An extended method of time-dependent fundamental solutions for inhomogeneous heat conduction

In this paper, the method of time-dependent fundamental solutions is extended to solving transient heat conduction problems in inhomogeneous media. The solution of the problem under investigation is split into two parts, namely the particular and homogenous solutions. The novelty of the proposed approach lies in that an approximation of the particular solution is derived by using the fundamental solutions of the associated eigenvalue equations. Numerical results for one- and two-dimensional geometries are presented to verify the efficacy of the proposed method. The effects of the numbers of source and collocation points, the eigenvalues and the parameter T on the accuracy of the numerical solution are also investigated.     

Experimental validation of a frequency domain BEM model to study 2D and 3D heat transfer by conduction

This article presents an experimental validation of 2D and 2.5D boundary element method (BEM) solutions for transient heat conduction in systems containing heterogeneities. The problem is formulated in the frequency domain. The responses in the time domain are obtained by means of an inverse Fourier transform. Complex frequencies with a small imaginary part are introduced to cope with aliasing. Their effect is taken into account by rescaling the results in the time domain.For validation purposes, the solutions provided by the proposed BEM formulation were first verified against analytical solutions and then compared with experimental results. In the laboratory tests a steel inclusion was embedded in a confined host medium and unsteady temperatures were applied to its boundary. Two host media were tested: molded expanded polystyrene and medium-density fiberboard. The systems were subjected to plane and point heat sources. The thermal properties of these materials have been previously defined experimentally. The results show that the BEM solutions agree well with the experimental results.     

A method of fundamental solutions for transient heat conduction in layered materials

In this paper we investigate an application of the method of fundamental solutions (MFS) to transient heat conduction in layered materials, where the thermal diffusivity is piecewise constant. Recently, in Johansson and Lesnic [A method of fundamental solutions for transient heat conduction. Eng Anal Boundary Elem 2008;32:697–703], a MFS was proposed with the sources placed outside the space domain of interest, and we extend that technique to numerically approximate the heat flow in layered materials. Theoretical properties of the method, as well as numerical investigations are included.     

一种非连续介质中热传导过程的数值模拟方法

传统热传导的分析基于连续模型,无法刻画热量在两个接触体之间的传递。该文提出了一种非连续介质中热传导过程的数值计算方法,并编制了相应的C++计算程序。该方法首先将计算域离散为一系列的块体,块体内部划分若干连续介质单元,块体边界设定为潜在接触界面,并利用半弹簧-半棱联合接触模型进行接触对的快速检索及标记。每个块体内部的热传导采用传统连续模型进行计算(该文采用有限体积法),每个接触界面采用点面接触型及棱棱接触型热传导模型进行描述。通过调整接触界面热传导系数中的刚度因子,可以实现接触界面对热传导过程不同的抵抗效应。数值算例表明,该文所述方法可以较为准确地模拟热量在非连续介质中的传递过程;接触界面上的刚度因子越大,界面对热传导过程的抵抗效应越小;当刚度因子大于100,界面抵抗效应基本消失,非连续介质的计算结果与连续介质的计算结果完全一致;此外,接触界面上的刚度因子仅影响热传导的瞬态过程,而不影响其稳态解。     

Axial symmetric stationary heat conduction analysis of non-homogeneous materials by triple-reciprocity boundary element method

The heat conduction problems in homogeneous media can be easily solved by the boundary element method. The spatial variations of heat sources as well as material coefficients gives rise to domain integrals in integral formulations for solution of boundary value problems in functionally gradient materials (FGM), since the fundamental solutions are not available for partial differential equations with variable coefficients, in general. In this paper, we present the development of the triple reciprocity method for solution of axial symmetric stationary heat conduction problems in continuously non-homogeneous media with eliminating the domain integrals. In this method, the spatial variations of domain “sources” are approximated by introducing new potential fields and using higher order fundamental solutions of the Laplace operator.     

Boundary element method analyses of transient heat conduction in an unbounded solid layer containing inclusions

The boundary element method (BEM) is used to compute the three-dimensional transient heat conduction through an unbounded solid layer that may contain heterogeneities, when a pointwise heat source placed at some point in the media is excited. Analytical solutions for the steady-state response of this solid layer when subjected to a spatially sinusoidal harmonic heat line source are presented when the solid layer has no inclusions. These solutions are incorporated into a BEM formulation as Greens functions to avoid the discretization of flat media interfaces. The solution is obtained in the frequency domain, and time responses are computed by applying inverse (Fast) Fourier Transforms. Complex frequencies are used to prevent the aliasing phenomena. The results provided by the proposed Greens functions and BEM formulation are implemented and compared with those computed by a BEM code that uses the Greens functions for an unbounded media which requires the discretization of all solid interfaces with boundary elements. The proposed BEM model is then used to evaluate the temperature field evolution through an unbounded solid layer that contains cylindrical inclusions with different thermal properties, when illuminated by a plane heat source. In this model zero initial conditions are assumed. Different simulation analyses using this model are then performed to evaluate the importance of the thermal properties of the inclusions on transient heat conduction through the solid layer.     

