Arbitrary Function Of Time Research Articles

Analysis of transient uncoupled thermoelastic problems involving moving point heat sources using the method of fundamental solutions

The method of fundamental solutions (MFS) is very effective for analysis of problems with moving domain loads. In this work, the MFS is formulated for solution of two dimensional transient uncoupled thermoelastic problems involving moving point heat sources. At first, the time-dependent temperature field is obtained by solving the transient heat conduction equation involving a moving heat source term. Then, at each time point, the temperature field is used in the constitutive and equilibrium equations to determine the displacement field. The intensity and location of the point heat source can be arbitrary functions of time. The particular solutions for temperature and stress are described by simple and closed-form expressions and they are used without considering any internal points. Two numerical example problems are provided to demonstrate the efficiency of the presented formulation. The obtained numerical results show that the presented MFS is very efficient and useful. Unlike the finite element method, only a small number of collocation and source points are sufficient to achieve very accurate results in the proposed MFS.

Exploring the effect of the first cycle on the economic production quantity repair and waste disposal model

The classical formulation of reverse logistics inventory systems assumes an equal returned amount in all cycles. This assumption necessitates that manufacturing and remanufacturing processes are static in all cycles, i.e. it ignores the impact of the first cycle. This implies that the result of such modelling may not be an optimal policy. Another implicit assumption embedded in the classical formulation is that input parameters remain static indefinitely. In practice, however, it is often desirable that input parameters are adjusted to be responsive to a new policy due to acquired new knowledge. This paper develops a general joint model for production of new items together with remanufacturing of returned items “as good as new” condition. The proposed model focuses on a single production cycle and a single remanufacturing cycle and, as part of its formulation, considers the first time interval. This implies that the decision variables should vary from cycle to cycle until the system plateaus. Adopting this approach ensures a more realistic tractability of the impact of previous cycles, including the first time cycle. The mathematical formulation considers arbitrary functions of time, which allows the decision maker to assess and compare the consequences of a diverse range of time-varying forms by employing a single inventory model. All functions and input parameters may change their values for subsequent cycles and/or periodic review or remain static. A rigorous method is utilised to show that the solution, if it exists, is unique and global optimal. The paper provides illustrative examples that demonstrate the application of the theoretical model in different settings.

MODELING OF THE HEAT TRANSFER PROCESS IN A MULTI-LAYER SPHERICAL CONSTRUCTION TAKING INTO ACCOUNT OF INTERNAL HEAT SOURCES

The proposed work is devoted to the application of the direct method to the study of heat transfer processes in a multilayer hollow spherical structure in the presence of internal (distributed) heat sources. It is assumed that there is an ideal thermal contact between the layers, and that the laws of changes in the temperatures of the media that wash the surface (inner and outer) layers of the structure are arbitrary functions of time and are evenly distributed over the surfaces. That is, convective heat exchange with the environment is assumed and the boundary conditions of the third are satisfied. Therefore, the isotherms inside this structure are concentric circles. The coefficients of the thermal conductivity equation are considered to be lump-constant functions with respect to the spatial coordinate.
 This problem is solved by applying the reduction method, when the original problem is divided into two or more simple ones. Analytical studies were conducted using the method of reduction, the concept of quasi-derivatives, modern theory of systems of linear differential equations, the Fourier method and a modified method of eigenfunctions with the active use of computer mathematical environments. The numerical implementation of the method was performed using the Maple 13 computer algebra system.
 To illustrate the proposed method, a model example of finding the distribution of the temperature field in a four-layer spherical structure with available internal heat sources under the influence of the temperature of the external fire is solved. The results of the calculations are presented as a graph of temperature change depending on time and spatial coordinate. It should be noted that 30 first roots of the characteristic equation were used to achieve the result with the given accuracy.
 The results obtained are directly applicable in a number of applications. This task describes the processes of heat exchange both heating and cooling.

Energy density and spectrum of single-cycle and sub-cycle electromagnetic pulses

Based on the exact solution of the Maxwell equations in the form of a collapsing spherical vector wave specified by an arbitrary function of time, we have calculated the maximum energy density that can be achieved when focusing extremely short pulses of various shapes. It is shown that our earlier formula expressing the maximum energy density in terms of the spectrum parameters for Gaussian quasi-monochromatic pulses is approximately (with an accuracy of ±40 %) valid for the main types of extremely short pulses.

