Applying the Hopfield Neural Network to the Solution of the Temperature Distribution Field in Heat Conduction Problems

This article employs the continuous-time analog Hopfield neural network (CHNN) to compute the temperature distribution in one- and two-dimensional transient heat conduction problems. The relationship between the CHNN synaptic connection weights and the governing equations of the problems is established and a corresponding network connectivity circuit design scheme proposed. The CHNN algorithm is initially applied to the solution of conventional problems and is then used to solve more complicated problems involving time-varying heat flux profiles. The results confirm that the CHNN scheme provides an accurate means of solving the transient temperature distributions of heat conduction problems on a real-time basis.     

TRANSIENT LINEAR AND NONLINEAR HEAT CONDUCTION IN WEAKLY MODULATED DOMAINS

Heat conduction in two-dimensional domains with spatially periodic boundary is addressed in this study. The periodic modulation is assumed to be weak, but is of arbitrary shape. A regular perturbation approach is implemented to determine the temperature and heat flux throughout the domain. It is observed that the validity of the perturbation approach extends to include geometries of practical importance. Transient linear as well as steady nonlinear heat conduction problems are examined. The periodic domain is mapped onto the rectangular domain. For both steady and transient linear heat conduction, a fully analytical spectral solution becomes possible. The nonlinear problem is shown to reduce to a set of ordinary differential equations of the two-point-boundary-value type, which is solved using a variable-step-size finite-difference scheme. The perturbation approach is validated upon comparison with conventional methods; excellent agreement is obtained against the boundary- and finite-element methods.     

Integral transform solution of integro-differential equations in conduction-radiation problems

A new approach for solving nonlinear integro-differential equations in conductive-radiative heat transfer has been developed. The method relies on eigenfunctions expansions for the unknown potentials, following the hybrid analytical-numerical framework provided by the generalized integral transform technique. The problem of conjugated conduction–radiation in a finned-tube radiator is selected for illustrating the method, and a traditional numerical solution of the problem is performed for comparing the proposed approach. A thorough error analysis demonstrates that the proposed scheme is very effective for handling integro-differential problems. Finally, a parametric analysis is provided, demonstrating the effects of the dimensionless groups in the temperature distribution.     

Inverse shape design for heat conduction problems via the ball spine algorithm

ABSTRACT

In this article, a novel iterative physical-based method is introduced for solving inverse heat conduction problems. The method extends the ball spine algorithm concept, originally developed for inverse fluid flow problems, to inverse heat conduction problems by employing a subtle physical-sense rule. The inverse problem is described as a heat source embedded within a solid medium with known temperature distribution. The object is to find a body configuration satisfying a prescribed heat flux originated from a heat source along the outer surface. Performance of the proposed method is evaluated by solving many 2-D inverse heat conduction problems in which known heat flux distribution along the unknown surface is directly related to the Biot number and surface temperature distribution arbitrarily determined by the user. Results show that the proposed method has a truly low computational cost accompanied with a high convergence rate.     

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.     

COMPUTATION OF TURBULENT NATURAL CONVECTION IN A RECTANGULAR CAVITY WITH THE k–ϵ– –f MODEL

A hybrid numerical method involving the Laplace transform technique and finite-difference method in conjunction with the least-squares method and actual experimental temperature data inside the test material is proposed to estimate the unknown surface conditions of inverse heat conduction problems with the temperature-dependent thermal conductivity and heat capacity. The nonlinear terms in the differential equations are linearized using the Taylor series approximation. In this study, the functional form of the surface conditions is unknown a priori and is assumed to be a function of time before performing the inverse calculation. In addition, the whole time domain is divided into several analysis subtime intervals and then the unknown estimates on each subtime interval can be predicted. In order to show the accuracy and validity of the present inverse scheme, a comparison among the present estimates, direct solution, and actual experimental temperature data is made. The effects of the measurement errors, initial guesses, and measurement location on the estimated results are also investigated. The results show that good estimation of the surface conditions can be obtained from the present inverse scheme in conjunction with knowledge of temperature recordings inside the test material.     

Application of the variational iteration method to nonlinear non‐Fourier conduction heat transfer equation with variable coefficient

In this paper, a variational iteration method (VIM) has been applied to nonlinear non‐Fourier conduction heat transfer equation with variable specific heat coefficient. The concept of the variational iteration method is introduced briefly for applying this method for problem solving. The proposed iterative scheme finds the solution without any discretization, linearization, or restrictive assumptions. The results of VIM as an analytical solution are then compared with those derived from the established numerical solution obtained by the fourth order Runge–Kutta method in order to verify the accuracy of the proposed method. The results reveal that the VIM is very effective and convenient in predicting the solution of such problems, and it is predicted that VIM can find a wide application in new engineering problems. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20362     

A semi-analytical boundary collocation solver for the inverse Cauchy problems in heat conduction under 3D FGMs with heat source

