Time-dependent Boundary Conditions Research Articles

One-dimensional nonlinear rheological consolidation analysis of soft ground under continuous drainage boundary conditions

This work presents an upgraded one-dimensional (1D) nonlinear rheological system of soft ground consolidation. In the system, the boundaries are characterized by the time-dependent drainage boundary conditions at the surface and bottom of the layer. Meanwhile, the modified UH relationship and non-Newtonian index flow model are used to simulate the viscous effect of clay and the flow of excess pore-water during consolidation, respectively. Then, two numerical discretization formats of the system, namely the finite volume method (FDM) and finite difference method (FVM), are given, and corresponding calculation programs are compiled. The model solutions have been validated by degraded analytical solutions, FDM and FVM comparison solutions, and available laboratory data. Further, the consolidation behaviour under different model parameters is illustrated. The calculated results indicate that the change of interface parameters and flow mode does not influence the final foundation settlement, but both significantly affect the overall dissipation process of EPWP of soils. At last, an efficient method for determining CDB parameters is proposed and verified by available test data.

The Search for a Generalized Analytical Solution for the Inverse Problem; Some Surprisingly Simple Approximate Methods for Problems of Practical Importance

Advances in numerical algorithms for Inverse problems has led to significant leaps in our ability to model complex situations of importance. Nonetheless, there is still a strong need for analytical approaches that can be used to calibrate/validate numerical simulations since the old adage of Trust, but Verify is sound advice. Moreover, access to Inverse codes is often limited despite a strong need. Fortunately, generalized Inverse formulations capable of good approximations can be obtained for linear problems by using a known Direct solution. Duhamel’s Integral (convolution) and a systems Unit Response are first employed to derive a Direct solution, with generality maintained via an arbitrary function with coefficients such as polynomials to described the time-dependent boundary condition. For the ensuing Inverse problem, the Direct solution in the form of a more complex polynomial with its coefficients now considered unknown are then used to enforce remotely measured data using the Levenberg–Marquardt algorithm. Once the coefficients are determined, the original function (polynomial or alternative) now describes the unknown boundary condition. Intriguingly, an approximate Inverse Laplace Transform method applied to the Convolution Integral allows for relatively simple formulations for both the Direct and Inverse problems; for this approximation, only the remotely measured data along with the Unit Response as the solitary system information are required. However, the predicted boundary temperature history tends to be shifted forward in time indicating the influence of the thermal thickness and will also reflect any errors in the data. Accuracy can be enhanced by correcting (back-shifting) for the penetration time to the measurement depth and/or using least-squares smoothing. When the various generalized solutions were used for a plate and a thick-cylinder with a time-dependent thermal boundary-condition on one surface and convection on the other, good-to-excellent Inverse predictions were observed; the methods can be adapted so that remotely measured and near instantaneous surface-strains can also be used to determine an unknown surface temperature history. Finally, the convolution-based approach can easily be adapted for 1-degree of freedom vibration problems.

Effect of Infiltration on Single-Pile and Monopile–Raft Foundation Embedded in Unsaturated Sand

The present study discusses the effect of hydraulic loading in terms of water infiltration caused by rainfall on load-mobilization mechanism of single-pile (SP) and monopile–raft foundation (MPRF) embedded in unsaturated sand by means of the finite element-based computer program Plaxis3D. Soil was modeled using the conventional Mohr–Coulomb model incorporating Bishop’s effective stress, and the hydromechanical behavior was incorporated using the van Genuchten model. A fully coupled flow deformation analysis based on Biot’s theory of consolidation was employed to model the behavior of soil during water infiltration. Water infiltration at different rates ranging from 5 to 864 mm/day was simulated using the time-dependent flow boundary condition at the ground surface. After successful validation of the developed numerical model using the experimental study available in literature, the evolution of suction, effective saturation, and suction stress during the different rates of infiltration was investigated and the responses of SP and MPRF in terms of settlement and axial forces were obtained. The results indicated the contribution of suction stress in increasing the stiffness of soil to an infiltration rate of 240 mm/day, thereby increasing the capacities of both SP and MPRF. The mobilization of shaft resistance was greatly dependent on the suction stress, and a reduction in the suction stress upon infiltration led to a lesser mobilization of shaft resistances (reduced around 6%). This caused a higher pile deformation, leading to further mobilization of pile base resistance. The rate of infiltration and the raft dimension has significant effect on mobilization of raft and pile resistance within MPRF due to raft–soil and pile–raft interactions. A higher rate of infiltration led to a greater mobilization of pile resistance due to positive pile–raft interaction brought by reduction in the contribution of raft base resistance that was associated with reduction in Bishop’s stress. The outcome of the present study would help in better understanding the effect of hydraulic loading on the behavior of SP and MPRF and can be extended to pile group and combined pile–raft foundation.