Three-dimensional transient heat conduction analysis by Laplace transformation and multiple reciprocity boundary face method

In this paper, a new multiple reciprocity formulation is developed to solve the transient heat conduction problem. The time dependence of the problem is removed temporarily from the equations by the Laplace transform. The new formulation is derived from the modified Helmholtz equation in Laplace space (LS), in which the higher order fundamental solutions of this equation are firstly derived and used in multiple reciprocity method (MRM). Using the new formulation, the domain integrals can be converted into boundary integrals and several non-integral terms. Thus the main advantage of the boundary integral equations (BIE) method, avoiding the domain discretization, is fully preserved. The convergence speed of these higher order fundamental solutions is high, thus the infinite series of boundary integrals can be truncated by a small number of terms. To get accurate results in the real space with better efficiency, the Gaver-Wynn-Rho method is employed. And to integrate the geometrical modeling and the thermal analysis into a uniform platform, our method is implemented based on the framework of the boundary face method (BFM). Numerical examples show that our method is very efficient for transient heat conduction computation. The obtained results are accurate at both internal and boundary points. Our method outperforms most existing methods, especially concerning the results at early time steps.     

Three-dimensional fundamental solutions for transient heat transfer by conduction in an unbounded medium,half-space,slab and layered media

This paper presents analytical Green's functions for the transient heat transfer phenomena by conduction, for an unbounded medium, half-space, slab and layered formation when subjected to a point heat source. The transient heat responses generated by a spherical heat source are computed as Bessel integrals, following the transformations proposed by Sommerfeld [Sommerfeld A. Mechanics of deformable bodies. New York: Academic Press; 1950; Ewing WM, Jardetzky WS, Press F. Elastic waves in layered media. New York: McGraw-Hill; 1957]. The integrals can be modelled as discrete summations, assuming a set of sources equally spaced along the vertical direction. The expressions presented here allow the heat field inside a layered formation to be computed without fully discretizing the interior domain or boundary interfaces.The final Green's functions describe the conduction phenomenon throughout the domain, for a half-space and a slab. They can be expressed as the sum of the heat source and the surface terms. The surface terms need to satisfy the boundary conditions at the surfaces, which can be of two types: null normal fluxes or null temperatures. The Green's functions for a layered formation are obtained by adding the heat source terms and a set of surface terms, generated within each solid layer and at each interface. These surface terms are defined so as to guarantee the required boundary conditions, which are: continuity of temperatures and normal heat fluxes between layers.This formulation is verified by comparing the frequency responses obtained from the proposed approach with those where a double-space Fourier transformation along the horizontal directions [Tadeu A, António J, Simões N. 2.5D Green's functions in the frequency domain for heat conduction problems in unbounded, half-space, slab and layered media. CMES: Computer Model Eng Sci 2004;6(1):43–58] is used. In addition, time domain solutions were compared with the analytical solutions that are known for the case of an unbounded medium, a half-space and a slab.     

Boundary element analysis of nonlinear transient heat conduction problems involving non-homogenous and nonlinear heat sources using time-dependent fundamental solutions

A new method for the boundary element analysis of unsteady heat conduction problems involving non-homogenous and/or temperature dependent heat sources by the time-dependent fundamental solution is presented. Nonlinear terms are converted to a fictitious heat source and implemented in the present formulation. The domain integrals are efficiently treated by the recently introduced Cartesian transformation method. Similar to the dual reciprocity method, some internal grid points are considered for the treatment of the domain integrals. In the present method, unlike the dual reciprocity method, there is no need to find particular solution for the shape functions in the interpolation computations and the form of the shape functions can be arbitrary and sufficiently complicated. In the present method, at each time step the temperature at boundary nodes and some internal grid points is computed and used as pseudo-initial values for the next time step. Most of the generated matrices are constant at all time steps and computations can be carried out fast. An example with different forms of heat sources is presented to show the efficiency and accuracy of the proposed method.     

