Nonlinear Heat Conduction Research Articles

Analytical solution of a one-dimensional thermistor problem with Robin boundary condition

A one-dimensional nonlinear heat conduction equation of steady-state Joule heating in the presence of an electric field in a metal with temperature-dependent conductivities is considered. A technique developed by Young (1986) is adapted and used to derive an analytical solution for the problem with a Robin boundary condition.

Метод прогнозирования температурного состояния цилиндра при термообработке в условиях неполной исходной информации

Япарова Наталья Михайловна, канд. физ.-мат. наук, зав. кафедрой вычислительной математики и высокопроизводительных вычислений, Южно-Уральский государственный университет, г. Челябинск; iaparovanm@susu.ru. N.M. Yaparova, iaparovanm@susu.ru South Ural State University, Chelyabinsk, Russian Federation

Determination of the heat state in the five-layer hollow sphere at convective and radiative heat exchange with the environment

The exact analytical expressions for the temperature field in a five-layer thermal sensitive hollow sphere using the model of the stationary nonlinear heat conduction with temperature-dependent coefficients of heat conduction and heat exchange has been obtained. The heat sources are present in the second and fourth layers. The inner surface has steady temperature and the outer one is subjected to the convective and radiative heat exchange. The influence of the thermal sensitivity of the material of sphere components on the temperature distribution has been investigated. Cite as: V. O. Horoshko, “Determination of the heat state in the five-layer hollow sphere at convective and radiative heat exchange with the environment,” Prykl. Probl. Mekh. Mat. , Issue 16, 141–146 (2018) (in Ukrainian), http://doi.org/10.15407/10.15407/apmm2018.16.141-146

Determination of the time-dependent reaction coefficient and the heat flux in a nonlinear inverse heat conduction problem

ABSTRACTDiffusion processes with reaction generated by a nonlinear source are commonly encountered in practical applications related to ignition, pyrolysis and polymerization. In such processes, determining the intensity of reaction in time is of crucial importance for control and monitoring purposes. Therefore, this paper is devoted to such an identification problem of determining the time-dependent coefficient of a nonlinear heat source together with the unknown heat flux at an inaccessible boundary of a one-dimensional slab from temperature measurements at two sensor locations in the context of nonlinear transient heat conduction. Local existence and uniqueness results for the inverse coefficient problem are proved when the first three derivatives of the nonlinear source term are Lipschitz continuous functions. Furthermore, the conjugate gradient method (CGM) for separately reconstructing the reaction coefficient and the heat flux is developed. The ill-posedness is overcome by using the discrepancy principle to stop the iteration procedure of CGM when the input data is contaminated with noise. Numerical results show that the inverse solutions are accurate and stable.

A new approach to speed up the conjugate gradient method by applying the complex variable differentiation method

ABSTRACTThis study represents a novel approach to speed up the solution of nonlinear inverse heat conduction problems (IHCPs) by the implementation of the complex variable differentiation method (CVDM). A difficulty appeared in the solution of nonlinear problems is the great amount of computational time. To handle this problem, a new scheme is introduced to improve the conjugate gradient method (CGM). The main contribution in the improvement of conventional CGM is the simultaneous solution of direct and sensitivity problems by applying the complex variable method. The derivation of sensitivity problem by utilizing the analytic derivative is difficult or impossible, since the direct problem is complicated mathematically in many cases. By this approach, the analytic derivation of sensitivity problem is circumvented, while the Jacobian matrix components are obtained accompanied by the solution of the direct problem. Therefore, the developed scheme results in the reduction of mathematical manipulations. Due to the high nonlinearity of moving boundary inverse problems, the ablation problem is considered as the benchmark to examine the accuracy and effectiveness of the proposed scheme. The simulated results illustrate that the developed scheme has the potential to significantly reducing of the computational expenses while maintaining the quality of numerical solution.

