Nonlinear Heat Conduction Problem Research Articles

Nonstationary Nonlinear Problems of Heat Conduction for a Half Space

We propose a method for the determination of temperature fields in a half space with regard for the thermal radiation, temperature dependence of the thermal characteristics, densities of the surface and volume heat sources in the case of nonuniform distribution of the initial temperature. By using the Kirchhoff transformation, Green function, and linear splines, the corresponding problems of heat conduction are reduced to the solution of a recurrent system of nonlinear algebraic equations. We also present examples of numerical analyses.

Uncoupled Quasistatic Problem of Thermoelasticity for a Two-Layer Hollow Thermally Sensitive Cylinder Under the Conditions of Convective Heat Exchange

We determine the unsteady temperature field depending on the radial coordinate and the thermoelastic state caused in a two-layer hollow thermally sensitive cylinder both by this field and the action of external force loads. The nonlinear boundary-value problem of heat conduction with discontinuous coefficients is reduced by the integro-interpolation method to the Cauchy problem for a system of ordinary differential equations, which is solved numerically. The stressed state is determined from the integral equations of the second kind and integral conditions obtained as a result of direct integration of the problem of quasistatic thermoelasticity in stresses. We study the influence of the temperature dependence of the thermal and mechanical characteristics of the layers of materials on the values and character of distributions of temperature and stresses caused by this dependence in a two-layer cylinder.

Temperature-constrained topology optimization of nonlinear heat conduction problems

Abstract This paper presents topology optimization of nonlinear heat conduction problems with multiple domains and multiple constraints, including regional temperature and material volume for reducing temperature. Maximum approximation temperatures in the constraint regions are accurately and dynamically calculated, though temperature and temperature-dependent thermal conductivity change with the update of material distribution. A temperature measure with constant error to approximate regional maximum temperature is adaptive to different temperature ranges. A strategy of hole nucleation generation combined with the regional temperature constraints is presented for the level set-based topology optimization. The solid isotropic material with penalization (SIMP) and parametrized level set methods are compared for the temperature-constrained topology optimization. Finally, several numerical examples are solved by the SIMP and parametrized level set methods. The results demonstrate that the proposed approach can obtain intricate topological details and reduce regional temperatures for the nonlinear heat conduction problems.

Numerical investigations on model order reduction to SEM based on POD-DEIM to linear/nonlinear heat transfer problems

The motivation in this article is to foster and conduct an in-depth investigation on reduced-order modeling (ROM) techniques via proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM) in conjunction with high-order Legendre spectral element method (SEM) applied for time-dependent linear and nonlinear heat conduction problems. This approach significantly relieves the CPU burden, yet preserves the numerical accuracy of high-order schemes; this is especially pronounced for linear transient problems. For nonlinear cases, we use DEIM to save CPU time for nonlinear counterparts. For this purpose, we first obtain training data by solving the system of full-order models (FOM) using snapshot techniques to construct the POD basis. The well-known generalized single step single solve (GSSSS-1) framework is next employed for the first-order system for the time discretization. Numerical results for both linear and nonlinear heat conduction problems such as in a nuclear reactor, etc., show excellent agreement between FOM solutions and ROM solutions, and the CPU advantages via POD ROM techniques are also prominent for high-order Legendre SEM.

Inverse estimation of hot-wall heat flux using nonlinear artificial neural networks

Compared with cold-wall heat flux measurement, hot-wall heat flux indicates more information from dynamic heat transfer between the aerothermodynamic boundary layer and the true model’s surface; thus, hot-wall heat flux measurement has become predominant in the aerothermodynamic measurement field. In harsh aerothermodynamic in-flight or ground tests, the hot-wall heat flux must be determined from time history temperature measurements at one or more interior locations. Therefore, accounting for the temperature dependence of the thermophysical properties, hot-wall heat flux measurement essentially results in the solution of the nonlinear inverse heat conduction problem (IHCP). In this paper, a novel inverse estimation of hot-wall heat flux using nonlinear artificial neural networks (ANN) is presented. First, motivated by the hybrid method proposed by Clayton A. Pullins, David O. Hubble, and Tom E. Diller et al., [2010], a modified hybrid heat flux measurement method using two in-depth thermocouples is proposed, which avoid to directly measure surface temperature of gauge or model under harsh aerodynamic heating environments; accounting for the unknown temperature dependent thermophysical properties, a new heat flux inverse estimation model of a nonlinear ANN is proposed and identified to approximate the modified hybrid measurement method through calibration experiment. This heat flux inverse estimation method does not need to solve a first kind Volterra integral equation and to obtain the information about the thermophysical properties of heat conduction body, and the thermal inertia, locations of thermocouples. This paper presents xenon lamp calibration and arc-heated wind tunnel experiments that validate the new inverse estimation method combined with the fabricated hot-wall heat flux sensor and its probe. These experimental results show that, in general, the dynamical hot-wall heat flux estimated based on the proposed method agree well with the known calibrated values and the stagnation heat fluxes of the classic slug calorimeter.