DETERMINATION OF TEMPERATURE FIELD IN A SOLID CIRCULAR CYLINDER IN THE FIRE

The proposed work is devoted to the application of the direct method to the study of heat transfer processes in a solid homogeneous cylindrical structure in the presence of convective heat exchange with the environment, that is, the boundary conditions of the third. It is assumed that the law of change of ambient temperature, which washes the surface layer of the structure, is an arbitrary function of time. This function is uniformly distributed over the surface such that the isotherms inside the structure are coaxial cylindrical surfaces, that is, the temperature field inside the cylinder depends only on the radius r and the time τ.This problem is solved by applying the reduction method, when the original problem is represented by the sum of two unknown but related functions. It is found that the solution of the corresponding quasistationary problem does not depend on the radius r and is equal to the law of change of the ambient temperature.In the following, the corresponding inhomogeneous problem is solved, the main stage of which is the eigenvalue problem obtained after applying the Fourier variable method. As a standard procedure, a characteristic equation is obtained and the eigenvalues are determined and their own functions are constructed. The Fourier method was used to construct a nonuniform problem solution.It is found that when at the time τ = 0 the initial distribution of the structural temperature field and the ambient temperature coincide, the solution of the original problem is greatly simplified.To illustrate the proposed method, a model example of finding the distribution of the temperature field in a reinforced concrete homogeneous column of a circular cross section under the conditions of the temperature of the hydrocarbon fire is solved. The results of the calculations are presented as a graph of temperature change as a function of time. The numerical implementation of the method was performed using the Maple 13. computer algebra system. It should be noted that the first 30 roots of the characteristic equation were used to achieve the result with the given accuracy. The results obtained are directly applicable in a number of applications.

Dynamic Axisymmetric Problem of Direct Piezoelectric Effect for A Bimorphic Plate of Stepwise Variable Thickness

The dynamic axisymmetric problem of direct piezoelectric effect for a circular two-layer transducer consisting of a metal substrate and a piezoceramic plate of smaller diameter is considered. The oscillations of the electroelastic system are carried out due to the action of a mechanical load on its front surface, which is an arbitrary function of time and radial coordinate. The presented mathematical model of calculation allows to take into account the connectedness of longitudinal-transverse oscillations of a structure that is asymmetric in height. A new analytical solution was obtained in the framework of the refined theory of Timoshenko by the method of finite integral transformations. The calculated ratios make it possible to choose the geometric dimensions of the bimorphic plate, which most effectively allow to convert the mechanical load into the generated electrical pulse. The considered designs are widely used as touch switches that regulate the operation of valves for various purposes.

The dynamics of particles around time conformal AdS-Schwarzschild black hole

Abstract In this article, we explore the effect of the cosmological constant ( Λ ) on the dynamics of neutral as well as charged particles in vicinity of the time conformal Schwarzschild black hole. The general time conformal term e ϵ f ( t ) is introduced in the AdS-Schwarzschild spacetime, where f(t) is an arbitrary function of time t and ϵ is a small parameter which causes perturbation in the spacetime. The corresponding Lagrangian is used in the Noether symmetry equation which gives a system of 19 partial differential equations (PDEs). The solution of the 19 PDEs give us the require time conformal AdS-Schwarzschild spacetime along with approximate Noether symmetries and conservation laws. The dynamics of neutral and charged particles are discuss over the fabric of time conformal Ads–Schwarzschild spacetime.

