A Precise Time-Domain Expanding Boundary-Element Method for Solving Three-Dimensional Transient Heat Conduction Problems with Variable Thermal Conductivity

A combined approach of the radial integration boundary-element method (RIBEM) and the precise algorithm in the time domain is presented for solving three-dimensional transient heat conduction problems with variable thermal conductivity. First, by expanding physical quantities at discrete time intervals, the recursive formulation of the governing equation is derived. Then, the recursive equation is solved by the RIBEM, and a self-adaptive check technique is carried out to estimate how many expansion terms are needed in a time step. Finally, three numerical examples show that the present approach can obtain very stable and accurate results for different time-step size.     

A Precise Integration Boundary-Element Method for Solving Transient Heat Conduction Problems with Variable Thermal Conductivity

In this article, a combined approach of the radial integration boundary element method (RIBEM) and the precise integration method is presented for solving transient heat conduction problems with variable thermal conductivity. First, the system of ordinary differential equations on the boundary integral equation can be obtained by the RIBEM. Then, the precise integration method is adopted to solve the system of ordinary differential equations. Finally, three numerical examples are presented to demonstrate the performance of the present method. The results show that the present approach can obtain satisfactory performance even for very large time-step size.     

Non-Fourier Heat Conduction Problems and the Use of Exponential Basis Functions

A solution method using exponential basis functions (EBFs) is proposed for transient one-/two-dimensional non-Fourier heat conduction problems having particular application in bio-heat fields. A summation of EBFs satisfying the governing differential equation is considered in time and space. The presented method uses a noniterative algorithm for the solution of direct/inverse problems. It is demonstrated that the use of extra EBFs in the form of enrichment functions significantly improves the results when some jumps are seen in the input data. Four numerical examples, including bio-heat conduction problems, are provided to investigate the accuracy and performance of the method presented.     

A Combined Approach of RIBEM and Precise Time Integration Algorithm for Solving Transient Heat Conduction Problems

A combined approach of the radial integration boundary-element method (RIBEM) and the precise algorithm in the time domain is presented for solving transient heat conduction problems with heat sources. First, by expanding physical quantities at discrete time intervals, the recursive formulations of the governing equation are derived. Then, the recursive equation is solved by the RIBEM. Finally, a self-adaptive check technique is carried out to estimate how many expansion terms are needed in a time-step size. Numerical results show that the present approach can obtain satisfactory performance.     

Improved Cuckoo Search Algorithm for Solving Inverse Geometry Heat Conduction Problems

Cuckoo Search (CS) algorithm has shortcomings of weak local search ability, slow convergence speed, and low accuracy. In order to overcome these disadvantages, an improved CS algorithm based on Broyden–Fletcher–Goldfarb–Shanno (BFGS) algorithm (CS-BFGS) is proposed for solving inverse geometry heat conduction problems, and the physical field is the steady-state heat conduction. Firstly, the unknown initial boundary is evolved by Lévy flights and elimination mechanism. Then the BFGS algorithm is applied to minimize the objective function. Finally, the influences of random errors, measurement point number, and measurement point position on the inverse results are investigated. The results show that the CS-BFGS algorithm has higher accuracy and faster convergence speed than BFGS and CS algorithm. With the decrease of measurement errors, the increase of measurement point number, and measurement point position closer to the inverse boundary, the results become more accurate.     

Equivalence Principle for Analyzing Steady-State Heat Conduction Problems

This article presents the concept of equivalence principle in the analysis of steady-state heat conduction problems. A surface integral equation formulation is established for homogeneous thermal media and a volume integral equation formulation for inhomogeneous isotropic/anisotropic thermal media. These formulations are analogous to those commonly used in electromagnetic scattering problems and can be solved with the same numerical algorithms used for electromagnetic analysis. This feature makes the proposed theory useful for simultaneous analysis of electromagnetic fields and heat conduction in an electronic system.     

Solving the Temperature Distribution Field in Nonlinear Heat Conduction Problems Using the Hopfield Neural Network

This article employs the continuous-time analog Hopfield neural network (CHNN) to compute the temperature distribution in nonlinear heat conduction problems. The relationship between the CHNN synaptic connection weights and the governing equations of the nonlinear heat conduction problems is established and a corresponding network connectivity circuit design scheme proposed. The CHNN algorithm is used to solve the heat equation for conduction problems with a power-law nonlinearity. The results confirm that the proposed CHNN scheme provides an accurate means of solving the transient temperature distributions of nonlinear heat conduction problems on a real-time basis.     

