Steady-state Heat Conduction Problems Research Articles

Hybrid ballistic–diffusive solution to the frequency-dependent phonon Boltzmann Transport Equation

The phonon Boltzmann Transport Equation (BTE) is appropriate for modeling heat conduction in semiconductor materials at the nanoscale. However, the BTE is difficult to solve on account of the directional and spectral nature of the phonon intensity, which necessitates angular and spectral discretization, and ultimately results in a large number (typically few hundreds) of four-dimensional partial differential equations. In the ballistic (large Knudsen number) regime, the phonon intensity is highly anisotropic, and therefore, angular resolution is desirable. However, in the diffusive (small Knudsen number) regime, the intensity is fairly isotropic, and hence, angular discretization is wasteful. In such scenarios, the method of spherical harmonics (PN approximation) may be effectively used to reduce the large number of directional BTEs to a few partial differential equations. Since the Knudsen number is frequency dependent, the decision to preserve or eliminate angular discretization may be made frequency by frequency based on whether the spectral Knudsen number is large or small. In this article, a hybrid method is proposed in which for some frequency intervals (bands), full angular discretization is used, while for others, the P1 approximation is invoked to reduce the number of directional BTEs. The accuracy and efficiency of the hybrid method is tested by solving several steady state and transient nanoscale heat conduction problems in two and three-dimensional geometries. Silicon is used as the candidate material. It is found that hybridization is effective in significantly improving the efficiency of solution of the BTE—sometimes by a factor of three—without significant penalty on the accuracy.

Hybrid asymptotic-numerical modeling of thin layers for dynamic thermal analysis of structures

Purpose – The purpose of this paper is to propose a computational model for thin layers, for problems of linear time-dependent heat conduction. The thin layer is replaced by a zero-thickness interface. The advantage of the new model is that it saves the need to construct and use a fine mesh inside the layer and in regions adjacent to it, and thus leads to a reduction in the computational effort associated with implicit or explicit finite element schemes. Design/methodology/approach – Special asymptotic models have been proposed for linear heat transfer and linear elasticity, to handle thin layers. In these models the thin layer is replaced by an interface with zero thickness, and specific jump conditions are imposed on this interface in order to represent the special effect of the layer. One such asymptotic interface model is the first-order Bövik-Benveniste model. In a paper by Sussmann et al., this model was incorporated in a FE formulation for linear steady-state heat conduction problems, and was shown to yield an accurate and efficient computational scheme. Here, this work is extended to the time-dependent case. Findings – As shown here, and demonstrated by numerical examples, the new model offers a cost-effective way of handling thin layers in linear time-dependent heat conduction problems. The hybrid asymptotic-FE scheme can be used with either implicit or explicit time stepping. Since the formulation can easily be symmetrized by one of several techniques, the lack of self-adjointness of the original formulation does not hinder an accurate and efficient solution. Originality/value – Most of the literature on asymptotic models for thin layers, replacing the layer by an interface, is analytic in nature. The proposed model is presented in a computational context, fitting naturally into a finite element framework, with both implicit and explicit time stepping, while saving the need for expensive mesh construction inside the layer and in its vicinity.

A meshless method based on the method of fundamental solution for solving the steady-state heat conduction problems

Motivated by the incompleteness of the method of fundamental solution (MFS) for the interior problem, we give another meshless method both theoretically and numerically for recovering the temperature and the heat flux based on the normal derivative of the fundamental solution in this paper. Although the problems under investigation are well-posed, we should note that the method presented here results in an ill-conditioned system and this is a feature of the numerical method employed in the present approach. The ill-posedness of this system is given by the potential function. In order to overcome the ill-posedness of the system, the Tikhonov regularization method, as well as Morozov’s discrepancy principle for selecting an appropriate regularization parameter, are used to increase the stability of this method. Then three kinds of boundary value problems are presented to show the effectiveness of this method with some examples, whilst the comparisons with the MFS is presented. The numerical convergence and stability of this method are also analyzed.

