A new algorithm for direct and backward problems of heat conduction equation

Direct heat conduction problem (DHCP) and backward heat conduction problem (BHCP) are numerically solved by employing a new idea of fictitious time integration method (FTIM). The DHCP needs to consider the stability of numerical integration in the sense that the solution may be divergent for a specific time stepsize and specific spatial stepsize. The BHCP is renowned as strongly ill-posed because the solution does not continuously depend on the given data. In this paper, we transform the original parabolic equation into another parabolic type evolution equation by introducing a fictitious time variable, and adding a fictitious viscous damping coefficient to enhance the stability of numerical integration of the discretized equations by employing a group preserving scheme. When 10 numerical examples are amenable, we find that the FTIM is applicable to both the DHCP and BHCP. Even under seriously noisy initial or final data, the FTIM is also robust against disturbance. More interestingly, when we use the FTIM, we do not need to use different techniques to treat DHCP and BHCP as that usually employed in the conventional numerical methods. It means that the FTIM can unifiedly approach both the DHCP and BHCP, and the gap between direct problems and inverse problems can be smeared out.     

The Method of Fundamental Solutions for Solving the Backward Heat Conduction Problem with Conditioning by a New Post-Conditioner

We consider a backward heat conduction problem (BHCP) in a slab, subject to noisy data at final time. The BHCP is known to be highly ill-posed. In order to stably solve the BHCP by a numerical method, we employ a new post-conditioner in the linear system obtained by the method of fundamental solutions (MFS), and then we use the conjugate gradient method (CGM) to solve the post-conditioned linear system to determine the unknown coefficients used in the expansion by the MFS. The method can retrieve the initial data rather well, with a certain degree of accuracy. Several numerical examples of the BHCP demonstrate that the present method is applicable, even for those of strongly ill-posed problems with a large value of final time and with large noise. We also demonstrate that the CGM alone is not enough to accurately recover the initial temperature.     

A highly accurate LGSM for severely ill-posed BHCP under a large noise on the final time data

We consider a backward heat conduction problem (BHCP) in a slab, by subjecting to data at a final time, and some different type boundary conditions at two ends of the slab. The BHCP is known to be highly ill-posed. In order to numerically solve the BHCP we develop a new Lie-group shooting method (LGSM) in the spatial direction. It can retrieve very well the initial data with a high order accuracy. Several numerical examples of the BHCP demonstrate that the LGSM is applicable, even for those of strongly ill-posed ones with a large value of final time. Under the noisy final data the LGSM is robust against the disturbance. The new method is applicable for a severely ill-posed case with a final data very small in the order of 10^?43, and the noise level is in the order of 10^?1, of which the numerical solution still has an accuracy in the order of 10^?2. The results are rather significant in the computations of BHCP.     

Estimation of front surface temperature and heat flux of a locally heated plate from distributed sensor data on the back surface

We present a new method of solving the three-dimensional inverse heat conduction (3D IHC) problem with the special geometry of a thin sheet. The 3D heat equation is first simplified to a 1D equation through modal expansions. Through a Laplace transform, algebraic relationships are obtained that express the front surface temperature and heat flux in terms of those same thermal quantities on the back surface. We expand the transfer functions as infinite products of simple polynomials using the Hadamard Factorization Theorem. The straightforward inverse Laplace transforms of these simple polynomials lead to relationships for each mode in the time domain. The time domain operations are implemented through iterative procedures to calculate the front surface quantities from the data on the back surface. The iterative procedures require numerical differentiation of noisy sensor data, which is accomplished by the Savitzky–Golay method. To handle the case when part of the back surface is not accessible to sensors, we used the least squares fit to obtain the modal temperature from the sensor data. The results from the proposed method are compared with an analytical solution and with the numerical solution of a 3D heat conduction problem with a constant net heat flux distribution on the front surface.     

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).     

