Static Equilibrium Conditions Research Articles

Vectorised graphics processing unit accelerated dynamic relaxation for bar and beam elements

Dynamic relaxation is an analysis method that can be used to find the static equilibrium state of a structure undergoing large displacements. Due to the algorithm's iterative nature, which derives from a damped dynamic oscillations approach, it can take considerable time when solving models with dense node counts. This research looks at a vectorised dynamic relaxation implementation, executed on a graphics card's processing unit (GPU) for the most computationally demanding calculations, and compared to traditional computations on the central processing unit (CPU), to determine if time savings can be found with the change in processor type. The programming language Python and module PyCUDA are utilised on a desktop graphics card with over two thousand shader cores, and compared to a quad-core processor using the same code layout. A vectorised formulation for bar elements and reduced degree-of-freedom beam elements is used, to analyse the absolute and relative run-times for three benchmark cases. It is found that for models with low node counts, the CPU is faster, while for medium to highly dense models, the GPU acceleration can reduce absolute run-times by a factor of three to just over eight. Most benefit is gained when the executed code allows the GPU cores to be overloaded, such that complex calculations involving large arrays of data can be spread over the many cores. This was evident in the GPU load demand, which increased steadily with increasing node count but remained below 100%, while the CPU was readily activated to maximum load for even simple models. The time savings are of interest to architects, engineers and researchers utilising the dynamic relaxation method with medium to complex models, as the benefits can be harnessed with minimal code alteration.

Asymptotic solutions of coupled equations of supercritically axially moving beam

In supercritical regime, the coupled model equations for the axially moving beam with simple support boundary conditions are considered. The critical speed is determined by linear bifurcation analysis, which is in agreement with the results in the literature. For the corresponding static equilibrium state, the second-order asymptotic nontrivial solutions are obtained through the multiple scales method. Meantime, the numerical solutions are also obtained based on the finite difference method. Comparisons among the analytical solutions, numerical solutions and solutions of integro-partial-differential equation of transverse which is deduced from coupled model equations are made. We find that the second-order asymptotic analytical solutions can well capture the nontrivial equilibrium state regardless of the amplitude of transverse displacement. However, the integro-partial-differential equation is only valid for the weak small-amplitude vibration axially moving slender beams.

A container loading algorithm with static mechanical equilibrium stability constraints

The Container Loading Problem (CLP) literature has traditionally guaranteed cargo static stability by imposing the full support constraint for the base of the box. Used as a proxy for real-world static stability, this constraint excessively restricts the container space utilization and has conditioned the algorithms developed for this problem. In this paper we propose a container loading algorithm with static stability constraints based on the static mechanical equilibrium conditions applied to rigid bodies, which derive from Newton’s laws of motion. The algorithm is a multi-population biased random-key genetic algorithm, with a new placement procedure that uses the maximal-spaces representation to manage empty spaces, and a layer building strategy to fill the maximal-spaces. The new static stability criterion is embedded in the placement procedure and in the evaluation function of the algorithm. The new algorithm is extensively tested on well-known literature benchmark instances using three variants: no stability constraint, the classical full base support constraint and with the new static stability constraint—a comparison is then made with the state-of-the-art algorithms for the CLP. The computational experiments show that by using the new stability criterion it is always possible to achieve a higher percentage of space utilization than with the classical full base support constraint, for all classes of problems, while still guaranteeing static stability. Moreover, for highly heterogeneous cargo the new algorithm with full base support constraint outperforms the other literature approaches, improving the best solutions known for these classes of problems.

Elastic-plastic secondary indeterminate problem for thin-walled pipe through the inner-wall loading by three-point bending

ABSTRACTThe inner-wall loading by three-point bending about thin-walled pipe is an elastic-plastic secondary indeterminate problem in the symmetrical three-roller setting round process. In this study, the shifting of the tangent point between the pipe and lower roller is ignored. The bilinear hardening material model is adopted, and the static equilibrium condition, physical relationship of elastic-plastic deformation, and deformation compatibility condition are taken into account. Based on the geometrical discrete idea, a semi-circular pipe is meshed equably into N micro-pipe-wall elements with same geometric parameters along the circumferential direction. Deformation characteristics of each element are calculated, and then the deformation history response of the whole pipe is resolved by the load increment method. The finite element model of static bending in three-roller setting round process is established by using the software package ABAQUS. The theoretical and simulated results show that the cross section of pipe has two positive bending regions and two reverse bending regions; the maximum bending curvature appears in the bottom section of pipe, the minimum bending curvature appears in the section corresponding to the tangent point of the pipe and lower roller. The quantitative relationships between the upper roller load, maximum(minimum) bending curvature and reduction are given. Finally, the reliability of theoretical calculation is proved by numerical simulation.

