Reconstruction of heat transfer coefficients using the boundary element method

This paper describes the reconstruction of the heat transfer coefficient (space, Problem I, or time dependent, Problem II) in one-dimensional transient inverse heat conduction problems from surface temperature or average temperature measurements. Since the inverse problem posed does not involve internal temperature measurements, this means that non-destructive testing of materials can be performed. In the formulation, convective boundary conditions relate the boundary temperature to the heat flux. Numerical results obtained using the boundary element method are presented and discussed.     

Numerical determination of boundary heat fluxes in an enclosure dynamically with natural convection through Fletcher-Reeves gradient method

An iterative Fletcher-Reeves conjugate gradient method (CGM) is adopted to estimate the boundary heat fluxes in a fluid-saturated enclosure, where the fluid flow is dynamically coupled with the heat convection. The sets of direct, sensitivity and adjoint equations required for the solution of the inverse problem are formulated in terms of an arbitrary domain in two dimensions. The methodology of conjugate gradient method solves the inverse natural convection problem satisfactorily without any a priori information about the unknown heat fluxes. The pressure-correction method is utilized to solve the continuum direct, sensitivity and adjoint problems by enforcing global mass and energy conservations. Effects of boundary heat flux profile and thermal Rayleigh number on the convective heat transport are investigated. The effects of position and number of temperature sensors on the inverse problem solution are also addressed in this paper. Inverse solutions of noise data are regularized with the Discrepancy Principle of Alifanov; otherwise, the high frequency components of the random noise were reproduced.     

Solvability of the Nonlocal Inverse Parabolic Problem and Numerical Results

In this paper, we consider the unique solvability of the inverse problem of determining the right-hand side of a parabolic equation whose leading coefficient depends on time variable under nonlocal integral overdetermination condition. We obtain sufficient conditions for the unique solvability of the inverse problem. The existence and uniqueness of the solution of the inverse parabolic problem upon the data are established using the fixed point theorem. This inverse problem appears extensively in the modelling of various phenomena in engineering and physics. For example, seismology, medicine, fusion welding, continuous casting, metallurgy, aircraft, oil and gas production during drilling and operation of wells. In addition, the numerical solution of the inverse problem is studied by using the Crank-Nicolson finite difference method together with the Tikhonov regularization to find a stable and accurate approximate solution of finite differences. The resulting nonlinear system of parabolic equation is solved computationally using the MATLAB subroutine lsqnonlin. Both analytical and numerically simulated noisy input data are inverted. The root mean square error values for various noise levels for both continuous and discontinuous time-dependent heat source term are compared. Numerical results presented for two examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data. Furthermore, the choice of the regularization parameter is also discussed based on the trial and error technique.     

A meshless method to the numerical solution of an inverse reaction–diffusion–convection problem

In this paper, the determination of the source term in a reaction–diffusion convection problem is investigated. First with suitable transformations, the problem is reduced, then a new meshless method based on the use of the heat polynomials as basis functions is proposed to solve the inverse problem. Due to the ill-posed inverse problem, the Tikhonov regularization method with a generalized cross-validation criterion is employed to obtain a numerical stable solution. Finally, some numerical examples are presented to show the accuracy and effectiveness of the algorithm.     

Reconstruction of the perfusion coefficient from temperature measurements using the conjugate gradient method

We consider the inverse bio-heat transfer problem to determine the space- and time-dependent perfusion coefficient from temperature measurements. In this formulation, the problem is fully determined and the coefficient is identifiable if and only if the temperature has dense support. However, the problem is still ill-posed since small errors in the measured temperature cause large errors in the output perfusion coefficient due to the numerical differentiation of noisy data involved which represents an unstable procedure. In order to overcome this difficulty and restore stability, we employ for the first time the conjugate gradient method (CGM) for solving the inverse problem under investigation. Regularization is achieved by stopping the iteration process at an appropriate threshold dictated by the discrepancy principle. Numerical results show that the CGM is accurate and reasonably stable in retrieving the perfusion coefficient. Moreover, comparison with other methods shows improved efficiency and stability in inverting noisy data.     

Determination of a time-dependent heat transfer coefficient from non-standard boundary measurements

In this paper the determination of the time-dependent heat transfer coefficient in one-dimensional transient heat conduction from a non-standard boundary measurement is investigated. For this inverse nonlinear ill-posed problem the uniqueness of the solution holds. Numerical results obtained using the boundary element method are presented and discussed.     

The truncation method for identifying the heat source dependent on a spatial variable

This paper deals with the ill-posed problem of determining an unknown source depending only on the spatial variable in the heat equation. In this problem, the truncation method is employed to get the regularization solution. Moreover, the Hölder type error estimate is also obtained between this regularization solution and the exact solution. Various typical examples are given to verify the efficiency and accuracy of the proposed method.     