Analytical thermal model of orthogonal cutting process for predicting the temperature of the cutting tool with temperature-dependent thermal conductivity

High temperatures generated in cutting processes significantly affect the surface integrity of machined parts and tool wear, leading to workpiece thermal damage, tensile residual stresses in the workpiece and a reduction in tool life. In recent years, different analytical thermal models to predict cutting temperatures have been developed in literature based on 2D modeling of the cutting process and the assumption that thermal conductivities of workpiece and chip are not dependent on temperature. However, this dependence of conductivity on temperature may have a significant influence on predicted temperatures and must be taken into account. In this paper, a thermal model of the orthogonal cutting process that considers thermal conductivity of materials (chip and tool) to be dependent on temperature is developed. A linear variation of thermal conductivity with temperature is assumed for chip (workpiece) and tool materials. The model is based on application of: (1) the Kirchhoff transformation in order to convert the nonlinear heat conduction problem into a linear one, (2) the theory of moving and stationary heat sources in semi-infinite and infinite mediums in order to model primary and secondary deformation zones and (3) imaginary heat sources to meet adiabatic boundary conditions in the chip and tool. Imaginary heat sources were defined in the thermal model proposed in this paper in such a way that the effect of the tool-chip interface dimensions and of cutting tool width on the tool temperature could be taken into account. This allows the temperature on the rake face and lateral faces of the tool to be predicted. To this end, a new methodology that considers the temperature-dependent thermal conductivity of materials was developed in order to estimate heat partition ratio along the secondary heat source (tool-chip interface), which is assumed to be non-uniform. Orthogonal cutting tests were also performed in order to verify model predictions by comparing them to tool temperature distributions measured using an IR camera.

Reduction of plane-radial freezing - thawing problems to plane-parallel ones

Construction in northern conditions is complicated by the processes of seasonal freezing and thawing of soils. This factor is of particular importance in the area of permafrost distribution. The paper deals with the problems that arise when laying communication networks. Any engineering solution must be justified by a heat engineering calculation. In one case, it is necessary to exclude the thawing of water or sewer networks, in the other case – to prevent the destructive effect of the heating main on the permafrost soil. The most common mathematical model of the heat transfer process in the presence of phase transitions is the Stefan problem. When calculating the processes of freezing-thawing near the pipes of communications, it is rational to set the problem in cylindrical coordinates, using the property of plane-radial symmetry. But the Stefan problem has an exact solution only in Cartesian coordinates, for the case of plane-parallel symmetry. For this case, many approximate dependencies are also obtained. The paper presents a method that allows using the results of solving the plane-parallel Stefan problem to solve the plane-radial problem with the same values of the input parameters. Approximate values of the coordinate of the phase transition front as a function of time for the plane-parallel and plane-radial areas are obtained using the method of sequential change of stationary states. A one-to-one relationship is established between the obtained values. If there is a solution obtained by any method for a plane-parallel area, and then using this dependence, it can be applied to a plane-radial area. This technique is suitable not only for the Stefan problem, but also for many nonlinear heat conduction problems. The results given in the paper are of practical importance: using the above method, almost any solution obtained for one type of symmetry directly extends to other types of symmetry.

Combustion Model of a Dual‐Fuel Diesel Engine

AbstractA mathematical model is presented for the volumetric combustion of a homogeneous fuel mixture in compression‐ignition and forced‐ignition engines. With careful consideration of all combustion parameters, the profile of the burning rate has a two‐peak structure. A simple nonlinear heat conduction problem was solved to illustrate the importance of including the isobaric process and heat conduction as functions of the temperature. The study shows that, when the pre‐exponential parameter was chosen as a calibrated parameter, the ignition delay time significantly depends on the start of combustion and the pressure in the common rail, while the effect of the engine speed seems less noticeable. The results can be extended to quasi‐sized combustion models.

Анализ живучести магистрального нефтепровода в зоне стыкового сварного соединения

The article describes the development of a technique for analyzing the survivability of the trunk pipeline in the zone of the transverse weld during operation, taking into account the residual welding stresses. The calculation of residual welding stresses was performed by solving the problem of thermoelasto-viscoplasticity for a material with a non-stationary structure by the finite element method. Solving the nonlinear nonstationary heat conduction problem was carried out by the finite difference method using boundary conditions of the third kind. Modeling the kinetics of transformation of austenite into ferrite-pearlite and bainite under non-isothermal conditions during welding was carried out on the basis of the theory of isokinetic reactions. The calculation of survivability is based on the Irwin criterion and the Paris formula.