Control of the radiative heating of a semi-transparent body

In a preceding paper, we have studied the radiative heating of a semi-transparent body $$\varOmega $$ (e.g., glass) by a black radiative source S surrounding it, black source at absolute uniform temperature u(t) at time t between time 0 and time $$t_\mathrm{f}$$ , the final time of the radiative heating. This problem has been modeled by an appropriate coupling between quasi-steady radiative transfer boundary value problems with nonhomogeneous reflectivity boundary conditions (one for each wavelength band in the semi-transparent electromagnetic spectrum of the glass) and a nonlinear heat conduction evolution equation with a nonlinear Robin boundary condition which takes into account those wavelengths for which the glass behaves like an opaque body. In the present paper, u being considered as the control variable, we want to adjust the absolute temperature distribution $$(x,t) \mapsto T(x,t)$$ inside the semi-transparent body $$\varOmega $$ near a desired temperature distribution $$T_\mathrm{d}(\cdot ,\cdot )$$ during the time interval of radiative heating $$]0,t_\mathrm{f}[$$ by acting on u, the purpose being to deform $$\varOmega $$ to manufacture a new object. In this respect, we introduce the appropriate cost functional and the set of admissible controls $$U_\mathrm{ad}$$ , for which we prove the existence of optimal controls. Introducing the state space and the state equation, a first-order necessary condition for a control $$u:t \mapsto u(t)$$ to be optimal is then derived in the form of a Variational Inequality by using the implicit function theorem and the adjoint problem. We close this paper by some numerical considerations.

A Multifront Problem of Freezing-Thawing Moist Soil

Construction activities in the North of Western Siberia is carried out in specific conditions of the cryolithozone. The study of heat transfer processes in freezing soils near buildings is necessary to ensure their successful operation. Heat exchange with phase transition is traditionally described by Stefan problem, which is a system of differential equations of parabolic type with standard boundary conditions and a subsidiary condition on the phase transition boundary. It is possible to formally pass on to one equation of the conductivity type, but in that case the delta function appears in one of the coefficients. The delta function reflects the Joule heat release at the phase transition temperature. A widespread “pass-through counting” method reduces the Stefan problem to a boundary problem for a nonlinear heat conductivity equation. With this approach, the calculation results in a temperature field. But it is difficult to identify the position of the phase transition boundary on the temperature field. A large number of methods for solving the Stefan problem are developed, in which the required value is the phase transition front coordinate. The common disadvantage of these methods is their unsuitability for situations with several fronts, when these fronts appear and disappear, change the movement direction, merge with each other. The article presents a method for solving the Stefan problem, which allows to obtain the front coordinate as a zero isotherm. As an example this method is used to solve the problem of freezing-thawing of moist soil under the influence of seasonal surface temperature fluctuations. The method eliminates the need to control the evolution of each front. The Stefan problem is considered as a limiting case of the general phase transition problem in a certain temperature range. Standard transformations and application of Green’s function allow to write down a problem in the form of an integral equation. The approximate solution is obtained in the form of a recurrent formula.

A Multifront Problem of Freezing-Thawing Moist Soil

Construction activities in the North of Western Siberia is carried out in specific conditions of the cryolithozone. The study of heat transfer processes in freezing soils near buildings is necessary to ensure their successful operation. Heat exchange with phase transition is traditionally described by Stefan problem, which is a system of differential equations of parabolic type with standard boundary conditions and a subsidiary condition on the phase transition boundary. It is possible to formally pass on to one equation of the conductivity type, but in that case the delta function appears in one of the coefficients. The delta function reflects the Joule heat release at the phase transition temperature. A widespread “pass-through counting” method reduces the Stefan problem to a boundary problem for a nonlinear heat conductivity equation. With this approach, the calculation results in a temperature field. But it is difficult to identify the position of the phase transition boundary on the temperature field. A large number of methods for solving the Stefan problem are developed, in which the required value is the phase transition front coordinate. The common disadvantage of these methods is their unsuitability for situations with several fronts, when these fronts appear and disappear, change the movement direction, merge with each other. The article presents a method for solving the Stefan problem, which allows to obtain the front coordinate as a zero isotherm. As an example this method is used to solve the problem of freezing-thawing of moist soil under the influence of seasonal surface temperature fluctuations. The method eliminates the need to control the evolution of each front. The Stefan problem is considered as a limiting case of the general phase transition problem in a certain temperature range. Standard transformations and application of Green’s function allow to write down a problem in the form of an integral equation. The approximate solution is obtained in the form of a recurrent formula.

