Dependent Boundary Research Articles

Quantum Otto cycle in a superconducting cavity in the nonadiabatic regime

We analyze the efficiency of the quantum Otto cycle applied to a superconducting cavity. We consider its description in terms of a full quantum scalar field in a one-dimensional cavity with a time dependent boundary condition that can be externally controlled to perform and extract work unitarily from the system. We study the performance of this machine when acting as a heat engine as well as a refrigerator. It is shown that, in a non-adiabatic regime, the efficiency of the quantum cycle is affected by the dynamical Casimir effect, that induces a sort of quantum friction that diminishes the efficiency. We also find regions of parameters where the effect is so strong that the machine can no longer function as an engine since the work that would be produced is completely consumed by the quantum friction. However, this effect can be avoided for some particular temporal evolutions of the boundary conditions that do not change the occupation number of the modes in the cavity, leading to a highly improved efficiency.

Numerical Predictions of Elastic and Stiff Characteristics of Marine Structures

Hydroelasticity of marine structures with and without forward speed is studied directly using time dependent Boundary Integral Equation Method with Neumann-Kelvin linearisation where the potential is considered as the impulsive velocity potential. The exciting and radiation hydrodynamic parameters are predicted in time with transient wave Green function whilst the structural analysis is solved with Euler-Bernoulli beam method at which modeshapes are defined analytically. The modal analysis is used to approximate the hydroelastic behaviour of the floating systems through fully coupling of the structural and hydrodynamic analyses. As it is expected, it is found with numerical experience that the effects of the rigid body modes are greater than elastic modes in the case of stiff structures. The predicted numerical results of the present in-house computational tool ITU-WAVE are compared with experimental results for validation purposes and show the acceptable agreements.

Delta derivatives of the solution to a third-order parameter dependent boundary value problem on an arbitrary time scale

Delta derivatives of the solution to a third-order parameter dependent boundary value problem on an arbitrary time scale

Spectral problems for Sturm-Liouville operator with eigenparameter boundary conditions on time scales

In this paper, we consider the inverse problem for Sturm-Liouville operators with eigenparameter dependent boundary conditions on time scales. We give new uniqueness theorems and investigate its some special cases.

Eigenvalues of fourth-order differential operators with eigenparameter dependent boundary conditions

<abstract><p>This paper is concerned with a fourth-order differential operator with eigenparameter dependent boundary conditions. We prove that each of the eigenvalues of the problem can be embedded in a continuous eigenvalue branch. Furthermore, the differential expressions of the eigenvalues with respect to each of parameters are given.</p></abstract>

Properties for fourth order discontinuous differential operators with eigenparameter dependent boundary conditions

<abstract><p>In this paper, a class of fourth order differential operators with eigenparameter-dependent boundary conditions and transmission conditions is considered. A new operator associated with the problem is established, and the self-adjointness of this operator in an appropriate Hilbert space $ H $ is proved. The fundamental solutions are constructed. Sufficient and necessary conditions of the eigenvalues are investigated. Then asymptotic formulas for the fundamental solutions and the characteristic functions are given. Finally, the completeness of eigenfunctions in $ H $ is given and the Green function is also involved.</p></abstract>

Stability analysis of functionals in variational data assimilation with respect to uncertainties of input data for a sea thermodynamics model

AbstractThe problem of stability and sensitivity of functionals of the optimal solution of the variational data assimilation of sea surface temperature for the model of sea thermodynamics is considered. The variational data assimilation problem is formulated as an optimal control problem to find the initial state and the boundary heat flux. The sensitivity of the response functions as functionals of the optimal solution with respect to the observation data is studied. Computing the gradient of the response function reduces to the solution of a non-standard problem being a coupled system of direct and adjoint equations with mutually dependent initial and boundary values. The algorithm to compute the gradient of the response function is presented, based on the Hessian of the original cost functional. Stability analysis of the response function with respect to uncertainties of input data is given. Numerical examples are presented for the Black and Azov seas thermodynamics model.