Deep learning or interpolation for inverse modelling of heat and fluid flow problems?

PurposeThe purpose of this study is to compare interpolation algorithms and deep neural networks for inverse transfer problems with linear and nonlinear behaviour.Design/methodology/approachA series of runs were conducted for a canonical test problem. These were used as databases or “learning sets” for both interpolation algorithms and deep neural networks. A second set of runs was conducted to test the prediction accuracy of both approaches.FindingsThe results indicate that interpolation algorithms outperform deep neural networks in accuracy for linear heat conduction, while the reverse is true for nonlinear heat conduction problems. For heat convection problems, both methods offer similar levels of accuracy.Originality/valueThis is the first time such a comparison has been made.

Application of the Protective Coating for Blade’s Thermal Protection

This paper presents an algorithm applied for determining temperature distribution inside the gas turbine blade in which the external surface is coated with a protective layer. Inside the cooling channel, there is a porous material enabling heat to be transferred from the entire volume of the channel. This algorithm solves the nonlinear problem of heat conduction with the known: heat transfer coefficient on the external side of the blade surface, the temperature of gas surrounding the blade, coefficients of heat conduction of the protective coating and of the material the blade is made of as well as of the porous material inside the channel, the volumetric heat transfer coefficient for the porous material and the temperature of the air flowing through the porous material. Based on these data, the distribution of material porosity is determined in such a way that the temperature on the boundary between the protective coating and the material the blade is made of is equal to the assumed distribution To. This paper includes results of calculations for various thicknesses of the protective coating and the given constant values of temperature on the boundary between the protective coating and the material the blade is made of.

Radial point interpolation method and high-order continuation for solving nonlinear transient heat conduction problems

Radial point interpolation method and high-order continuation for solving nonlinear transient heat conduction problems

Numerical solution of two-dimensional (2D) nonlinear heat conductivity problem on moving grids

For the numerical solution of two-dimensional (2D) nonlinear heat conduction problem a transition to a moving coordinate system is used. A local speed of the latter is chosen in such a way that the considered process is close to stationary. Such coordinate system guarantees a concentration of grid nodes in a range of the solution features. This allows for a quality improvement of the obtained numerical solution with a small number of computational grid nodes. The developed numerical algorithm is tested on the heat wave propagation problem which has an exact solution.

Topology optimization of transient nonlinear heat conduction using an adaptive parameterized level-set method

ABSTRACT In this article, topology optimization of transient nonlinear heat conduction problems is solved by a parameterized level-set method. Nonlinear density-based and continuum shape sensitivity analyses are conducted for topology generation and shape optimization. Meanwhile, the backward time transient adjoint structures are derived. To suppress dependence of the initial guess for the parameterized level-set-based topology optimization, the solid isotropic material with penalization method is adopted to generate the initial topology design. Then, the parameterized level-set method is further performed to obtain the optimal shape. The material volume is strictly controlled by restricting the coefficients of radial basis functions within a dynamic bound, which can avoid incorrect topological results due to topological changes. Finally, three numerical examples are solved with the parameterized level-set method, which demonstrates that the proposed approach can obtain complex topological details with smooth boundaries for transient nonlinear heat conduction problems.

Effect of Temperature Field on Mechanical Properties of Direct Laser Deposited Ti-6Al-4V Alloy

The mechanical and service properties of Ti-6Al-4V alloy parts produced by direct laser deposition (DLD) depend on the thermal cycle parameters. The temperature field during deposition is significantly affected not only by the parameters of the process, but also by the interpass dwell time and length of the deposited layers. The aim of the article is to establish the relationship between mechanical properties of deposited Ti-6Al-4V samples and DLD thermal cycle parameters. Numerical simulation was used in order to establish relationship between temperature field parameter and the process parameters. The nonlinear three-dimensional heat conduction problem was solved by the finite element method. It is shown that an increase in the dwell time between passes from 5 to 10 seconds leads to a significant decrease in the inter-pass temperature and an increase in the cooling rate. This leads to the metastable structure formation of the deposited layers of Ti-6Al-4V that consists mainly of a nonequilibrium a’-phase which hardness is higher than 390 HV. Without dwell time an equilibrium a + (3 structure with hardness of 360 HV and higher elongation is formed.

