Finite Equilibrium Research Articles

On effects of track random irregularities on random vibrations of vehicle–track interactions

As the integrated reflection of track substructure deformations and the most important excitation for vehicle–track interactions, track irregularities show random nature, and generally being regarded as weak stationary random processes. To better expose the full statistical characteristics of track random irregularities on amplitude and wavelength, a time–frequency transform and probability theory based model is developed to simulate representative and realistic track irregularity sets by combining with random sampling methods. Moreover, a three-dimensional (3-D) vehicle–track coupled model is established by finite element method and dynamic equilibrium equations, where the nonlinearity of wheel/rail interaction is considered. Finally, a probability density evolution method (PDEM) is introduced to solve the probabilistic transmission issues between track irregularity sets and dynamic responses of vehicle–track coupled systems. There is a clear demonstration that the results derived by proposed methods are comparable to the experimental measurements. Through effectively applying the above methodologies, the probabilistic and random characteristics of vehicle–track interaction can be properly revealed.

Lead and selenite adsorption at water–goethite interfaces from first principles

The complexation of toxic and/or radioactive ions on to mineral surfaces is an important topic in geochemistry. We apply periodic-boundary-conditions density functional theory (DFT) molecular dynamics simulations to examine the coordination of Pb(II), , and their contact ion pairs to goethite (1 0 1) and (2 1 0) surfaces. The multitude of Pb(II) adsorption sites and possibility of Pb(II)-induced FeOH deprotonation make this a complex problem. At surface sites where Pb(II) is coordinated to three FeO and/or FeOH groups, and with judicious choices of FeOH surface group protonation states, the predicted Fe–Pb distances are in good agreement with EXAFS measurements. Trajectories where Pb(II) is in part coordinated to only two surface Fe–O groups exhibit larger fluctuations in Pb–O distances. Pb(II)/ contact ion pairs are at least metastable on goethite (2 1 0) surfaces if the has a monodentate Se–O–Fe bond. Our DFT-based molecular dynamics calculations are a prerequisite for calculations of finite temperature equilibrium binding constants of Pb(II) and Pb(II)/ ion pairs to goethite adsorption sites.

Modified viscosity implicit rules for nonexpansive mappings in Hilbert spaces

In this paper, we study a modified viscosity implicit rule of a nonexpansive mapping in Hilbert spaces. Strong convergence theorems are obtained under some suitable assumptions imposed on the parameters. As applications, we obtain some strong convergence theorems for solving fixed-point problems of strict pseudocontractive mappings and finite equilibrium problems in Hilbert spaces. We also give some numerical examples to support our main results.

Enhanced hyperuniformity from random reorganization

Diffusion relaxes density fluctuations toward a uniform random state whose variance in regions of volume [Formula: see text] scales as [Formula: see text] Systems whose fluctuations decay faster, [Formula: see text] with [Formula: see text], are called hyperuniform. The larger [Formula: see text], the more uniform, with systems like crystals achieving the maximum value: [Formula: see text] Although finite temperature equilibrium dynamics will not yield hyperuniform states, driven, nonequilibrium dynamics may. Such is the case, for example, in a simple model where overlapping particles are each given a small random displacement. Above a critical particle density [Formula: see text], the system evolves forever, never finding a configuration where no particles overlap. Below [Formula: see text], however, it eventually finds such a state, and stops evolving. This "absorbing state" is hyperuniform up to a length scale [Formula: see text], which diverges at [Formula: see text] An important question is whether hyperuniformity survives noise and thermal fluctuations. We find that hyperuniformity of the absorbing state is not only robust against noise, diffusion, or activity, but that such perturbations reduce fluctuations toward their limiting behavior, [Formula: see text], a uniformity similar to random close packing and early universe fluctuations, but with arbitrary controllable density.

