Coupled Boundary Conditions Research Articles

Positive Solutions for a System of Coupled Semipositone Fractional Boundary Value Problems with Sequential Fractional Derivatives

We study the existence and multiplicity of positive solutions for a system of Riemann–Liouville fractional differential equations with sequential derivatives, positive parameters and sign-changing singular nonlinearities, subject to nonlocal coupled boundary conditions which contain Riemann–Stieltjes integrals and various fractional derivatives. In the proof of our main existence results we use the nonlinear alternative of Leray–Schauder type and the Guo–Krasnosel’skii fixed point theorem.

Stability analysis of a rotor system with electromechanically coupled boundary conditions under periodic axial load

The parametric instability of a rotor system with electromechanically coupled boundary conditions under periodic axial loads is studied. Based on the current flowing piezoelectric shunt damping technique, the detailed rotor model is established by the finite element (FE) method. In the matrix assembly procedure, a novel simple process is proposed to make the equations of shunt circuits more conveniently to be introduced into the global FE equations. The discrete state transition matrix method which is used for determining the influence of circuit parameters on instability regions in this paper has also been presented. The numerical simulation shows that only the combination instability regions exist when the shaft is rotating. The mechanical damping has different effect on the simple and combined instability regions. These two points are consistent with the previous references, which verifies the obtained FE model. In addition, the simulated results also reveal that the introduction of shunt circuits has little influence on the rotor’s original whirling frequencies. It gives rise to the appearance of new synchronous whirl modes. The new whirling frequencies are combined with the original ones to form the new combination instability regions. Furthermore, the resistance of shunt circuits has the same performance as the mechanical damping has. That is, moving up the start points of instability regions and expanding its width.

A 3-Dimentional Framework for Studying the Effects of Structural and Physical Properties of the Renal Outer Medulla on Urine Concentrating Mechanism

Background: Many theories and mathematical simulations have been proposed concerning urine concentrating mechanism (UCM). Due to significant effect of the tubule and vessel architecture in concentrating mechanism, the numerical analysis of UCM through a 3-Dimensional structure might be the answer to find a better consistency between the experimental and theoretical results. Methods: In this paper we have investigated the effects of structural characteristics of the tubules and vessels on the urine concentrating mechanism in the outer medulla (OM) by developing a simple three-dimensional mathematical model. This model is a framework to attain a converged numerical solution for the momentum and species transport equations along with their stiff and coupled boundary conditions based on standard expressions for trans-tubular solutes and water transports on tubule’s membrane. Results: The model structure and the number of the involved tubules have been assumed to be as simple as possible. The effects of slip boundary condition on membrane, Darcy permeability and solute’s diffusivity of the intermediate media on UCM have been studied. It has been shown that this approach can simply simulate preferential interactions and tubule’s confinement by radial diffusion coefficients. Conclusions: All in all, the feasibility of the idea of completely 3-D modeling by employing the concept of diffusion coefficient and Darcy permeability has been explored and validated.

Bloch waves in an array of elastically connected periodic slender structures

This paper proposes the methodology to carry out the analysis of Bloch wave propagation in an array of vertically aligned and elastically connected structural elements such as beams, strings, plates, or other slender structures. The suggested approach is based on the Galerkin approximation and Floquet-Bloch theorem used in defining the eigenvalue problem and obtaining the band structure of the periodic systems. Special attention is devoted to the case of elastically connected Rayleigh beams with attached concentrated masses and wave propagation in the direction normal to the beam’s length. A validation study is performed by using the finite element model and the frequency response function to confirm the accuracy of the solution obtained via the Galerkin approximation. Two configurations of unit cells, having two and three elastically connected beams with different geometrical and material properties, are considered in the numerical study. The effects of various parameters are investigated to reveal their influence on the frequency band structure and emergence of the zero-frequency bandgap. The results of this study demonstrates the tunability properties of the proposed periodic systems due to changes in values of concentrated masses, stiffness of the coupling medium or boundary conditions on structural elements within the unit cell.