A meshless radial basis function method for 2D steady-state heat conduction problems in anisotropic and inhomogeneous media

The paper presents a new meshless numerical method for solving 2D steady-state heat conduction problems in anisotropic and inhomogeneous media. The coefficients of the governing PDEs are spatially dependent functions including the main operator part. The boundary conditions of a most general form for the temperature and the heat flux are considered. The key idea of the method is the use of the basis functions which satisfy the homogeneous boundary conditions of the problem. Each basis function used in the algorithm is a sum of a RBF and a special correcting function which is chosen to satisfy the homogeneous BC of the problem. The conical radial basis functions, the Duchon splines and the multiquadric RBFs are used in approximation of the PDE. This allows us to seek an approximate solution in the form which satisfies the boundary conditions of the initial problem with any choice of the free parameters. As a result we separate the approximation of the boundary conditions and the approximation of the PDE inside the solution domain. The numerical experiments are carried out for accuracy and convergence investigations. The comparison of the numerical results obtained in the paper with the exact solutions and with the data obtained with the use of other numerical techniques is performed. The numerical examples demonstrate that the present method is accurate, convergent, stable, and computationally efficient in solving this kind of problems.

A steady state heat conduction problem in a thick annular disc due to arbitrary axisymmetric heat flux

In this paper attempt has been made to discuss steady state quasi static thermal stresses in a thick annular disc subjected to arbitrary heat flux at upper and lower surface of annular disc while inner and outer circular surface of thick annular disc are maintained at zero degree temperature. The governing heat conduction have been solved by using integral transform technique .The results are obtained in series form in term of Bessel’s functions, The result of displacement and stresses have been computed numerically and illustrated graphically.

マルチスケール解析SDMの非定常問題への拡張

The Seamless-Domain Method (SDM) is the multiscale analysis method, whose solution can represent complicated behaviors of heterogeneous materials with microscopic constituents. It is achieved not by homogenization, but by a special interpolation procedure. In previous studies, the SDM had been applied to steady-state heat conduction problems and linear elastic problems. In this paper, the way how to extend the SDM to transient problems of heterogeneous materials is discussed. Numerical results of an example problem of a transient heat conduction is presented. The results of the proposed method are compared with those of widely-used Finite Difference Method.

An invariant method of fundamental solutions for two-dimensional steady-state anisotropic heat conduction problems

We investigate both theoretically and numerically the so-called invariance property, see e.g. Sun and Ma (2015a,b), of the solution of boundary value problems associated with the anisotropic heat conduction equation (or Laplace–Beltrami’s equation) in two dimensions with respect to elementary transformations of the solution domain, e.g. dilations or contractions. We also show that the standard method of fundamental solutions (MFS) does not satisfy the invariance property. Motivated by these reasons, we introduce, in a natural manner, a modified version of the MFS that remains invariant under elementary transformations of the solution domain and is referred to as the invariant MFS (IMFS). Five two-dimensional examples are thoroughly investigated to assess the numerical accuracy, convergence and stability of the proposed IMFS, in conjunction with the Tikhonov regularization method (Tikhonov and Arsenin, 1986) and Morozov’s discrepancy principle (Morozov, 1966), for Laplace–Beltrami’s equation with perturbed boundary conditions.

Voronoi based discrete least squares meshless method for heat conduction simulation in highly irregular geometries