Predicting the popularity growth of online content: Model and algorithm

The evolution of the popularity of online content is analyzed, and two characteristic patterns pertaining to linear and non-linear growth periods are detected. While the former characterizes the propagation of online content through a dynamical process in a state of statistical equilibrium, the latter appears when this state is perturbed by exogenous intervention events. Such episodes increase the susceptibility of higher threshold individuals who opportunistically adopt the propagating content. To capture the dynamics of both diffusion modes, the popularity of online content is modeled by interlacing linear and non-linear growth terms, reduced to 1st-degree polynomial and logistic functions corresponding respectively to stationary and non-stationary adoption phases. The precise fit of the model to empirical popularity patterns verifies its suitability as prediction tool. The proposed model is employed to generate forecasts about the popularity of online content through extrapolation. Highly accurate prediction results surpassing existing methods in terms of precision and predictive capacity are demonstrated. The prediction method is formulated into an algorithm, applicable to real time forecasting of the popularity of online content without training, using minimal, macroscopic, publicly available information.

Postbuckling and postbuckled vibration analysis of sandwich plates under non-uniform mechanical edge loadings

Postbuckling and postbuckled vibration characteristics of sandwich plate subjected to non-uniform in-plane loadings is investigated in this article. The core compressibility effects are considered in the model by assuming fourth and fifth order expansions for the transverse and tangential displacement of the core. The exact stress distributions within the panel are determined by panel prebuckling analysis for the applied parabolic and partial edge loadings. Multi-term Galerkin's method is employed to reduce the governing partial differential equation to a set of non-linear ordinary differential equations in the case of dynamic analysis and to a set of non-linear algebraic equations in the case of static stability analysis. Newton-Raphson method in conjunction with Riks approach is used to trace the postbuckling path. Free vibration frequencies of the sandwich plate about a static equilibrium state are obtained by solving the associated linear eigenvalue problem. From numerical study it is observed that the postbuckling path becomes more softening type with increase of aspect ratio. Moreover, unloaded fixed edge of SCSC sandwich plate gives more rigidity than unloaded simply supported edge of CSCS. The prebuckling and postbuckling frequency increases with increase in edge restraint, and in postbuckling range, frequency increases with increase of in-plane load.

Instabilities of wrinkled membranes with pressure loadings

Wrinkling can affect the functionality of thin membranes subjected to various loadings or boundary conditions. The concept of relaxed strain energy was studied for isotropic, hyperelastic, axisymmetric membranes pressurized by gas or fluid. Non-intuitive instabilities were observed when axisymmetric wrinkled membranes were perturbed with angle dependent displacement fields. A linearized theory showed that static equilibrium states of pressurized membranes, modelled by a relaxed strain energy formulation, are unstable, when the wrinkled surface is subjected to pressure loadings. The theory is extended to the non-axisymmetric membranes and it is shown that these instabilities are local phenomena. Simulations for the pressurized cylindrical membranes with non-uniform thickness and hemispherical membranes support the claims in both theoretical and numerical contexts including finite element simulations.

An analytic model for tube bending springback considering different parameter variations of Ti-alloy tubes

Springback after unloading is an issue that directly reduces the accuracy of bent tubes, especially for Ti-alloy tubes which are of high strength and low Young’s modulus. The Young’s modulus, E; wall thickness, t; and neutral layer, De, of a tube vary during the bending process. These variations may influence the bending deformation of components, thus on springback. Considering these variations, an analytic elastic-plastic tube bending springback model was established in this study based on the static equilibrium condition. When these variations were considered individually or combined, the resulting springback angles were all larger and closer to the experimental results than the results when variations were not considered for a D6mm×t0.6mm Ti-3Al-2.5V Ti-alloy tube. The t variation contribution is the largest and decreases the prediction error by 41.2%–45.3%. De variation ranks second and decreases the error by 21.2%-25.3%. E variation is the least significant, decreasing the error by only 2.4%. Furthermore, the influence of the stable Young’s modulus Ea on the springback is larger than the initial Young’s modulus E0. Therefore, for the bending springback of tubes with a small difference between E0 and Ea and under a normal bending radius, E variation effects can be neglected. While for tubes with large differences between E0 and Ea, and high springback prediction requirements, the E variation should be replaced by Ea. The influences of the initial tube sizes, material properties and bent tube sizes of the Ti-3Al-2.5V tube on springback were obtained using the newly developed model.