Finite element analysis of the three-dimensional crack and defects in piezoelectric materials under the electro-mechanical coupling field

This paper presents three-dimensional planar crack and non-planar defects analysis of piezoelectric materials by the finite element method (FEM). Based on practical engineering, a series of reasonable three-dimensional defects models are established. Different from the usual two-dimensional plane as a three-dimensional defect form, the present paper takes into account the different causes, concrete position, and various shapes of typically three-dimensional non-planar defects in practical engineering. By changing the geometric size of the defect and the electro-mechanical boundary conditions, the effects of three-dimensional defect geometry and different electro-mechanical coupling boundary conditions on the distribution of stress, electric field strength, and electric displacement around the defect are investigated. The calculation results show that the electro-mechanical parameters are strongly affected by the geometry of defect and boundary conditions. Furthermore, the distribution of electro-mechanical parameters in the detailed position of the different three-dimensional defect model is also studied, which can accurately assess the location where the defect is prone to fracture. It is concluded that this paper can be used as a reference of three-dimensional defects and the service life of piezoelectric materials in practical engineering structures.

Existence and uniqueness results for a nonlinear coupled system involving Caputo fractional derivatives with a new kind of coupled boundary conditions

This paper is concerned with the existence and uniqueness of solutions for a nonlinear fractional-order coupled system involving Caputo fractional derivatives of different orders equipped with a new kind of coupled boundary conditions. We transform the given system into an equivalent fixed point problem and solve it by applying the standard fixed point theorems. Examples are constructed for the illustration of the obtained results.

On the Parametrization of Nonlinear Impulsive Fractional Integro–Differential System With Non-Separated Integral Coupled Boundary Conditions

We give a new investigation of periodic solutions of nonlinear impulsive fractional integro-differential system with different orders of fractional derivatives with non-separated integral coupled boundary conditions. Uniformly Converging of the sequence of functions according to the main idea of the Numerical-analytic technique, from creating a sequence of functions. An example of impulsive fractional system is also presented to illustrate the theory.

Numerical analysis of the heat transfer of radiant tubes and the slab heating characteristics in an industrial heat treatment furnace with pulse combustion

For a heat treatment furnace, the precise study of the flow characteristics, combustion and heat transfer in the furnace is essential to know the temperature distribution in a slab during the heating process. In this context several different three-dimensional CFD (Computational Fluid Dynamics) models have been developed to simulate the combustion and heat transfer process in the furnace. These models were developed based on direct fired reheating furnaces with continuous fuel input. In this paper, a novel three-dimensional transient CFD model used to investigate the slab heating characteristics in the heat treatment furnace with pulse combustion was developed. The transient calculation in the furnace was performed based on the steady-state simulation of the furnace temperature, and the movement of slabs on the rollers in the furnace was simulated using dynamic mesh. By comparison, the results from the EDC with two different reaction mechanisms showed good consistency. However, the EDC with the two-step reaction mechanism required one fifth the calculation time of the EDC with the GRI-Mech3.0, and was thus the preferred model. In contrast to direct fired reheating furnaces, the outer radiant tube walls were treated as the coupled boundary conditions in the model to realize indirect heat transfer between the process gas in the furnace and the combustion gas in the radiant tubes. Furthermore, a pulse combustion method was proposed to control the working states of the burners based on UDF (User-Defined Function), in order to realize the pulse combustion process in the heat treatment furnace. By the validation of “Black Box”, the temperature variation of the slab and the gas temperature in the furnace predicted by the model agreed well with the experimental results. The simulation revealed that the temperature distribution of the radiant tubes and furnace gas was uniform and the maximum temperature difference on the outer radiant tube walls was less than 84 K. During the slab transient heating process, the slab temperature distribution became more uniform in the soaking zone and the maximum temperature difference along the thickness of the slab was below 13 K at discharge.

Theory and calculation of the mixed-mode fracture for coupled chemo-mechanical fracture mechanics

