Steady-state Heat Conduction Problems Research Articles

Two-dimensional numerical manifold method for heat conduction problems

The numerical manifold method (NMM) is a calculation method based on Galerkin's variation and contains a dual covering system. Therefore, the NMM can easily deal with the construction of high-order manifold elements and is convenient for adaptive analysis. This study employed the NMM to simulate steady-state and transient heat conduction problems. The system equations were derived and a penalty function method was applied to deal with the boundary conditions. By simulating calculation examples for a one-dimensional bar, cuboid rock specimens, and thick-walled cylinders, the process of solving steady-state and transient heat-conduction problems and the influence law of the temperature pattern are demonstrated. Moreover, the convergence and effectiveness of the NMM in handling two-dimensional (2D) steady-state and 2D transient heat conduction problems was verified.

Isogeometric Boundary Element Method for Two-Dimensional Steady-State Non-Homogeneous Heat Conduction Problem

Isogeometric Boundary Element Method for Two-Dimensional Steady-State Non-Homogeneous Heat Conduction Problem

The Improved Element-Free Galerkin Method for Anisotropic Steady-State Heat Conduction Problems

In this paper, we considered the improved element-free Galerkin (IEFG) method for solving 2D anisotropic steady-state heat conduction problems. The improved moving least-squares (IMLS) approximation is used to establish the trial function, and the penalty method is applied to enforce the boundary conditions, thus the final discretized equations of the IEFG method for anisotropic steady-state heat conduction problems can be obtained by combining with the corresponding Galerkin weak form. The influences of node distribution, weight functions, scale parameters and penalty factors on the computational accuracy of the IEFG method are analyzed respectively, and these numerical solutions show that less computational resources are spent when using the IEFG method.

Distributed optimal control problems for a class of elliptic hemivariational inequalities with a parameter and its asymptotic behavior

In this paper, we study optimal control problems on the internal energy for a system governed by a class of elliptic boundary hemivariational inequalities with a parameter. The system has been originated by a steady-state heat conduction problem with non-monotone multivalued subdifferential boundary condition on a portion of the boundary of the domain described by the Clarke generalized gradient of a locally Lipschitz function. We prove an existence result for the optimal controls and we show an asymptotic result for the optimal controls and the system states, when the parameter, like a heat transfer coefficient, tends to infinity on a portion of the boundary.

Existence, Comparison, and Convergence Results for a Class of Elliptic Hemivariational Inequalities

In this paper we study a class of elliptic boundary hemivariational inequalities which originates in the steady-state heat conduction problem with nonmonotone multivalued subdifferential boundary condition on a portion of the boundary described by the Clarke generalized gradient of a locally Lipschitz function. First, we prove a new existence result for the inequality employing the theory of pseudomonotone operators. Next, we give a result on comparison of solutions, and provide sufficient conditions that guarantee the asymptotic behavior of solution, when the heat transfer coefficient tends to infinity. Further, we show a result on the continuous dependence of solution on the internal energy and heat flux. Finally, some examples of convex and nonconvex potentials illustrate our hypotheses.

Conduction in Rings With Internal Heat Generation

Abstract The basic problem of steady-state heat conduction in a ring with internal heat generation and convective boundary conditions is considered. An exact solution is found for the ring with a rectangular cross section and an efficient Ritz method is presented for general cross sections. The latter is applied to the torus or the ring with a circular cross section. Hot spots and cold spots are determined.

A steady-state heat conduction problem of a nonhomogeneous conical body

Numerous studies and textbooks deal with the steady-state thermal conduction of radially nonhomogeneous circular cylinder. In contrast, there are relatively few studies on the thermal conduction problems of conical solid bodies. This study is intended as a modest contribution to the solution of thermal conductance problems of nonhomogeneous conical bodies. A one-dimensional steady-state heat conduction in nonhomogeneous conical body is considered. The thermal conductivity of the hollow conical body in a suitable chosen spherical coordinate system depends on the polar angle but is independent of the radial coordinate and azimuthal angle coordinate. A functionally graded type of material inhomogeneity is considered. All results of the paper are based on Fourier’s theory of heat conduction in solid bodies.