A two-scale homogenization framework for nonlinear effective thermal conductivity of laminated composites

In this study, we formulate the effective temperature-dependent thermal conductivity of laminated composites. The studied laminated composites consist of laminas (plies) made of unidirectional fiber-reinforced matrix with various fiber orientations. The effective thermal conductivity is obtained through a two-scale homogenization scheme. A simplified micromechanical model of a unidirectional fiber-reinforced lamina is formulated at the lower scale. Thermal conductivities of fiber and matrix constituents are allowed to change with temperature. The upper scale uses a sublaminate model to homogenize temperature-dependent thermal conductivities of only a representative lamina stacking sequence in laminated composites. The effective thermal conductivity of each lamina, in the sublaminate model, is obtained using the simplified micromechanical model. The thermal conductivities from the micromechanical and sublaminate models represent average nonlinear properties of fictitiously homogeneous composite media. Interface conditions between fiber and matrix constituents and within laminas are assumed to be perfect. Experimental data available in the literature are used to verify the proposed multi-scale framework. We then analyze transient heat conduction in the homogenized composites. Temperature profiles, during transient heat conduction, in the homogenized composites are compared to the ones in heterogeneous composites. The heterogeneous composites, having different fiber arrangements and sizes, are modeled using finite element (FE) method.     

Three-dimensional unsteady heat conduction analysis by triple-reciprocity boundary element method

The conventional boundary element method (BEM) requires a domain integral in heat conduction analysis with heat generation or an initial temperature distribution. In this paper it is shown that the three-dimensional heat conduction problem can be solved effectively using the triple-reciprocity BEM without internal cells. In this method, the distributions of heat generation and initial temperature are interpolated using integral equations and time-dependent fundamental solutions are used. A new computer program was developed and applied to solving several problems.     

Acoustic analysis with absorbing finite elements and far-field computations using free-space Green''s functions

A family of absorbing-boundary finite elements (FEs) is recently developed for the acoustic time-harmonic, transient and modal analyses of problems having fluid domains of infinite extent. These elements can be easily ‘glued’ to the truncated exterior surface of an underlying FE mesh. The desirable characteristics of the absorbing elements are that they are local and produce banded symmetric element matrices. Since the use of the absorbing FEs reduces the size of the FE model through the premature truncation of the mesh, a post-processing procedure must be employed to obtain solutions far away from FE domain in the case of transient and time-harmonic analyses. In this paper free-space Green's functions are utilized, in the context of time-harmonic analysis, in association with the computed solutions at the exterior boundary where absorbing-boundary FEs are placed in order to compute pressure distribution outside the FE computational domain. The paper demonstrates that the computation of accurate pressure gradients at the absorbing-boundary element nodes plays a crucial role in obtaining accurate pressure solutions outside the FE computational domain.     

Transient Heat Conduction in a Functionally Graded Cylindrical Panel Based on the Dual Phase Lag Theory

The transient heat conduction in a functionally graded cylindrical panel is investigated based on the dual phase lag (DPL) theory in this article. Except for the phase lags which are assumed to be constant, all the other material properties of the panel are assumed to change continuously along the radial direction according to a power-law formulation with different non-homogeneity indices. The heat conduction equations based on the DPL theory in the cylindrical coordinate system are written in a general form which are then used for the analyses of four different geometries: (1) a hollow cylinder of an infinite length; (2) a hollow cylinder of a finite length; (3) a cylindrical panel of an infinite length; and (4) a cylindrical panel of a finite length. Using the Laplace transform, the analytical solutions for temperature and heat flux are obtained in the Laplace domain. The solutions are then converted into the time domain by employing the fast Laplace inversion technique. The exact expressions for the radial thermal wave speed are obtained for the four different geometries. The numerical results are displayed to reveal the effect of different approximations of the DPL theory on the temperature distribution for various non-homogeneity indices. The results are verified with those reported in the literature.     

Application of the Laplace transform dual reciprocity boundary element method in the modelling of laser heat treatments

The Laplace Transform Dual Reciprocity Boundary Element Method (LTDRM or LT-DRBEM) provides with an alternative numerical technique to finite difference (FDM) or finite element methods (FEM) for solving transient diffusion problems. With this method, solutions are calculated directly at any specific time thus avoiding the use of time-stepping schemes. Besides, domain integrals are removed from the problem formulation.In this work we study the applicability of the LT-DRBEM method for laser heat treatment modelling purposes. A simple model was developed based on a two dimensional transient heat conduction equation, in which the laser beam is included as a heat flux boundary condition of gaussian shape. Results corresponding to a stationary and a moving beam are presented and discussed. Non-linear formulations of the problem as those given by temperature dependent material properties are also considered. Good accuracy results were obtained for the stationary beam approach, whereas severe limitations were found for the moving beam case.     

Numerical analysis of two-dimensional lagging thermal behavior under short-pulse-laser heating on surface

In this work, the dual-phase-lag (DPL) model of heat conduction is introduced in treating the transient heat conduction problems in finite rigid mediums under short-pulse-laser heating. Two-dimensional numerical solutions in a rectangular and an axially symmetric system are given by finite difference method. Calculations are performed to exhibit various two-dimensional lagging thermal behavior of conduction heat transfer, such as wavy, wavelike, and diffusive behavior.     