Stability of cracked concrete hydraulic structures by nonlinear quasi-static explicit finite element and 3D limit equilibrium methods

Several hydraulic concrete structures suffer from severe tridimensional (3D) discrete cracking, producing an assembly of concrete blocks resting one on top of the other. It is important to consider the 3D particularities of the cracked surfaces in the nonlinear sliding safety evaluation of these structures. A methodology to assess a sliding safety factor (SSF) and a sliding direction D for any structure with an a priori known 3D discrete crack surface geometry using the quasi-static explicit nonlinear finite elements method (QSE-FEM) is presented herein in the context of the strength reduction approach. QSE-FEM is known for its efficiency in solving highly nonlinear problems compared to implicit FEM. However, in QSE-FEM, the determination of the incipient failure motion is challenging. In addition to the ratio of kinetic to internal strain energy, a new criterion is proposed to identify sliding initiation based on absolute displacements of a control point. As part of the proposed methodology, a complementary simple tool, 3D-LEM, has been developed as a 3D extension of the classical limit equilibrium method (LEM). It is useful in preliminary sliding analyses to estimate the critical friction coefficient inducing sliding and the corresponding direction D. Three benchmark examples, of increasing complexities, are presented to verify the performance of the proposed methodology. Finally, a case study is adapted from an existing cracked hydraulic structure to evaluate its sliding stability using the QSE-FEM and 3D-LEM approaches. In the three benchmark examples, the same results are found for SSF and D for the two approaches. In the hydraulic structure example, SSFLEM, computed with 3D-LEM, is a lower bound of SSFFEM from QSE-FEM, and a strong correlation is found between the sliding directions computed with the two approaches.

Stresses associated with multiple holes whose individual stresses interact

The stress consequences of adding an auxiliary hole in the neighborhood of an original hole are determined from the load-induced thermal information when the geometric discontinuities are sufficiently close together that their respective stresses interact. The particular situation involves a smaller round hole near a larger round hole in a finite, isotropic tensile member where the holes are aligned in the direction of loading. Introducing a second hole reduces the stresses at each hole had it alone been present. The results correlate well with those from finite elements, strain gage and load equilibrium. The developed technology has relevance for isotropic, orthotropic or functionally graded materials.

Oscillatory instability of convection in ferromagnetic nanofluid and in transformer oil

Stability of a mechanical equilibrium of ferromagnetic nanofluid and transformer oil in a spherical cavity carved inside a Plexiglas block heated from below is investigated experimentally. It is shown that in a fluid left at rest prior to the start of experiment the onset of convection is delayed and the convective motion arises abruptly acquiring a finite equilibrium amplitude at supercritical temperature differences. Convection has an oscillatory character associated with a precession of a vortex axis in the equatorial plane. A hysteresis is detected in the reverse transition to a quiescent state. Oscillations in the investigated ferro-nanofluid have been observed during the complete experimental runs (up to several weeks). Similar oscillations in transformer oil have been detected only in the beginning of experimental runs. They were observed to eventually decay resulting in a stationary convection.

Classification of global phase portraits and bifurcation diagrams of Hamiltonian systems with rational potential

In this paper we study the global dynamics of the Hamiltonian systems x˙=Hy(x,y), y˙=−Hx(x,y), where the Hamiltonian function H has the particular form H(x,y)=y2/2+P(x)/Q(x), P(x),Q(x)∈R[x] are polynomials, in particular H is the sum of the kinetic and a rational potential energies. Firstly, we provide the normal forms by a suitable μ-symplectic change of variables. Then, the global topological classification of the phase portraits of these systems having canonical forms in the Poincaré disk in the cases where degree(P)=0,1,2 and degree(Q)=0,1,2 are studied as a function of the parameters that define each polynomial. We use a blow-up technique for finite equilibrium points and the Poincaré compactification for the infinite equilibrium points. Finally, we show some applications.