Meshless Analysis of Nonlocal Boundary Value Problems in Anisotropic and Inhomogeneous Media

In this work, meshless methods based on a radial basis function (RBF) are applied for the solution of two-dimensional steady-state heat conduction problems with nonlocal multi-point boundary conditions (NMBC). These meshless procedures are based on the multiquadric (MQ) RBF and its modified version (i.e., integrated MQ RBF). The meshless method is extended to the NMBC and compared with the standard collocation method (i.e., Kansa’s method). In extended methods, the interior and the boundary solutions are approximated with a sum of RBF series, while in Kansa’s method, a single series of RBF is considered. Three different sorts of solution domains are considered in which Dirichlet or Neumann boundary conditions are specified on some part of the boundary while the unknown function values of the remaining portion of the boundary are related to a discrete set of interior points. The influences of NMBC on the accuracy and condition number of the system matrix associated with the proposed methods are investigated. The sensitivity of the shape parameter is also analyzed in the proposed methods. The performance of the proposed approaches in terms of accuracy and efficiency is confirmed on the benchmark problems.

An extended finite element method formulation for modeling multi-phase boundary interactions in steady state heat conduction problems

The present paper proposes an XFEM formulation for heat transfer analysis of multi-phase materials with explicit treatment of boundary interactions. The existence of interfacial resistance at the boundaries of the material phases produces discontinuities in the temperature field and a standard finite element treatment would require complex domain discretizations and additional surface elements to capture the jumps of the temperature values. To overcome this problem, the proposed method captures these jumps by enriching the temperature field around the phase boundaries with appropriate discontinuous functions. Specifically, a new XFEM enrichment scheme is developed to address the issue of multiple-phase junctions, that is, areas where multiple boundaries with different properties intersect. This approach offers the advantage of bypassing the need for complex meshes required by the standard FE method and thus it significantly simplifies the analysis procedure. The elaborated methodology is first validated with existing results from the literature on heat conduction in polycrystalline materials. Then, a detailed model for heat conduction analysis of polymers reinforced with carbon-nanotubes is introduced, which takes into account the role of the interfacial resistance between different material phases. Even though the proposed method is demonstrated in heat conduction problems, it can be straightforwardly extended to other similar problem types, such us electrical conduction or equivalent mechanical properties.

Solving direct and inverse heat conduction problems in functionally graded materials using an accurate and robust numerical method

In this study we present a numerical approach to solve steady-state heat conduction problems in functionally graded materials (FGMs). Two different types of material gradations are considered for spatially varying thermal conductivity of FGM: (1) Quadratic material gradation; (2) Exponential material gradation. The proposed numerical procedure is based on finite-difference method and is developed to solve the steady-state heat conduction equation over a general two-dimensional (irregular) heat conducting body (FGM) with Dirichlet, Neumann, and Robin boundary conditions. In addition to presenting an accurate heat conduction equation solution considering an irregular shape and a variety of boundary conditions, the other novel aspect of this study is to identify the constant parameters in the material gradations accurately by an inverse analysis thereby determining the accurate form of gradation. The novelty of the inverse analysis lies in proposing an accurate and efficient explicit sensitivity analysis scheme. The main advantage of the sensitivity analysis scheme is that it is not involved with an adjoint equation and all the sensitivity coefficients can be explicitly computed in only one direct solution. The conjugate gradient method (CGM) is used to reduce the mismatch between the computed temperature on part of the boundary and the simulated measured temperature distribution. The accuracy, efficiency, and robustness of the proposed numerical approach are demonstrated through presenting two test cases.

Analyze the singularity of the heat flux with a singular boundary element

In the steady state heat conduction problem, the heat flux could be infinite at the re-entrant corner, at the point where the boundary condition is discontinuous, or at the place where the material properties are changing abruptly. The conventional numerical methods, such as the finite element method (FEM) and the boundary element method (BEM), have difficulties in analyzing the singular heat flux field. Herein, a new singular element employed in the boundary integral equation is developed to interpolate the heat flux field near the singular point. The shape function of the singular element is set as an asymptotic expansion model with respect to the distance from the singular point. The singular order in the asymptotic expansion can be determined by solving the singularity eigen equation on the basis of the interpolating matrix method. The self-adaptive co-ordinate transformation method is introduced to deal with the weakly singular integral in the proposed method. Benefited from the singular element, more accurate temperature and heat flux results near the singular point are obtained with fewer elements. In addition, the proposed method is easy to be coupled with the conventional BEM without too much modifications.