Ferrofluidic wavy Taylor vortices under alternating magnetic field.

Many natural and industrial flows are subject to time-dependent boundary conditions and temporal modulations (e.g. driving frequency), which significantly modify the dynamics compared with their static counterparts. The present problem addresses ferrofluidic wavy vortex flow in Taylor-Couette geometry, with the outer cylinder at rest in a spatially homogeneous magnetic field subject to an alternating modulation. Using a modified Niklas approximation, the effect of frequency modulation on nonlinear flow dynamics and appearing resonance phenomena are investigated in the context of either period doubling or inverse period doubling. This article is part of the theme issue 'Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical Transactions paper (part 1)'.

Toward the Design of a Representative Heater for Boiling Flow Characterization under PWR’s Prototypical Thermalhydraulic Conditions

In order to improve the understanding of the phenomena underlying the boiling occurrence, CEA Cadarache (France) is designing a new experimental setup, intended to operate for pressures ranging from atmospheric to PWR conditions. This will allow optical access to the convective boiling flow as well as thermal imaging (infrared thermography) of the heated surface. A two-step methodology for designing the heater (particularly its thickness since it directly influences the boiling mechanisms) was developed. This approach is based on solving the heat conduction problem within the heater, considering realistic time-dependent boundary conditions representative of the boiling process. Since those boundary conditions are measured on the external face of the heater, this heat transfer problem is known as an inverse problem that is difficult to solve because of its ill-posedness and high sensitivity to boundary condition uncertainties. In the first stage, we considered one-dimensional modeling to determine the order of magnitude of the heater’s thickness that guaranteed a correct reconstruction of the wet temperature from the measured dry temperature in terms of uncertainties. This value was confirmed in the second stage using a two-dimensional model that accounted for the presence of multiple bubbles on the wet side.

Comparison and Verification of Gas-Bearing Parameter Evaluation Methods for Deep Shale Based on the Pressure Coring Technique

Gas-bearing properties, such as gas-in-place (GIP) content and adsorbed gas ratio (AGR), have drawn considerable attention because they are essential for shale gas resource evaluation and sweet spot forecasting. The above-mentioned metrics have been determined using a variety of approaches, but they lack systematic comparison and accuracy verification, particularly for deep shale gas, where conventional methods frequently fall short. A total of 30 shales were taken for the investigation utilizing pressure and conventional coring methods from Wufeng and Longmaxi Formations in the Sichuan Basin, China. For pressure coring samples, we developed the experimental protocols and a GIP content classification scheme that categorizes the GIP content into four categories: lost gas (estimated using the temperature-corrected pressure-holding ratio), extracted gas from depressurization, degassed gas, and residual gas. The four pressure coring samples from well Y206 with measured GIP content range from 2.84 to 5.58 m3/tonne, with an average of 4.72 m3/tonne. In contrast to the calculated GIP content of the pressure coring samples, the validity of the current GIP content evaluation methodologies has been confirmed. The comparison results demonstrate that the GIP content assessed using the United States Bureau of Mines (USBM) method is frequently underestimated for samples from conventional coring and notably overstated for samples from pressure coring. The polynomial fitting method dramatically overestimated the GIP content. With its accurate time-dependent boundary conditions and strict theoretical foundation, the simplified carbon isotope fractionation (s-CIF) model exhibits the highest level of accuracy in determining GIP content. Additionally, the s-CIF model can find yet another crucial parameter of AGR. Shale samples from well Y206 had an average AGR of 20.4% but a range from 12.2 to 33.2%. The examination of the influencing factors demonstrates that the lost gas ratio, AGR, and diffusion coefficient are responsible for controlling the calculation error of the USBM method. The lost gas ratio prior to core retrieval increases with decreasing AGR (or increasing diffusion coefficient), which increases the computation error of the USBM method.

