Rigid Boundary Conditions Research Articles

Circular flows for the Euler equations in two-dimensional annular domains, and related free boundary problems

In this paper, we first consider steady Euler flows in two-dimensional bounded annuli, as well as in complements of disks, in punctured disks and in the punctured plane. We prove that if the flow does not have any stagnation point and satisfies rigid wall boundary conditions together with further conditions at infinity in the case of unbounded domains and at the center in the case of punctured domains, then the flow is circular, in the sense that the streamlines are concentric circles. In other words, the flow then inherits the radial symmetry of the domain. We then show two classification results for the steady Euler equations in simply or doubly connected bounded domains with free boundaries. Here, the flows are further assumed to have constant norm on each connected component of the boundary, and the domains are then proved to be disks or annuli. On the one hand, the proofs use ODE and PDE arguments to establish some geometric properties of the trajectories of the flow and the orthogonal trajectories of the gradient of the stream function. On the other hand, we also show some comparison results of independent interest for a derived semilinear elliptic equation satisfied by the stream function. These last results, which are based on the method of moving planes, adapted here to some almost circular domains located between some streamlines of the flow, lead with a limiting argument to the radial symmetry of the stream function and the streamlines of the flow.

Investigation on gas migration behaviours in saturated compacted bentonite under rigid boundary conditions

Investigation on gas migration behaviours in saturated compacted bentonite under rigid boundary conditions

Gravity modulation effect on weakly nonlinear thermal convection in a fluid layer bounded by rigid boundaries

Abstract This paper investigates the effect of gravity modulation on Rayleigh–Bénard convection using the rigid isothermal boundary conditions. We calculate heat transfer results using the Nusselt and mean Nusselt numbers through the finite-amplitude of convection, which we got from the Ginzburg–Landau equation (GLE). The Ginzburg–Landau equation is derived analytically from the Fredholm solvability condition at third order. The finite amplitude equation (GLE) is a function of system parameters and solved numerically. The gravity modulation considered in terms of steady and sinusoidal parts. The sinusoidal part defines gravity modulation in terms of amplitude and frequency. Our study shows that gravity modulation controls the heat transfer results. The amplitude of modulation enhances heat transfer for low frequencies and diminishes for high frequencies. Further, we found that rigid isothermal boundary conditions are diminishing heat transfer than free and isothermal boundaries. Finally, we concluded that rigid isothermal boundary conditions and gravity modulation controls heat transfer results.

Unstable spectrum of a Rayleigh–Bénard system with variable viscosity

This Note studies a Rayleigh–Bénard system in an infinite layer, in the case of temperature-dependent viscosity, with rigid boundary conditions for the velocity at the bottom and free-slip at a top of the layer. It states the linearized problem in the relevant functional operator set-up and identifies, for each nonzero transverse frequency k and Rayleigh number R the (finite) number of modes which are unstable in time. This number is equal to the number of eigenvalues of a particular operator which are smaller than R.

Longitudinal vibration analysis of FG nanorod restrained with axial springs using doublet mechanics

In the current paper, the free longitudinal vibration response of axially restrained functionally graded nanorods is presented for the first time based on the doublet mechanics theory. Size dependent nanorod is considered to be made of functionally graded material consist of ceramic and metal constituents. It is assumed that the material properties of the functionally graded nanorod are assumed to vary in the radial direction. The aim of this study is that to investigate the influences of various parameters such as functionally graded index, small size parameter, length of the nanorod, mode number and spring stiffness on vibration behaviors of functionally graded nanorod restrained with axial springs at both ends. For this purpose, Fourier sine series are used to define the axial deflection of the functionally graded nanorod. Then, an eigenvalue approach is established for longitudinal vibrational frequencies thanks to Stokes’ transformation to deformable axial springs. Thus, the presented eigenvalue solution method is attributed to both rigid and deformable boundary conditions for the axial vibration of the functionally graded nanorod. With the help of the results obtained with the presented eigenvalue problem, it is observed that the parameters examined cause significant changes in the frequencies of the functionally graded nanorod.

