Application of Monte Carlo Method to Steady State Heat Conduction Problems

This paper presents a multi-domain boundary face method (MD-BFM) for solving steady-state heat conduction problems on large-scale engineering structures. In the BFM, both boundary integration and variable approximation are performed in the 2-D parametric space of the boundary surfaces of the body in question. The surface parameters are obtained through the boundary representation data structure (B-rep) implanted in all CAD packages. The geometric quantities in the boundary integrals of the BIE are computed directly in the parametric spaces of the surfaces to avoid geometric errors. On the other hand, a MDBFM is necessary for structures of complex shapes or constituted of different kinds of materials. In this work, the multi-domain BFM is developed and applied for heat conduction analysis of a real gravity dam. Numerical results have demonstrated that our method can achieve comparable accuracy than other methods (e.g. the FEM), but at much lower costs in terms of both computer

Recent research has identified Adini’s rectangular element as an efficient higher order element for solving second order elliptic partial differential equations such as Poisson’s equation, which governs the steady state heat conduction problem. This type of element includes both the primary field variable and its spatial derivatives as nodal degrees of freedom. Compared to the conventional cubic elements of the Serendipity and Lagrange families, Adini’s element includes the minimum number of nodes per element and has the advantage that the nodal values of the spatial derivatives of the temperature field are directly retrieved from the FEM solution. As a result, the differentiation and averaging procedures that are typically used to obtain the nodal values of the temperature gradients are avoided. In this paper a generalized version of Adini’s element for solving two-dimensional steady state heat transfer problems in non-rectangular geometries is presented. Also, the traditional finite element formulation is modified to allow the application of essential boundary conditions without having to constrain the nodal values of the tangential derivative of the temperature. The resulting higher order element and modified FEM formulation are used to solve an example problem and the accuracy of the solution is compared with solutions obtained using the traditional linear, quadratic, and cubic Serendipity elements to show the efficiency, in terms of accuracy per number of degrees of freedom, of the proposed approach for finding the nodal values of the temperature gradients, which are required to compute the nodal values of the heat flux vector.

Numerical solution of an inverse steady state heat conduction problem

MLS-based numerical manifold method for steady-state three-dimensional heat conduction problems of functionally graded materials

ABSTRACTAn inverse analysis is used to simultaneously estimate the thermal conductivity and the boundary shape in steady-state heat conduction problems. The numerical scheme consists of a body-fitted grid generation technique to mesh the heat conducting body and solve the heat conduction equation – a novel, efficient, and easy to implement sensitivity analysis scheme to compute the sensitivity coefficients, and the conjugate gradient method as an optimization method to minimize the mismatch between the computed temperature distribution on some part of the body boundary and the measured temperatures. Using the proposed scheme, all sensitivity coefficients can be obtained in one solution of the direct heat conduction problem, irrespective of the large number of unknown parameters for the boundary shape. The obtained results reveal the accuracy, efficiency, and robustness of the proposed algorithm.