Numerical Simulation of Dual-Phase-Lag Model and Inverse Fractional Single-Phase-Lag Problem for the Non-Fourier Heat Conduction in a Straight Fin

In recent years, many studies have been done on heat transfer in the fin under unsteady boundary conditions using Fourier and non-Fourier models. In this paper, unsteady non-Fourier heat transfer in a straight fin having an internal heat source under periodic temperature at the base was investigated by solving numerically Dual-Phase-Lag and Fractional Single-Phase-Lag models. In this way, the governing equations of these models were presented for heat conduction analysis in the fin, and their re...     

Temperature-Constrained Topology Optimization of Transient Heat Conduction Problems

A regional temperature measure model is constructed to obtain a small number of temperature constraints for local transient temperature control. The temperature sensitivity is derived using the adjoint variable method. The multiple temperature criteria and three-phase topology optimization are further investigated for transient heat conduction design. The material layout design of transient heat conduction is replaced by a static optimization problem, which is subsequently solved by the method of moving asymptotes. Finally, several numerical examples are provided to demonstrate the feasibility and validity of the proposed topology optimization for transient heat conduction problems.     

Lattice Boltzmann Method Applied to the Solution of Energy Equation of a Radiation and Non-Fourier Heat Conduction Problem

This article concerns the application of the lattice Boltzmann method (LBM) to solve the energy equation of a combined radiation and non-Fourier conduction heat transfer problem. The finite propagation speed of the thermal wave front is accounted by non-Fourier heat conduction equation. The governing energy equation is solved using the LBM. The finite-volume method (FVM) is used to compute the radiative information. The formulation is validated by taking test cases in 1-D planar absorbing, emitting, and scattering medium whose west boundary experiences a sudden rise in temperature, or, with adiabatic boundaries, the medium is subjected to a sudden localized energy source. Results are analyzed for the various values of parameters like the extinction coefficient, the scattering albedo, the conduction-radiation parameter, etc., on temperature distributions in the medium. Radiation has been found to help in facilitating faster distribution of energy in the medium. Unlike Fourier conduction, wave fronts have been found to reflect from the boundaries. The LBM-FVM combination has been found to provide accurate results.     

ADI Method for Heat Conduction Problems in an Orthogonal Coordinate System

A mathematical model has been developed for the time-dependent heat conduction process in a region of arbitrary geometry. An alternating direction implicit (ADI) scheme has been worked out for the computer solution of the problem in three space dimensions. The time and nodal propagation of the errors are depicted graphically. Also included in the computer software is a subroutine for plotting the isotherms in various geometries.     

Resolution of Unknown Heat Source Inverse Heat Conduction Problems Using Particle Swarm Optimization

The main objective of this article is to solve inverse heat conduction problems with the particle swarm optimization method. An enhanced particle swarm optimization (EPSO) algorithm is proposed to overcome the shortcoming of earlier convergence of standard PSO algorithms. The EPSO is used to estimate the unknown time-dependent heat source in complex regions. Numerical experiments indicate the validity and stability of the EPSO method.     

Heterogeneous Heat Conduction Problems by an Improved Element-Free Galerkin Method

In this article an improved element-free Galerkin method is proposed to solve heat conduction problems with heterogeneous media. Because the method almost possesses interpolation property, the implementation of essential boundary condition is as simple as that in the finite-element method. In order to validate the proposed method, several heat conduction problems with different degrees of heterogeneity are presented. In these test problems, we focus on the influence of nodal distribution to the proposed method for heat conduction problems with heterogeneous media. It is shown that, for different degrees of heterogeneity, regardless of matter whether the node is located on the interface, accurate solutions can be obtained by the proposed method for heterogeneous heat conduction problems without a source term.     

Level Set-based Topological Shape Optimization of Nonlinear Heat Conduction Problems

A level set-based topological shape optimization method is developed for nonlinear heat conduction problems. While minimizing the objective function of instantaneous thermal compliance and satisfying the constraint of allowable volume, solution of the Hamilton-Jacobi equation leads the initial boundary to an optimal one according to the normal velocity field determined from the descent direction of the Lagrangian. To overcome the convergence difficulty in nonlinear problems resulting from introduction of an approximate boundary, an actual boundary is identified by tracking the level set functions and remeshing using Delaunay triangulation. The velocity field outside the actual domain is determined through a velocity extension scheme.     