Numerical Analysis of a Semi-Infinite Solid with Temperature Dependent Thermal Conductivity using Truly Meshfree Method

This article presents Meshless Local Petrov-Galerkin (MLPG) method to obtain the numerical solution of linear and non-linear heat conduction in a semi-infinite solid object with specific heat flux. Moving least square approximants are used to approximate the unknown function of temperature T(x) with Th(x). These approximants are constructed by using a linear basis, a weight function and a set of non-constant coefficients. Essential boundary condition is imposed by the penalty function method. A predictor-corrector scheme based on direct substitution iteration has been applied to address the non-linearity and two-level  method for temporal discretization. The accuracy of MLPG method is verified by comparing the results for the simplified versions of the present model with the exact solutions. Once the accuracy of MLPG method is established, the method is further extended to investigate the effects of temperature-dependent properties.

A Model of Incompressible-Fluid Flow with a Free Surface in a Highly Porous Medium

A model is proposed for modeling a slow (inertialess) flow of a thin layer of viscous incompressible fluidwith a free surface inside a highly porousmediumabove an impermeable surface. The fluid flow is described by the Brinkman filtration law, on the bottom boundary the Navier slip condition is prescribed. The equation for the fluid layer thickness is obtained, which has the form of a special case of a nonlinear heat conduction equation. Solutions in the form of traveling waves and self-similar solutions describing the fluid spreading are investigated.

Reduced order model analysis method via proper orthogonal decomposition for nonlinear transient heat conductionproblems

A reduced order analysis method is proposed to solve the nonlinear transient heat conduction problems with temperature-dependent conductivity of material by using the proper orthogonal decomposition (POD) model reduction method and the finite element method. Firstly, the POD basis vectors are developed by calculating eigenvectors of an autocorrelation matrix composed of snapshots, which clustered by the given results of obtained from experiments or numerical methods for the linear transient heat transfer problem. Secondly, the finite element discrete scheme of the nonlinear transient heat conduction problem is reduced the order by using the POD basis vectors of the linear problem, and a lower order ordinary differential equation system of the reduced order model of the nonlinear transient heat conduction problem is obtained. Finally, the value of temperature field at each time is obtained by solving the reduced-order ordinary differential equations. Several numerical examples are given to verify the effectiveness of the proposed method. The results show that the proposed method has a good accuracy and the computational efficiency can be improved by 1–2 orders of magnitude.

Numerical investigation on global dynamics for nonlinear stochastic heat conduction via global random attractors theory

Abstract In term of the global random attractors theory, global dynamics of nonlinear stochastic heat conduction driven by multiplicative white noise with a variable coefficient are investigated numerically. It is shown that global 𝒟-bifurcation, secondary global 𝒟-bifurcation and complex dynamical behavior occur in motion of the system with increasing the intensity of linear component in the heat source. Furthermore, the results obtained here indicate that Hasudorff dimension which is relevant to global Lyapunov exponent can be used to describe global dynamics of the associated system qualitatively.

FEATURES OF THE APPLICATION OF THE IQLAB PROGRAM FOR SOLVING THE INVERSE HEAT CONDUCTION PROBLEM FOR CHROMIUM-NICKEL CYLINDRICAL THERMOSONDES

The paper presents an experimental-computational study of the results of using the IQLab program to solve inverse heat conduction problem and restore the surface temperature of cylindrical thermosondes from heat-resistant chromium-nickel alloys while cooling them in liquid media. The purpose of this paper is to verify the correct operation of the IQLab program when restoring the surface temperature of thermosondes with 1-3 thermocouples. The IQLab program is also designed to solve one-dimensional nonlinear direct lines and inverse heat conduction problems with constant initial and boundary conditions specified as a function of time in a tabular form with a constant and variable time step. A finite-difference method is used to solve the heat equation. Experiments were carried out on samples D = 10-50 mm in liquids with different cooling capacities such as aqueous solutions of NaCl and Yukon-E polymer, rapeseed oil and I-20A mineral oil. For the calculation we used the readings of thermocouples installed at internal points of cylindrical thermosondes. The advantages of solving inverse heat conduction problems with the IQLab program include the possibility of restoring the surface temperature for cylindrical samples with a diameter of 10 mm to 50 mm with practical accuracy according to the indications of a single thermocouple located in the geometrical center of the thermosonde, which simplifies the manufacture of the probe. For larger dimensions with a diameter D ≥ 50 mm, it is necessary to install control intermediate thermocouples and perform additional tests. The solution of inverse heat conduction problems and restoration of the surface temperature of the sample makes it possible to calculate other important characteristics of the cooling process: the heat flux density and the heat transfer coefficient.