A new technique is used in Discrete Least Square Meshfree(DLSM) method to remove the common existing deficiencies of meshfree methods in handling of the problems containing cracks or concave boundaries. An enhanced Discrete Least Squares Meshless method named as VDLSM(Voronoi based Discrete Least Squares Meshless) is developed in order to solve the steady-state heat conduction problem in irregular solid domains including concave boundaries or cracks. Existing meshless methods cannot estimate precisely the required unknowns in the vicinity of the above mentioned boundaries. Conducted researches are limited to domains with regular convex boundaries. To this end, the advantages of the Voronoi tessellation algorithm are implemented. The support domains of the sampling points are determined using a Voronoi tessellation algorithm. For the weight functions, a cubic spline polynomial is used based on a normalized distance variable which can provide a high degree of smoothness near those mentioned above discontinuities. Finally, Moving Least Squares(MLS) shape functions are constructed using a varitional method. This straight-forward scheme can properly estimate the unknowns(in this particular study, the temperatures at the nodal points) near and on the crack faces, crack tip or concave boundaries without need to extra backward corrective procedures, i.e. the iterative calculations for modifying the shape functions of the nodes located near or on these types of the complex boundaries. The accuracy and efficiency of the presented method are investigated by analyzing four particular examples. Obtained results from VDLSM are compared with the available analytical results or with the results of the well-known Finite Elements Method(FEM) when an analytical solution is not available. By comparisons, it is revealed that the proposed technique gives high accuracy for the solution of the steady-state heat conduction problems within cracked domains or domains with concave boundaries and at the same time possesses a high convergence rate which its accuracy is not sensitive to the arrangement of the nodal points. The novelty of this paper is the use of Voronoi concept in determining the weight functions used in the formulation of the MLS type shape functions.

Estimation of thermal conductivity, heat transfer coefficient, and heat flux using a three dimensional inverse analysis

This paper presents a numerical inverse analysis to estimate the thermal conductivity, the heat transfer coefficient, and the heat flux in three dimensional irregular bodies in steady state heat conduction problems. In this study, a 3-D elliptic grid generation technique is used to mesh the irregular body. The 3-D Laplace equation is solved in the computational domain to compute the temperature at any grid point in the meshed body. A novel and very efficient sensitivity analysis scheme is introduced to compute the sensitivity coefficients in gradient based optimization method. Using this sensitivity analysis scheme, one can solve the inverse problem without need to the solution of adjoint equation. The main advantages of the sensitivity analysis scheme are its simplicity, accuracy, and independency of the number of the direct problem solution from the number of the unknown variables which makes the numerical inverse analysis presented here very accurate and efficient. The conjugate gradient method (CGM) is used to minimize the objective function which is the difference between the computed temperature on part of the boundary and the measured temperature. The obtained results confirm that the proposed algorithm is very accurate, robust, and efficient.

Parallel ant colony optimization for the determination of a point heat source position in a 2-D domain

The Ant Colony Optimization (ACO) and its parallel implementations are applied to determine the unknown position of a point heat source in a two-dimensional steady-state heat conduction problem. The heuristic value, the determination of path and the objective function are all studied based on the inverse heat source problem. Some of the standard steps of the ACO are also modified according to the features of the heat conduction problem. In order to accelerate the speed of solving the inverse problem, two kinds of parallel ACO strategies, the fine-grained master-slave strategy and the coarse-grained strategy combined with the above modification are studied in this paper. The results show that the fine-grained strategy has a higher speedup than the coarse-grained strategy whiles the coarse-grained strategy has a higher accuracy rate. This means that the coarse-grained strategy is more proper during the parallel implementation for solving the inverse heat source problem and the parallel implementation of the ACO can improve the computational efficiency.

Isogeometric configuration design optimization of heat conduction problems using boundary integral equation

The shape variation of a domain naturally results in both shape and orientation variations, so called configuration variation, when employing a boundary integral equation (BIE) method. A configuration design sensitivity analysis (DSA) method is developed for steady state heat conduction problems using the boundary integral equations in an isogeometric approach, where NURBS basis functions in a CAD system are directly utilized in the response analysis. Thus, we can accomplish a seamless incorporation of exact geometry and the higher continuity into a computational framework. To enhance the accuracy of configuration design sensitivity, the CAD-based higher-order geometric information such as normal and tangent vectors is exactly embedded in the design sensitivity expressions. The necessary velocity field for configuration design obtained from the NURBS is analytically decomposed into shape and orientation velocity fields. It is shown to be essential to consider orientation variations and significant for accurate configuration sensitivity through comparison with finite differencing conventional BIE method. The developed isogeometric configuration DSA method turns out to be accurate compared with the analytic solution and the conventional DSA method. During the optimization, a mesh regularization scheme is employed to avoid excessive mesh distortion, which comes from significant design changes.