Assessment of assembly shape and geometric effects on amplification of shear waves using discrete element method

The discrete element method (DEM) is a capable tool used to simulate shear wave propagation in granular assemblies for many years. Researchers have studied assembly shapes such as rectangles (in 2D simulations) or cylinders and cubes (in 3D simulations). This paper aimed to qualify the effect of assembly shape on the shear wave propagation and maximum amplification in the vertical plane (horizontal and vertical directions) caused by this propagation. To this end, shear wave propagations in different assembly shapes such as rectangle, trapezium, and triangle with rigid boundary conditions were simulated. A sine wave pulse was applied with a point source by moving a particle as the transmitter particle. To evaluate the shear wave velocity of the assemblies, the transmitter and receiver particles were simulated. All the simulations were performed with 2D DEM which is a useful tool to determine the amount and location of the maximum amplification factor of the assembly in both horizontal and vertical directions. An advantage of this study was assessing the effect of parameters such as input wave frequency, assembly height, shape, and aspect ratios on the amplification of the input waves.

A semi-analytical study of the three-dimensional liquid sloshing in a horizontal cylindrical tank with an arbitrary liquid depth

In this paper, a semi-analytical study is presented to solve the linear liquid sloshing frequencies and mode shapes of a three-dimensional partially filled horizontal cylindrical container. The internal liquid of the tank is regarded as the incompressible, inviscid fluid, whose velocity potential function satisfies the Laplace function and the free surface gravity wave function. According to the sloshing modal shapes, the velocity potential function is expressed as symmetric and antisymmetric modes, respectively. The normal velocities of the fluid are set to be zero to model the rigid boundary condition on the fluid-structure interface of the container. Two coordinate systems are introduced in the model. The fluid field coordinate system is set at the midpoint of the free liquid surface. The other coordinate system, the structure coordinate system, is established at the geometric center of the tank cross-section. Through the coordinate transformation and the continuous boundary condition, the motion equation of the system could be obtained. The Galerkin method is employed to solve the governing equation and acquire the sloshing frequencies. Simultaneously, a series of experiments are carried out to investigate the sloshing movement with different liquid depths, and the low-order sloshing frequencies and mode shapes are obtained through the experiments. The accuracy of the semi-analytical method is verified by comparing the results with the published results, the finite element results, and the experimental results. The effects of the liquid depths and the tank length on the sloshing motion are investigated as well.

A membrane equivalent method to reproduce the macroscopic mechanical responses of steel wire-ring nets under rockfall impact

Steel wire-ring nets, which have complex nonlinear mechanical behaviour, are widely used in rockfall protection engineering. At present, the circular beam model with a discrete contact boundary is the most accurate method to simulate steel wire-ring nets’ mechanical responses, but its computational efficiency is low. This paper hereby proposes a membrane equivalence-based continuum simulation method: the membrane equivalent method (MEM)11MEM: abbreviation of the membrane equivalent method proposed in this paper to reproduce the macroscopic mechanical behaviour of steel wire-ring nets.. This study’s aim is to greatly improve the computational efficiency of numerical models through the MEM without compromising computational accuracy in macroscopic mechanical behaviour, compared to the circular beam model. According to the out-of-plane loading experimental results of the net panel and membrane analytical model of orthogonal tension belts, an exponential hardening constitutive model of the equivalent membrane was derived. The MEM’s material parameters were corrected in numerical models for a less than 10% error between the simulated and experimental results. Under the rigid boundary condition, the net panel impact simulation results established by the MEM agreed with extant literature’s experimental results, the peak acceleration error was 4.3%, and the rebound characteristic of the net was reproduced. Under the flexible boundary condition, the MEM’s obtained maximum impact displacement and impact force errors were 10.3% and 7.7%, respectively, compared to 3000 kJ full-scale impact experimental results. The MEM’s computational efficiency can be increased almost tenfold compared to that of the circular beam model.

A sound source reconstruction approach based on the machine vision and inverse patch transfer functions method