Nonlinear buckling behaviours of thin-walled functionally graded open section beams

In this paper, nonlinear buckling responses of functionally graded (FG) thin-walled open section beams based on Euler–Bernoulli–Vlasov theory is presented. The finite element incremental equilibrium equations are developed by updated Lagrangian formulation using the non-linear displacement cross-section field that accounts for large rotation effects. Young’s modulus of FG beams are varied continuously through the wall thickness based on the power-law distribution. Numerical results are obtained for thin-walled FG beams with symmetric and mono-symmetric I-section and channel-section for various configurations such as boundary conditions, geometry, skin-core-skin ratios and power-law index to investigate the flexural–torsional and lateral buckling loads and post-buckling responses. The accuracy and reliability of proposed model are proved by comparison with previous research and analytical solutions. The importance of above-mentioned effects on buckling results is demonstrated on benchmark examples.

Comprehensive analyses of the initiation and landslide-generated wave processes of the 24 June 2015 Hongyanzi landslide at the Three Gorges Reservoir, China

Reservoir landslides pose a great threat to shipping safety, human lives and properties, and the operation of the hydropower station. In this paper, the 24 June 2015 Hongyanzi landslide at the Three Gorges Reservoir is considered as an example to study the initiation mechanism and landslide-generated wave process of a reservoir landslide. The finite difference method and limit equilibrium analysis are used to analyze the deformation and failure characteristics of the Hongyanzi slope. Simulation results show that a large deformation (about 358 mm) happens in the shallow deposits under intermittent rainfall condition, and the slope is in a limit state. At the same time, continuous rapid drawdown of the water level (about −0.55 m/day during 8–24 June 2015) reduced the support and accelerated the drainage of the water for the bank slope. A coupling effect of intermittent rainfall and rapid drawdown of the water level was the triggering factor of the 24 June Hongyanzi landslide. Landslide-generated wave process was simulated using a fluid–solid coupling method by integrating the general moving object collision model. Simulation results show that the landslide-generated wave is dominated by the impulse wave, which is generated by sliding masses entering the river with high speed. The maximum wave height is about 5.90 m, and the wave would decay gradually as it spreads because of friction and energy dissipation. To prevent reservoir landslides, the speed for the rising or drawdown of the water level should be controlled, and most importantly, rapid drawdown should be avoided.

Bounded memory Folk Theorem

We show that the Folk Theorem holds for n-player discounted repeated games with bounded memory (recall) strategies. Our main result demonstrates that any payoff profile that exceeds the pure minmax payoff profile can be approximately sustained by a pure strategy finite memory subgame perfect equilibrium of the repeated game if the players are sufficiently patient. We also show that the result can be extended to any payoff profile that exceeds the mixed minmax payoff profile if players can randomize at each stage of the repeated game. Our results requires neither time-dependent strategies, nor public randomization, nor any communication. The type of strategies we employ to establish our result turn out to have new features that may be important in understanding repeated interactions.

Reconciling Elastic and Equilibrium Methods for Static Analysis

We examine two widely used classes of methods for static analysis of masonry buildings: linear elasticity analysis using finite elements and equilibrium methods. It is often claimed in the literature that finite element analysis is less accurate than equilibrium analysis when it comes to masonry analysis; we examine and qualify this claimed inaccuracy, provide a systematic explanation for the discrepancy observed between their results, and present a unified formulation of the two approaches to stability analysis. We prove that both approaches can be viewed as equivalent, dual methods for getting the same answer to the same problem. We validate our observations with simulations and physical tilt experiments of structures.

Stability Analysis of Anchored Soil Slope Based on Finite Element Limit Equilibrium Method

Under the condition of the plane strain, finite element limit equilibrium method is used to study some key problems of stability analysis for anchored slope. The definition of safe factor in slices method is generalized into FEM. The “true” stress field in the whole structure can be obtained by elastic-plastic finite element analysis. Then, the optimal search for the most dangerous sliding surface with Hooke-Jeeves optimized searching method is introduced. Three cases of stability analysis of natural slope, anchored slope with seepage, and excavation anchored slope are conducted. The differences in safety factor quantity, shape and location of slip surface, anchoring effect among slices method, finite element strength reduction method (SRM), and finite element limit equilibrium method are comparatively analyzed. The results show that the safety factor given by the FEM is greater and the unfavorable slip surface is deeper than that by the slice method. The finite element limit equilibrium method has high calculation accuracy, and to some extent the slice method underestimates the effect of anchor, and the effect of anchor is overrated in the SRM.