Effect of in-plane compression on the non-harmonic resonance of moderately thick time-dependent plates

This study investigates the nonlinear p-delta analysis and introduces a viscoelastic plate non-harmonic resonance in which the critical displacement will be several times more than the displacement at time zero. An innovative closed-form solution of Mindlin viscoelastic plate, under a dynamic out-of-plane exponential load decreasing over time and a constant in-plane compressive load is presented. The displacement vector of the Mindlin viscoelastic plate is explicitly formulated in space and time domains. Also, the critical time in which the non-harmonic resonance occurs is calculated. The stress–strain relationship of the viscoelastic material is defined according to the Boltzmann integral law. The displacement field is expressed by the product of a determinate geometrical function and a time function with two indeterminate coefficients. The time function coefficients are calculated with the help of the Laplace transform, satisfying the equilibrium equation at two asymptotic times. Abaqus software is used to confirm the results. The numerical results consider the effect of time-dependent material parameters, loads, boundary conditions, thickness-to-side ratios, and aspect ratios on the maximum displacements of moderately thick time-dependent plates.

Time-dependent boundary conditions for data-driven coronal global and spherical wedge-shaped models

ABSTRACT The development of an efficient and accurate method for boundary condition treatments is of fundamental importance to data-driven magnetohydrodynamic (MHD) modelling of the global solar corona and solar active region. Particularly, in a 3D spherical wedge-shaped volume, suitable to the numerical study of solar active region, the transverse terms calls for a delicate treatment at the computational domain’s edges and corners, and properly prescribed conditions for boundaries joining regions of different flow properties, so as to take account of the joint effect of incoming and outgoing waves. To provide a solution to the determination of boundary conditions, in this paper a systematic tactics is formulated for handling edges and corners and the prescribed conditions for inner/outer/edge/corner boundaries are proposed through the combination (CBC-ILW) of the time-dependent characteristic boundary conditions (CBCs) and the inverse Lax-Wendroff (ILW) procedure. First, a data-driven 3D MHD simulation has been carried out to study the dynamic evolution of the solar corona from 1Rs to 6.7Rs during the period between 2018 May 16 and August 6. The simulated results of the global coronal evolution provide a good comparison with observed coronal images during the period investigated. Then, the validity of 3D MHD-CBC-ILW is verified for a 3D spherical wedge model, by producing almost the same results as those taken out of the global model on a 3D spherical wedge-shaped volume.

A finite-difference ghost-point multigrid method for multi-scale modelling of sorption kinetics of a surfactant past an oscillating bubble

We propose a method for the numerical solution of a multiscale model describing sorption kinetics of a surfactant around an oscillating bubble. The evolution of the particles is governed by a convection-diffusion equation for the surfactant concentration c, with suitable boundary condition on the bubble surface, which models the action of the short range attractive-repulsive potential acting on them when they get sufficiently close to the surface [1]. In the domain occupied by the fluid, the particles are transported by the fluid motion generated by the bubble oscillations.The method adopted to solve the equation for c is based on a finite-difference scheme on a uniform Cartesian grid and implemented in 2D and 3D axisymmetric domains. We use a level-set function to define the region occupied by the bubble, while the boundary conditions are discretized by a ghost-point technique to guarantee second order accuracy at the curved boundary. The sparse linear system is finally solved with a geometric multigrid technique designed ad-hoc for this specific problem. Several accuracy tests are provided to prove second order accuracy in space and time.The fluid dynamics generated by the oscillating bubble is governed by the Stokes equation solved with a second order accurate method based on a monolithic approach, where the momentum and continuity equations are solved simultaneously. Since the amplitude of the bubble oscillations are very small, a simplified model is presented where the computational bubble is actually steady and its oscillations are represented purely with time-dependent boundary conditions. A numerical comparison with the moving domain model confirms that this simplification is perfectly reasonable for the class of problems investigated in this paper.

Hygro-thermal coupling in earth building materials

Raw earth is emerging as a viable building material with lower carbon emissions than conventional concrete and fired bricks. Raw earth is as an excellent passive hygro-thermal regulator, which improves occupants’ comfort while reducing the need for active heating/cooling installations. The coupled hygro-thermal response of earth materials is investigated by exploiting the principles of the thermodynamics of porous media and unsaturated soil mechanics. The degree of coupling between temperature and relative humidity (or water content) depends on the adopted simplifying assumptions. Some of these assumptions are valid for traditional building materials but may not be applicable to raw earth characterised by relatively high levels of liquid/gas permeability. The validity of current approaches is here assessed with reference to earth building via a simple one-dimensional transfer model, which simulates the behaviour of an unbounded earth wall subjected to time-dependent boundary conditions on the two faces. For typical values of water and vapour permeability, the complexity of the governing equations can be greatly reduced by neglecting variations of vapour mass and the dependency of suction on temperature without significantly reducing accuracy. Results are also strongly influenced by both initial state and water retention properties of the earth material.