A new stochastic computing paradigm for the dynamics of nonlinear singular heat conduction model of the human head

Bio-inspired computing approaches are effective to solve a variety of dynamical problems. The strength of these stochastic solvers is exploited for the numerical treatment of a nonlinear heat conduction model of the human head using artificial neural networks (ANNs), genetic algorithms (GAs), active-set technique (AST), and their hybrids. The universal function approximation competencies of unsupervised ANNs are utilized in constructing the mathematical model of the problem by defining an error function in the mean squared sense. The training of the design parameters of ANN models is made with global search brilliance of GAs, viably local search with AST and hybrid approach GA-AST. The results of the proposed schemes are determined in terms of temperature profiles by considering variants of the problem with different Biot numbers, metabolic thermogenesis slope parameters and thermogenesis heat production factors. The correctness, effectiveness and convergence of the proposed approaches are also ascertained through statistics.

An extended approach of a Kalman smoothing technique applied to a transient nonlinear two-dimensional inverse heat conduction problem

An extended Kalman smoother (EKS) introducing the concept of future measurement information is developed to handle nonlinear inverse heat conduction problems. In the present study, the formulation of the EKS algorithm is generalized in the case of a two-dimensional problem in order to reconstruct a non-homogeneous heating condition on the front surface of a cylindrical sample. This leads to apply the new method in a multiple sensor case for the simultaneous reconstruction of several parameters at each time. The analyzed inverse problem is nonlinear due to the variability of the thermophysical properties with temperature and to the presence of radiation boundary conditions. Inverse estimation is successfully performed where temporal and spatial variations of the front surface heat flux are recovered based on non intrusive transient temperature measurements made on the back surface. Numerical experiments show that the use of an optimal number of future data greatly improves the solution of the EKS compared to the extended Kalman filter (EKF) estimates, where a noticeable reduction of time lag and sensitivity to measurement errors is observed. Inversion results show that the optimal number of future data, chosen on the basis of a minimum measure of the heat flux bias, depends on the modeling error, the measurement time step, the distance between the sensors and the front heated surface and the measurement noise level. The proposed algorithm is robust in recovering different heat flux profiles and provides for all the examined patterns symmetric and stable solutions that superposes well with the exact functions. The EKS formulation based on an augmented state vector allows to efficiently recover the front surface temperature simultaneously with the heat flux.

Solution of an inverse transient heat conduction problem in a part of a complex domain

The purpose of this work is to formulate two simple methods which can be used to solve nonlinear inverse heat conduction problems in a part of a complex-shaped component in the on-line mode. The proposed methods can be useful if temperature measurements are carried out only in a selected part of an outsize element or if the whole element cannot be analysed due to the inverse heat conduction problem being ill-conditioned. It is assumed that the conductive heat transfer occurs through the surfaces separating the domain from the rest of the component. It is shown that the simplifying assumption of thermal insulation on these surfaces, which is often presented in literature, can cause significant errors. Compared to works published previously, if two additional unknown boundary conditions are introduced on the separating surfaces, the inverse problem conditioning deteriorates substantially. Despite this, stable solutions are achieved for “noisy measured data”.The presented methods can be used to optimize the power unit start-up and shutdown operation. They may also enable a reduction in the heat loss arising during the process and extend the power unit life. The methods presented herein can be applied in monitoring systems working both in conventional and nuclear power plants.