DETERMINATION OF NON-STATIONARY TEMPERATURE FIELD IN THE SYSTEM OF TWO SPHERICAL SHELL

The main classification indicator, in terms of fire safety, is the degree of fire resistance of the house. Depending on this indicator normalize its surface, the area of development and distance to other buildings and structures. The de-gree of fire resistance of the house is determined by the limit of fire resistance of its building structures and the limit of the fire spread by these structures. Therefore, the value of the fire resistance limit of building constructions, which con-sists of a house, significantly affect its architectural solution and the parameters of construction in general. On this ba-sis, taking into account the approaches to ensuring normalized fire resistance limits of the design and the features of their behavior under high-temperature (fire) influence is very relevant.Most research on building constructions. The proposed work is devoted to the application of the direct method to the study of heat transfer processes in the system of two embedded spherical bodies – a ball in a sphere. It is assumed that there is an ideal thermal contact between the balls, and the law of temperature change on the outer surface is an arbitrary function of time, and evenly distributed over the surface of the ball. Consequently, isotherms inside this construction are concentric fields, that is, the problem is symmetric and is solved for the first time in such a statement. To solve such a problem, in parallel, the auxil-iary problem of determining the distribution of a non-stationary temperature field in a two-layer hollow spherical structure with a "extracted" sphere of sufficiently small radius is raised. In this case the symmetry condition of the original problem is replaced by the condition of the second kind on the inner surface of this construction. The implementation of the solution of the auxiliary problem is carried out by applying a reduction method using the concept of quasiderivatives. In the future, the Fourier scheme is used with the use of the modified eigenfunctions method. To find the solution of the original problem, the idea of the boundary transition is used by passing the radius of the withdrawn bullet to zero. It is established that in this approach all the eigenfunctions of the corresponding problem on the eigenvalues have no singularities at zero, which means that the solutions of the original problem are constrained throughout the design. The solution of this problem at zero temperature on the outer surface coincides with those known in the literature. To illustrate the proposed method, a model example of finding the temperature field distribution in a system of two spherical bodies with different thermophysical properties of materials is solved. The results of the calcu-lations are presented in the form of a table and a three-dimensional graph of temperature change, depending on the time and spatial coordinates. The generalization of the results obtained in the event of any finite number of nested balls is a purely technical problem, and not a fundamental one. Note that while changing the boundary condition of the first kind to any other boundary condition (for example, the third kind) does not affect the scheme of solving similar tasks. Since the general scheme of studying the distribution of temperature fields in multi-layered structures with an arbitrary number of layers in the presence of internal sources of heat is studied in detail, the setting and solving of such problems for the system of nested balls does not cause any difficulty.

New Solutions of Dynamical Equations of Ideal Plasticity

Point symmetries allowed by plasticity equations in the dynamical case are used to construct solutions for the dynamical equations of ideal plasticity. These symmetries make it possible to convert the exact solutions of stationary dynamical equations to nonstationary solutions. The so-constructed solutions include arbitrary functions of time. The solutions allow us to describe the plastic flow between the plates changing their shape under the action of dynamical loads. Some new spatial self-similar solution is also presented.

DETERMINATION OF THE NON-STATIONARY TEMPERATURE FIELD IN THE SYSTEM OF TWO CYLINDRICAL SHELL UNDER THE FIRE CONDITIONS

The proposed work is devoted to the application of the direct method to the study of heat transfer processes in the system "solid cylinder inside a cylindrical shell". It is assumed that there is an ideal thermal contact between them, and the law of changing the ambient temperature, which rinses the surface of the structure, is an arbitrary function of time, and evenly distributed over the surface. Consequently, isotherms inside this construction are concentric circles, that is, the problem is symmetric and is solved for the first time in such a statement. To solve such a problem, the auxiliary problem of determining the distribution of a non-stationary temperature field in a two-layer hollow cylindrical structure with a "withdrawn" cylinder of sufficiently small radius is raised in parallel. In this case the symmetry condition of the original problem is replaced by the condition of the second kind on the inner surface of this construction. The implementation of the solution of the auxiliary problem is carried out by applying a reduction method using the concept of quasi-derivatives. In the future, the Fourier scheme is used with the use of the modified eigenfunctions method. To find the solution of the original problem, the idea of the boundary transition is used by passing the radius of the withdrawn cylinder to zero. It is established that in this approach all the eigenfunctions of the corresponding problem on the eigenvalues have no singularities at zero, which means that the solutions of the original problem are constrained throughout the design. In order to illustrate the proposed method, a model example of finding the temperature field distribution in a column of a circular cross-section (concrete in a steel shell) is solved under the influence of the standard temperature regime of the fire. The results of the calculations are presented in a bulk schedule of temperature changes, depending on time and spatial coordinates. The generalization of the results obtained in the case of any finite number of cylindrical shells is a purely technical problem, and not a fundamental one. Note that while changing the boundary condition of the third kind to any other boundary condition (for example, the first kind) does not affect the scheme of solving similar tasks. Since the general scheme of studying the distribution of temperature fields in multi-layered structures with an arbitrary number of layers in the presence of internal sources of heat is studied in detail, the setting and solving of such problems for the system of "solid cylinder inside a cylindrical shell" is not without difficulty.

Exact Solutions of Stationary Equations of Ideal Magnetohydrodynamics in the Natural Coordinate System

Equations of ideal magnetohydrodynamics that describe stationary flows of an inviscid ideally electrically conducting fluid are considered. Classes of exact solutions of these equations are described. With the use of the natural curvilinear coordinate system, where the streamlines and magnetic force lines play the role of the coordinate curves, the model equations are partially integrated and converted to the form that is more convenient for the description of the magnetic lines and streamlines of particles. As the coordinate system used is related to the initial coordinate system by a nonlocal transformation, the group admitted by the system can change. An infinite-dimensional (containing three arbitrary functions of time) group of symmetries is calculated for the system in the natural coordinates. An optimal system of subgroups of dimensions 1 and 2 is constructed for this group. For one of the optimal system subgroups, an invariant exact solution is found, which describes the electrically conducting fluid flow of the vortex source type with swirling magnetic lines and streamlines.