Effect of a Moving Mirror on the Free Fall of a Quantum Particle in a Homogeneous Gravitational Field

We investigate the effect of time-dependent boundary conditions on the dynamics of a quantum bouncer—a particle falling in a homogeneous gravitational field on a moving mirror. We examine more particularly the way a moving mirror modifies the properties of the entire wavefunction of a falling particle. We find that some effects, such as the fact that a quantum particle hitting a moving mirror may bounce significantly higher than when the mirror is fixed, are in line with classical intuition. Other effects, such as the change in relative phases or in the current density in spatial regions arbitrarily far from the mirror are specifically quantum. We further discuss how the effects produced by a moving mirror could be observed in link with current experiments, in particular with cold neutrons.

Quasistatic crack growth in elasto-plastic materials with hardening: The antiplane case

Abstract We study a variational model for crack growth in elasto-plastic materials with hardening in the antiplane case. The main result is the existence of a solution to the initial value problem with prescribed time-dependent boundary conditions.

Analytical solution of non-Fourier heat conduction in a 3-D hollow sphere under time-space varying boundary conditions

In the current research, a comprehensive analytical technique is presented to evaluate the non-Fourier thermal behavior of a 3-D hollow sphere subjected to arbitrarily-chosen space and time dependent boundary conditions. The transient hyperbolic thermal diffusion equation is driven based upon the Cattaneo-Vernotte (C-V) model and nondimensionalized using proper dimensionless parameters. The conventional procedure of separation of variables is applied for solving the 3-D hyperbolic heat conduction equation with general boundary conditions. In order to handle the time dependency of the boundary conditions, Duhamel's theorem is employed. Furthermore, for the purpose of demonstrating the applicability of the obtained general solution, two particular cases with different time-space varying boundary conditions are considered. Subsequently, their respective non-Fourier thermal characteristics are elaborately discussed and compared with the Fourier case. The quantitative analysis is carried out, including the profiles of the time-dependent temperature and 3-D distributions of temperature at different time frames. Eventually, the influences of the controlling factors such as Fourier number and Vernotte number on the temperature field distributions within a hollow sphere for both cases are assessed. The findings reveal that the lag time in the hyperbolic thermal propagation diminishes with a decrement of Vernotte number, and it asymptotically vanishes for the Fourier case. Also, the number and severity of the jump points that occurred in the non-Fourier cases decrease by increasing the Fourier number, and these points finally vanish at particular Fourier numbers.

Existence of quasi-static crack evolution for atomistic systems

We consider atomistic systems consisting of interacting particles arranged in atomic lattices whose quasi-static evolution is driven by time-dependent boundary conditions. The interaction of the particles is modeled by classical interaction potentials where we implement a suitable irreversibility condition modeling the breaking of atomic bonding. This leads to a delay differential equation depending on the complete history of the deformation at previous times. We prove existence of solutions and provide numerical tests for the prediction of quasi-static crack growth in particle systems.

Implementation of phase change material for cooling load reduction: a case study for Cairo, Egypt

Integrating a phase change material (PCM) into building envelopes can reduce energy needs in the built environment, and the consequent greenhouse emissions. This research examines the impact of PCM integrated into a traditional wall in Egypt on peak and average cooling energy consumption. A MATLAB code based on the finite volume technique using the Crank-Nicolson method for discretization is implemented. Several benchmark cases and experimental results validate the code. The time-dependent boundary conditions of the cases examined were based on the irradiance and ambient temperatures measured in Cairo, Egypt. Simulations are performed on eight different PCMs, using their real published DSC curve. The study aims to investigate the performance of each PCM at different positions, thicknesses, and wall orientations. The calculations revealed that using the proper PCM type and the proper position could decrease the average by 38.14%, Also the peak heat flux could be decreased by 58.53%.

Impact of Geometry on a Thermal-Energy Storage Finned Tube during the Discharging Process

