A finite element analysis for inverse heat conduction problems

Our purpose is to establish a numerical method meeting the requirements of efficiency, stability, accuracy and easy-using. Faced with these demands, in this paper, a hybrid method combining the advantages of 2 weak-form meshless methods is proposed for solving steady and transient heat conduction problems with temperature-dependent thermophysical properties in heterogeneous media. In the process of the calculation for an arbitrary model discretized by the hybrid method, two different weak-form methods are used for the nodes categorized as: (1) boundary nodes, (2) near boundary interior nodes and (3) interior nodes. A Galerkin free element method (GFREM) is proposed for the interior nodes of the Cartesian grid; and the local radial point interpolation method (LRPIM) is used for the other two types of nodes. The GFREM using the Cartesian grid can greatly improve the accuracy and efficiency of the hybrid scheme. Special decomposition technology to determine the irregular integral domain for LRPIM can increase the flexibility for complex models. In addition, the full implicit finite difference scheme and Newton–Raphson method are used for solving transient nonlinear problems. And non-crossing interface integral domain and support domain are employed for heterogeneous media. Compared with other methods, numerical results show better accuracy, stability and efficiency from the numerical experiments.

The solution of direct and inverse transient heat conduction problems with layered materials using exponential basis functions

: This study discusses the analytical and implementational difficulties associated with transient scalar boundary element methods. The fundamental solutions to steady and transient heat conduction problems are derived from first principles. Their implementation in the solutions to seven representative heat conduction problems is detailed. Keywords: boundary value problems; heat conduction problems; boundary element; first principles; Poisson Equation; Helmholtz Equation.

In this paper, we consider a three-dimensional transient inverse heat conduction problem arising in pool boiling experiments, i.e., the reconstruction of the surface heat flux from pointwise temperature observations inside a heater. We show that the inverse problem is ill-posed and utilize Tikhonov regularization and conjugate gradient methods together with a discrepancy stopping rule for a stable solution. We investigate the proper choice of regularization terms, which not only affects stability of the reconstructions but also greatly influences the quality of reconstructions in the case of limited observations. For the numerical solution of the governing partial differential equation, a space-time finite element method is used. This allows us to compute exact gradients for the discretized Tikhonov functional, and enables the use of conjugate gradient methods for the solution of the regularized inverse problem. We discuss further aspects of an efficient implementation, including a multilevel optimization strategy, together with an implementable stopping criterion. Finally, the proposed algorithms are applied to the reconstruction of local boiling heat fluxes from experimental data.

A novel approach to solve inverse heat conduction problems: Coupling scaled boundary finite element method to a hybrid optimization algorithm

A wide variety of inverse heat conduction problems have been studied in the last two decades for the estimation of boundary or initial conditions, thermophysical properties, geometrical parameters, or heat source intensities. In most transient heat conduction problems, the mathematical models were cast in dimensionless forms, by using a diffusion time scale. As the thermal diffusivities are usually small, the physical time scales turn to be rather long. In this way, most works show that the inverse analysis yields satisfactory results, without addressing the implications of the physical time scale. The physical time scale, in fact, influences significantly the quality of the inverse solution. We present here a unified treatment for one-dimensional, linear inverse heat conduction problems using the conjugate gradient method with an adjoint equation, and also show that there are physical limitations by the time scale on the inverse solutions.

Shape reconstruction in transient heat conduction problems based on radial integration boundary element method

Solution of an inverse transient heat conduction problem in a part of a complex domain

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.

This paper develops a numerical treatment of classical boundary value problems for ar- bitrarily shaped plane heat conducting solids obeying Fourier's law. An exact integral formula defined on the boundary of an arbitrary body is obtained from a fundamental singular solu- tion to the governing differential equation. This integral formula is shown to be a means of numerically determining boundary data, complementary to given data, such that the Laplace transformed temperature field may subsequently be generated by a Green's type integral identity. The final step, numerical transform inversion, completes the solution for a given problem. All operations are ideally suited for modern digital computation. Three illustra- tive problems are considered. Steady-state problems, for which the Laplace transform is un- necessary, form a relatively simple special case. A FORMULATION of the various transient boundary value problems associated with isotropic solids obeying Fourier's law of heat conduction is developed. An exact in- tegral formula is derived relating boundary heat flux and boundary temperature, in the Laplace transform space, that corresponds to the same admissible transformed temperature field throughout the body. Part of the boundary data in the formula is known from the description of a well posed bound- ary value problem. As is shown, the remaining part of the boundary data is obtainable numerically from the formula it- self regarded as a singular integral equation. Once both trans- formed temperature and heat flux are known everywhere on the boundary, the transformed temperature throughout the body is obtainable by means of a Green's type integral identity. This identity yields the field directly in terms of the mentioned boundary data. The final step, transform in- version, although done approximately also, is accomplished by a technique particularly well suited to the class of problems under investigation. The main feature of the solution procedure suggested is its generality. It is applicable to solids occupying domains of rather arbitrary shape and connectivity. Boundary data may be prescription of temperature, or heat flux, or parts of each corresponding to a mixed type problem. Also, a linear combination of temperature and flux may be given corre- sponding to the so-called convection boundary condition. The same boundary formula described previously is applicable in every case. Approximations in the transform space are made only on the boundary, in contrast to finite difference procedures, and the approximations made are conceptually simple, natural to make, and give rise, as is shown, to very ac- curate data for a relatively crude boundary approximation pattern. Problems posed for composite bodies, i.e., two or more heat conducting solids bonded together, are particularly amenable to the present treatment. One computer program is employed which utilizes only data describing the domain geometry, boundary temperature or flux, material properties, and a sequence of values of the transform parameter neces- sary for the inversion scheme. Output is the transformed temperature at any desired field point. A second program in-