A novel inverse method for identification of 3D thermal conductivity coefficients of anisotropic media by the boundary element analysis

This paper presents a new numerical inverse method for identifying the six components of the thermal conductivity tensor of a 3D anisotropic medium with an arbitrary shape. The unknowns are inversely calculated using heat conduction problems with extra information at some boundary sampling points. For better stability, the inverse method uses data supplied from more than one steady-state heat conduction problem. Since all sampling points are taken to be on boundary surfaces for the sake of easy access, the boundary element method (BEM) for direct calculation is employed for the sensitivity analyses. The off-diagonal components of the thermal conductivity tensor must satisfy three nonlinear inequality constraints, which make the inverse analysis even more challenging. To overcome this difficulty, the inverse problem is formulated in terms of the principal thermal conductivities along with the rotation angles of principal axes, by which the three constraints disappear. For the inverse analysis, the damped Gauss–Newton method is adopted for the optimization process. In the end, numerical examples are presented, showing that the proposed method can yield reliable solutions even in cases with relatively large measurement error and with initial guesses far from the exact solution.

Optimizing of the underground power cable bedding using momentum-type particle swarm optimization method

Thermal performance optimization of underground power cable system is presented in this paper. The analyzed system consists of three underground power cables situated in an in-line arrangement. The HDPE (High-Density Polyethylene) casing pipes, filled with SBM (Sand-Bentonite Mixture), covers the cables to protect them from heavy mechanical loads (e.g. vibrations). The FTB (Fluidized Thermal Backfill) layer is applied to prevent the cables from overheating. Due to the substantial costs of FTB backfill material (in relation to the native soil or dry sand), the cross-sectional area of FTB bedding layer has to be minimized. Furthermore, the maximum cable conductor temperature is expected not to exceed the optimum operating temperature. Therefore, the optimization procedure i.e. momentum-type PSO (Particle Swarm Optimization) is applied. The FEM (Finite Element Method) is used to solve the two-dimensional steady-state heat conduction problem. As a result, temperature distribution is determined for the native soil, FTB bedding, and cables. The performed computations considered the temperature dependent current rating and volumetric heat generation rate from cable conductor. The applied optimization procedure resulted in determination of the optimum cable spacing and cross-sectional area of the rectangular-shaped FTB bedding layer. Moreover, the obtained maximum temperature for the cable core do not exceed the allowable value.

A finite element method for prediction of unknown boundary conditions in two-dimensional steady-state heat conduction problems

This paper presents a finite element method in predicting unknown boundary conditions of homogeneous and two-layered materials subjected to steady-state heat conduction. Firstly, a finite element formulation is introduced to solve a steady-state boundary inverse heat conduction problem of homogeneous material. Effects of bias error of temperature specified on interior nodes and locations of such nodes on accuracy of predicted temperature distribution are examined. Then, modified cubic spline is specified on material boundary to stabilize predicted temperature distribution. Cubic spline functions using different numbers of control points are used in examining their effects on accuracy of predicted temperature distribution and computing time when specifying no bias temperatures. Finally, the formulation with cubic spline function specification is employed in predicting unknown boundary conditions of two-layered materials with thermal conductivity ratio of 0.1, 1, and 10. Concept of coincident nodes is applied in handling physical condition characterized by thermal contact resistance and heat source strength at layer interface. Effect of bias error of temperatures specified on nodes within thicker layer is examined under three interface conditions. Cubic spline function with five control points can predict temperature distributions accurately for all interface conditions when specifying no bias temperatures. RMS errors vary linearly with bias errors for interface conditions with no heat source but are drastically affected by bias error when heat source exists at the interface.