Heat conduction [1] is usually modeled as a diffusion process embodied in heat conduction equation. The traditional numerical methods [2, 3] for heat conduction problems such as the finite difference or finite element are well developed. However, these methods are based on discretized mesh systems, thus they are inherently limited in the geometry treatment. This chapter describes the Monte Carlo method that is based on particle transport simulation to solve heat conduction problems. The Monte Carlo method is “meshless” and thus can treat problems with very complicated geometries. The method is applied to a pebble fuel to be used in very high temperature gas-cooled reactors (VHTGRs) [4], which is a next-generation nuclear reactor being developed. Typically, a single pebble contains ~10,000 particle fuels randomly dispersed in graphite– matrix. Each particle fuel is in turn comprised of a fuel kernel and four layers of coatings. Furthermore, a typical reactor would house several tens of thousands of pebbles in the core depending on the power rating of the reactor. See Fig. 1. Such a level of geometric complexity and material heterogeneity defies the conventional mesh–based computational methods for heat conduction analysis. Among transport methods, the Monte Carlo method, that is based on stochastic particle simulation, is widely used in neutron and radiation particle transport problems such as nuclear reactor design. The Monte Carlo method described in this chapter is based on the observation that heat conduction is a diffusion process whose governing equation is analogous to the neutron diffusion equation [5] under no absorption, no fission and one speed condition, which is a special form of the particle transport equation. While neutron diffusion approximates the neutron transport phenomena, conversely it is applicable to solve diffusion problems by transport methods under certain conditions. Based on this idea, a new Monte Carlo method has been recently developed [6-8] to solve heat conduction problems. The method employs the MCNP code [9] as a major computational engine. MCNP is a widely used Monte Carlo particle transport code with versatile geometrical capabilities. Monte Carlo techniques for heat conduction have been reported [10-13] in the past. But most of the earlier Monte Carlo methods for heat conduction are based on discretized mesh systems, thus they are limited in the capabilities of geometry treatment. Fraley et al[13] uses a “meshless” system like the method in this chapter but does not give proper treatment to the boundary conditions, nor considers the “diffusion-transport theory correspondence” to be described in Section 2.2 in this chapter. Thus, the method in this chapter is a transport theory treatment of the heat conduction equation with a methodical boundary correction. The

We propose an efficient boundary collocation method for the solution of certain two- and three-dimensional problems of steady-state heat conduction in isotropic bimaterials. In particular, in two dimensions we consider the case where a circular region composed of one material is coated with an annular region of another material. In three dimensions, we examine the corresponding case for axisymmetric domains. The proposed method involves the use of a domain decomposition technique in conjunction with a matrix decomposition algorithm. The circulant structure of the matrices appearing in this method is exploited by using fast Fourier transforms. The method is tested numerically on several problems.

The effective thermal conductivity of composite materials can be calculated when the temperature distribution in the body is known.The present paper deals with a numerical analysis on the effective thermal conductivity of filament assembly, where the filaments lie in parallel and at regular intervals with each other. In this case, a filament of circular cross section surrounded by matrix can be considered as a unit model of rectangular profile. Further, it is assumed that two surfaces of the model are isothermal and opposite to each other in parallel to the filament axis, and that residual others are insulated. Then a quarter of the model can be used as an element. The mumerical analysis is applied to this model as the two-dimentional steady state heat conduction problem, and the procedure is as follows:(1) To replace the element under consideration by a various rectangular-grid network.(2) Calculation of the temperature distribution in the element by means of the so-called iteration method.(3) Calculation of the effective thermal conductivity using the rates of heat flow in and out across the isothermal surfaces of the element.From the method described above, a new result is obtained. The value K/Km of the filament assembly composed of circular filaments lying in a line and in contact with two opposite isothermal surfaces may be expressed by the following equation:K: Effective thermal conductivity of filament assembly Ks: Thermal conductivity of a filament Km: Thermal conductivity of matrix f: Filament volume fractionThis equation will be confirmed in the experiments in the succeeding paper.

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

We studied the problem of one-dimensional steady-state heat conduction in metallic foams used as thermal protection systems with varying density in the thickness direction. The thermal conductivity of the foam is a function of temperature as well as the density, and it has a minimum value in the range of densities of our interest. An optimality criterion in the form of a differential equation was derived in order to minimize the total mass of the insulation for a given heat input. The heat-conduction equation and the optimality equation were solved numerically to obtain optimum density profiles for various values of thickness of the insulation. It is shown that for a given thickness using functionally graded foams can reduce the mass of the insulation panel. Acknowledgments

