WELL-CONDITIONED NUMERICAL METHOD FOR THE NONLINEAR INVERSE HEAT CONDUCTION PROBLEM

A numerical procedure recently presented by the authors for linear inverse heat conduction problems is extended to nonlinear cases. The method is well conditioned in the sense that it always generates bounded solutions and never generates heat fluxes oscillating with increasing amplitude. The convergence of the algorithm is studied, and a set of examples shows convergence for a wide range of nonlinear problems. In the case of perfect data the accuracy of the estimated heat fluxes increases as the time step is decreased without need for additional stabilization or regularization. Filters allow perturbed data to be treated and facilitate attaining a balance between accuracy and resolving power.     

SOLUTION TO INVERSE HEAT CONDUCTION PROBLEM IN NANOSCALE USING SEQUENTIAL METHOD

An inverse heat conduction problem for nanoscale structure is studied. The conduction phenomenon is modeled using the Boltzmann transport equation. Phonon-mediated heat conduction in one dimension is considered. One boundary is exposed to an unknown temperature and the other boundary, where temperature observation takes place, is subject to a known boundary condition. A sequential scheme with constant function specification is employed for inverse estimation of the unknown temperature. Sample results are presented and discussed.     

A THREE-DIMENSIONAL INVERSE PROBLEM OF ESTIMATING THE VOLUMETRIC HEAT GENERATION FOR A COMPOSITE MATERIAL

A 3-D transient inverse heat conduction problem is solved in this study by using the conjugate gradient method (CGM) and the general-purpose commercial code CFX4.2-based inverse algorithm to estimate the strength of the unknown heat generation in a 3-D irregular composite medium. The advantage of calling the CFX4.2 code as a subroutine in the present inverse calculation is that many difficult but practical 3-D inverse problem can be solved under this construction because the general-purpose commercial code has the ability to solve the direct problem easily. Results obtained by using the CGM to solve this 3-D inverse problems are justified based on the numerical experiments with the simulated exact and inexact measurements. It is concluded that accurate heat generation can be estimated by the CGM except for the final time. The reason and improvement of this singularity are addressed.     

NUMERICAL SOLUTION OF 2-D NONLINEAR INVERSE HEAT CONDUCTION PROBLEMS USING FINITE-ELEMENT TECHNIQUES

A general method is presented for solving different classes of nonlinear inverse heat conduction problems (IHCP) for two-dimensional, arbitrarily shaped bodies. It is based on the systematic use of a finite-element library. It is shown that, following this approach, the conjugate gradient method can be easily implemented. The method offers a very wide field of practical applications in inverse thermal analysis, while reducing very significantly the amount of work which remains specific for each particular IHPC. Two numerical experiments illustrate the influence of data errors and the iterative regularization principle.     

INVERSE HEAT CONDUCTION PROBLEM OF SIMULTANEOUSLY ESTIMATING SPATIALLY VARYING THERMAL CONDUCTIVITY AND HEAT CAPACITY PER UNIT VOLUME

An inverse heat conduction method for simultaneously estimating spatially varying thermal conductivity and heat capacity per unit volume under the conditions of a flash method type of experiment is developed. The unknown thermal properties are assumed to vary only in the space dimension normal to the slab sample and are modeled with piecewise linear representations. Lacking in the literature are specific requirements that must be satisfied by the number of measurements in the spatial domain in order to ensure uniqueness of the inverse solution. We prepared a series of numerical experiments to provide a better understanding of this issue. Multiple temperature sensors are shown to be necessary to determine spatially varying properties. The effectiveness of the method is illustrated through simulated experimental applications of the method.     

SLOWLY DIVERGENT SPACE MARCHING SCHEMES IN THE INVERSE HEAT CONDUCTION PROBLEM

Abstract

Recently developed “slowly divergent” space marching difference schemes, coupled with Tikhonov regularization, can solve the one-dimensional inverse heal conduction problem at values of the nondimensional lime step δt⁺ as low as δt⁺ = 0.0003. A Lax-Richtmyer analysis is used to demonstrate dramatic differences in error amplification behavior among various space marching algorithms, for the same problem, on the same mesh; maximum error amplification factors may differ by more than 10 orders of magnitude at parameter values that are of interest in rocket nozzle applications. Slowly divergent schemes are characterized by their damping behavior at high frequencies. A widely used benchmark problem, where the surface temperature gradient is a step function, provides a basis for evaluating Tikhonov-regularized marching computations. With standard marching procedures, relatively high values of the regularization parameter r are found to be necessary; the resulting loss of resolution leads to erroneous solutions. When slowly divergent schemes are used, much lower values of r are possible, leading to reasonably accurate reconstruction of thermal histories at the active surface.     