Analytical solutions to the axisymmetric elasticity and thermoelasticity problems for an arbitrarily inhomogeneous layer

In this paper, we present a technique for constructing an analytical solution to the axisymmetric elasticity and thermoelasticity problems in terms of stresses for an inhomogeneous layer, whose elastic and thermophysical properties vary arbitrarily within the thickness-coordinate. By making use of the direct integration method, the equilibrium and compatibility equations are reduced to the governing Volterra-type integral equations accompanied with both integral and local boundary conditions for the key functions. To solve the obtained governing equations, we employed the resolvent-kernel technique which results in closed-form analytical expressions for the key functions. Having determined the key functions, the stress-tensor components are found through the relationship established by the integration of equilibrium equations. The same solution procedure is employed for solving the steady-state heat-conduction problem in an inhomogeneous layer. Typical numerical examples are discussed.

Shape optimizations of inhomogeneities of two dimensional (2D) and three dimensional (3D) steady state heat conduction problems by the boundary element method

The shape optimizations of inhomogeneities for 2D and 3D steady state heat conductions in an infinite medium are respectively studied by the boundary element method (BEM). Interest in the shape optimization by the BEM is mainly due to its high computation accuracy and simplicity in meshing. The boundary integral equations and the heat energy formulations in this paper only contain the temperature on each inhomogeneity–matrix interface. The heat energy increment in the inhomogeneous medium is taken as the objective function. The method of moving asymptotes (MMA) is adopted to carry out numerical implementation of the shape optimizations of 2D and 3D inhomogeneities.

Design of bi-metallic devices based on the topological derivative concept

In this work the topological derivative concept is applied in the context of topology design of thermo-mechanical devices, where the linear elasticity system (modeled by the Navier equation) is coupled with the steady-state heat conduction problem (modeled by the Laplace equation). The mechanical coupling term comes out from the thermal stress induced by the temperature field. We consider the topology design of bi-metallic devices. The idea is to maximize the displacement in a given direction defined on the boundary of the thermo-elastic body with respect to a bi-metallic material distribution. The topological derivative is obtained by considering the nucleation of a small circular inclusion with different thermal expansion coefficients. A level-set domain representation method is used, together with the derived topological sensitivity formula, to devise a topology design algorithm. Finally, some numerical experiments regarding the conceptual design of thermo-mechanical devices are presented.

Finite-Element Discretized Symplectic Method for Steady-State Heat Conduction with Singularities in Composite Structures

A new-finite element discretized symplectic method for solving the steady-state heat conduction problem with singularities in composite structures is presented. The model with a singularity is divided into two regions, near and far fields, and meshed using conventional finite elements. In the near field, the temperature and heat flux densities are expanded in exact symplectic eigensolutions. After a matrix transformation, the unknowns in the near field are transformed to coefficients of the symplectic series, while those in the far field are as usual. The exact local solutions for temperature and heat flux densities are obtained simultaneously without any post-processing.

Modified Quadrature Method for Solving BIEs of Steady-State Anisotropic Heat Conduction Problems

In this paper, steady-state anisotropic heat conduction equation can be converted into the first kind integral equation, then modified quadrature formula based on trapezoidal rule is used to deal the integrals with singular kernels. In addition, Sidi transformation is applied to remove the singularities at concave points in concave polygons. This technique improves the accuracy of numerical solutions of the heat conduction equation. Numerical results show the convergence rate of the proposed method is the order three.

Hybrid uncertain analysis for steady-state heat conduction with random and interval parameters

Abstract A new numerical technique named stochastic interval finite volume method is proposed for the steady-state temperature field prediction with hybrid uncertainties in material properties, external loads and boundary conditions. Random variables are used to treat the uncertain parameters with sufficient information; whereas, interval variables are used to quantitatively describe the uncertain parameters with limited information. Based on the perturbation theory and random interval moment method, the expectation and variance of the interval bounds of the temperature field are calculated effectively. By comparing the results with traditional Monte Carlo simulation, three numerical examples are given to verify the feasibility and effectiveness of the proposed method for solving steady-state heat conduction problems with hybrid uncertain parameters, pure random parameters and pure interval parameters, respectively.