A radial integration boundary element method for solving transient heat conduction problems with heat sources and variable thermal conductivity

A new radial integration boundary element method (RIBEM) for solving transient heat conduction problems with heat sources and variable thermal conductivity is presented in this article. The Green’s function for the Laplace equation is served as the fundamental solution to derive the boundary-domain integral equation. The transient terms are first discretized before applying the weighted residual technique that is different from the previous RIBEM for solving a transient heat conduction problem. Due to the strategy for dealing with the transient terms, temperature, rather than transient terms, is approximated by the radial basis function; this leads to similar mathematical formulations as those in RIBEM for steady heat conduction problems. Therefore, the present method is very easy to code and be implemented, and the strategy enables the assembling process of system equations to be very simple. Another advantage of the new RIBEM is that only 1D boundary line integrals are involved in both 2D and 3D problems. To the best of the authors’ knowledge, it is the first time to completely transform domain integrals to boundary line integrals for a 3D problem. Several 2D and 3D numerical examples are provided to show the effectiveness, accuracy, and potential of the present RIBEM.     

Chebyshev collocation spectral domain decomposition method for coupled conductive and radiative heat transfer in a 3D L-shaped enclosure

ABSTRACT

This paper presents a Chebyshev collocation spectral domain decomposition method (CSDDM) to study the coupled conductive and radiative heat transfer in a 3D L-shaped enclosure. The partitioned 3D L-shaped enclosure is subdivided into rectangular subdomains based on the concept of domain decomposition. The radiative transfer equation is angularly discretized by the discrete ordinate method with the SRAP_N quadrature scheme and then solved by the CSDDM using the same grid system as in solving the conduction. The effects of the conduction–radiation parameter, the optical thickness, the scattering albedo, and the aspect ratio on thermal behavior of the system are investigated. The results indicate that the 3D CSDDM has a good accuracy and can be considered as a good alternative approach for the solution of the coupled conduction and radiation problems in 3D partitioned domains.     

A new meshless “fragile points method” and a local variational iteration method for general transient heat conduction in anisotropic nonhomogeneous media. Part I: Theory and implementation

Abstract

A new and effective computational approach is presented for analyzing transient heat conduction problems. The approach consists of a meshless Fragile Points Method (FPM) being utilized for spatial discretization, and a Local Variational Iteration (LVI) scheme for time discretization. Anisotropy and nonhomogeneity do not give rise to any difficulties in the present implementation. The meshless FPM is based on a Galerkin weak-form formulation and thus leads to symmetric matrices. Local, very simple, polynomial and discontinuous trial and test functions are employed. In the meshless FPM, Interior Penalty Numerical Fluxes are introduced to ensure the consistency of the method. The LVIM in the time domain is a combination of the Variational Iteration Method (VIM) and a collocation method in each finitely large time interval. The present methodology represents a considerable improvement to the current state of science in computational transient heat conduction in anisotropic nonhomogeneous media. In this first part of the two-paper series, we focus on the theoretical formulation and implementation of the proposed methodology. Numerical results and validation are then presented in Part II.     

THREE-DIMENSIONAL TRANSIENT THERMAL STRESS ANALYSIS OF A NONHOMOGENEOUS HOLLOW CIRCULAR CYLINDER DUE TO A MOVING HEAT SOURCE IN THE AXIAL DIRECTION

In this paper, a theoretical analysis of a three-dimensional transient thermal stress problem is developed for a nonhomogeneous hollow circular cylinder due to a moving heat source in the axial direction from the inner and /or outer surfaces. Assuming that the hollow circular cylinder has nonhomogeneous thermal and mechanical material properties in the radial direction, the heat conduction problem and the associated thermoelastic behaviors for such nonhomogeneous medium are developed by introducing the theory of laminated composites as one of theoretical approximation. The transient heat conduction problem is treated with the help of the methods of Fourier cosine transformation and Laplace transformation, and the associated thermoelastic field is analyzed making use of the thermoelastic displacement potential, Michell's function, and the Boussinesq's function. Some numerical results for the temperature change and the stress distributions are shown in figures, and the effect of relaxing the thermal stress in the nonhomogeneous hollow circular cylinder and the influence of the velocity of a moving heat source are briefly discussed     