The inverse patch transfer functions (iPTF) method can realize local reconstruction of a vibrating structure in non-anechoic environments, and the geometrical shape of the reconstruction area could be irregular as long as it is known. However, the shape of the non-planar reconstruction area is not easy to be obtained in practice, which is adverse to its application in the situ tests. To achieve the sound source reconstruction of the local area with unknown shape in non-anechoic environments, a hybrid method that combines the machine vision and the iPTF method is proposed. The machine vision technology is applied to facilitate the boundary modeling of a target radiation segment. It makes use of the BundleFusion method with an RGB-D camera to accomplish the 3D reconstruction of the target area, as well as the localization of microphones. A virtual cavity consisting of the target area, virtual measurement surface, and the gap between them could be constructed automatically. Consequently, the impedance matrix of the virtual cavity is obtained based on the boundary integral equation with the adoption of the free field Green’s function. A microphone array mounted up with a rigid masker is applied in the iPTF method, which can form the rigid Neumann boundary condition on the masker. A simulation is first carried out to investigate the influence of 3D reconstruction errors of the machine vision technology on the sound source reconstruction. Then, two experiments are performed to validate the feasibility and effectiveness of the reconstruction in noisy environments. One is conducted in the semi-anechoic room with a cylindrical radiator, and a loudspeaker is put aside as the interference source. The other is conducted in the cabin of an aircraft under cruising condition. It is demonstrated that the proposed method can realize the normal velocity reconstruction without prior geometry knowledge of the target area, and the magnitude and distribution of the dominant sound source could be reconstructed accurately.

Compressed stiffened composite panels design with skin postbuckling behavior

The aim of this paper is development of stiffened composite panels design method taking thin skin postbuckling behavior and stringers elastic behavior in case of longitudinal compression into account. Statement of the problem takes geometrically non-linear correlations into account. Variant of rigid boundary conditions is considered for both stiffened panels and skin. Analytic solution of non-linear problem using Ritz method which is reduced to solution of non-linear system of two equations in skin and stringer deflection amplitudes is shown. Obtained analytical solution is used for development of load-bearing panels design method based on postbuckling state. A design method based on thin skin postbuckling state not taking stringers influence into account is shown as an example. In this case following variants of strength criteria are considered: reaching longitudinal normal or tangential transverse critical stress. Besides stiffened composite panels design method taking skin postbuckling behavior in case of stringers elastic behavior into account is shown. With known stringer parameters design task is reduced to solution of non-linear system of two equations in skin thickness and stringer deflection amplitude. In this case stipulation of reaching critical normal stress in the skin is taken as strength criterion.

Study of shear behavior of granular materials by 3D DEM simulation of the triaxial test in the membrane boundary condition

This paper studies the shear behavior of granular materials by using a three-dimensional (3D) discrete element method (DEM) simulation of the triaxial test. The experimental triaxial tests were conducted on glass beads samples for verification. DEM simulations of the triaxial test were carried out in the membrane boundary condition consisting of 37,989 membrane particles. A new method that divides the irregular sample shape into two parts of cones and parts of three-dimensional simplexes is used to follow the volume change of irregular deformation of samples. The free rotatable upper platen is considered during the shearing process, which influences the shear behavior of samples especially in the residual stage and formations of a single shear band or X-shape shear band. The confining pressures have been demonstrated to influence the rotation angle and angular velocity of the upper platen. Moreover, the timing of replacing a rigid wall boundary condition with the membrane boundary condition is investigated, which affects the porosity of samples before shearing and the mechanical strength. The DEM model in the membrane boundary condition reflects well the evolution of irregular sample deformation and shear band in the shearing process. From the perspective of micro structures, the normal force decreases and the tangential stress increases during the shearing stage. This study greatly improves the accuracy of DEM simulations of the triaxial test in the membrane boundary condition.

Buckling analysis of restrained nanobeams using strain gradient elasticity

Size-dependent stability analysis of nanobeams under axial loading is investigated by strain gradient elasticity. An effective general model is proposed for different deformable boundary conditions by combining Fourier sine series and Stokes’ transformation. The theoretical model presented here bridges the gap between clamped and pinned-pinned supporting conditions, which is of great significance for the application of the strain gradient small scale effects to nanobeam. Analytical solutions are obtained and comparisons with results obtained by the researchers for rigid boundary conditions are carried out. The outcomes can be useful for the optimization and design of nano-sized beams and provide new benchmarks for nanomechanical analyses.

Numerical analysis of a method for solving 2D linear isotropic elastodynamics with traction free boundary condition using potentials and finite elements

When solving 2D linear elastodynamic equations in homogeneous isotropic media, a Helmholtz decomposition of the displacement field decouples the equations into two scalar wave equations that only interact at the boundary. It is then natural to look for numerical schemes that independently solve the scalar equations and couple the solutions at the boundary. The case of rigid boundary condition was treated by Burel [Ph.D. thesis, Universite Paris Sud-Paris XI (2014)] and Burel et al. [Numer. Anal. Appl. 5 (2012), pp. 136— 143]. However the case of traction free boundary condition was proven by Martinez et al. [J. Sci. Comput. 77 (2018), pp. 1832-1873] to be unstable if a straightforward approach is used. Then an adequate functional framework as well as a time domain mixed formulation to circumvent these issues was presented. In this work we first review the formulation presented by Martinez et al. [J. Sci. Comput. 77 (2018), pp. 1832-1873] and propose a subsequent discretised formulation. We provide the complete stability analysis of the corresponding numerical scheme. Numerical results that illustrate the theory are also shown.