A sequential conjugate gradient method to estimate heat flux for nonlinear inverse heat conduction problem

In this paper, a sequential conjugate gradient method is presented to reconstruct the undetermined surface heat flux for nonlinear inverse heat conduction problem (IHCP). This modified method combines the merits of sequential function specification method (SFSM) and conjugate gradient method (CGM). The proposed method can effectively eliminate the leading error which is inevitably generated in SFSM due to the existence of temporary assumption of the unknown parameters and reduce the computing time of nonlinear IHCP. The square wave and sine wave heat flux are used to evaluate the proposed method and the influence of the number of future time steps and measurement errors on the predicted results is studied. The numerical experiments verify the effectiveness and timeliness of the proposed nonlinear IHCP algorism in reconstructing the heat flux. The comparisons with the typical SFSM and CGM are conducted to demonstrate the superiority of the proposed method. The results show that proposed method is more accurate compared with that of the SFSM under the condition of large measurement errors and the computing time is much faster than that of CGM.

A new alternating iteration strategy based on the proper orthogonal decomposition for solving large-scaled transient nonlinear heat conduction problems

In this paper, the application of Free Element Method (FREM) is extended to the transient nonlinear heat conduction problems, and the characteristics of anisotropic, heterogeneous, temperature-dependent thermophysical properties and heat generation are also involved. In order to improve the efficiency during the nonlinear iterations, an alternating iteration strategy, FREM-ROM, is proposed, in which the proper orthogonal decomposition (POD) technique is used to establish a sectionalized extrapolation algorithm with lower dimensions and sufficiently high accuracy. This iteration strategy reduces the computational scale by using the Reduced-Order Model (ROM) technique and can guarantee the accuracy by continually correcting the database and POD bases every few time steps during the process of iterations. It also has the potential to be applied to other numerical methods for improved the efficiency. The accuracy and efficiency are examined by two numerical examples, in which the 2D example illustrates the relationship between truncated basis and accuracy in detail, and the heat transfer performance of a large-scale 3D cross-fin heat sink(CFHS) is investigated to highlight the efficiency advantages of solving large-scaled practical problems.

A new approach to solve multi-medium nonlinear transient heat conduction problems using interface integration BEM

In order to solve multi-medium nonlinear transient heat conduction problems, based on interface integration boundary element method, a new approach, which can convert the multi-domain integrals to the boundary integrals with high precision, is firstly proposed in this paper. The boundary element method with the internal and exterior source is used to establish the interface integral equation of the multi-medium heat conduction problems, and then there are some new multi-domain integrals that should be converted to the boundary integrals. However, different from the domain integral in the single medium problems, the conventional transformation technique cannot be directly applied to convert multi-domain integrals. To maintain the advantage of only boundary discretization for the BEM and take the discontinuity of integrands in multi-domain integrals into consideration, a uniformly continuous function over additional topology domains is defined. Therefore, an improved method is presented to transform the multi-domain integrals to the boundary and interface integrals based on the conventional radial integration method (RIM). What's more, the idea of the proposed approach can theoretically prove the correctness of applying RIM to the complex models with multi-connected domain, which has been only confirmed from some numerical examples. Finally, two numerical examples of multi-medium nonlinear heat conduction problems are shown to demonstrate the correctness and feasibility of the new proposed method.

Sequential particle filter estimation of a time-dependent heat transfer coefficient in a multidimensional nonlinear inverse heat conduction problem

In the applied mathematical modelling of heat transfer systems, the heat transfer coefficient (HTC) is one of the most important parameters. This paper proposes a combination of the Method of Fundamental Solutions (MFS) with particle filter Sequential Importance Resampling (PF-SIR) to estimate the time-dependent HTC in two-dimensional transient inverse heat conduction problems from non-standard boundary integral measurements. These measurements ensure the unique solvability of the boundary coefficient identification problem. Numerical results show high performance on several test cases with both linear and nonlinear Robin boundary conditions, supporting the synergy between the MFS and simulation-based particle filter sequential analysis methods.

An efficient and accurate hybrid weak-form meshless method for transient nonlinear heterogeneous heat conduction problems

Our purpose is to establish a numerical method meeting the requirements of efficiency, stability, accuracy and easy-using. Faced with these demands, in this paper, a hybrid method combining the advantages of 2 weak-form meshless methods is proposed for solving steady and transient heat conduction problems with temperature-dependent thermophysical properties in heterogeneous media. In the process of the calculation for an arbitrary model discretized by the hybrid method, two different weak-form methods are used for the nodes categorized as: (1) boundary nodes, (2) near boundary interior nodes and (3) interior nodes. A Galerkin free element method (GFREM) is proposed for the interior nodes of the Cartesian grid; and the local radial point interpolation method (LRPIM) is used for the other two types of nodes. The GFREM using the Cartesian grid can greatly improve the accuracy and efficiency of the hybrid scheme. Special decomposition technology to determine the irregular integral domain for LRPIM can increase the flexibility for complex models. In addition, the full implicit finite difference scheme and Newton–Raphson method are used for solving transient nonlinear problems. And non-crossing interface integral domain and support domain are employed for heterogeneous media. Compared with other methods, numerical results show better accuracy, stability and efficiency from the numerical experiments.