade–offs between alternative configurations. We demonstrate that advanced exhaust solutions, simultaneously applied to both inboard and outboard divertors, are accessible with feasible coil sets under conditions relevant for STEP. We show that the most promising inner–X geometry, paired with the outer super–X configuration, can significantly enhance divertor’s power handling capability, allowing access to stable detached regimes. The coil set feasibility is further assessed considering its compatibility with the assumed plasma initiation scenario, and with the most demanding plasma current density profiles utilising alternative heating and current drive schemes.

Dynamics of weighted flexible ribbons in a uniform flow

This study explores the dynamics of flexible ribbons with an added weight $G$ at the tail in uniform flow, considering key parameters like inflow Reynolds number ( $Re_u$ ), mass ratio ( $M_t$ ) and aspect ratio ( ${A{\kern-4pt}R}$ ). For two-dimensional ribbons, a simplified theoretical model accurately predicts equilibrium configurations and forces. Inspired by Barois & De Langre (J. Fluid Mech., vol. 735, 2013, R2), we introduce an important control parameter ( $C_G$ ) that effectively collapses normalized forces and angle data. Vortex-induced vibration is observed, and Strouhal number ( $St$ ) scaling laws with $C_G$ are identified. In three-dimensional scenarios, the model effectively predicts lift, but its accuracy in predicting drag is limited to situations with small $Re_u$ values. The flow along the side edges mitigates pressure differences, thereby suppressing vibration and uplift, particularly noticeable in the case of narrow ribbons. This study offers new insights into the dynamics of flexible bodies in uniform flow.

Compliant frame geometry for DEMES-based gripper and flapping wing actuators: A comprehensive design study

Dielectric elastomer minimum energy structures (DEMES) have attracted significant attention in the recent past because of their ability to switch between multiple equilibrium states. The DEMES is formed when a pre-stretched elastomer film adheres to an inextensible frame and is further allowed to attain an equilibrium configuration as a result of energy minimization. While several researchers have investigated; both theoretically and experimentally, the underlying mechanics of DEMES, a majority of them deploy simplistic boundary frames (square/rectangular/circular) to obtain the requisite minimum energy configuration. In this paper, we demonstrate that this seemingly restrictive choice of using simplistic frame geometries can be given away by designing more complex-shaped compliant frames capable of producing useful modes of deformation. To demonstrate this idea, two prototypes; namely, the four-arm gripper actuator and the flapping-wing actuator, are studied numerically and experimentally. In both cases, a single compliant frame is used to realize the respective configuration, thus circumventing the need to assemble multiple MES on a common platform. In tandem, we investigate the role of reinforcements in (a) controlling the warping in MES and (b) maximizing actuation in the desired mode of deformation. Finite element analysis are carried out using the ABAQUS to determine the equilibrium configuration of the actuators, and subsequently their electromechanical behavior. Experimental investigations involve the utilization of the commercially available VHB 4910 acrylic tape in conjunction with PET frames and 3D-printed reinforcements. Excellent qualitative agreement is achieved between the numerical predictions and the experimental observations. Finally, we allude to a few innovative frame architectures leading the MES of engineering interest.

Equilibria and dynamics of two coupled chains of interacting dipoles.

We explore the energy transfer dynamics in an array of two chains of identical rigid interacting dipoles. Varying the distance b between the two chains of the array, a crossover between two different ground-state (GS) equilibrium configurations is observed. Linearizing around the GS configurations, we verify that interactions up to third nearest neighbors should be accounted to accurately describe the resulting dynamics. Starting with one of the GS, we excite the system by supplying it with an excess energy ΔK located initially on one of the dipoles. We study the time evolution of the array for different values of the system parameters b and ΔK. Our focus is hereby on two features of the energy propagation: the redistribution of the excess energy ΔK among the two chains and the energy localization along each chain. For typical parameter values, the array of dipoles reaches both the equipartition between the chains and the thermal equilibrium from the early stages of the time evolution. Nevertheless, there is a region in parameter space (b,ΔK) where even up to the long computation time of this study, the array does neither reach energy equipartition nor thermalization between chains. This fact is due to the existence of persistent chaotic breathers.

Coupled Spacecraft Charging Due to Continuous Electron Beam Emission and Impact

Spacecraft charge naturally in orbit due to the plasma environment and the electromagnetic radiation from the sun. By emitting an electron beam, a servicer spacecraft can control its electric potential and also the potential of a neighboring target if the electron beam is aimed at the target spacecraft. In addition, the impacting electron beam excites secondary electrons and x-rays, providing a way to touchlessly sense the potential of the target. Because of the electron beam, the charging dynamics of the two spacecraft are coupled. This paper studies the effects of the beam on the electric potentials using a numerical charging model. It is found that multiple equilibria may exist due to the electron beam. Jumps between equilibrium configurations are possible when the electron beam energy is quickly reduced or when current fluctuations are present. Being aware of multiple equilibrium configurations is important for feedback-based charge control but also enables a new open-loop charge control around one of the equilibria. The effect of the electron beam on the spacecraft potentials is studied for geostationary Earth orbit and cislunar space. It is found that the current applied by the beam to the target may influence remote electric potential sensing methods.

Implications of vertical stability control on the SPARC tokamak

To achieve its performance goals, SPARC plans to operate in equilibrium configurations with a strong elongation of κareal∼1.75 , which in turn will destabilize the n = 0 vertical instability. However, SPARC also features a relatively thick conducting wall that is designed to withstand disruption forces, leading to lower vertical instability growth rates than usually encountered. In this work, we use the TokSyS framework to survey families of accessible shapes near the SPARC baseline configuration, finding maximum growth rates in the range of γ≲100 s−1. The addition of steel vertical stability plates has only a modest ( ∼25% ) effect on reducing the vertical growth rate and almost no effect on the plasma controllability when the full vertical stability system is taken into account, providing flexibility in the plate conductivity in the SPARC design. Analysis of the maximum controllable displacement on SPARC is used to inform the power supply voltage and current limit requirements needed to control an initial vertical displacement of 5% of the minor radius. From the expected spectra of plasma disturbances and diagnostic noise, requirements for filter latency and vertical stability coil heating tolerances are also obtained. Small modifications to the outboard limiter location are suggested to allow for an unmitigated vertical disturbance as large as 5% of the minor radius without allowing the plasma to become limited. Further, investigations with the 3D COMSOL code reveal that strategic inclusion of insulating structures within the VSC supports are needed to maintain sufficient magnetic response. The workflows presented here help to establish a model for the integrated predictive design for future devices by coupling engineering decisions with physics needs.