BFree Finite Element Approach for Saturated Porous Media: Consolidation

TheBfree finite element approach is applied to the governing equations describing the consolidation process in saturated poroelastic medium with intrinsically incompressible solid and fluid phases. Under this approach, where Voigt notation is avoided, the finite element equilibrium equations and the linearization of the coupled governing equations are fully derived using tensor algebra. In order to assess theBfree approach for the consolidation equations, direct comparison with analytical solution of the response of a homogeneous and isotropic water-saturated poroelastic finite column under harmonic load is presented. The results illustrate the capability of this finite element approach of reproducing accurately the response of quasistatic phenomena in a saturated porous medium.

MODELLING THE EFFECT OF WIND FORCES ON LANDSLIDE OCCURRENCE IN BUDUDA DISTRICT, UGANDA

The hilly area of Bududa District, in Eastern Uganda has suffered heavily from landslide occurrence, making the local populace unsafe. Although the area is well covered by vegetation, a catastrophic landslide occurred on the 26th June, 2012 and killed more than 18 people and wiped out the nearby community. The fallen trees due wind effect prior to the landslide coupled with intense rainfall are believed to trigger that landslide. Therefore, the effect of commonly known causative factor of rainfall and an uncommon landslide causal factor of wind forces on landslide occurrence were investigated in this study. The geometry of the modelled slope consists of 111 m height, 480 m length, and an inclination of 21°. Finite element seepage software, Seep/W and limit equilibrium software, Slope/W were used for modelling the landslide occurrence using five (5) slope models with and without additional shear strength in form of tree root cohesion. Each of the slope models was subjected to four failure criteria (FC). The first two failure criteria (FC1 and FC2) are assigned for group of tree along the slope (FC1) and a single tree at the middle (FC2) of the modelled slope. The other two failure criteria (FC3 and FC4) are assigned for the modelled slope with rainfall infiltration alone and rainfall plus several tree weights, respectively. The wind speed during which the landslide occurred was depicted using Beaufort wind scale 11 and was applied to the appropriate models. The results obtained show that with eucalyptus trees of Diameters at Breast Heights (DBH) of 25 cm, a minimum factor of safety (FOS) of 1.012 and 1.273 were recorded for FC2 in Slope model 1 and slope model 4 with and without increased cohesion, respectively. This shows an increase in the FOS of 23.81%. However, when the DBH was increased to a maximum of 60 cm (i.e. in slope model 3), the safety of the modelled slope diminishes and the FOS reduces to 0.601 for FC2 on the 25th day, without cohesion, just some hours before the fateful day of the 26th June 2012. This FOS remained below 1.0 (0.800 to 0.601) for FC2 with roots cohesion, although a gain in the FOS of 33.11% was realized. Hence, it can be concluded that the uncommon natural causative factors of wind have a great influence on landslide occurrence in the hilly areas of Bududa district

Uncertain temperature field prediction of heat conduction problem with fuzzy parameters

This paper proposes a first-order fuzzy perturbation finite element method (FFPFEM) and a modified fuzzy perturbation finite element method (MFPFEM) for the uncertain temperature field prediction with fuzzy parameters in material properties, external loads and boundary conditions. Fuzzy variables are used to represent the subjective uncertainties associated with the expert opinions. Based on the α-cut method, the finite element equilibrium equation is equivalently transformed into groups of interval equations. The interval matrices and vectors in both methods are expanded by the first-order Taylor series. First-order Neumann series is employed in FFPFEM for the inversion of interval matrix, while some high-order terms of the Neumann series are retained in MFPFEM. Based on the fuzzy decomposition theorem, the fuzzy solutions to the original uncertain heat conduction problem are eventually derived. By comparing the results with traditional Monte Carlo simulation and vertex method, two numerical examples evidence the feasibility and effectiveness of the proposed methods at predicting the fuzzy temperature field in engineering.