Thermal stress analysis of a thermosensitive functionally graded rectangular plate due to thermally induced resultant moments

Purpose The purpose of this paper is to determine the temperature distribution of a thin rectangular plate made of thermosensitive functionally graded (FG) material. By finding out thermal deflection and stress resultants, the thermal stresses have been obtained and analyzed. Design/methodology/approach Initially, the rectangular plate is kept at the surrounding temperature. The upper, lower and two parallel sides (y=0, b and z=0, c) are thermally insulated, while other parallel sides (x=0, a) are given convective-type heating, that is, the rate of change of the temperature of the rectangular plate is proportional to the difference between its own temperature and the surrounding temperature. The non-linear heat conduction equation has been converted to linear form by introducing Kirchhoff’s variable transformation and the resultant heat conduction equation is solved by integral transform technique with hyperbolic varying point heat source. Findings A mathematical model is prepared for FG ceramic–metal-based material, in which alumina is selected as the ceramic and nickel as the metal. The thermal deflection and thermal stresses have been obtained for the homogeneous and nonhomogeneous materials. The results are illustrated numerically and depicted graphically for comparison. During this study, one observed that variations are seen in the stresses, due to the variation in the inhomogeneity parameters. Research limitations/implications The paper is constructed purely on theoretical mathematical modeling by considering various parameters and functions. Practical implications This type of theoretical analysis may be useful in high-temperature environments like nuclear components, spacecraft structural members, thermal barrier coatings, etc., as the effect of temperature and evaluation of temperature-dependent and nonhomogeneous material properties plays a vital role for accurate and reliable structural analysis. Originality/value In this paper, the authors have used thermal deflection and resultant stresses to determine the thermal stresses of a thin rectangular plate with temperature- and spatial variable-dependent material properties which is a new and novel contribution to the field.

Numerical solution of multi-dimensional transient nonlinear heat conduction problems with heat sources by an extended element differential method

In this paper, the element differential method is extended to solve a transient nonlinear heat conduction problem with a heat source and temperature-dependent thermophysical properties for the first time. The transient term is discretized by employing a finite difference scheme. An iterative methodology is developed to deal with the nonlinearity caused by temperature-dependent thermophysical properties. Examples of two-dimensional (2D) and three-dimensional (3D) problems are given to validate the present method for solving multi-dimensional transient nonlinear heat conduction problems. The results show that the present EDM provides a promising way that is effective and with high accuracy for solving multi-dimensional transient nonlinear heat conduction problems.

An isogeometric independent coefficients (IGA-IC) reduced order method for accurate and efficient transient nonlinear heat conduction analysis

This article develops an isogeometric independent coefficients (IGA-IC) reduced order method for transient nonlinear heat conduction analysis. Herein, we first exactly represent the geometric model via isogeometric analysis (IGA), and therein provide an accurate solution for the semi-discretized equations. Next, our proposed GSSSS-1 time-stepping framework is employed to solve the transient nonlinear temperature in space and time domains. We advance our independent coefficients (IC) reduced order method to efficiently solve IGA-based transient nonlinear heat conduction problems. We extend the IC method to significantly reduce the original full IGA-discretized formulations and calculate the reduced equilibrium formulations in each Newton–Raphson iteration. Thereby, hugely improving the efficiency and guaranteeing the accuracy simultaneously. Illustrative numerical examples validate this proposed IGA-IC method is reliable, accurate, and efficient; especially, the larger the scale of the problem, the more advantages the proposed IGA-IC will inherit.

An iterative finite-element algorithm for solving two-dimensional nonlinear inverse heat conduction problems

It is often useful to determine temperature and heat flux in multidimensional solid domains of arbitrary shape with inaccessible boundaries. In this study, an effective algorithm for solving boundary inverse heat conduction problems (IHCPs) is implemented: transient temperatures on inaccessible boundaries are estimated from redundant simulated measurements on accessible boundaries. A nonlinear heat equation is considered, where some of the material properties are dependent on temperature. The IHCP is reformulated as an optimization problem. The resulting functional is iteratively minimized using a conjugate gradient method together with an adjoint (dual) problem approach. The associated partial differential equations are solved using the finite-element package FEniCS. Tikhonov regularization is introduced to mitigate the ill-posedness of the IHCP. The accuracy of the implemented algorithm is assessed by comparing the solutions to the IHCP with the correct temperature values, on the inaccessible boundaries. The robustness of our method is tested by adding Gaussian noise to the initial conditions and redundant boundary data in the inverse problem formulation. A mesh independence study is performed.