Abstract

In this article, the inverse Cauchy problems in heat conduction under 3D functionally graded materials (FGMs) with heat source are solved by using a semi-analytical boundary collocation solver. In the present semi-analytical solver, the combined boundary particle method and regularization technique is employed to deal with ill-pose inverse Cauchy problems. The domain mapping method and variable transformation are introduced to derive the high-order general solutions satisfying the heat conduction equation of 3D FGMs. Thanks to these derived high-order general solutions, the proposed scheme can only require the boundary discretization to recover the solutions of the heat conduction equations with a heat source. The regularization technique is used to eliminate the effect of the noisy measurement data on the accessible boundary surface of 3D FGMs. The efficiency of the proposed solver for inverse Cauchy problems is verified under several typical benchmark examples related to 3D FGM with specific spatial variations (quadratic, exponential and trigonometric functions).     

Temperature-Constrained Topology Optimization of Transient Heat Conduction Problems

A regional temperature measure model is constructed to obtain a small number of temperature constraints for local transient temperature control. The temperature sensitivity is derived using the adjoint variable method. The multiple temperature criteria and three-phase topology optimization are further investigated for transient heat conduction design. The material layout design of transient heat conduction is replaced by a static optimization problem, which is subsequently solved by the method of moving asymptotes. Finally, several numerical examples are provided to demonstrate the feasibility and validity of the proposed topology optimization for transient heat conduction problems.     

Numerical solution of three-dimensional backward heat conduction problems by the time evolution method of fundamental solutions

The time evolution method of fundamental solutions (MFS) is proposed to solve three-dimensional backward heat conduction problems (BHCPs). The time evolution MFS is obtained through the linear superposition of diffusion fundamental solutions. Through a correct treatment of temporal evolution, the MFS can be implemented to solve strongly ill-posed problems. The numerical results demonstrate the accuracy and stability of the MFS for three-dimensional BHCPs with high levels of noise. This represents the first implementation of MFS to solve three-dimensional BHCPs, and demonstrates that time evolution MFS is a stable and powerful numerical scheme which has the potential to significantly improve the solution of three-dimensional backward heat conduction problems.     

An integral equation approach to diffusion

A method of solution of transient diffusion, e.g. heat conduction, problems in homogeneous and isotropic media with internal sources and arbitrary (including nonlinear) boundary conditions and initial conditions is proposed. The method is based on the reduction of the problem to one only involving surface values of temperature and/or heat flux in the form of an integral equation through the introduction of fundamental solutions and the use of Green's theorem. The integral equation is solved numerically for a specific example.     

Level Set-based Topological Shape Optimization of Nonlinear Heat Conduction Problems

A level set-based topological shape optimization method is developed for nonlinear heat conduction problems. While minimizing the objective function of instantaneous thermal compliance and satisfying the constraint of allowable volume, solution of the Hamilton-Jacobi equation leads the initial boundary to an optimal one according to the normal velocity field determined from the descent direction of the Lagrangian. To overcome the convergence difficulty in nonlinear problems resulting from introduction of an approximate boundary, an actual boundary is identified by tracking the level set functions and remeshing using Delaunay triangulation. The velocity field outside the actual domain is determined through a velocity extension scheme.     

Immersed boundary method for the simulation of flows with heat transfer

In the present paper a direct heat source scheme is proposed to let the temperature at the immersed boundary satisfy the temperature Dirichlet boundary condition. And the explicit interactive process of the direct heat source scheme called multi-direct heat source scheme is applied to ensure the satisfaction of the temperature Dirichlet boundary condition at the immersed boundary. The second-order spacial accuracy of the solver is confirmed by simulating the Taylor–Green vortices. The simulations of natural convection between concentric cylinders, and flow past a stationary circular cylinder are conducted to validate the accuracy of present method on solving heat transfer problems. And the computation of flow past a staggered tube bank with heat transfer is conducted to verify the capability of present method on solving complex geometries problems.     

AN EXPLICIT TVD SCHEME FOR HYPERBOLIC HEAT CONDUCTION IN COMPLEX GEOMETRY

Two-dimensional hyperbolic heat conduction problems of complex geometry are investigated numerically. A second-order total variation diminishing (TVD) scheme is introduced and its application to the hyperbolic heat conduction is developed in detail using the knowledge of characteristics. In current work primitive variables, rather than characteristic variables, are used as the dependent variables. The governing equations of two-dimensional heat conduction are transformed from the physical coordinates to the computational coordinates, so that the hyperbolic heat conduction problems of irregular geometry can be solved numerically by the present TVD scheme. Three examples with different geometry are used to verify the accuracy of the present numerical scheme. Results show the explicit TVD scheme can predict the thermal wave without oscillation.     