A 2-D INVERSE METHOD FOR SIMULTANEOUS ESTIMATION OF THE INLET TEMPERATURE AND WALL HEAT FLUX IN A LAMINAR CIRCULAR DUCT FLOW

A two-dimensional inverse analysis is presented for the estimation of the inlet temperature of the fluid flow and wall heat flux in a thermally developing hydrodynamically developed laminar flow in a duct. The inverse analysis is based on the temperature reading located at a stream inside the duct at several different points. At the beginning of the study, finite difference methods are employed to discretize the problem, and then a linear inverse model is constructed to identify the unknown conditions. The present approach is to rearrange the matrix forms of the differential governing equation and estimate the inlet temperature of the fluid and unknown surface conditions of the duct. The linear least squares method is adopted to find the solution. The advantage of applying this method in inverse analysis is that no prior information is needed on the functional form of the unknown quantities, no initial guess is required, and the number of iterations in the calculation process is limited to one. The effects of sensor position, magnitude of measurement error, and number of measurements on the accuracy of estimates are examined. The results show that the preferred position of the sensor is closer to the inlet region and only few measuring points are sufficient to estimate the wall heat flux and inlet temperatures of the fluid when the measurement errors are neglected. When the measurement errors an considered, more measuring points are needed in order to increase the congruence of the estimated results to exact solutions.     

THERMAL STRESS PROBLEM FOR MIXED HEAT CONDUCTION BOUNDARY AROUND AN ARBITRARILY SHAPED HOLE WITH CRACK UNDER UNIFORM HEAT FLUX

This article deals with a thermal stress problem for thermal conduction around an arbitrarily shaped hole with a crack under uniform heat flux. Two cases for the hole edge and the crack faces are assumed: adiabatic and isothermal conditions or vice versa (isothermal and adiabatic). A closed-form solution is obtained using conformal mapping, dislocation functions, and the complex variable method. Results of temperature, heat flux, stress, and stress intensity factor are illustrated.     

SOLUTION OF DISPLACEMENT BOUNDARY VALUE PROBLEM UNDER UNIFORM HEAT FLUX

The general solution of the displacement boundary value problem is obtained for an infinite plate with an arbitrary shaped hole under uniform heat flux in any direction. The complex stress functions, the dislocation method, and a rational mapping function are used and the closed solution is obtained. An infinite plate with a circular hole and a slit is analyzed under the condition of the constrained displacements. The singularity at the tip of the slit of the constrained displacement is investigated     

A COMPARISON OF TIME DISCRETIZATION SCHEMES FOR THE GREEN ELEMENT SOLUTION OF THE TRANSIENT HEAT CONDUCTION EQUATION

We present three time discretization schemes for the Green element solution of the linear conduction equation. Numerical results obtained from the three methods are assessed by their convergence and stability-related properties, namely, numerical amplitude and amplification factor through Fourier analysis. The main emphasis is the ability of any of the schemes to handle conduction problems typified by those properties that are known to be taxing to most numerical schemes. This was found in some cases to be related to how accurately the harmonics of the Fourier waves are propagated.     

APPLICATION OF THE ADAPTIVE MATRIX FILTER TO INVERSE HEAT CONDUCTION PROBLEMS

A matrix filter has been developed to remove errors from temperature measurements in the field of heat conduction. It has been shown that the matrix filter substantially reduces the errors in measurements. The filter extended to a new adaptive matrix filter (AMF) enables identification of some thermal properties based on the temperature measurements of the system involved. The approach allows elimination of many problems associated with the solution of nonlinear equations. AMF can use data containing errors; this allows for a simultaneous estimation of unknown properties and removal of errors from measurements. It has been proved that the algorithm is very efficient.     

A BOUNDARY CONDITION DISSECTION METHOD FOR THE SOLUTION OF BOUNDARY-VALUE HEAT CONDUCTION PROBLEMS WITH POSITION-DEPENDENT CONVECTIVE COEFFICIENTS

A boundary condition dissection method is developed on the postulation that a condition imposed on the boundary of a heat conduction problem may be realized in practice by using conditions not of the imposed kind. Thus, a Robin condition imposed on the boundary may be dissected as a linear combination of the boundary heal flux and temperature, arid by doing so, heat conduction problems with position-dependent convective coefficients can be solved by the separation-of-variables technique. The method leads to the solution of a Fredholm integral equation of the second kind with a degenerate kernel, and this equation may be solved by using simultaneous algebraic equations. The method is superior to the finite-difference method and the methods of weighted residuals, which have been conventionally used in solving suck problems. Extension of the method to the solution of other heat conduction problems is also possible and mentioned in the paper.     

SOLUTION OF STEADY PERIODIC HEAT CONDUCTION BY THE FINITE-ELEMENT METHOD

This paper describes the use of the finite-element technique to solve problems of steady periodic heal conduction. The concept of a complex variable can be used to reduce the governing unsteady heat conduction equation to two noncoupled Poisson equations. The validity of the present method is confirmed with a one-dimensional problem for which an analytic solution exists. Numerical solutions for a three-dimensional problem are presented to illustrate the capability of the method.     