Analytical simulation of time dependent electromagneto-hydrodynamic flow of Williamson fluid due to oscillatory curved convectively heated Riga surface with variable thermal conductivity and diffusivity

The main objective of the present article is to provide an analytical simulation for time dependent boundary layer flow of non-Newtonian Williamson fluid due to stretchable curved oscillatory Riga surface. Also the characteristics of heat and mass transport are studied under the influence of variable thermal conductivity and diffusivity along with convective heat and mass boundary conditions. Additionally, energy equation is also characterized with the impact of heat production. Curvilinear coordinate scheme is followed to attain the boundary layer expressions for the flow model. The governing nonlinear partial differential equations are solved analytically via homotopy analysis method (HAM). Graphs are plotted to examine comprehensively the consequences of various concerned parameters like modified magnetic parameter and radius of curvature, Williamson fluid parameter, relation of the surface's oscillating frequency to its stretching rate constant, Prandtl number, variable conductivity and heat production parameters, Schmidt number and variable diffusivity parameter on concentration, temperature, pressure and velocity profile. Also the outcomes of afore said variables on surface drag force, rate of temperature and mass transmission (Nusselt and Sherwood numbers) are shown in tabular form. The liquid velocity amplitude is enhanced with modified magnetic parameter and shows opposite behavior for Williamson fluid parameter.

English

In this paper, we consider the direct and inverse problem for time-fractional diffusion in a domain with an impenetrable subregion. Here we assume that on the boundary of the subregion the solution satisfies a generalized impedance boundary condition. This boundary condition is given by a second order spacial differential operator imposed on the boundary. A generalized impedance boundary condition can be used to model corrosion and delimitation. The well-posedness for the direct problem is established where the Laplace Transform is used to study the time dependent boundary value problem. The inverse impedance problem of determining the parameters from the Cauchy data is also studied provided the boundary of the subregion is known. The uniqueness of recovering the boundary parameters from the Neumann to Dirichlet mapping is proven.

Hydrodynamic Model of Decaying Radial Oscillations in RU Cam

Calculations of stellar evolution up to the early white dwarf stage were carried out for stars with mass on the main sequence $M_0=0.82M_\odot$, $0.85M_\odot$, $0.9M_\odot$ and with initial abundances of helium and heavier elements $Y=0.25$ and $Z=10^{-3}$, respectively. For each value of $M_0$ the AGB and post--AGB evolutionary phases were computed with three values of the mass loss parameter in the Bl\"ocker formula: $\eta_B=0.02$, 0.05 and 0.1. The variable star RU~Cam with pulsation period $\Pi\approx 22$ day is shown to be in the post--AGB stage and the pulsation amplitude decrease in years 1962--1963 is due to movement of the star across the HR diagram beyond the pulsation instability region. Theoretical estimates of the mass and the luminosity of RU~Cam are $0.524M_\odot\le M\le 0.532M_\odot$ and $2.20\times 10^3L_\odot\le L\le 2.33\times 10^3L_\odot$, respectively. Hydrodynamic calculations of nonlinear stellar pulsations show that while the star approaches the instability boundary a significant reduction ($\approx 90\%$) in the pulsation amplitude occurs for nearly two years with subsequent slow decay of low--amplitude oscillations. Solution of the equations of hydrodynamics with time--dependent inner boundary conditions describing evolutionary changes in the radius and the luminosity at the bottom of the pulsating envelope allows us to conclude that decay of radial oscillations in RU~Cam is accompanied by the effect of oscillation hysteresis. In particular, the stage of large--amplitude limit cycle oscillations extends by $\approx 12$ years and the subsequent stage of decaying small--amplitude oscilations spreads beyond the formal boundary of pulsation instability.

Sensitivity of functionals in variational data assimilation for a sea thermodynamics model

The sensitivity of functionals of the optimal solution to a variational data assimilation problem for the sea thermodynamics model is studied. The variational data assimilation problem is formulated as an optimal control problem to find the initial state and the boundary condition. The sensitivity of the response functions as functionals of the optimal solution with respect to the observation data is determined by the gradient of the response function and reduces to the solution of a non-standard problem being a coupled system of direct and adjoint equations with mutually dependent initial and boundary values. The algorithm to compute the gradient of the response function is presented, based on the Hessian of the original cost functional. The sensitivity analysis of the response function with respect to errors of observation data is carried out. Numerical examples are presented for the Black Sea thermodynamics model.