This work focused on the modelling of latent heat thermal energy storage systems. The mathematical modelling of a melting and solidification process has time-dependent boundary conditions because the interface between solid and liquid phases is a moving boundary. The heat transfer analysis needs the interface position over time to predict the temperature inside the liquid and the solid regions. This work started by solving the classical two-phase (one-dimensional) Stefan problem through a Matlab implementation of the analytical model. The same physical problem was numerically simulated using ANSYS FLUENT, and the good match of analytical and numerical results validated the numerical model, which was used for a more interesting problem: comparing three different latent heat TES configurations during the discharging process to evaluate the most efficient in terms of maximum average discharging power. The three axial heat conduction structures changed only for the fin shape (rectangular, trapezoidal and fractal), keeping constant the volume fractions of steel, aluminium and PCM to perform a proper comparison. Results showed that the trapezoidal fin profile performs better than the rectangular one, and the fractal fin profile geometry was revealed as the best for faster thermal exchange when the solidifying frontier moves away from the steel ring. In conclusion, the average discharging power for the three configurations was evaluated for a time corresponding to a reference value (10%) of the liquid fraction: the rectangular fin profile provided 950.8 W, the trapezoidal fin profile 979.4 W and the fractal fin profile 1136.6 W, confirming its higher performance compared with the other two geometries.

HUXt—An open source, computationally efficient reduced-physics solar wind model, written in Python

HUXt is an open source numerical model of the solar wind written in Python. It is based on the solution of the 1D inviscid Burger’s equation. This reduced-physics approach produces solar wind flow simulations that closely emulate the flow produced by 3-D magnetohydrodynamic (MHD) solar wind models at a small fraction of the computational expense. While not intended as a replacement for 3-D MHD, the simplicity and computational efficiency of HUXt offers several key advantages that enable experiments and the use of techniques that would otherwise be cost prohibitive. For example, large ensembles of 102–105 members can easily be run with modest computing resources, which are useful for exploring and quantifying the uncertainty in space weather predictions, as well as for the application of some data assimilation methods. In this article we present the developments in the latest version of HUXt, v4.0, and discuss our plans for future developments and applications of the model. The three key developments in v4.0 are: 1) a restructuring of the models solver to enable fully time-dependent boundary conditions, such that HUXt can in principle be initialised with in-situ observations from any of the fleet of heliospheric monitors; 2) new functionality to trace streaklines through the HUXt flow solutions, which can be used to track features such as the Heliospheric Current Sheet; 3) introduction of a small test-suite so that we can better ensure the reliability and reproducibility of HUXt simulations for all users across future versions. Other more minor developments are discussed in the article. Future applications of HUXt are discussed, including the development of both sequential and variational data assimilation schemes for assimilation of both remote sensing and in-situ plasma measures. Finally, we briefly discuss the progress of transitioning HUXt into an operational model at the UK’s Met Office Space Weather Operations Center as part of the UK governments SWIMMR programme.

Energy extraction from AdS black holes via superradiance

Superradiance is known as a wave amplification process caused by rotating or charged black holes. We argue that the superradiance of stationary black holes in asymptotically AdS spacetimes can be characterized by the ability of energy extraction. Specifically, we demonstrate that energy can be extracted from Reissner-Nordström-AdS4 and Kerr-AdS4 under appropriate time-dependent boundary conditions at conformal boundaries. This indicates that energy can be extracted from thermal states dual to these black holes by applying appropriate time-dependent sources. We also show that the energy extraction can be realized as a reversible process.

Global stability under dynamic boundary conditions of a nonlinear PDE model arising from reinforced random walks

This paper is devoted to the study of the global stability of classical solutions to an initial and boundary value problem (IBVP) of a nonlinear PDE system, converted from a model of reinforced random walks, subject to time-dependent boundary conditions. It is shown that under certain integrability conditions on the boundary data, classical solutions to the IBVP exist globally in time and the differences between the solutions and their corresponding ansatz, determined by the initial and boundary conditions, converge to zero, as time goes to infinity. Though the final states of the boundary functions are required to match, the boundary values do not necessarily equal to each other at any finite time. In addition, there is no smallness restriction on the magnitude of the initial perturbations. Furthermore, numerical simulations are performed to investigate the long-time dynamics of the IBVP when the final states of the boundary functions do not match or the boundary functions are periodic in time.

High-order pressure estimates for Navier-Stokes Runge-Kutta solvers using stage pseudo-pressures

This note addresses the question of whether stage pseudo-pressures can be used to build high-accuracy instantaneous pressure estimates in the context of projection-based Navier-Stokes solvers. The construction of such estimates using inter-stage pseudo-pressures is not straightforward, especially when time-dependent boundary conditions are present. New formulas are introduced that circumvent the limitation imposed by this type of boundary condition. While additional numerical costs might be involved in the construction of such estimates, the formulas are robust and show how one is able to recover high-order estimates of the pressure from within a Runge-Kutta pressure-projection scheme. The proposed formulas are then numerically tested on a two-dimensional channel flow simulation with unsteady inflow condition and show expected convergence rates.