Acoustic radiation force and torque on spheroidal particles in an ideal cylindrical chamber.

In this article, the acoustic radiation force and torque exerted on a small spheroidal particle immersed in a nonviscous fluid inside an ideal cylindrical chamber is theoretically investigated. The ideal chamber comprises a hard top and bottom (rigid boundary condition) and a soft or hard lateral wall. By assuming that the particle is much smaller than the acoustic wavelength, analytical expressions of the radiation force and torque caused by an acoustic wave of arbitrary shape are presented. Unlike previous results, these expressions are given relative to a fixed laboratory frame. The model is showcased for analyzing the behavior of an elongated metallic microspheroid (with a 10:1 aspect ratio) in a half-wavelength acoustofluidic chamber with a diameter of a few millimeters. The results show that the radiation torque aligns the microspheroid along the nodal plane, and the radiation force causes a translational motion with a speed of up to one body length per second. Finally, the implications of this study on propelled nanorods by ultrasound are discussed.

A simplified approximate analytical model for Rayleigh–Taylor instability in elastic–plastic solid and viscous fluid with thicknesses* *Project supported by of the Science Challenge Project of China (Grant No. TZ2018001).

A simplified theoretical model for the linear Rayleigh–Taylor instability of finite thickness elastic–plastic solid constantly accelerated by finite thickness viscous fluid is performed. With the irrotational assumption, it is possible to consider viscosity, surface tension, elasticity or plasticity effects simultaneously. The model considers thicknesses at rigid wall boundary conditions with the velocity potentials, and deals with solid elastic–plastic transition and fluid viscosity based on the velocity continuity and force equilibrium at contact interface. The complete analytical expressions of the amplitude motion equation, the growth rate, and the instability boundary are obtained for arbitrary Atwood number, viscosity, thicknesses of solid and fluid. The thicknesses effects of two materials on the growth rate and the instability boundary are discussed.

Instability of electroconvection in viscoelastic fluids subjected to unipolar injection

In this paper, a two-dimensional numerical study on the nonlinear behaviors of electrohydrodynamic flows of Oldroyd-B viscoelastic dielectric liquid subjected to unipolar injection is performed via the finite volume method. The entire set of coupled equations, which includes the Navier–Stokes equations, simplified Maxwell’s equations, and conformation transport equations, is solved for the first time. The effects of elasticity on the nonlinear evolution of electroconvection and instability patterns are mainly investigated. Various physical models including free and rigid boundary cases are simulated entirely, and detailed analyses of stability parameters are performed. Convection and fluid motion instability are investigated and explained in detail, with a focus on the onset of motion transitions from a purely conducted state to losing its stability. It is found that the coupling of the electric field with the elasticity field gives rise to new instability and completely new mechanisms. In addition to instabilities such as subcritical bifurcation in electroconvection of Newtonian fluids, supercritical bifurcation and Hopf bifurcation are also possible as the first instability in electroconvection of viscoelastic fluids under free boundary conditions. Under rigid boundary conditions, the system with a large Weissenberg number can also lose its stability via earlier Hopf bifurcation. The stability threshold is not affected by the elastic effect if the Weissenberg number is small enough but decreases when the first instability of the system becomes Hopf bifurcation. Moreover, elasticity promotes the transition from a steady state flow to unsteady convection after the onset of convection. These phenomena are closely related to the elastic parameters.

Unconditional nonlinear stability for double‐diffusive convection with temperature‐ and pressure‐dependent viscosity

AbstractA linear and nonlinear stability analyses are carried out for a double‐diffusive chemically reactive fluid layer with viscosity being a function of temperature and pressure. The linear stability analysis is studied when the stabilizing salt gradient acts against the destabilizing thermal gradient. The effect of reaction parameters and variable viscosity on the stability of the system is studied for heated below, salted above, and the heated and salted below models with Rigid–Rigid boundary conditions. Chebyshev pseudospectral method is applied to determine the numerical solutions.