Transient heat conduction in anisotropic and functionally graded media by local integral equations

Reliable computational techniques are developed for the solution of two-dimensional (2-d) transient heat conduction problems in anisotropic media with continuously variable material coefficients. Two kinds of the domain-type interpolation, namely the standard domain elements and the meshless point interpolation, are adopted for the approximation of the spatial variation of the temperature field or its Laplace-transform. The coupling among the nodal values of the approximated field is given by integral equations considered on local sub-domains. Three kinds of local integral equations are derived from physical principles instead of using a weak-form formulation. The accuracy and the convergence of the proposed techniques are tested by several examples and compared with exact benchmark solutions, which are derived too.     

A boundary/domain element method for analysis of building raft foundations

In this paper, a new boundary/domain element method is developed to analyse plates resting on elastic foundations. The developed formulation is then used in analysing building raft foundations. For more practical representation, the considered raft plate is treated as thick plate with free edge boundary conditions. The soil or the elastic foundation is represented as continuous media (follows the Winkler assumption). The boundary element method is employed to model the raft plate; whereas the soil is modelled using constant domain cells or elements. Therefore, in the present formulation both the domain and the boundary of the raft plate are discretized. The associate soil domain integral is replaced by equivalent boundary integrals along each cell contour. The necessary matrix implementation of such formulation is carried out and explained in details. The main advantage of the present formulation is the ability of analysing rafts on non-homogenous soils. Two examples are presented including raft on non-homogenous soil and raft for practical building applications. The results are compared with those obtained from other finite element and alternative boundary element methods to verify the validity and accuracy of the present formulation.     

An analytical solution for hydraulic conductivity of concrete considering properties of the Interfacial Transition Zone (ITZ)

As a composite material, hydraulic conductivity of concrete depends on conductivity of its components that are the mortar, aggregates and the Interfacial Transition Zone (ITZ). Since hydraulic conduction is analogous to heat and electrical conduction, analytical models from these analogous areas relating effective conductivity of composite to conductivity of its components can be used to find the effective hydraulic conductivity of concrete as a function of properties of its components, i.e., aggregate, mortar and the ITZ. However, effect of the conduction in the ITZ has not been considered in these models. This paper presents an analytical solution for the hydraulic conductivity of concrete as a three-phase composite material. The solution is an extension to the model originally proposed for conduction of composite media with randomly suspended spheres. Results of the proposed model compare well against the experimental results and those obtained from rigorous numerical analysis using the Finite Element (FE) method. The principal significance of this study lies in the development of a versatile analytical model that can be employed as a quick tool for assessment of hydraulic conductivity of concrete without the need for sophisticated FE models at the meso-scale level. It offers more insight into effect of different components of concrete on its overall conductivity.     

Local BIEM for transient heat conduction analysis in 3-D axisymmetric functionally graded solids

An advanced computational method for transient heat conduction analysis in 3-D axisymmetric continuously nonhomogeneous functionally graded materials (FGM) is proposed. The analysed domain is covered by small circular subdomains. On each subdomain local boundary integral equations for the transient heat conduction problem are derived in the Laplace transform domain. The meshless approximation based on the moving least-squares method is employed for the numerical implementation. The Stehfest algorithm is applied for the numerical Laplace inversion to obtain the temporal variation. Numerical results are presented for finite full and hollow cylinders with an exponential variation of material parameters with spatial coordinates. The authors acknowledge the support by the Slovak Science and Technology Assistance Agency registered under number APVT-51-003702, and the Project for Bilateral Cooperation in Science and Technology supported jointly by the International Bureau of the German BMBF and the Ministry of Education of Slovak Republic under the project number SVK 01/020.     

A finite element method for non-Fourier heat conduction in strong thermal shock environments

Non-Fourier effect is important in heat conduction in strong thermal environments. Currently, generally-purposed commercial finite element code for non-Fourier heat conduction is not available. In this paper, we develop a finite element code based on a hyperbolic heat conduction equation, which includes the non-Fourier effect in heat conduction. The finite element space discretization is used to obtain a system of differential equations for the time. The transient responses are obtained by solving the system of differential equations, based on the finite difference, mode superposition, or exact time integral. The code is validated by comparing the numerical results with exact solutions for some special cases. The stability analysis is conducted and it shows that the finite difference scheme is an ideal method for the transient solution of the temperature field. It is found that with mesh refining (decreasing mesh size) and/or high-order elements, the oscillation in the vicinity of sharp change vanishes, and can be essentially suppressed by the finite difference scheme. A relationship between the time step and the space length of the element was identified to ensure that numerical oscillation vanishes.