Estimation of heat flux imposed on the rake face of a cutting tool: A nonlinear,complex geometry inverse heat conduction case study

In this paper the sequential function specification method is used to estimate the transient heat flux imposed on the rake face of a cutting tool during the cutting operation with two different assumptions. In one of them the thermal conductivity is taken to be constant, and in the other one it varies with temperature. The cutting tool is modeled as a three dimensional object. The capabilities of the geometric modeling, mesh generation as well as solver of the commercial software ANSYS are utilized in order to reduce the time expended for modeling and direct heat conduction solution, in both linear and nonlinear problems. This way the inverse heat conduction algorithm employs ANSYS as a subprogram through the ANSYS Parametric Design Language (APDL). The stability as well as accuracy is compared for cases of linear and nonlinear heat conductions. The effect of nonlinearity, as well as different sensor locations is investigated in order to arrive at an optimal experimental procedure. Finally, a typical temperature data during the working condition are used to recover the heat flux at the cutting tool surface using linear as well as nonlinear solutions.     

Two-dimensional transient conduction and radiation heat transfer with temperature dependent thermal conductivity

This paper deals with the effect of the temperature dependent thermal conductivity on transient conduction and radiation heat transfer in a 2-D rectangular enclosure containing an absorbing, emitting and scattering medium. The thermal conductivity of the medium is assumed to vary linearly with temperature. The radiative part of the energy equation was solved using the collapsed dimension method. To facilitate solution of the energy equation, which is a highly nonlinear one, time linearization was done first and then the equation was solved using the alternating direction implicit scheme. Results for the effects of the variable thermal conductivity were found for temperature and heat flux distributions.     

Numerical solution of hyperbolic heat conduction in thin surface layers

The purpose of the present paper is to propose a new hybrid method investigating the effect of the surface curvature of a solid body on hyperbolic heat conduction. The difficulty encountered in the numerical solutions of hyperbolic heat conduction problems is the numerical oscillation in vicinity of sharp discontinuities. In the present study, we have developed a new hybrid method combined the Laplace transform, the weighting function scheme [Shong-leih Lee, Weighting function scheme and its application on multidimensional conservation equations, Int. J. Heat Mass Transfer 32 (1989) 2065–2073], and the hyperbolic shape function for solving time dependent hyperbolic heat conduction equation with a conservation term. Four different examples have been analyzed by the present method. It is found from these examples that the present method is in good agreement in the analytical solutions [Tsai-tse Kao, Non-Fourier heat conduction in thin surface layers, J. Heat Transfer 99 (May) (1977) 343–345] and does not exhibit numerical oscillations at the wave front and the surface temperature is modified by the surface curvature during the short period when the non-Fourier effect is significant. The curvature will increase or decrease the temperature of the wave front, depending on whether the surface is concave or convex.     

LATTICE BOLTZMANN SCHEME FOR HYPERBOLIC HEAT CONDUCTION EQUATION

An extended lattice Boltzmann (LB) equation, the lattice Boltzmann equation with a source term, is developed for the system of equations governing the hyperbolic heat conduction equation. Mathematical consistence between the proposed extended LB equation and the governing equations are accomplished by the Chapman-Enskog expansion. Four illustrative examples, with both finite and semi-infinite computational domains and subjected to linear and nonlinear boundary conditions, are simulated. All numerical predications agree very well with the existing solutions in the literature. It is also demonstrated that the present scheme is stable and free of numerical oscillations especially around the wave front, where sharp change in temperature occurs.     

An axisymmetric bioheat transfer analysis through a unified model

A unified model is developed for the analysis of heat transfer (radiation and non-Fourier conduction) in an axisymmetric participating medium. The proposed model includes three different variants of hyperbolic–parabolic heat conduction models, that is, the single phase lag model, dual phase lag model, and the Fourier (no phase lag) model. The radiating-conducting medium is radiatively absorbing, emitting, and isotropically scattering. Significance of all the above mentioned models on the heat transfer characteristics is investigated in a two-dimensional axisymmetric geometry. The equation of transfer and the coupled non-Fourier conduction-radiation equation are solved via finite volume method. A fully implicit scheme is used to resolve the transient terms in the energy equation. For spatial resolution of radiation information, the STEP scheme is applied. Tri-diagonal-matrix-algorithm is used to solve the resulting set of linear discrete equations. Effects of two important influencing parameters: the scattering albedo and the radiation- conduction parameter are studied on the temporal evolution of temperature field in the radiatively participating medium. The non-Fourier effect of heat transport captured well with the proposed unified model. A good agreement can be found between the proposed model predictions and those available in the literature. It is also found that when the phase lag of the temperature gradient and the heat flux are the same, it reduces to conventional Fourier conduction-radiation and the wave behavior diminishes. However, the reduction to this Fourier model fails in the presence of constant blood perfusion and metabolic heat generation.     

Adaptive finite element analysis of microwave driven convection

This study describes an adaptive finite element methodology for heat transfer by convection applied to microwave heating of liquids. This is the first attempt to model such type of problems employing the concepts of error estimation and mesh adaptivity. The proposed methodology is generic and can be applied to steady-state, transient, linear and nonlinear problems involving heat transfer by conduction and convection. There was very good agreement between simulation and experimental results.