Inverse heat conduction analysis provides an efficient approach for estimating the thermophysical properties of materials, the boundary conditions, or the initial conditions. In this paper, two-dimensional transient nonlinear inverse heat conduction problems are investigated, for estimating time- and space-dependent boundary heat flux, as well as the temperature-dependent thermal conductivities. Modifications are carried out to extend the previous one-dimensional inversion algorithm to solve the two-dimensional transient nonlinear heat conduction problem, to overcome the frequently occurring divergence issues, and to improve the stability of the inversion algorithm. Boundary-only measurements are used as additional information, and a dimensionless objective function is adopted. In the direct problem, formulations for solutions to the two-dimensional transient nonlinear heat conduction problem are derived and validated. Numerical examples show that the inversion algorithm is effective, efficient, accurate, and robust, for recovering multiple parameters, with and without a functional form.

Surface heat flux is an important parameter in various industrial applications, which is often estimated based on measured temperature by solving inverse heat conduction problem (IHCP). In this chapter, the available IHCP methods including sequential function specification (SFS), transfer function (TF) and Duhamel’s theorem were compared, taking the example of surface heat flux estimation during spray cooling. The Duhamel’s theorem was improved to solve 1D multi-layer ICHP. Considering the significant nonuniformity of heat transfer, the 2D filter solution method was proposed to estimate surface heat flux for 2D multi-layer mediums. The maximum heat flux calculated by the 1D method was underestimated by 60% than that calculated by 2D filter solution, indicating that the lateral heat transfer cannot be ignored. The cooling performances based on 2D filter solution demonstrated that substituting the environment friendly R1234yf for R134a can remarkably reduce global warming potential to <1, but its cooling capacity is insufficient. The effective heat flux of R1234yf can be enhanced by 18.8% by reducing the nozzle diameter and decreasing the back pressure, providing the theoretical basis for the clinical potential substitution of R1234yf with low global warming potential (GWP) for commercial R134a with high GWP in laser dermatology.

A new method for identifying temperature-dependent thermal conductivity in transient heat conduction problems based on element differential method

An analytical technique is developed for solving the inverse problem of steady heat conduction in a two-dimensional flat plate. In the problem, the temperature and heat flux on one of the plate boundaries are known functions of the space variable. Explicit closed-form expressions are derived for calculating the two-dimensional distributions of temperature and heat flux in the solid plate for arbitrary over-specified boundary temperature and heat flux profiles. The solution does not require knowledge of the thermal conditions on the other plate boundaries. The effect of uniform heat generation inside the plate is exactly modeled in the general solution. Test problems of known exact solutions are used to examine the validity of the proposed analytical technique.

Cryogen spray cooling (CSC) is an effective technique to protect the epidermis during cutaneous laser therapies. Spraying a cryogen onto the skin surface creates a time-varying heat flux, effectively cooling the skin during and following the cryogen spurt. In previous studies mathematical models were developed to predict the human skin temperature profiles during the cryogen spraying time. However, no studies have accounted for the additional cooling due to residual cryogen left on the skin surface following the spurt termination. We formulate and solve an inverse heat conduction (IHC) problem to predict the time-varying surface heat flux both during and following a cryogen spurt. The IHC formulation uses measured temperature profiles from within a medium to estimate the surface heat flux. We implement a one-dimensional sequential function specification method (SFSM) to estimate the surface heat flux from internal temperatures measured within an in vitro model in response to a cryogen spurt. Solution accuracy and experimental errors are examined using simulated temperature data. Heat flux following spurt termination appears substantial; however, it is less than that during the spraying time. The estimated time-varying heat flux can subsequently be used in forward heat conduction models to estimate temperature profiles in skin during and following a cryogen spurt and predict appropriate timing for onset of the laser pulse.