In this paper, a J-integral formula and numerical calculation method are presented, which confirm the path-independence of a J-integral, considering the mixed-mode fracture in a chemo-mechanical coupled medium. The Jk-integral denotes a two-dimensional vector consisting of the components J1 and J2. Using the Equivalent Domain Integral (EDI) method, the chemo-mechanical coupled J2-integral, with an equivalent domain form that includes both a surface integral and a line integral, is deduced using a poroelastical model. Based on a semi-analytical method, an auxiliary J¯2-integral is derived to develop a finite element calculation program. By choosing two characteristic lengths, near the crack tip, the mixed-mode fracture parameters, KI, KII and Ts (T stress) at the crack tip, can be calculated. The mixed-mode fracture problems, which are related to center-cracked and edge-cracked plates, are numerically studied for certain chemo-mechanical coupled boundary conditions. The accuracy of this method is verified with numerical examples, which shows the path-independence of the J-integral for chemo-mechanical coupling. The stress fields of two different cracked hydrogel plates are analyzed with different boundary conditions under the chemo-mechanical coupled mixed-mode fracture. This work provides new insights into the mixed-mode fracture mechanics of a chemo-mechanical coupled medium.

Dynamic modeling for rotor-bearing system with electromechanically coupled boundary conditions

In this paper, a method based on Green's function is proposed for the dynamic analysis of a rotor-bearing system with electromechanically coupled boundary conditions. The rotor system is supported by two ring-shaped piezoelectric dampers, which has been manufactured in our previous research. Based on the piezoelectric shunt resonant circuits, e.g., current flowing shunt circuit, the rotor system's boundary condition will become complicated and electromechanically coupled. The Laplace transform method is applied to solve the partial governing equations with such boundary conditions and the steady-state whirl Green's function is derived subsequently. Since the Green's functions are fundamental solutions of the system, the steady-state whirl solutions can be obtained by applying the superposition principle. Owing to the solutions’ concise form, it is convenient and suitable for analysis and computation. Validation of the proposed method is demonstrated by comparison with the finite element method (FEM). The simulated results show that the analytical solutions possess high accuracy and the piezoelectric damper has significant damping performance.

Seismic response of monopiles to vertical excitation in offshore engineering

This paper studies the dynamic response of a large diameter and thin-walled monopile, embedded in a partially saturated porous seabed with saturation degrees larger than 95%, under a seawater layer, subjected to vertically incident P wave excitations. The ground is a semi-infinite homogenous half-space with a horizontal surface. The solution of the coupled water-pile-soil vibration problem is constructed using a rigorous semi-analytical method. By considering the coupled boundary conditions at the pile-soil and water-soil interfaces, the earthquake induced response is reduced to integral equations and solved numerically. Results for the free field soil displacement and kinematic interaction factor are obtained, which are different from the results reported in the literature for onshore earthquake engineering applications as these do not have the effects of the water layer and the vibration induced dynamic water pressures. The existence of a water layer will always decrease the vertical response of the seabed, but may amplify the monopile's displacement near some resonant frequencies, compared to the case of no water. These results are useful in understanding the kinematic interaction of soil and monopile in offshore engineering applications.

Nonlinear interaction between clustered unstable thermoacoustic modes in can-annular combustors

A can-annular combustor consists of a set of nominally identical cans, in which the flames burn in an essentially isolated manner. However, adjacent cans are able to communicate acoustically, which provides dynamic coupling of the entire can-annular arrangement. Recently, it was shown that the acoustic coupling is not negligible and can cause clustering of eigenfrequencies. In this study, we present a low-order modeling framework for self-excited thermoacoustic oscillations in generic can-annular combustors consisting of N identical cans. The dynamics of the flames are modeled with the nonlinear G-equation; the acoustic model accounts for plane acoustic waves inside the cans and can-to-can communication. The latter is enabled through a coupling boundary condition that is based on conservation of mass and a Rayleigh conductivity. For weak coupling between adjacent cans, the thermoacoustic feedback cycle shows clusters of linearly unstable modes of different azimuthal order, which are close in frequency and growth rate. Their interaction in the nonlinear regime is investigated using time-domain simulations. Two simulations for generic can-annular combustors consisting of 4 and 6 cans with weak acoustic coupling are discussed in this study. We observe a strong interaction between the modes, which can cause long transition times and allows modes that do not dominate the system dynamics in the linear regime to be dominant in the nonlinear regime. While the N=6 case converges to a periodic oscillation pattern with one dominant frequency, the N=4 case converges to a quasi-periodic oscillation involving modes of different azimuthal order. Moreover, we observe a synchronization of these modes. These results raise the questions whether it is possible to predict which mode(s) will dominate the system in the saturated state and under which conditions synchronization of clustered modes can occur.