THE VALIDATION OF THE ROBUST INPUT ESTIMATION APPROACH TO TWO-DIMENSIONAL INVERSE HEAT CONDUCTION PROBLEMS

A novel adaptive and robust input estimation inverse methodology of estimating the time-varying unknown heat flux, named as the input, on the two active boundaries of a 2-D inverse heat conduction problem is presented. The algorithm includes using the Kalman filter to propose a regression model between the residual innovation and the two thermal unknown boundaries flux through given 2-D heat conduction state-space models and noisy measurement sequence. Based on this regression equation, a recursive least-square estimator (RLSE) weighted by the forgetting factor is proposed to on-line estimate these unknowns. The adaptive and robust weighting technique is essential since unknowns input are time-varied and have unpredictable changing status. In this article, we provide the bandwidth analysis together with bias and variance tests to construct an efficient and robust forgetting factor as the ratio between the standard deviation of measurement and observable bias innovation at each time step. Herein, the unknowns are robustly and adaptively estimated under the system involving measurement noise, process error, and unpredictable change status of time-varying unknowns. The capabilities of the proposed algorithm are demonstrated through the comparison with the conventional input estimation algorithm and validated by two benchmark performance tests in 2-D cases. Results show that the proposed algorithm not only exhibits superior robust capability but also enhances the estimation performance and highly facilitates practical implementation.     

SENSITIVITY ANALYSIS AND OPTIMAL DESIGN FOR STEADY CONDUCTION PROBLEM WITH RADIATIVE HEAT TRANSFER

ABSTRACT

A steady heat conduction problem is considered, that is described by the heat conduction equation and the thermal boundary conditions (i.e., Dirichlet, Neumann, Henkel, and radiation conditions on the external boundary, and radiation condition on the hole boundary). An arbitrary behavioral functional is defined and its first-order sensitivity is derived using both the direct and the adjoint approaches. The shape optimization problem is next formulated and two optimization functionals are discussed. The simple numerical example is presented.     

A NUMERICAL SCHEME FOR NON-FOURIER HEAT CONDUCTION,PART II: TWO-DIMENSIONAL PROBLEM FORMULATION AND VERIFICATION

A flux-splitting algorithm based on the Godunov numerical scheme developed for the solution of the one-dimensional non-Fourier heat conduction equation by Yeung and Lam [1] is extended for the investigation of thermal wave propagation in rectangular media. The derivation of the solution method and the stability criteria are presented in detail. Physical problems subjected to various boundary conditions (e.g., first, second, and third kinds) can be studied with the numerical scheme. A comparison of the exact solution with the one calculated by the proposed procedure is presented to confirm the validity of the numerical procedure. The numerical scheme is applicable for the study of short-pulse heating in advanced materials, microstructures, thin films, semiconductor devices, and superconductors.     

SOLUTION OF THE INVERSE PROBLEMS IN HEAT TRANSFER AND THERMAL STRESS ANALYSIS

The stable method of soloing inverse problems in heat transfer and inverse problems concerning thermal stress control in elements of complex shapes is presented. Based on the discrete form of Duhamel's integral and on so-called future time steps, a simple and accurate method of solving one- and multi-dimensional inverse problems is developed. Two examples illustrating the application of the described method are presented.     

太阳入射热流对吸热器换热的影响

空间太阳能热动力发电系统是非常有前景的未来空间能源供应系统。吸热器的入射热流分布将影响到换热管的传热以及系统的寿命，采用焓法处理相变区的传热，建立了太阳能热动力发电系统吸势器换热管三维换热模型，计算得到在轨道周期内对应三种入射热流的换热管的温度场、工质的出口温度变化、相变材料熔化率等重要的结果，并进行了比较、分析。     

BOUNDARY INTEGRAL SOLUTION METHOD FOR THE GRAETZ PROBLEM

This article describes a new method for the numerical solution of the generalized Graetz problem. The method involves the use of boundary integral concepts over subintervals or elements in the radial direction to eliminate the radial heat conduction operator. This leads to a new discretization scheme whereby the problem is reduced to a set of ordinary differential equations for the nodal temperatures and their radial gradients as a function of the axial distance. This set of equations is then solved by axial marching using the implicit Euler method. The method yields accurate solutions both in the entry region as well as for the fully developed regions of the pipe, and an example problem of heat transfer with a constant wall temperature is used for numerical testing of the computational scheme.     

NONUNIQUENESS AND STABILITY FOR HEAT CONDUCTION THROUGH A DUPLEX HEAT EXCHANGER TUBE

An analysis is given for radial heat flow through the wall of a duplex heat exchanger tube with a pressure or gap-dependent contact resistance at the interface. Thermal expansion of the tube changes the value of this resistance, and it is shown that for certain thermal conditions there will be more than one steady-state solution. The stability of these multiple solutions is then investigated using a perturbation method, and it is found that there is always an odd number of solutions, alternately stable and unstable.

This kind of behavior has been observed in other thermoelastic contact problems, but this is the first example in which it has been exhibited in a case of contact between similar materials.