Analysis of steelmaking kinetics from IMPHOS pilot plant data

It is widely accepted that understanding the kinetics of steelmaking is a complex task, and reliable and validated kinetics models are required for developing successful steelmaking process models. Therefore, as an initial attempt, this paper analyses the applicability of first order kinetics to explain the steelmaking reaction kinetics using the published data in the IMproving Phosphorus Refining research report. The process data for 20 heats in a 6 tonne pilot plant were analysed for the removal of carbon, silicon, manganese and phosphorus using first order kinetics with static and dynamic equilibrium conditions. It was observed that the removal behaviour of silicon closely followed a first order kinetics relationship, while that of carbon only approximately followed a first order kinetics relationship. The removal of manganese did not show a good degree of fit with first order kinetics using static equilibrium condition, but a clear improvement was observed when calculated using dynamic equilibrium condition. In contrast, the kinetics of phosphorus oxidation did not follow any first order relationship.

Shear Stress Calculation and Distribution in Variable Cross Sections of Box Girders with Corrugated Steel Webs

AbstractBased on the condition of static equilibrium and the equivalent law of shearing stress of an infinitesimal segment, this paper provides a strict derivation of the general formula for shearing stress in a nonuniform box girder with corrugated steel webs in the elastic stage. The derived formula is applied to investigate the stress distribution in concrete flanges and corrugated steel webs. Additional shear stress was found to be caused by the bending moment, and the axial force should be included in calculations from the effect of variable cross sections, which are quite different from that of the uniform cross sections. Moreover, because the additional shear stress of the bending moment is self-balanced, the shear force distribution can be adjusted between the concrete slabs and steel webs. Research shows that shear stress decreases markedly in the webs, whereas the shear stress apparently increases in the inclined bottom flange in the action of the bending moment. This study also found and explai...

Edge-state transport in Floquet topological insulators

Floquet topological insulators are systems in which the topology emerges out of equilibrium when a time periodic perturbation is applied. In these systems one can define quasi-energy states which replace the quilibrium stationary states. The system exhibits its non-trivial topology by developing edge localized quasi-energy states which lie in a gap of the quasi energy spectrum. These states represent a non-equilibrium analogue of the topologically protected edge-states in equilibrium topological insulators. In equilibrium these edge-states lead to very specific transport properties, in particular the two-terminal conductivity of these systems is $2e^2/h$. Here we explore the transport properties of the edge-states in a Floquet topological insulator. In stark contrast to the equilibrium result, we find that the two terminal conductivity of these edge states is significantly different from $2e^2/h$. This fact notwithstanding, we find that for certain external potential strengths the conductivity is smaller than $2e^2/h$ and robust to the effects of disorder and smooth changes to the Hamiltonian's parameters. This robustness is reminiscent of the robustness found in equilibrium topological insulators. We provide an intuitive understanding of the reduction of the conductivity in terms of scattering by photons. This leads us to consider a previously proposed Floquet sum rule which recovers the equilibrium value of $2e^2/h$ for the conductivity when edge states are present. We show that this sum rule holds in our system using both numerical and analytic techniques.

Processing of Deployment Using an Adaptive Self-Spreading Algorithm Based on Electrostatic Field for Mobile Sensor

The deployment of mobile sensor networks is a hotspot issue. This paper is dedicated to proposing a distributed, adaptive, and scalable mobile sensor network algorithm based on the theory in electrostatic field. In the algorithm, the obstacles and nodes in the deployment region were regarded as the charged particles, which were driven by the Coulomb Force exerted by the other particles and obstacles, and eventually all the nodes were distributed throughout the entire network while repelled by the obstacles and other nodes simultaneously. Based on the classical virtual potential field algorithm, the boundary conditions, the static equilibrium conditions and the controllable coverage rate were expanded in the present algorithm, which are also the primary innovation points and contributions of this paper. The efficiency of the proposed algorithm was evaluated by performing simulations under four frequently-used scenarios, namely, deployment under normal conditions, deployment with obstacles, re-deployment due to unpredicted node failures, and deployment with regions of interest. Moreover, the performance of the algorithm was assessed using coverage and uniformity as the criteria. The simulation results indicate that the algorithm runs well under all these four frequently-used scenarios. In a completely unknown deployment area, the designated goal can also be achieved using the proposed algorithm owing to its adaptivity.