Estimation of the validity domain of hyper-reduction approximations in generalized standard elastoviscoplasticity

We propose an a posteriori estimator of the error of hyper-reduced predictions for elastoviscoplastic problems. For a given fixed mesh, this error estimator aims to forecast the validity domain in the parameter space, of hyper-reduction approximations. This error estimator evaluates if the simulation outputs generated by the hyper-reduced model represent a convenient approximation of the outputs that the finite element simulation would have predicted. We do not account for the approximation error related to the finite element approximation upon which the hyper-reduction approximation is introduced. We restrict our attention to generalized standard materials. Upon use of incremental variational principles, we propose an error in constitutive relation. This error is split into three terms including a tailored norm of the hyper-reduction approximation error. This error norm is defined by using the convexity of an incremental potential introduced to state the constitutive equations. The second term of the a posteriori error is related to the stress recovery technique that generates stresses fulfilling the finite element equilibrium equations. The last term is a coupling term between the hyper-reduction approximation error at each time step and the errors committed before this time step. Unfortunately, this last term prevents error certification. In this paper, we restrict our attention to outputs extracted by a Lipschitz function of the displacements. In the proposed numerical examples, we show very good preliminary results in predicting the validity domain of hyper-reduction approximations. The average computational time of the predictions obtained by hyper reduction, is accelerated by a factor of 6 compared to that of finite element simulations. This speed-up incorporates the computational time devoted to the error estimation. The numerical implementation of the proposed error estimator is straightforward. It does not require the computation of the incremental potential. In the numerical results, the estimated validity domain of hyper-reduced approximations is inside the reference validity domain. This paper is a first attempt for a posteriori error estimation of hyper-reduction approximations.

Dynamic systems behaviour analysis and design based on the qualitative theory of differential equations: the Boost power converter case

This paper uses the qualitative theory of differential equations to analyse/design the dynamic behaviour of control systems. In particular, the Poincaré compactification and the Poincaré--Hopf theorem are used for analysing the local dynamics near the finite and infinite equilibrium points. As an application, a large signal characterisation of a Boost type power converter in closed loop, including its equilibrium/bifurcation points and its global dynamics, which depends upon the value of the load resistance, is studied.

The Master equation in mean field theory

In his lectures at College de France, P.L. Lions introduced the concept of Master equation, see [8] for Mean Field Games. It is introduced in a heuristic fashion, from the prospective as a system of partial differential equations, that the equation is associated to a Nash equilibrium for a large, but finite, number of players. The method, also explained in [3], composed of a formalism of derivations. The interest of this equation is that it contains interesting particular cases, which can be studied directly, in particular the system of HJB–FP (Hamilton–Jacobi–Bellman, Fokker–Planck) equations obtained as the limit of the finite Nash equilibrium game, when the trajectories are independent, see [6]. Usually, in mean field theory, one can bypass the large Nash equilibrium, by introducing the concept of representative agent, whose action is influenced by a distribution of similar agents, and obtains directly the system of HJB–FP equations of interest, see for instance [1]. Apparently, there is no such approach for the Master equation. We show here that it is possible. We first do it for the Mean Field type control problem, for which we interpret completely the Master equation. For the Mean Field Games itself, we solve a related problem, and obtain again the Master equation.

Impact of spatial variability in undrained shear strength on active lateral force in clay

This study explores the mechanism of the active lateral force (Pa) in clay when there is spatial variability in the undrained shear strength. It is shown that the effect of such spatial variability cannot be fully explained by considering the spatial averaging over a prescribed area or line only; the mechanism to seek the favorable failure path is also important. Ignoring this important mechanism is risky, rendering the Pa estimate smaller than the actual Pa value. The mechanism of seeking the favorable failure path is explored, and a probability distribution model for Pa is proposed to characterize this sophisticated mechanism. Furthermore, a simplified procedure is proposed to simulate the Pa samples without the use of the finite element method or limit equilibrium method.