Resolving the Final Time Singularity in Gradient Methods for Inverse Heat Conduction Problems

This work investigates overall performance of the modification techniques for resolving the singularity at the final time in the gradient method for the inverse heat conduction problem. Four representative methods are selected based on the literature and analyzed for the same case. They are the regularization term method, the differential equation method, the gradient integration method, and the sequential gradient method. All four methods are reproduced and tested for the same test case. Based on the test results, a two-step method that can both alleviate the systematic bias and at the same time resolve the singularity is proposed.     

Inverse Transient Heat Conduction Problems of a Multilayered Functionally Graded Cylinder

Inverse transient heat conduction problems of a multilayered functionally graded (FG) cylinder are presented. The approach is based on measurement of temperature on the outer surface of the cylinder to estimate the heat flux and convection heat transfer coefficient on its inner surface. The non-Fourier heat transfer equation is employed to accurately formulate the problem. The conjugate gradient method (CGM) is used for the optimization procedure and the incremental differential quadrature method (DQM) is applied to solve the direct, sensitivity, and adjoint problems. The accuracy of the presented approach is examined by simulating the exact and noisy data through different examples. Good accuracy of the obtained results validates the presented approach.     

Topological Shape Optimization of Heat Conduction Problems using Level Set Approach

ABSTRACT

A topological shape optimization method for heat conduction problems is developed using a level set method. The level set function obtained from the “Hamilton-Jacobi type” equation is embedded into a fixed initial domain to implicitly represent thermal boundaries and obtain the finite-element response and adjoint sensitivity. The developed method minimizes the thermal compliance, satisfying the constraint of allowable volume by varying the implicit boundary. During optimization, the boundary velocity to integrate the Hamilton-Jacobi equation is obtained from the optimality condition. The newly developed method shows no numerical instability and makes it easy to represent topological shape variations.     

A New High-Precision Boundary-Type Meshless Method for Calculation of Multidomain Steady-State Heat Conduction Problems

This article presents the idea for calculating 2-D steady-state heat conduction problems with multidomain combination by employing the virtual boundary meshless least-square method. Being different from the conventional virtual boundary-element method (VBEM), this method incorporates the point interpolation method (PIM) with the compactly supported radial basis function (CSRBF) to approximately construct the virtual source function of the VBEM. Thus, the proposed method has the advantages of both the boundary-type meshless method and the virtual boundary element method. Since the configuration of the virtual boundary requires a certain preparation, the integration along the virtual boundary can be carried out over the smooth simple curve that can be structured beforehand (for 2-D problems) to reduce the complexity and difficulty of calculus without loss of accuracy, while the “vertex question” existing in the BEM can be avoided. Numerical examples show that the proposed method is more precise than several other numerical methods while selecting fewer degrees of freedom. In addition, its numerical stability is also verified by computing several cases.     

无机热传导技术用于烟气余热再利用

无机热传导技术是以无机元素为导热介质,将其注入类金属或非金属管内,经密封形成导热元件。受热时间利用分子震荡、摩擦,将热能快速激发、传递。唐钢高速线材厂二车间就是将这种传热元件制成蒸汽发生器和煤气预热器,利用烟气二次余热产生蒸汽同时预热煤气,设备小巧,传热效率高。     

The Method of Fundamental Solutions for Solving the Backward Heat Conduction Problem with Conditioning by a New Post-Conditioner

We consider a backward heat conduction problem (BHCP) in a slab, subject to noisy data at final time. The BHCP is known to be highly ill-posed. In order to stably solve the BHCP by a numerical method, we employ a new post-conditioner in the linear system obtained by the method of fundamental solutions (MFS), and then we use the conjugate gradient method (CGM) to solve the post-conditioned linear system to determine the unknown coefficients used in the expansion by the MFS. The method can retrieve the initial data rather well, with a certain degree of accuracy. Several numerical examples of the BHCP demonstrate that the present method is applicable, even for those of strongly ill-posed problems with a large value of final time and with large noise. We also demonstrate that the CGM alone is not enough to accurately recover the initial temperature.