Numerically solving twofold ill-posed inverse problems of heat equation by the adjoint Trefftz method

The inverse problem endowing with multiple unknown functions gradually becomes an important topic in the field of numerical heat transfer, and one fundamental problem is how to use limited minimal data to solve the inverse problem. With this in mind, in the present article we search the solution of a general inverse heat conduction problem when two boundary data on the space-time boundary are missing and recover two unknown temperature functions with the help of a few extra measurements of temperature data polluted by random noise. This twofold ill-posed inverse heat conduction problem is more difficult than the backward heat conduction problem and the sideways heat conduction problem, both with one unknown function to be recovered. Based on a stable adjoint Trefftz method, we develop a global boundary integral equation method, which together with the compatibility conditions and some measured data can be used to retrieve two unknown temperature functions. Several numerical examples demonstrate that the present method is effective and stable, even for those of strongly ill-posed ones under quite large noises.     

Recovering both the space-dependent heat source and the initial temperature by using a fast convergent iterative method

In this paper, we solve two types of inverse heat source problems: one recovers an unknown space-dependent heat source without using initial value, and another recovers both the unknown space-dependent heat source and the initial value. Upon inserting the adjoint Trefftz test functions into Green’s second identity, we can retrieve the unknown space-dependent heat source by an expansion method whose expansion coefficients are derived in closed form. We assess the stability of the closed-form expansion coefficients method by using the condition numbers of coefficients matrices. Then, numerical examples are performed, which demonstrates that the closed-form expansion coefficient method is effective and stable even when it imposes a large noise on the final time data. Next, we develop a coupled iterative scheme to recover the unknown heat source and initial value simultaneously, under two over specified temperature data at two different times. A simple regularization technique is derived to overcome the highly ill-posed behavior of the second inverse problem, of which the convergence rate and stability are examined. This results in quite accurate numerical results against large noise.     

Identifying heat conductivity and source functions for a nonlinear convective-diffusive equation by energetic boundary functional methods

Abstract

In the article, we solve the inverse problems to recover unknown space-time dependent functions of heat conductivity and heat source for a nonlinear convective-diffusive equation, without needing of initial temperature, final time temperature, and internal temperature data. After adopting a homogenization technique, a set of spatial boundary functions are derived, which satisfy the homogeneous boundary conditions. The homogeneous boundary functions and zero element constitute a linear space, and then a new energetic functional is derived in the linear space, which preserves the time-dependent energy. The linear systems and iterative algorithms to recover the unknown parameters with energetic boundary functions as the bases are developed, which are convergent fast at each time marching step. The data required for the recovery of unknown functions are parsimonious, including the boundary data of temperatures and heat fluxes and the boundary data of unknown functions to be recovered. The accuracy and robustness of present methods are confirmed by comparing the exact solutions with the identified results, which are obtained under large noisy disturbance.     

To recover heat source G(x) + H(t) by using the homogenized function and solving rectangular differencing equations

ABSTRACT

For recovering an unknown heat source F(x, t) = G(x) + H(t) in the heat conduction equation, we develop a homogenized function method and the expansion methods by polynomials or eigenfunctions, which can solve the inverse heat source recovery problem by using collocation technique. Because the initial condition/boundary conditions/supplementary condition are satisfied automatically and a rectangular differencing technique is developed, a middle-scale linear system is sufficient to determine the expansion coefficients. After deriving a multiscale postconditioning matrix, the present methods converge very quickly, and are accurate and stable against large noise, as verified by numerical tests.     

Solution in Elementary Functions to a BVP of Thermoelasticity: Green's Functions and Green's-Type Integral Formula for Thermal Stresses within a Half-Strip

This article presents new elementary Green's functions for displacements and stresses created by a unit heat source applied in an arbitrary interior point of a half-strip. We also obtain the corresponding new integration formulas of Green's and Poisson's types which directly determine the thermal stresses in the form of integrals of the products of internal distributed heat source, temperature, or heat flux prescribed on boundary and derived thermoelastic influence functions (kernels). All these results are presented in terms of elementary functions in the form of a theorem. Based on this theorem and on derived early by author general Green's type integral formula, we obtain a new solution to one particular boundary value problem of thermoelasticity for half-strip. The graphical presentation of thermal stresses created by a unit point heat source and of thermal stresses for one particular boundary value problem of thermoelasticity for half-strip is also included. The proposed method of constructing thermoelastic Green's functions and integration formulas are applicable not only for a half-strip but also for many other two- and three-dimensional canonical domains of Cartesian system of coordinates.     

Exact Elementary Green's Functions and Poisson-Type Integral Formulas for a Thermoelastic Half-Wedge with Applications

In this paper new exact Green's functions and new exact Poisson-type integral formula for a boundary value problem (BVP) in thermoelastostatics for a half-wedge with mixed homogeneous mechanical boundary conditions (the boundary angle is free of loadings and normal displacements and tangential stresses are prescribed on the boundary quarter-planes) are derived. The thermoelastic displacements are produced by a heat source applied in the inner points of the half-wedge and by mixed non-homogeneous boundary heat conditions (the temperature is prescribed on the boundary angle and the heat fluxes are given on the boundary quarter-planes). When thermoelastic Green's function is derived the thermoelastic displacements are generated by an inner unit point heat source, described by δ-Dirac's function. All results are obtained in terms of elementary functions and they are formulated in a special theorem. Analogous results for an octant and for a quarter-space as particular cases of the angle of the thermoelastic half-wedge also are obtained. The main difficulties to obtain these results are in deriving the functions of influence of a unit concentrated force onto elastic volume dilatation Θ^(q) and, also, in calculating a volume integral of the product of function Θ^(q) and Green's function in heat conduction. Exact solutions in elementary functions for two particular BVPs of thermoelasticity for a quarter-space and a half-wedge, using the derived Poisson-type integral formula and the influence functions Θ^(q) also are included. The proposed approach may be extended not only for many different BVPs for half-wedge, but also for many canonical cylindrical and other orthogonal domains.     