Nonlinear finite element analysis of temperature-dependent functionally graded porous micro-plates under thermal and mechanical loads

In the present study, a displacement based nonlinear finite element model for functionally graded porous micro-plates is developed based on the general third-order shear deformation plate theory and the modified couple stress theory. The developed finite element model accounts for von Kármán nonlinear strains, a power-law variation of two material constituents through the plate thickness, and different distributions of porosity with a constant volume of voids on static bending of micro-plates are analyzed. The length scale dependency is captured using a single parameter of the modified couple stress theory. A power-law distribution is assumed to model the variation of the two material constituents, while the porosity distributions vary according to cosine functions. The temperature-dependent properties are obtained using a cubic-spline interpolation from experimental data. A one-dimensional steady-state heat conduction problem is solved using the effective thermal conductivity of the porous material based on the Maxwell–Eucken model to obtain temperature distribution through the plate thickness. The Newton–Raphson iteration scheme is used to solve the nonlinear system of equations. A parametric study is conducted to demonstrate the effects of material and porosity parameters, temperature and length scale dependencies, and boundary conditions on the deflections and stress distributions.

Heat conduction in rectangular solids with internal heat generation

A representative steady-state heat conduction problem in rectangular solids with uniformly distributed heat generation has been investigated analytically. An analytical solution is provided by solving a non-homogeneous PDE. A simple and accurate model is proposed to predict the dimensionless shape factor parameter for the first time. The dimensionless shape factor is obtained in the light of the solution of Poisson equation with constant wall temperature boundary conditions. The area-mean temperature is found by integration on the rectangular cross-section. The model is very concise and nice for quick real world approximations, and it provides acceptable accuracy for engineering practice.

Isogeometric boundary element for analyzing steady-state heat conduction problems under spatially varying conductivity and internal heat source

This paper proposes an isogeometric boundary element (IGABEM) with the aim to solve the 2D steady-state heat conduction problems under spatially varying conductivity and internal heat source. The IGABEM boundary-domain integral equation is derived underlying the divergence theorem of Gauss and the Laplace equation. The non-uniform rational B-spline (NURBS) basis used in the CAD/CAE industries is applied to approximate both the boundary of problem domain and the unknown physical quantities, and the radial integration method (RIM) is employed to deal with the domain integrals induced by the non-homogeneous thermal conductivity and the internal heat source. The present boundary-domain integral equation is firstly divided into several NURBS patches, upon which the IGABEM can be further defined. The main advantage of the NURBS basis in comparison with the standard piecewise polynomial basis is that there is almost no error in the boundary approximation with IGABEM, which plays a significant role in controlling the accuracy of numerical calculation. Four numerical examples consisting of square regions with cubic and exponential thermal conductivities (with or without internal heat source), annulus region with constant and quadratic conductivities as well as complex region with varying thermal conductivity and complex heat source are provided. The accuracy and convergence of the proposed method are assessed by comparing the present solution with the existing results obtained by several other researchers or ANSYS results, and excellent performance is achieved.

Bounds for the Thermal Conductance of Body of Rotation

A mathematical heat transfer model is developed for the steady-state heat transfer through a homogeneous isotropic body of rotation. The developed model is used to obtain estimations for two-sided thermal heat transfer conductance in case of a special type of body of rotation. The body of rotation considered is bounded by the coordinate surfaces of an orthogonal curvilinear coordinate system. The equations of the Fourier’s theory of heat conduction in solid body are used to formulate the corresponding thermal boundary value problem. The studied steady-state heat conduction problem is axisymmetric. The determination of the heat flow is based on the concept of the thermal conductance the inverse of which is the thermal resistance. The exact (strict) value of thermal conductance is known only for bodies with very simple shapes, therefore, the principles and the methods that can be used for creating lower and upper bounds to the numerical value of thermal conductance are important. The derivation of the upper and the lower bound formulae for the heat conductance of axisymmetric ring-like body is based on the two types of Cauchy–Schwarz inequality. The condition of equality of the derived lower and upper bounds is examined. Several examples illustrate the applications of the derived upper and lower bound formulae. The presented method can be used not only for the estimation of thermal conductance. For example, similar formulation can be given to obtain two-sided bounds for the electrical resistance of solid body conductor in the case of spatially steady flow of electric charges.