An Inverse Spectral Problem of Sturm–Liouville Problems with Transmission Conditions

In this paper, a class of inverse Sturm–Liouville problems of Atkinson type with transmission conditions is investigated. Based on the method of well-known theory for inverse matrix eigenvalue problems and the spectral properties of Sturm–Liouville problems with transmission conditions of Atkinson type, the corresponding Sturm–Liouville problems can be constructed using two sets of interlacing real numbers and some other conditions. We solve the problems not only under the separated boundary conditions, but also under the real coupled boundary conditions and the general real self-adjoint transmission conditions.

Multi-scale elastoplastic mechanical model and microstructure damage analysis of solid expandable tubular**Project supported by the National Major Science & Technology Project of China (Grant No. 2016ZX05020-003).

We present an in-depth study of the failure phenomenon of solid expandable tubular (SET) due to large expansion ratio in open holes of deep and ultra-deep wells. By examining the post-expansion SET, lots of microcracks are found on the inner surface of SET. Their morphology and parameters such as length and depth are investigated by use of metallographic microscope and scanning electron microscope (SEM). In addition, the Voronoi cell technique is adopted to characterize the multi-phase material microstructure of the SET. By using the anisotropic elastoplastic material constitutive model and macro/microscopic multi-dimensional cross-scale coupled boundary conditions, a sophisticated and multi-scale finite element model (FEM) of the SET is built successfully to simulate the material microstructure damage for different expansion ratios. The microcrack initiation and growth is simulated, and the structural integrity of the SET is discussed. It is concluded that this multi-scale finite element modeling method could effectively predict the elastoplastic deformation and the microscopic damage initiation and evolution of the SET. It is of great significance as a theoretical analysis tool to optimize the selection of appropriate tubular materials and it could be also used to substantially reduce costly failures of expandable tubulars in the field. This numerical analysis is not only beneficial for understanding the damage process of tubular materials but also effectively guides the engineering application of the SET technology.

Interdiffusion in many dimensions: mathematical models, numerical simulations and experiment

Over the last two decades, there have been tremendous advances in the computation of diffusion and today many key properties of materials can be accurately predicted by modelling and simulations. In this paper, we present, for the first time, comprehensive studies of interdiffusion in three dimensions, a model, simulations and experiment. The model follows from the local mass conservation with Vegard’s rule and is combined with Darken’s bi-velocity method. The approach is expressed using the nonlinear parabolic–elliptic system of strongly coupled differential equations with initial and nonlinear coupled boundary conditions. Implicit finite difference methods, preserving Vegard’s rule, are generated by some linearization and splitting ideas, in one- and two-dimensional cases. The theorems on the existence and uniqueness of solutions of the implicit difference schemes and the consistency of the difference methods are studied. The numerical results are compared with experimental data for a ternary Fe-Co-Ni system. A good agreement of both sets is revealed, which confirms the strength of the method.

Dynamics and power flow control of irregular elastic coupled plate systems: Precise modeling and experimental validation

The vibration characteristics of irregular elastic coupled plate systems are studied by Chebyshev-Ritz method in this paper, including dynamics and power flow control. Firstly, the accurate geometric model is established, and the coupling conditions and boundary conditions are simulated by artificial virtual springs. In order to facilitate integral operations, the irregular integral area is mapped into the regular integral area by a special coordinate transformation. Then all energy equations of the coupled system are established according to the first order shear deformation theory. The Chebyshev polynomial is used as the allowable displacement functions. Finally, the unknown coefficients in the allowable displacement functions are determined by the Rayleigh-Ritz technique. On the basis of the free vibration study, the dynamic behavior is studied by applying the external load to the system surface, including the forced response, the admittance and the power flow. While deriving the method, some modal verification experiments about coupled plate systems are designed. The innovation of the study is the establishment of the vibration analysis model of irregular elastic coupled plate systems based on the elastic theory, the coordinate transformation criterion and the two-dimensional Chebyshev-Ritz principle. The mechanism and control of energy transfers of the coupled plate system are studied by the power flow intensity analysis. The numerical results show that both the coupling condition and the boundary condition are important parameters in the vibration. This study provides some new insights into the vibration characteristics and energy transfer of the irregular elastic coupled plate system.