Asymmetric mechanical and thermal stresses in 2D-FGPPMs hollow cylinder

The general solution of steady-state asymmetric mechanical and thermal stresses of a hollow thick cylinder made of fluid-saturated functionally graded porous piezoelectric materials based on two-dimensional equations of thermoelasticity is considered. The general form of thermal and mechanical boundary conditions is considered on the inside and outside surfaces. A direct method is used to solve the heat conduction equation and the nonhomogeneous system of partial differential Navier equations, using the complex Fourier series and the power law functions method. The material properties are assumed to depend on the radial and circumferential variables and are expressed as power law functions along the radial and circumferential direction.     

Study on heat conduction of functionally graded plate with the variable gradient parameters under the H(t) heat source

Abstract

A hybrid numerical method for heat conduction of functionally graded plate with the variable gradient parameters under the H(t) heat source was studied. A weighted residual equation of heat conduction was considered under thermal boundary conditions. In order to calculate temperature distribution of functionally graded plate with variable gradient parameters, the Fourier transform and inverse Fourier transform were applied and the temperature field was obtained under the H(t) heat source. Results show that the influences of the gradient parameters on temperature distribution are dramatic. But with the increase of gradient parameters, the influences of parameters on the temperature distribution are gradually reduced. When the gradient parameters reach a certain critical value, the temperature does not change anymore. By comparing the temperature distribution of the upper and lower surfaces, it is seen that the temperature presents a gentle downward trend with the increase of the heat source distance, while the temperature does not change with the time in farther distance from heat source. Also, the results show that the influence of the heat source has only partial and limited influence on the temperature, which is in accordance with St. Venant’s Principle. The law of the temperature distribution of the lower surface varies with the gradient parameters, which is also discussed, an optimal gradient parameter with the thermal insulation effect of the functionally graded plate is obtained.     

Non-Fourier heat conduction in a finite medium subjected to arbitrary periodic surface disturbance

The non-Fourier transient heat conduction in a finite medium under arbitrary periodic surface thermal disturbance is investigated analytically. In order to obtain the desired temperature field from the known solution for non-Fourier heat conduction under a harmonic disturbance, the principle of superposition along with the Fourier series representation of an arbitrary periodic function is employed. The developed method can be applied for more realistic periodic boundary conditions occurred in nature and technology.     

Optimal Boundary Control of Thermal Stresses in a Plate Based on Time-Fractional Heat Conduction Equation

This article presents an optimal control problem for a fractional heat conduction equation that describes a temperature field. The main purpose of the research was to find the boundary temperature that takes the thermal stress under control. The fractional derivative is defined in terms of the Caputo operator. The Laplace and finite Fourier sine transforms were applied to obtain the exact solution. Linear approximation is used to get the numerical results. The dependence of the solution on the order of fractional derivative and on the nondimensional time is analyzed.     

A new meshless “fragile points method” and a local variational iteration method for general transient heat conduction in anisotropic nonhomogeneous media. Part II: Validation and discussion

Abstract

In the first part of this two-paper series, a new computational approach is presented for analyzing transient heat conduction problems in anisotropic nonhomogeneous media. The approach consists of a truly meshless Fragile Points Method (FPM) being utilized for spatial discretization, and a Local Variational Iteration (LVI) scheme for time discretization. In the present article, extensive numerical results are provided as validations, followed by a discussion on the recommended computational parameters. The FPM?+?LVIM approach shows its capability in solving 2?D and 3?D transient heat transfer problems in complex geometries with mixed boundary conditions, including preexisting cracks. Both functionally graded materials and composite materials are considered. It is shown that, with appropriate computational parameters, the FPM?+?LVIM approach is not only accurate, but also efficient, and has reliable stability under relatively large time intervals.     