The problem of determination of the stationary temperature field in a body with plane sheet of heat sources is reduced to the solution of an integral equation of the first kind. A method for the determination of the set of solutions to this equation is proposed. The components of the vector of elastic displacements and the components of the tensor of temperature stresses are found according to the known temperature field by using the equations of thermoelasticity. We study the distributions of thermal displacements and stresses for any axially symmetric distribution of temperature in a circular domain. Numerous elements of contemporary engineering constructions operate under conditions of nonuniform heating that promote the formation of temperature gradients and thermal stresses. For the comprehensive analysis of the strength of structures, it is necessary to know the levels of thermal stresses and the character of their action. The investigation of the thermoelastic state of heat-generating elements is of high importance for practical purposes [9]. The presence of thin heat-generating inclusions in a body leads to the formation of tensile or compressive stresses. In the first case, the action of these stresses, together with the action of the other factors, may lead to crack initiation and propagation. In the second case, the action of stresses may decrease the intensity of stresses in the vicinity of the existing crack under force loading and promote the formation of conditions inhibiting the process of crack growth [10]. In the investigation of the stressed state of a body with thermally active cracks (with temperature or heat fluxes given on these cracks), the determination of stresses acting at the sites of the cracks is an intermediate stage and then the stresses acting upon the crack surfaces are removed. The major part of investigations of the stress-strain state of bodies with circular thermally active or heatinsulated cracks are performed for axially symmetric problems by the method of integral Hankel transformation and dual integral equations [11–16]. In [3], a method of two-dimensional boundary integral equations based on the theory of the Newton potential is developed for the solution of three-dimensional problems of stationary heat conduction and thermoelasticity for bodies with cracks. According to this method, the densities of potentials in the problem of heat conduction are the intensities of heat sources at the sites of the cracks determined, for given temperatures on the cracks, from the integral equations with polar kernels. The temperature field and stressed state of a body with given temperature or heat flow in a circular domain described by polynomials of the third degree are investigated in [4]. In the axially symmetric case, for polynomials of any degree, this problem is studied in [5]. According to the classic statement of the problems of stationary heat conduction with thermally active inclusions [4, 5], the presence of heat sources is postulated solely in the domain of the inclusion, which leads to singular distributions of the heat flows on its edges.

This paper proposes a noniterative analytical thermal model for 3-D integrated circuits (3-D-ICs) based on the solution of the governing energy equations using Green’s function, separation of variables, and superposition methods. The model is applicable to both transient and steady-state heat conduction problems. Any finite number of dies equally sized in plane can be taken into consideration in current analysis. In addition, the analytical solution presented here is very general, so that it takes into account the different interface boundary conditions between the dies. Importantly, the thermal properties (i.e., $k$ , $\rho $ , $c$ ) and thickness of different dies can be different, and the spatial and time-varying volumetric heat generation can be applied in each of the different dies. The results compared with the finite-element simulations show that the temperature fields predicted by the proposed model can offer an accurate and efficient solution to compute the temperature fields in the representative 3-D-ICs, with a maximum deviation between the two being around 2.0% in the transient heat conduction. In the steady-state case, the maximum deviation can reach even less than 1.0%.

Introduction.- Mathematical Preliminaries.- Regularization of moment problems by trancated expansion and by the Tikhonov method.- Backus-Gilbert regularization of a moment problem.- The Hausdorff moment problem: regularization and error estimates.- Analytic functions: reconstruction and Sinc approximations.- Regularization of some inverse problems in potential theory.- Regularization of some inverse problems in heat conduction.- Epilogue.- References.- Index.

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

A new coarse-mesh computational method for the numerical solution of heat conduction and fluid flow problems is formally developed and applied to sample problems. The method is based upon formal use of Green's functions, which are defined locally over subdomains of the original system under consideration. The formal development of the local Green's function method for the solution of heat conduction problems is presented and discussed. Numerical solutions of sample problems for one-dimensional heat conduction with constant thermal conductivity, one-dimensional heat conduction with temperature-dependent thermal conductivity, and two-dimensional heat conduction with constant thermal conductivity are given, and these results are compared with results obtained using the finite difference and finite element methods. The formal development of the local Green's function method for the solution of fluid flow problems is then also presented and discussed; the numerical solution of a sample problem for simple one-dimensional incompressible fluid flow with viscous heating is also given.