Inverse resistivity problem: Geoelectric uncertainty principle and numerical reconstruction method

Mathematical model of vertical electrical sounding by using resistivity method is studied. The model leads to an inverse problem of determination of the unknown leading coefficient (conductivity) of the elliptic equation in R²$R^2$ in a slab. The direct problem is obtained in the form of mixed BVP in axisymmetric cylindrical coordinates. The additional (available measured) data is given on the upper boundary of the slab, in the form of tangential derivative. Due to ill-conditionedness of the considered inverse problem the logarithmic transformation is applied to the unknown coefficient and the inverse problem is studied as a minimization problem for the cost functional, with respect to the reflection coefficient. The Conjugate Gradient method (CGM) is applied for the numerical solution of this problem. Computational experiments were performed with noise free and random noisy data.     

应用QPSO算法求解二维热传导反问题

热传导反问题在国内研究起步较晚,研究方法有很多,但通常方法很难较好地接近全局最优。在经典的微粒群优化算法(PSO)的基础上,通过研究基于量子行为的微粒群优化算法(QPSO)提出了应用基于量子行为的微粒群优化算法进行二维热传导参数优化,具体介绍依据目标函数如何利用上述的算法去寻找最优参数组合。在具体应用中为了提高算法的收敛性和稳定性对算法进行了改进,并进行了大量实验,结果显示在解决热传导反问题优化问题中,基于QPSO算法的性能优越,证明QPSO在热传导领域具有很大的实际应用价值。     

Simultaneous space diffusivity and source term reconstruction in 2D IHCP

A numerical marching scheme is presented for the simultaneous recovery of the diffusivity coefficient, spatial source term, temperature, and heat flux distributions in the two-dimensional inverse heat conduction problem when noisy data at the active boundary and the initial measured temperature distribution are given. No information about the quality and/or quantity of the noise is assumed. A proof of stability and convergence of the algorithm are provided together with several numerical examples of interest.     

Simultaneous identification of Robin coefficient and heat flux in an elliptic system

In this paper, we shall derive and propose an efficient algorithm for simultaneously reconstructing the Robin coefficient and heat flux in an elliptic system from part of the boundary measurements. The uniqueness of the simultaneous identification is demonstrated. The ill-posed inverse problem is formulated into an output least-squares nonlinear and non-convex minimization with Tikhonov regularization, while the regularizing effects of the regularized system are justified. The Levenberg–Marquardt method is applied to change the non-convex minimization into convex minimization, which will be solved by surrogate functional method so as to get the explicit expression of the minimizer. Numerical experiments are provided to show the accuracy and efficiency of the algorithm.     

Source term identification for an axisymmetric inverse heat conduction problem

We consider an inverse heat source problem of determining the heat source term from the final temperature history of a cylinder. This problem is ill-posed. A simplified Tikhonov regularization method is applied to formulate regularized solution, which is stably convergent to the exact one with a logarithmic type error estimate.     

A method of fundamental solutions for radially symmetric and axisymmetric backward heat conduction problems

We propose and investigate an application of the method of fundamental solutions (MFS) to the radially symmetric and axisymmetric backward heat conduction problem (BHCP) in a solid or hollow cylinder. In the BHCP, the initial temperature is to be determined from the temperature measurements at a later time. This is an inverse and ill-posed problem, and we employ and generalize the MFS regularization approach [B.T. Johansson and D. Lesnic, A method of fundamental solutions for transient heat conduction, Eng. Anal. Boundary Elements 32 (2008), pp. 697–703] for the time-dependent heat equation to obtain a stable and accurate numerical approximation with small computational cost.     

滴状冷凝传热的理论模型与计算

滴状冷凝传机理已有一些报道，作者提出了一个新的观点并建立了一个计算滴状冷凝传热的新模型，模型通过计算液滴之间的液膜传热和液滴传热并结合液滴分布，获得了滴状冷凝的总传热系数。该模型仅含少量几个主要参数，如温差，液膜厚度，脱离液膜半径及液滴尺寸分布，滴状冷凝的传热系数和热通量的理论计算结果与文献值符合较好，说明该模型能较好地描述滴状冷凝传热的实际行为。     

Some consequences on the planar three-index transportation problem

In this study, some algebraic characterizations of the coefficient matrix A of the planar three-index transportation problem are derived and the equivalent formulation of this problem is obtained using the Kronecker product. It is shown that eigenvectors of the matrix G ⁺ G are characterized in terms of eigenvectors of the matrix A ⁺ A , where G ⁺ is the Moore–Penrose inverse of the coefficient matrix G of the equivalent problem.     