A locally refined adaptive isogeometric analysis for steady-state heat conduction problems

This paper presents an isogeometric analysis with adaptivity using locally refined B-splines (LR B-splines) for steady-state heat conduction simulations in solids. Within this framework, the LR B-splines, which have an efficient and simple local refinement algorithm, are used to represent the geometry, and are also employed for spatial discretization, thus providing a seamless interaction between the CAD models and the numerical analysis. A Zienkiewicz-Zhu a posteriori error estimator in terms of the temperature gradient recovery is used to identify the regions for local mesh refinement. The accuracy and convergence properties of the proposed framework are demonstrated through several two-dimensional isotropic examples. Numerical results indicate good performance of the present method as the adaptive refinement technique yields more accurate solution compared to the uniform global refinement with an improved convergence rate.

Explicit solutions for distributed, boundary and distributed-boundary elliptic optimal control problems

We consider a steady-state heat conduction problem in a multidimensional bounded domain $$\Omega $$ for the Poisson equation with constant internal energy g and mixed boundary conditions given by a constant temperature b in the portion $$\Gamma _1$$ of the boundary and a constant heat flux q in the remaining portion $$\Gamma _2$$ of the boundary. Moreover, we consider a family of steady-state heat conduction problems with a convective condition on the boundary $$\Gamma _1$$ with heat transfer coefficient $$\alpha $$ and external temperature b. We obtain explicitly, for a rectangular domain in $${\mathbb {R}}^{2}$$ , an annulus in $${\mathbb {R}}^{2}$$ and a spherical shell in $${\mathbb {R}}^{3}$$ , the optimal controls, the system states and adjoint states for the following optimal control problems: a distributed control problem on the internal energy g, a boundary optimal control problem on the heat flux q, a boundary optimal control problem on the external temperature b and a distributed-boundary simultaneous optimal control problem on the source g and the flux q. These explicit solutions can be used for testing new numerical methods as a benchmark test. In agreement with theory, it is proved that the system state, adjoint state, optimal controls and optimal values corresponding to the problem with a convective condition on $$\Gamma _1$$ converge, when $$\alpha \rightarrow \infty $$ , to the corresponding system state, adjoint state, optimal controls and optimal values that arise from the problem with a temperature condition on $$\Gamma _1$$ . Also, we analyze the order of convergence in each case, which turns out to be $$1/\alpha $$ being new for these kind of elliptic optimal control problems.

Calculating 3-D multi-domain heat conduction problems by the virtual boundary element-free Galerkin method

A new boundary-type meshfree collection method, namely the virtual boundary element-free Galerkin method (VBEFGM), is demonstrated for three-dimension (3-D) multi-domain steady-state heat conduction problems. The virtual source functions of the virtual nodes are constructed by the radial basis function interpolation. And the virtual and real boundaries are discreted into the virtual and real nodes. 3-D multi-domain steady-state heat conduction problems are solved without the background integral element by VBEFGM. Using the Galerkin method to obtain the formula of VBEFGM, the proposed method has the advantages of the virtual boundary element-free method, the meshfree collection method and the Galerkin method. It is easily programed and extended to solve other more complicated 3-D heat conduction problems, since the detailed calculation scheme is deduced and shown. Two numerical examples of the boundary conditions varying with the space are computed. The computational results are compared with other numerical methods. The accuracy and stability of the proposed method are validated for 3-D multi-domain steady-state heat conduction problems.

Two-dimensional steady-state heat conduction problem for heat networks

The method of determining the level of heat losses through structures of heat networks and the two-dimensional steady-state heat conduction problem for multilayer structures of heat networks have been considered, given that each of the layers has its own physical properties and heat transfer coefficients depend on r and z coordinates. The definition of heat losses is an important practical task, since they are one of the main indicators of energy efficiency in the operating heat networks and are included in heat tariffs.

Singular boundary method for 2D and 3D heat source reconstruction

This paper presents the singular boundary method (SBM) in conjunction with the regularization method to recover the heat source in steady-state heat conduction problems from boundary temperature and heat flux measurements. To avoid the internal measurements, the heat source is assumed to satisfy a second-order partial differential equation on a physical basis. As a result, the source inverse problems can be transformed to a direct fourth-order governing equation, which can be tackled by the SBM. The SBM utilizes the fundamental solutions to approximate the solutions and desingularizes the source singularities of fundamental solutions with origin intensity factors (OIFs), which are derived for both 2D and 3D cases. The regularization method is introduced to deal with noisy boundary. The feasibility and accuracy of the proposed method is validated by both 2D and 3D cases.