Long-time dynamics of the Cahn–Hilliard equation with kinetic rate dependent dynamic boundary conditions

We consider a Cahn–Hilliard model with kinetic rate dependent dynamic boundary conditions that was introduced by Knopf et al. (2021) and will thus be called the KLLM model. In the aforementioned paper, it was shown that solutions of the KLLM model converge to solutions of the GMS model proposed by Goldstein et al. (2011) as the kinetic rate tends to infinity. We first collect the weak well-posedness results for both models and we establish some further essential properties of the weak solutions. Afterwards, we investigate the long-time behavior of the KLLM model. We first prove the existence of a global attractor as well as convergence to a single stationary point. Then, we show that the global attractor of the GMS model is stable with respect to perturbations of the kinetic rate. Eventually, we construct exponential attractors for both models, and we show that the exponential attractor associated with the GMS model is robust against kinetic rate perturbations.

Time-domain impedance boundary conditions for acoustic reduced order finite element simulations

Transient responses impose additional restrictions concerning model order reduction of acoustic finite element systems. Time-stable model order reduction methods achieve model compaction while guaranteeing frequency domain models transform in a physically meaningful way. Efficiency and stability are of course of little consequence if the model is rendered inaccurate. Krylov subspaces inherently include system input and/or output behavior in the reduction basis making them ideal reduction bases for investigating system behavior outside of steady state. Realistic boundary conditions are demanded and must be preserved in the reduction basis. Frequency dependent impedance boundary conditions help in this regard but complicate both model reduction and time-integration strategies. Multiplications to enforce system damping in the frequency domain become time-domain convolutions. Recursively calculated minimal memory convolution formulations have long proven useful in lowering the associated computational burden. Complex frequency-dependent damping matrices create a challenge for Krylov subspace based model reduction due to the way the reduction basis is constructed. Arnoldi iterations implicitly match the moments of the system transfer function to span a Krylov subspace. This paper demonstrates how to maintain compatibility with such algorithms while including frequency dependent damping. This work proposes combining projection based model order reduction with an efficient time domain-impedance boundary condition formulation. An important benefit of working in the time domain is the ability to directly output binaural audio signals. To this end, discrepancies are discussed in the perceptual context of audibility. A reduction of system degrees of freedom from NDOF=13125 to RDOF=63 and the inclusion of time-domain impedance boundary conditions are shown to enable computational speedups by a factor of 11–36 without introducing audible differences.

A superconvergent B-spline technique for second order nonlinear boundary value problems

In the present work, a high-order numerical scheme based on B-spline functions is developed for solving a class of nonlinear derivative dependent singular boundary value problems (DDSBVP). To derive the method, we first generate a high order perturbation of the original problem by using spline alternate relations. Then, we determine the approximate solution by forcing it to satisfy the resulting perturbed problem at the grid points of the spline. Convergence analysis of the method is established through matrix approach. Four nonlinear examples are considered to demonstrate the accuracy and robustness of the method. The proposed method provides O(h6) superconvergent approximation to the solution of the problem under consideration, where h is the step size. This method produces significantly more accurate results than the two newly developed numerical schemes using the same B-spline functions as used in the present method, namely UCS method and NCS method. Moreover, the computational time of present method is compared with that of NCS method.

Quasi-static limit for a hyperbolic conservation law

We study the quasi-static limit for the $L^\infty$ entropy weak solution of scalar one-dimensional hyperbolic equations with strictly concave or convex flux and time dependent boundary conditions. The quasi-stationary profile evolves with the quasi-static equation, whose entropy solution is determined by the stationary profile corresponding to the boundary data at a given time.

An expansion of the Fisher model for concentration dependent grain boundary diffusion

In accounting for concentration dependent grain boundary diffusion, an extra term is derived and added to the boundary condition in Fisher’s Model for atomic transport in fast diffusing boundaries under the condition of low diffusant concentration resulting in an expanded model. The validity of this is evaluated using an implicit finite difference model which shows that a minimum of around 0.5–0.6 order of magnitude difference between the minimum and maximum grain boundary diffusion values is necessary for a significant effect to be observed. This was investigated for a grain boundary diffusion coefficient that exponentially increased and decreased with concentration. Whereas effects on measured grain boundary diffusion coefficients are evaluated to be negligible, the concentration dependency of the grain boundary diffusion substantially affects the elemental distribution in the surrounding of the grain boundary. The latter directly influences grain boundary related metallurgical phenomena such as grain growth or phase transformations in nanocrystalline materials.