Design and test of an apparatus for transient measurements of the thermalconductivity and volumetric heat capacity of solids

Knowledge of the thermal conductivity (κ) of solids is crucial for all thermoelectric devices. A new approach in transient measurement to determine this value as well as electric conductivity (σ) and the Seebeck coefficient (S) is presented here. This approach can be combined with current steady-state methods. A cylindrical sample is mounted between two heat-flux sensors that can be heated or cooled at their outer ends. The input signals defining the heat fluxes at the sensors can be any arbitrary function of time, although some waveforms yield more valid results. The method is evaluated by employing a one-dimensional numerical model and finding the best fit to extract the thermal conductivity (κ) of the sample as well as its volumetric heat capacity (cρ) . Trial measurements on an insulating (σ = 1 × 1013 Sm-1) and a conducting sample (σ = 1 × 105 Sm-1) are presented and the results are in good agreement with the literature and data obtained by a commercial laser-flash analysis system. Improvements in comparison to present measurement methods are the direct determination of κ compared to other transient methods like laser-flash analysis, shorter measurement times by acquiring κ(T) data in a single temperature approach, and simultaneous S and σ measurements.

Numerical simulation of fatigue failure of composite materials under compression

The process of fatigue fracture of carbon fibres reinforced plastic under compressive loads is considered from the standpoint of the theory of reaction rates. Numerical simulation is based on rheological structural models of the material, which reproduce thermodynamic processes of local plastic deformation and failure occurring in time. The prediction of longevity of composite materials under compressive loads is similar to the solution of the problem for metal alloys under tensile loads when temperature and stresses are arbitrary functions of time.

Models of Anisotropic Self-Gravitating Source in Einstein-Gauss-Bonnet Gravity

In this paper, we have studied gravitational collapse and expansion of nonstatic anisotropic fluid in 5D Einstein-Gauss-Bonnet gravity. For this purpose, the field equations have been modeled and evaluated for the given source and geometry. The two metric functions have been expressed in terms of parametric form of third metric function. We have examined the range of parameter β (appearing in the form of metric functions) for which Θ, the expansion scalar, becoming positive/negative leads to expansion/collapse of the source. The trapped surface condition has been explored by using definition of Misner-Sharp mass and auxiliary solutions. The auxiliary solutions of the field equations involve a single function that generates two types of anisotropic solutions. Each solution can be represented in term of arbitrary function of time; this function has been chosen arbitrarily to fit the different astrophysical time profiles. The existing solutions forecast gravitational expansion and collapse depending on the choice of initial data. In this case, wall to wall collapse of spherical source has been investigated. The dynamics of the spherical source have been observed graphically with the effects of Gauss-Bonnet coupling term α in the case of collapse and expansion. The energy conditions are satisfied for the specific values of parameters in both solutions; this implies that the solutions are physically acceptable.

Traveling Waves and Structures of a Film Flow with Phase Transitions in the Nakoryakov–Ostapenko–Bartashevich Model

To describe the motion of the film flowing downward the vertical wall in the mode of condensation or evaporation into the surrounding space, the model proposed in [1] is used. It is reduced to one equation for the film thickness. The model comprises two governing parameters. The first one is proportional to the difference in the wall temperature, assumed to be constant, and the saturation temperature, and the second is proportional to the surface tension coefficient. In a series of publications [1–4] the authors studied solutions of thementioned equation at zero surface tension. Their characteristic feature is the presence of strong and weak discontinuities of layer thickness. In this paper we studied the regularizing effect of surface tension on the film flow structure with phase transitions. Numerical and asymptotic analysis of the resulting structures is carried out. Also, the case where the wall temperature is an arbitrary function of time is considered.