Numerically solving twofold ill-posed inverse problems of heat equation by the adjoint Trefftz method

The inverse problem endowing with multiple unknown functions gradually becomes an important topic in the field of numerical heat transfer, and one fundamental problem is how to use limited minimal data to solve the inverse problem. With this in mind, in the present article we search the solution of a general inverse heat conduction problem when two boundary data on the space-time boundary are missing and recover two unknown temperature functions with the help of a few extra measurements of temperature data polluted by random noise. This twofold ill-posed inverse heat conduction problem is more difficult than the backward heat conduction problem and the sideways heat conduction problem, both with one unknown function to be recovered. Based on a stable adjoint Trefftz method, we develop a global boundary integral equation method, which together with the compatibility conditions and some measured data can be used to retrieve two unknown temperature functions. Several numerical examples demonstrate that the present method is effective and stable, even for those of strongly ill-posed ones under quite large noises.     

3-Omega Method for Thin Films with the Non-Fourier Boundary Condition

The effect of non-Fourier boundary condition on the 3-omega method for measuring the thermal conductivity of microscale thin films using the hyperbolic heat conduction equation and the Fourier equation is examined. Non-Fourier boundary condition with the Fourier equation leads to 80% error in the temperature oscillations and increases the error to 85% in the case of non-Fourier boundary condition with the hyperbolic heat conduction equation. The solution of the non-Fourier boundary condition with the hyperbolic heat conduction equation gives the most accurate thermal conductivity expression. The analysis also provides a method for determining the relaxation time of thin films.     

四边对流换热的内含热源各向异性矩形域稳态热传导解析

应用复级数方法给出的含热源各向异性矩形域稳态传导解析解。结合引入板角流角点条件,首次解析分析了四边与流体换热的含热源各向异性矩形域温度场。讨论了铺设角,各向异性程度,边界对流换热系数及跨宽对比温度场分布的影响。     

Memory response on thermal wave propagation emanating from a cavity in an unbounded elastic solid

The present paper deals with the thermal memory response of wave propagation in an unbounded, homogeneous, isotropic elastic body, emanating from a spherical cavity. To analyze the memory response, the generalized heat conduction model with the fractional order as well as memory-dependent-derivatives (MDDs) concepts are considered. The solution space is obtained in Laplace transform domain by using the eigenfunction expansion method to the vector-matrix form of the corresponding governing equations. Finally, a comparison study is furnished for thermal displacement, stresses, and temperature changes in the space-time domain and is presented graphically.     

Hyperbolic model of thermal interactions in a system biological tissue—protective clothing subjected to an external heat source

Abstract

The numerical modeling of thermal processes in domain of biological tissue (the male thigh) secured by multilayered protective clothing being in the thermal contact with the environment is discussed. The thigh is treated as the nonhomogeneous domain in which the sub-domains of skin tissue, fat, muscle, bone and blood vessels are distinguished. Between the protective clothing and skin tissue the air gap is taken into account. The heat transfer is described by the system of hyperbolic Cattaneo–Vernotte equations (for the tissue sub-domains) and parabolic Fourier equations (for the remaining sub-domains). The process of external heating is determined by the appropriate boundary condition and the internal heat source (in the fabric sub-domain) related to the absorption of incident thermal radiation. The mathematical model is solved numerically using the control volume method, while the considered sub-domains (the 2?D problem) are covered by the Voronoi meshes. In the final part of the article, the example of computations is presented.     

TRANSIENT THERMOELASTIC ANALYSIS OF AN ANNULAR FIN WITH COUPLING EFFECT AND VARIABLE HEAT TRANSFER COEFFICIENT

A hybrid numerical method of the Laplace transformation and the finite difference method is applied to solve the transient thermoelastic problem of an annular fin, in which the thermomechanical coupling effect is taken into account in the governing equation of heat conduction and the heat transfer coefficient is a function of the radius of the fin. The general solutions of the governing equations are first solved in the transform domain. Then the inversion to the real domain is completed via the method of matrix similarity transformation and Fourier series technique. The transient distributions of temperature increment and thermal stresses of the fin in the real domain are calculated numerically. The presented method is more efficient in computing time and is applicable to other types of boundary conditions.