The role of SiO2 in semiconducting BaTiO3-based ceramics: Extension of the Jonker model and effect on the electrical properties

BaTiO3-based ceramics with positive temperature coefficient of resistance (PTCR) recently gained in significance due to their application as cabin heaters in electrical vehicles. Although used as a sintering aid in BaTiO3 ceramics for decades, the effect of SiO2 on the electrical properties as well as the underlying physical mechanisms have not been investigated in detail. Here, we systematically study the influence of SiO2 on PTCR properties, but also on the resistance development at temperatures below the PTC regime, which has hardly been given attention to so far. Using capacitance-voltage measurements, we determine temperature dependent grain boundary potential barrier parameters. By combining these results with x-ray diffraction measurements and resistance-temperature properties, we propose a model describing the resistance evolution throughout the whole temperature range from room-temperature up to the PTCR regime. Further, we show that electrical field strength increases with higher SiO2 content, accompanied by a reduced voltage sensitivity of resistance.

A practical method for recovering Sturm–Liouville problems from the Weyl function

In the paper we propose a direct method for recovering the Sturm–Liouville potential from the Weyl–Titchmarsh m-function given on a countable set of points. We show that using the Fourier–Legendre series expansion of the transmutation operator integral kernel the problem reduces to an infinite linear system of equations, which is uniquely solvable if so is the original problem. The solution of this linear system allows one to reconstruct the characteristic determinant and hence to obtain the eigenvalues as its zeros and to compute the corresponding norming constants. As a result, the original inverse problem is transformed to an inverse problem with a given spectral density function, for which the direct method of solution from Kravchenko and Torba (2021 Inverse Problems 37 015015) is applied. The proposed method leads to an efficient numerical algorithm for solving a variety of inverse problems. In particular, the problems in which two spectra or some parts of three or more spectra are given, the problems in which the eigenvalues depend on a variable boundary parameter (including spectral parameter dependent boundary conditions), problems with a partially known potential and partial inverse problems on quantum graphs.

Faceting diagram for Ag segregation induced nanofaceting at an asymmetric Cu tilt grain boundary

In this work, we experimentally establish the isothermal nanofacet evolution at an asymmetric ∑5 tilt grain boundary in the Cu-Ag system using a diffusion couple approach. We investigate the nanofacet formation along the grain boundary in dependence of the Ag solute excess concentration. The initial grain boundary dissociates into asymmetric Ag-lean segments and Ag-rich symmetric (210) segments. Increasing Ag excess leads to an increase in Ag-rich facet segment length, while the length of the asymmetric facets remains constant. From this, we construct a grain boundary nanofaceting diagram deduced from our experiments relating local atomic structure, overall inclination and Ag solute excess.

Vapor flow control in dune sediments under dry bare soil conditions

The study presents daily values of inward and outward vapor flux crossing the soil-atmosphere boundary in dependence on meteorological parameters and soil moisture. Measurements of vapor flux, soil moisture and meteorological parameters were evaluated during days without rainfall between 2015 and 2019. Vapor flux data were obtained at 1 min intervals by a precision lysimeter installed in the Doñana National Park, southwest Spain. Meteorological data were measured on-site at 10-min intervals, including temperature, relative humidity, wind velocity and net radiation. After manual and automatic noise elimination of the lysimeter data, time series of a summer and winter period were generated for selected relevant parameters. Furthermore, daily cumulative data of inward and outward vapor flux were calculated for the entire period and analyzed for their control by meteorological parameters and soil moisture. Results show moderate correlations of daily outward vapor flux on the diurnal amplitudes of temperature and humidity whereas daily inward vapor indicates moderate correlations with temperature and soil moisture. A correction coefficient for potential evaporation to estimate actual evaporation was determined based on soil moisture and air temperature.