Chaos in a non-autonomous nonlinear system describing asymmetric water wheels

We derive a water wheel model from first principles under the assumption of an asymmetric water wheel for which the water inflow rate is in general unsteady (modeled by an arbitrary function of time). Our model allows one to recover the asymmetric water wheel with steady flow rate, as well as the symmetric water wheel, as special cases. Under physically reasonable assumptions, we then reduce the underlying model into a non-autonomous nonlinear system. In order to determine parameter regimes giving chaotic dynamics in this non-autonomous nonlinear system, we consider an application of competitive modes analysis. In order to apply this method to a non-autonomous system, we are required to generalize the competitive modes analysis so that it is applicable to non-autonomous systems. The non-autonomous nonlinear water wheel model is shown to satisfy competitive modes conditions for chaos in certain parameter regimes, and we employ the obtained parameter regimes to construct the chaotic attractors. As anticipated, the asymmetric unsteady water wheel exhibits more disorder than does the asymmetric steady water wheel, which in turn is less regular than the symmetric steady state water wheel. Our results suggest that chaos should be fairly ubiquitous in the asymmetric water wheel model with unsteady inflow of water.

Transient thermoelastic analysis of infinite plates with time-periodic heat transfer coefficient: Frequency response of thermal stress

ABSTRACTAn analytical method is presented for one-dimensional transient heat conduction and associated thermal stress problems of a layered infinite plate whose heat transfer coefficient (HTC) on one external boundary is an arbitrary function of time. A shifting function method and an orthogonal expansion technique are used to obtain analytical solutions to the problems. Using the solutions, the structural response to sinusoidal HTC oscillation is investigated in single and bilayered plates, the other boundary of which is kept at constant temperature. Numerical results demonstrate the effects of average HTC (measured by the Biot number) and mechanical boundary conditions on the frequency responses of temperature and thermal stress. It is found that the range of stress oscillation decreases with an increase in the frequency of HTC oscillation except for single-layered plates without surface tractions.

Nonlinear dynamics of mushy layers induced by external stochastic fluctuations.

The time-dependent process of directional crystallization in the presence of a mushy layer is considered with allowance for arbitrary fluctuations in the atmospheric temperature and friction velocity. A nonlinear set of mushy layer equations and boundary conditions is solved analytically when the heat and mass fluxes at the boundary between the mushy layer and liquid phase are induced by turbulent motion in the liquid and, as a result, have the corresponding convective form. Namely, the 'solid phase-mushy layer' and 'mushy layer-liquid phase' phase transition boundaries as well as the solid fraction, temperature and concentration (salinity) distributions are found. If the atmospheric temperature and friction velocity are constant, the analytical solution takes a parametric form. In the more common case when they represent arbitrary functions of time, the analytical solution is given by means of the standard Cauchy problem. The deterministic and stochastic behaviour of the phase transition process is analysed on the basis of the obtained analytical solutions. In the case of stochastic fluctuations in the atmospheric temperature and friction velocity, the phase transition interfaces (mushy layer boundaries) move faster than in the deterministic case. A cumulative effect of these noise contributions is revealed as well. In other words, when the atmospheric temperature and friction velocity fluctuate simultaneously due to the influence of different external processes and phenomena, the phase transition boundaries move even faster. This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.This article is part of the theme issue 'From atomistic interfaces to dendritic patterns'.

Weighted Cox Regression Using the R Package coxphw

Cox's regression model for the analysis of survival data relies on the proportional hazards assumption. However, this assumption is often violated in practice and as a consequence the average relative risk may be under- or overestimated. Weighted estimation of Cox regression is a parsimonious alternative which supplies well interpretable average effects also in case of non-proportional hazards. We provide the R package coxphw implementing weighted Cox regression. By means of two biomedical examples appropriate analyses in the presence of non-proportional hazards are exemplified and advantages of weighted Cox regression are discussed. Moreover, using package coxphw, time-dependent effects can be conveniently estimated by including interactions of covariates with arbitrary functions of time.

An Analytical Solution for Transient Heat Conduction in a Composite Slab with Time-Dependent Heat Transfer Coefficient

An analytical solution is derived for one-dimensional transient heat conduction in a composite slab consisting of n layers, whose heat transfer coefficient on an external boundary is an arbitrary function of time. The composite slab, which has thermal contact resistance at n-1 interfaces, as well as an arbitrary initial temperature distribution and internal heat generation, convectively exchanges heat at the external boundaries with two different time-varying surroundings. To obtain the analytical solution, the shifting function method is first used, which yields new partial differential equations under conventional types of external boundary conditions. The solution for the derived differential equations is then obtained by means of an orthogonal expansion technique. Numerical calculations are performed for two composite slabs, whose heat transfer coefficient on the heated surface is either an exponential or a trigonometric function of time. The numerical results demonstrate the effects of temporal variations in the heat transfer coefficient on the transient temperature field of composite slabs.