A Mass Transport Model for Flows in Channels and Porous Media of Fuel Cells

Low-temperature fuel cells are a promising alternative technology to conventional energy systems such as internal combustion engines. However, their large-scale commercialization is still limited due to several challenges, such as mass transport losses at high power densities [1]. Estimation of gas transport properties in porous materials of fuel cells, such as gas diffusion layers (GDLs), is therefore critical. Several experimental studies in literature estimated the permeability of different GDL samples using a one-dimensional model, which does not account for channel effects [1-6]. Some authors in literature have assumed that the flow is incompressible in their numerical studies [7-10]. This hypothesis is not justified because the error in permeability estimation with an incompressible fluid flow model compared to the estimation with a compressible fluid flow model can be as high as 20% [2]. Numerical models for channels and porous media require volume-averaged formulations, and an in-depth discussion on the physical meaning of density and velocity in these models is seldom found in literature. Also, most of the existing numerical studies neglect the anisotropic nature of GDLs.In this work, a volume-averaged form of the steady-state, compressible, and isothermal Navier-Stokes equations for flows in channels and porous materials is developed and implemented in the open-source framework OpenFCST [11]. Particular attention is given to flows in channels and GDLs of polymer electrolyte fuel cells. The fluid flow model in porous materials is derived by means of the method of volume averaging [12]. A continuous Galerkin finite element method is used to discretize and numerically solve the resulting system of governing equations in the framework of a single domain approach. The coupling boundary conditions at the internal interface between channels and porous media are discussed and a stable non-oscillatory pair of solution variables in the porous domain is obtained. The study reveals that for volume-averaged formulations, the permeability obtained in experiments has to be corrected with the sample porosity. The model is used to estimate in-plane and through-plane permeabilities of fuel cell diffusion media, which is compared to experimental data. Three-dimensional simulations show that channel effects cannot be neglected and therefore one-dimensional models for permeability estimation are limited. Assuming that the fluid is incompressible is only valid for through-plane permeability, and a compressible formulation should be used for in-plane simulations, even at moderate gas flow rates. The suitability of the mathematical model for fuel cell applications is illustrated by estimating the change in pressure drop in a serpentine channel in contact with either a solid wall or a gas diffusion media. An interdigitated channel design is also considered in order to compare the pressure drop and the velocity in the GDL with the results observed with a serpentine channel.

Solutions fornonlinear System of Fractional Integro–differential Equations with Non-separated Integral Coupled Boundary Conditions

Solutions fornonlinear System of Fractional Integro–differential Equations with Non-separated Integral Coupled Boundary Conditions

Periodic Solutions for nonlinear Fractional Integro– differential System with Non-separated Integral Coupled Boundary Conditions

Periodic Solutions for nonlinear Fractional Integro– differential System with Non-separated Integral Coupled Boundary Conditions

Simulating the transport of brine in the strata of bedded salt cavern storage with a fluid–solid coupling model

The majority of salt mines in China are located in the strata of lacustrine deposition, which are interbedded structure consisting of rock salts and non-salt interlayers. In the solution mining stage, rock salts will be dissolved to form a cavity for storing natural gas or petroleum. But the non-salt interlayers will soften or even collapse under the immersion of brine (dissolved rock salt), some of which will sink to the cavern bottom to form insolubles, and the other part will become the surrounding rock of this cavity. The retention time of brine in a salt cavern depends on the cavern size and efficiency of solution mining, which is generally about three years. During this period, the brine can seep into the fractures and joints of interlayers, and then inevitably has physical and chemical interactions with the rock minerals (typically montmorillonite). To reveal the affected range and results of brine on the interlayers, this paper takes poro-elasticity theory into account convection–diffusion equation to form a new fluid–solid coupling equation, and then establishes an actual geomechanical simulation model for numerical solution. The calculation results show that the penetration of brine in interlayers is very obvious. The penetration range is 28.37 m within three years, which is about 5.6 times that of rock salt. As the penetration range increases, the brine pressure gradually decreases and eventually stabilizes. In addition, brine transport has complex coupling boundary conditions composed of concentration, pressure and geostress, which interact with each other. This research contents have important reference and guidance for the study on brine crystallization and halite self-healing, and can lay a foundation for the subsequent safe operation and tightness evaluation of salt cavern.