A mechanical model of symmetrical three-roller setting round process: The static bending stage

The static bending deformation of three-roller setting round process is an elastic–plastic secondary indeterminate problem, and belongs to the plane bending deformation. This paper adopts a bilinear material model. In consideration of shifting of the tangent points between the pipe and the lower rollers, the static equilibrium conditions, the physical relationships of elastic–plastic deformation, the deformation compatibility conditions, and the constraint conditions are analyzed comprehensively. By applying the discrete method, a semi-circular pipe is meshed equably into N micro-pipe-wall elements with the same geometric parameters. Deformation characteristics of each micro-pipe-wall element are calculated, and then the deformation history responses of the whole pipe are resolved by applying the load increment method, based on the variable step size, and the experimental apparatus is further designed to verify the theoretical model. Finally, the effects of various process parameters on the upper roller load and bending curvature are studied. The results show that the static bending deformation can be assumed to be a pure bending process; the cross section of the pipe has two positive bending regions and two reverse bending regions; the upper roller diameter and the lower rollers spacing are the major factors for the upper roller load and the bending curvature; the dynamic roll bending process can be simplified into a static to-and-fro bending process.

Numerical algorithm for solving of nonlinear problems of structural mechanics based on the continuation method in combination with the dynamic relaxation method

A numerical algorithm of strength and stability analysis of nonlinear deformable bar systems and thin-walled spatial structures is proposed. A numerical technique is based on the continuation method and the dynamic relaxation method. When using the method of dynamic relaxation the state of static equilibrium of structures is defined after the damped oscillations by integrating over leading parameter. The solution of the initial equation system describing the motion of the mechanical system is reduced to the solution of Cauchy problem for systems of ordinary differential equations. At an each step in the leading parameter the vector of nodal displacements and the time parameter are defined. Several examples of numerical analysis for bar, shell and plate are given.

Investigation of Thermostressed State of Coating Formation at Electric Contact Surfacing of “Shaft” Type Parts

The forming of coating at electric contact surfacing is considered. The mathematical model of the coating formation is developed. The method of numerical recurrent solution of the finite-difference form of static equilibrium conditions of the selected elementary volume of coating is used. This model considers distribution of thermal properties and geometric parameters along the thermal deformation zone during the process of electric contact surfacing by compact material. It is found that the change of value of speed asymmetry factor leads to increasing of the friction coefficient in zone of surfacing. This provides the forming of the coating of higher quality. The limitation of the technological capabilities of equipment for electric contact surfacing is related to the size of recoverable parts and application of high electromechanical powers. The regulation of the speed asymmetry factor allows for expanding the technological capabilities of equipment for electric contact surfacing. The nomograms for determination of the stress on the roller electrode and the finite thickness of the coating as the function of the initial thickness of the compact material and the deformation degree are shown.

Continual distribution with screw modes

Explicit approximate solution of the Boltzmann equation for the hardsphere model are built. It has the kind of continual distribution in the case of local Maxwellians of special form describing the screw-shaped stationary equilibrium states of a gas. Some limited cases, in which this distribution minimized the uniform-integral remainder between the sides of this equation are obtained.

Gust response analysis and wind tunnel test for a high-aspect ratio wing

A theoretical nonlinear aeroelastic response analysis for a flexible high-aspect ratio wing excited by harmonic gust load is presented along with a companion wind tunnel test. A multidisciplinary coupled numerical calculation is developed to simulate the flexible model wing undergoing gust load in the time domain via discrete nonlinear finite element structural dynamic analysis and nonplanar unsteady vortex lattice aerodynamic computation. A dynamic perturbation analysis about a nonlinear static equilibrium is also used to determine the small perturbation flutter boundary. A novel noncontact 3-D camera measurement analysis system is firstly used in the wind tunnel test to obtain the spatial large deformation and responses. The responses of the flexible wing under different static equilibrium states and frequency gust loads are discussed. The fair to good quantitative agreements between the theoretical and experimental results demonstrate that the presented analysis method is an acceptable way to predict the geometrically nonlinear gust response for flexible wings.

