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The purpose of this paper is to examine the transient heat conduction in a two-dimensional anisotropic substrate coated with a thin layer of thermal barrier coating (TBC). Nowadays, materials with anisotropic properties have been extensively applied in various engineering applications for enhanced strength. However, under an extreme operating environment of high temperature, the strength of the materials may largely decline. As a common practice in engineering, TBC are usually applied to thermally insulate the substrates so as to allow for higher operating temperature. This research provides engineers a numerical approach for properly designing the TBC to protect the anisotropic substrate.

For this investigation, a finite difference scheme using the domain mapping technique, transforming the anisotropic domain into isotropic one, is employed. The analysis considers three respective boundary conditions, namely Dirichelete condition, Neumann condition, and also forced convection, and studies the effect of various variables on the heat conduction in the coated system. Additionally, formulas for the steady-state temperature drop across the coating layer at the center are analytically derived. By comparing the numerical results with the analytical solutions, the veracity of the formulas is verified.

A few interesting phenomena are observed from the numerical results. First, the rotation of the substrate's principal axes affects the temperature on the TBC front surface in a more obvious manner for the Neumann condition than that for convection. Second, the temperature profile of the Dirichelete condition rises faster than the other cases, although all their profiles present a similar pattern. Third, the transient temperature drop across the TBC under the convection condition presents a complicated pattern, depending on the TBC thickness. Finally, the increase of TBC thickness under the Dirichelete condition may provide better insulation than the other cases. In this paper, approximate analytical formulations for the steady-state temperature drop across the TBC are also presented. Numerical results by the finite difference method indicate excellent agreements with the analytical solutions.

In the past, the finite element method (FEM) is usually applied for analyzing the heat conduction problem of TBC. However, one serious deficiency of applying the FEM to the TBC problem lies in the demand for a vast amount of elements (or cells) when the TBC thickness is far smaller than the substrate dimension. For ultra-thin coating, an enormous amount of elements are required that may lead to an extremely heavy computational burden. The paper presents an innovative finite difference approach that can be applied to analyze the heat conduction across the TBC coated on an anisotropic substrate. On the interface between the TBC and the substrate, a special heat equilibrium condition and the compatibility condition of identical temperature on the adjacent materials are used to propose three new models to predict the temperature drop across the TBC.

A numerical formulation for solving homogeneous anisotropic heat conduction problems based on the use of an isotropic fundamental solution is presented in detail. The analysis is carried out assuming a generic position of the coordinate axes, which may not coincide with the principal directions of orthotropy of the material. The two primary integral equations of the method are derived from the governing differential equation of the problem. Then, the numerical procedure is developed by rewriting the internal degrees of freedom that arise from the domain discretization in terms of the boundary nodes and solving the resulting system of linear equations for the boundary unknowns only. Special attention is given to the differentiation of singular integrals which yields additional terms as well as to the evaluation of the resulting Cauchy principal value integral. The main feature of the proposed formulation is its generality, which makes possible its direct extension to solve the problem of three‐dimensional heat conduction in anisotropic media and, foremost, to three‐dimensional orthotropic and anisotropic elasticity or elastoplasticity.

In this paper, various regularization methods are numerically implemented using the boundary element method (BEM) in order to solve the Cauchy steady‐state heat conduction problem in an anisotropic medium. The convergence and the stability of the numerical methods are investigated and compared. The numerical results obtained confirm that stable numerical results can be obtained by various regularization methods, but if high accuracy is required for the temperature, or if the heat flux is also required, then care must be taken when choosing the regularization method since the numerical results are substantially improved by choosing the appropriate method.

Non‐linear reaction‐diffusion processes with cross‐diffusion in two‐dimensional, anisotropic media are analyzed by means of an implicit, iterative, time‐linearized approximate factorization technique as functions of the anisotropy of the heat and species diffusivity tensors, the Soret and Dufour cross‐diffusion effects, and five types of boundary conditions. It is shown that anisotropy and cross‐diffusion deform the reaction front and affect the front velocity, and the magnitude of these effects increases as the magnitude of the off‐diagonal components of the heat and species diffusivity tensors is increased. It is also shown that the five types of boundary conditions employed in this study produce similar results except when there is either strong anisotropy in the species or heat diffusivity tensors and there are no Soret and Dufour effects, or the species and heat diffusivity tensors are isotropic, but the anisotropy of the Soret and Dufour effects is important. If the species and heat diffusivity tensors are isotropic, the effects of either the Soret or the Dufour cross‐diffusion effects are small for the cases considered in this study. The time required to achieve steady state depends on the anisotropy of the heat and diffusivity tensors, the cross‐diffusion effects, and the boundary conditions.

The purpose of this paper is to depict the effect of thermal and diffusion phase-lags on plane waves propagating in thermoelastic diffusion medium with different material symmetry. A generalized form of mass diffusion equation is introduced instead of classical Fick's diffusion theory by using two diffusion phase-lags, one phase-lag of diffusing mass flux vector, represents the delayed time required for the diffusion of the mass flux and the other phase-lag of chemical potential, represents the delayed time required for the establishment of the potential gradient. The basic equations for the anisotropic thermoelastic diffusion medium in the context of dual-phase-lag heat transfer (DPLT) and dual-phase-lag diffusion (DPLD) models are presented. The governing equations for transversely isotropic and isotropic case are also reduced. The different characteristics of waves like phase velocity, attenuation coefficient, specific loss and penetration depth are computed numerically. Numerically computed results are depicted graphically for anisotropic, transversely isotropic and isotropic medium. The effect of diffusion and thermal phase-lags are shown on the different characteristic of waves. Some particular cases of result are also deduced from the present investigation.