Evaluation of heat transfer coefficients during upward and downward transient directional solidification of Al–Si alloys

Aluminum alloys with silicon as a major alloying element consist of a class of alloys which provides the most significant part of all shaped castings manufactured. This is mainly due to the outstanding effect of silicon in the improvement of casting characteristics, combined with other physical properties such as mechanical properties and corrosion resistance. In general, an optimum range of silicon content can be assigned to casting processes. For slow cooling rate processes (sand, plaster, investment), the range is 5 to 7 wt%; for permanent molds, 7 to 9%; and for die castings, 8 to 12%. Since most casting parts are produced considering there is no dominant heat flow direction during solidification, it seems to be adequate to examine both upward and downward growth directions to better understand foundry systems. The way the heat flows across the metal/mold interface strongly affects the evaluation of solidification and plays a remarkable role in the structural integrity of castings. Gravity or pressure die casting, continuous casting, and squeeze casting are some of the processes where product quality is more directly affected by the interfacial heat transfer conditions. Once information in this area is accurate, foundrymen can effectively optimize the design of their chilling systems to produce sound castings. The present work focuses on the determination and evaluation of transient heat transfer coefficients from the experimental cooling curves during solidification of Al 5, 7, and 9 wt% Si alloys. The method used is based on comparisons between experimental data and theoretical temperature profiles furnished by a numerical solidification model, which applies finite volume techniques. In other words, the resulting data were compared with a solution for the inverse heat conduction problem. The necessary solidification thermodynamic input data were obtained by coupling the software ThermoCalc Fortran interface with the solidification model. A comparison between upward and downward transient metal/mold heat transfer coefficients is conducted.     

QPSO算法与PSO算法求解二维热传导反问题

热传导反问题在国内研究起步较晚,研究方法有很多,但通常方法很难较好地接近全局最优.在介绍经典的微粒群优化算法(PSO)的基础上,研究基于量子行为的微粒群优化算法(QPSO)的二维热传导参数优化方法,具体介绍依据目标函数如何利用上述的算法去寻找最优参数组合.为了提高算法的收敛性和稳定性,在具体应用中对算法进行了改进,并进行了大量实验,结果显示在解决热传导反问题优化问题中,基于QPSO算法的性能比经典PSO算法更加优越,证明QPSO在热传导领域具有很大的实际应用价值.     

Recovering temperature-dependent heat conductivity in 2D and 3D domains with homogenization functions as the bases

The paper solves the parameters identification problem in a nonlinear heat equation with homogenization functions as the bases, which are constructed from the boundary data of the temperature in the 2D and 3D space-time domains. To satisfy the over-specified Neumann boundary condition, a linear equations system is derived and then used to determine the expansion coefficients of the solution. Then, after back substituting the solution and collocating points to satisfy the governing equations, the space-time-dependent and temperature-dependent heat conductivity functions in 2D and 3D nonlinear heat equations are identified by solving other linear systems. The novel methods do not need iteration and solving nonlinear equations, since the unknown heat conductivities are retrieved from the solutions of linear systems. The solutions and the heat conductivity functions recovered are quite accurate in the entire space-time domain. We find that even for the inverse problems of nonlinear heat equations, the homogenization functions method is easily used to recover 2D and 3D space-time-dependent and temperature-dependent heat conductivity functions. It is interesting that the present paper makes a significant contribution to the engineering and science in the field of inverse problems of heat conductivity, merely solving linear equations and without employing iteration and solving nonlinear equations to solve nonlinear inverse problems.

     

Determination of inner boundaries in modified Helmholtz inverse geometric problems using the method of fundamental solutions

In this paper, an inverse geometric problem for the modified Helmholtz equation arising in heat conduction in a fin, which consists of determining an unknown inner boundary (rigid inclusion or cavity) of an annular domain from a single pair of boundary Cauchy data is solved numerically using the method of fundamental solutions (MFS). A nonlinear minimisation of the objective function is regularised when noise is added into the input boundary data. The stability of numerical results is investigated for several test examples.     

Nonlinear convective flow with variable thermal conductivity and Cattaneo-Christov heat flux

An analysis is introduced to investigate the salient features of nonlinear convective flow of thixotropic fluid in the version of Cattaneo-Christov heat flux theory. The stagnation point flow is present. The flow phenomenon is by an impermeable stretching sheet. The energy expression is modeled through the theory of Cattaneo-Christov heat flux. Characteristics of heat transfer phenomenon are described within the frame of variable thermal conductivity. Suitable variables reduced to the nonlinear partial differential expressions to the ordinary differential expressions. Series solutions of resulting systems are acquired within the frame of homotopy theory. Convergence analysis is achieved and suitable values are determined by capturing the so-called ℏ−curves. Graphical results for velocity and temperature are displayed and argued for sundry physical variables. Expression of skin friction coefficient is calculated through numerical values. Higher values of mixed convection parameter, Prandtl number, and thermal relaxation time lead to decay the temperature and layer thickness.