Models of dark matter halos based on statistical mechanics: The fermionic King model

We discuss the nature of phase transitions in the fermionic King model which describes tidally truncated quantum self-gravitating systems. This distribution function takes into account the escape of high energy particles and has a finite mass. On the other hand, the Pauli exclusion principle puts an upper bound on the phase space density of the system and stabilizes it against gravitational collapse. As a result, there exists a statistical equilibrium state for any accessible values of energy and temperature. We plot the caloric curves and investigate the nature of phase transitions as a function of the degeneracy parameter in both microcanonical and canonical ensembles. We consider stable and metastable states and emphasize the importance of the latter for systems with long-range interactions. Phase transitions can take place between a "gaseous" phase unaffected by quantum mechanics and a "condensed" phase dominated by quantum mechanics. The phase diagram exhibits two critical points, one in each ensemble, beyond which the phase transitions disappear. There also exist a region of negative specific heats and a situation of ensemble inequivalence for sufficiently large systems. We apply the fermionic King model to the case of dark matter halos made of massive neutrinos. The gaseous phase describes large halos and the condensed phase describes dwarf halos. Partially degenerate configurations describe intermediate size halos. We argue that large dark matter halos cannot harbor a fermion ball because these nucleus-halo configurations are thermodynamically unstable (saddle points of entropy). Large dark matter halos may rather contain a central black hole resulting from a dynamical instability of relativistic origin occurring during the gravothermal catastrophe.

VERTICAL EQUILIBRIUM, ENERGETICS, AND STAR FORMATION RATES IN MAGNETIZED GALACTIC DISKS REGULATED BY MOMENTUM FEEDBACK FROM SUPERNOVAE

Recent hydrodynamic (HD) simulations have shown that galactic disks evolve to reach well-defined statistical equilibrium states. The star formation rate (SFR) self-regulates until energy injection by star formation feedback balances dissipation and cooling in the interstellar medium (ISM), and provides vertical pressure support to balance gravity. In this paper, we extend our previous models to allow for a range of initial magnetic field strengths and configurations, utilizing three-dimensional, magnetohydrodynamic (MHD) simulations. We show that a quasi-steady equilibrium state is established as rapidly for MHD as for HD models unless the initial magnetic field is very strong or very weak, which requires more time to reach saturation. Remarkably, models with initial magnetic energy varying by two orders of magnitude approach the same asymptotic state. In the fully saturated state of the fiducial model, the integrated energy proportions E_kin:E_th:E_mag,t:E_mag,o are 0.35:0.39:0.15:0.11, while the proportions of midplane support P_turb:P_th:\Pi_mag,t:\Pi_mag,o are 0.49:0.18:0.18:0.15. Vertical profiles of total effective pressure satisfy vertical dynamical equilibrium with the total gas weight at all heights. We measure the "feedback yields" \eta_c=P_c/\Sigma_SFR (in suitable units) for each pressure component, finding that \eta_turb~4 and \eta_th~1 are the same for MHD as in previous HD simulations, and \eta_mag,t~1. These yields can be used to predict the equilibrium SFR for a local region in a galaxy based on its observed gas and stellar surface densities and velocity dispersions. As the ISM weight (or dynamical equilibrium pressure) is fixed, an increase in $\eta$ from turbulent magnetic fields reduces the predicted \Sigma_SFR by ~25% relative to the HD case.

Do intertidal flats ever reach equilibrium?

Abstract Various studies have identified a strong relation between the hydrodynamic forces and the equilibrium profile for intertidal flats. A thorough understanding of the interplay between the hydrodynamic forces and the morphology, however, concerns more than the equilibrium state alone. We study the basic processes and feedback mechanisms underlying the long‐term behavior of the intertidal system, restricting ourselves to unvegetated intertidal flats that are controlled by cross‐shore tidal currents and wind waves and applying a 1‐D cross‐shore morphodynamic model. The results indicate that by an adjustment of the profile slope and shape, an initial imbalance between deposition and erosion is minimized within a few decades. What follows is a state of long‐term seaward progradation or landward retreat of the intertidal flat, in which the cross‐shore profile shape is largely maintained and the imbalance between deposition and erosion is not further reduced. These long‐term trends can be explained by positive feedbacks from the morphology onto the hydrodynamic forces over the flat: initial accretion (erosion) decreases (increases) the shear stresses over the flat, which induces further accretion (erosion). This implies that a static equilibrium state cannot exist; the flat either builds out or retreats. The modeled behavior is in accordance with observations in the Yangtze Estuary. To treat these unbalanced systems with a one‐dimensional numerical model, we propose a moving (Lagrangian) framework in which a stable cross‐sectional shape and progradation speed can be derived for growing tidal flats, as a function of the wave climate and the sediment concentration in deeper water.