The governing equations of thermoelastic diffusion are presented using DPLT model and a new model of DPLD. Effect of phase-lags of thermal and diffusion is presented on different characteristic of waves.

The effect of diffusion and thermal phase-lags on the different characteristic of waves is appreciable. Also the use of diffusion phase-lags in the equation of mass diffusion gives a more realistic model of thermoelastic diffusion media as it allows a delayed response between the relative mass flux vector and the potential gradient.

Introduction of a new model of DPLD in the equation of mass diffusion.

The purpose of this paper is to present a micromechanical model based on a new truly local meshless method for analysis of heat transfer in composite materials.

The presented meshless method is based on the integral form of energy equation in the sub‐particles in the material. In the presented meshless method due to elimination of domain integration the computational efforts are decreased substantially.

Numerical results are presented for temperature distribution, heat flux and thermal conductivity. Numerical results show that the presented meshless method is simple, effective, accurate and less costly method in micromechanical modeling of heat conduction in heterogeneous materials.

A small area of the composite system called representative volume element is considered as the solution domain. The fully bonded fiber‐matrix interface is considered and contact thermal resistant is neglected from the fiber matrix interface and so the continuity of temperature and reciprocity of heat flux is satisfied in the fiber‐matrix interface.

For the first time a new truly local meshless method based on the integral form of energy equation for the sub‐particles in the materials is presented for analysis of heat transfer in composite materials.

The purpose of this paper is to develop a hybrid‐Trefftz (HT) finite element model (FEM) for simulating heat conduction in nonlinear functionally graded materials (FGMs) which can effectively handle continuously varying properties within an element.

In the proposed model, a T‐complete set of homogeneous solutions is first derived and used to represent the intra‐element temperature fields. As a result, the graded properties of the FGMs are naturally reflected by using the newly developed Trefftz functions (T‐complete functions in some literature) to model the intra‐element fields. The derivation of the Trefftz functions is carried out by means of the well‐known Kirchhoff transformation in conjunction with various variable transformations.

The study shows that, in contrast to the conventional FEM, the HT‐FEM is an accurate numerical scheme for FGMs in terms of the number of unknowns and is insensitive to mesh distortion. The method also performs very well in terms of numerical accuracy and can converge to the analytical solution when the number of elements is increased.

The value of this paper is twofold: a T‐complete set of homogeneous solutions for nonlinear FMGs has been derived and used to represent the intra‐element temperature; and the corresponding variational functional and the associated algorithm has been constructed.

3D steady heat conduction analysis considering heat source is conducted on the fundamental of the fast multipole method (FMM)-accelerated line integration boundary element method (LIBEM).

Due to considering the heat source, domain integral is generated in the traditional heat conduction boundary integral equation (BIE), which will counteract the well-known merit of the BEM, namely, boundary-only discretization. To avoid volume discretization, the enhanced BEM, the LIBEM with dimension reduction property is introduced to transfer the domain integral into line integrals. Besides, owing to the unsatisfactory performance of the LIBEM when it comes to large-scale structures requiring massive computation, the FMM-accelerated LIBEM (FM-LIBEM) is proposed to improve the computation efficiency further.

Assuming N and M are the numbers of nodes and integral lines, respectively, the FM-LIBEM can reduce the time complexity from O(NM) to about O(N+ M), and a full discussion and verification of the advantage are done based on numerical examples under heat conduction.

(1) The LIBEM is applied to 3D heat conduction analysis with heat source. (2) The domain integrals can be transformed into boundary integrals with straight line integrals by the LIM. (3) A FM-LIBEM is proposed and can reduce the time complexity from O(NM) to O(N+ M). (4) The FM-LIBEM with high computational efficiency is exerted to solve 3D heat conduction analysis with heat source in massive computation successfully.

This paper presents a computationally efficient numerical solution scheme to solve transient heat conduction problems using the boundary element method (BEM) without volume discretization. Traditionally, a transient solution using BEM is very computer intensive due to the excessive numerical integration requirements at each time increment. In the present work a numerical solution scheme based on the separation of time and space integrals in the boundary integral equation through the use of an appropriate series expansion of the integrand (incomplete gamma function) is presented. The space integrals are evaluated only once in the beginning and within each time increment the additional integrals are obtained from the previously evaluated space integrals by a simple calculation. Three‐dimensional applications are provided to compare the proposed strategy with that used traditionally. The CPU requirements are reduced substantially. The solution scheme presented allows for dynamically changing the time step size as the solution evolves. This feature is not practical in the traditional schemes based on boundary discretization only.

The purpose of this paper is to present an efficient algorithm based on a multi‐level adaptive mesh refinement strategy for the solution of ill‐posed inverse heat conduction problems arising in pool boiling using few temperature observations.

The stable solution of the inverse problem is obtained by applying the conjugate gradient method for the normal equation method together with a discrepancy stopping rule. The resulting three‐dimensional direct, adjoin and sensitivity problems are solved numerically by a space‐time finite element method. A multi‐level computational approach, which uses an a posteriori error estimator to adaptively refine the meshes on different levels, is proposed to speed up the entire inverse solution procedure.

This systematic approach can efficiently solve the large‐scale inverse problem considered without losing necessary detail in the estimated quantities. It is shown that the choice of different termination parameters in the discrepancy stopping conditions for each level is crucial for obtaining a good overall estimation quality. The proposed algorithm has also been applied to real experimental data in pool boiling. It shows high computational efficiency and good estimation quality.

The high efficiency of the approach presented in the paper allows the fast processing of experimental data at many operating conditions along the entire boiling curve, which has been considered previously as computationally intractable. The present study is the authors' first step towards a systematic approach to consider an adaptive mesh refinement for the solution of large‐scale inverse boiling problems.