Effect of light elements impurity on process of nickel crystallization near the triple interface of grain boundaries: a molecular dynamics simulation

Molecular dynamics method was used to study the effect of impurities of light elements of carbon, nitrogen and oxygen on crystallization process near the triple interface of grain boundaries in nickel. Tilt boundaries with misorientation axis <111> were considered as the grain boundaries. Interactions of nickel atoms with each other were described by many-particle Clery-Rosato potential constructed within the framework of the tight binding model. To describe interactions of atoms of light elements impurities with nickel atoms and atoms of impurities with each other, Morse pair potentials were used. Calculation cell had a shape of cylinder, axis of which coincided with the line of triple interface and the axis of grain misorientation. Periodic boundary conditions were imposed along the cylinder axis, and the atoms on side surface of cylinder were motionless. To simulate crystallization, calculation cell was melted by heating to a temperature well above the melting temperature of nickel. After the simulated polycrystal become liquid, the thermostat was turned on and held at a constant temperature below the melting temperature. Rigid boundary conditions on the lateral surface of cylindrical calculation cell in this case simulated crystallization fronts from three crystallization centers. The area near the triple interface had crystallized the last. In this area, defects and free volume were concentrated. Presence of impurities led to a significant slowdown in the rate of crystallization. With introduction of 10 % of impurity atoms, the rate of motion of crystallization front decreased several times. The effect of impurities on crystallization rate was enhanced in C – N – O direction, which is due to difference in crystal lattice deformation caused by impurity atoms. The greater this deformation was, the stronger was impurity atoms inhibit crystallization front. Formation of aggregates at fairly high concentrations was typical for impurity carbon atoms. Crystallization front had impeded on these aggregates. The oxygen and nitrogen atoms did not form aggregates. However, due to distortions of crystal lattice caused by them, they also strongly slowed down the crystallization front.

On the Blast Mitigation Ability of Multiple V-Shape Deflectors

This paper presents a 2D numerical study of v-shape deflectors subjected to high explosive charge detonation. The literature provides many papers concerning the appropriate geometry of blast protection deflectors. Such structures require a relatively high distance between the ground and the vehicle chassis. In most cases, the placement of a deflector is not possible or it cannot cover the whole area under a chassis. An interesting question that arises is to what extent a set of small deflectors would be able to mitigate blast effects. The content of this paper constitutes an answer to this question. Three analyses were conducted: (1) multiplication of triangle components, (2) effect of deflector size, and (3) geometry-induced shock dynamics. The problem was solved with the use of modelling and simulation methods, in particular, CFD-FEM implemented in the LS-DYNA code. It was considered a plain issue in computational fluid dynamics, where space discretization for each option was built with two-dimensional elements to ensure efficient calculations. Deflectors were described using a rigid wall boundary condition, and an adequate simplified detonation model was assumed. The primary measure of the results was reaction force history, with momentum transfer and pressure distribution maps considered supplementary. The studies performed showed that both minimizing the v-shape deflector size and surrounding it with adjacent structures had a negative impact on its blast mitigation effectiveness. However, for each multi-v-shape deflector, some improvement was present, and, therefore, in situations where the installation of a typical protector is not possible due to dimensional requirements, it may offer a compromise solution.

Load bearing and settlement characteristics of Geosynthetic Encased Columns under seismic loads

Ordinary stone columns and Geosynthetic Encased Stone columns are used in remediation of soft soils with low bearing capacities. Primary objective of granular columns is to provide structural support to the overlying foundation layers. Given that the granular columns implemented in soft soil conditions support a large portion of the surcharge pressure, developing an understanding of settlement response and pressure bearing characteristics of granular columns under seismic loading conditions is of paramount importance. This study aims to present experimental findings of granular column load bearing response under simulated seismic loads and the effects of lateral boundary conditions on column behavior. In order to mimic the seismic behavior of columns supporting a sustained surcharge load, two different large-scale experimental assemblies are used in 1-g shaking table tests. The assemblies used were a laminar box and a rigid box which were intended as physical analogies for free-field and rigid boundary conditions. The reinforcement strains, pressure concentration on column heads, and settlement of the unit cells enhanced with granular columns with varying reinforcement stiffnesses are studied. The relationship between unit cell settlement and two important earthquake intensity measures namely, Arias Intensity and Shaking Intensity Rate, are also investigated.