Inverse determination of a heat source from a solute concentration generation model in porous medium

The conjugate gradient method with adjoint equations is applied to the natural convection problem in a porous medium for the determination of an unknown heat source which is dependent on a solute concentration generation rate. The direct, sensitivity and adjoint equations are given for a Boussinesq fluid, over an arbitrary domain in two dimensions. Solutions by control volumes are presented for a square enclosure under known temperature and concentration boundary conditions, assuming a source term proportional to the vertical average generation rate of a solute concentration governed by a Monod model. Reasonably accurate solutions are obtained at least up to Ra=10⁵.     

Simultaneously estimating the contact heat and mass transfer coefficients in a double-layer hollow cylinder with interface resistance

Based on the conjugate gradient method, this study presents a means of solving the inverse boundary value problem of coupled heat and moisture transport in a double-layer hollow cylinder. While knowing the temperature and moisture history at the measuring positions, the unknown time-dependent contact heat and mass transfer coefficients can be simultaneously determined. It is assumed that no prior information is available on the functional form of the unknown coefficients. The accuracy of this inverse heat and moisture transport problem is examined by using the simulated exact and inexact temperature and moisture measurements in the numerical experiments. Results show that excellent estimation on the time-dependent contact heat and mass transfer coefficients can be simultaneously obtained with any arbitrary initial guesses.     

NEW FORMULAE FOR DYNAMICAL THERMAL STRESSES

A generalization of the function of influence of a unit heat source to the displacements is suggested for the boundary value problems in the dynamical uncoupled thermoelasticity. This generalization is a convolution over time and bulk of two influence functions. One of them is a Green's function for the heat conduction problem. The other is a function of influence of unit concentrated forces onto bulk dilatation. Broad possibilities are shown in constructing these influence functions. In particular, the theorem on dilatation constructing is proved. To calculate the convolutions successfully the following properties of the introduced function are found to be useful. (1) In coordinates of the point of observation, the function satisfies the equations used to find the Green's functions in the problem of heat conduction, with the unit heat source being replaced by the influence function of concentrated force onto dilatation; and (2) in coordinates of the point of heat source application, it satisfies the boundary value problem used to find Green's matrix, with the unit concentrated forces being replaced by derivatives of Green's function in the problem of heat conduction. Based on the introduced influence function, some new integral formulae for displacements and stresses are obtained, which are a generalization of Mysel's formula in the theory of dynamical thermal stresses. The proposed formulae have certain advantages allowing us to unite the two-staged process of finding the solutions for boundary value problems in thermoelasticity in a single stage. It is established that, based on the obtained results it becomes possible to compile a whole handbook on the influence functions and integral solutions for boundary value problems in dynamical thermoelasticity. As examples, the solutions for two boundary value problems in the theory of dynamical thermal stresses for the half-space and quarter-space are presented.     

Inverse analysis with integral transformed temperature fields: Identification of thermophysical properties in heterogeneous media

The objective of this work is to introduce the use of integral transformed temperature measured data for the solution of inverse heat transfer problems, instead of the common local transient temperature measurements. The proposed approach is capable of significantly compressing the measured data through the integral transformation, without losing the information contained in the measurements and required for the solution of the inverse problem. The data compression is of special interest for modern measurement techniques, such as the infrared thermography, that allows for fine spatial resolutions and large frequencies, possibly resulting on a very large amount of measured data. In order to critically address the use of integral transformed measurements, we examine in this paper the simultaneous estimation of spatially variable thermal conductivity and thermal diffusivity in one-dimensional heat conduction within heterogeneous media. The direct problem solution is analytically obtained via integral transforms and the related eigenvalue problem is solved by the Generalized Integral Transform Technique (GITT). The inverse problem is handled with Bayesian inference by employing a Markov Chain Monte Carlo (MCMC) method. The unknown functions appearing in the formulation are expanded in terms of eigenfunctions as well, so that the unknown parameters become the corresponding series coefficients. Such projection of the functions in an infinite dimensional space onto a parametric space of finite dimension also permits that several quantities appearing in the solution of the direct problem be analytically computed. Simulated measurements are used in the inverse analysis; they are assumed to be additive, uncorrelated, normally distributed, with zero means and known covariances. Both Gaussian and non-informative uniform distributions are used as priors for demonstrating the robustness of the estimation procedure.     

Determination of a heat source in porous medium with convective mass diffusion by an inverse method

A formulation is given of the inverse natural convection problem by conjugate gradient with adjoint equations in a porous medium with mass diffusion for the determination, from temperature measurements by sensors located within the medium, of an unknown volumetric heat source which is a function of the solute concentration. The direct, sensitivity and adjoint set of equations are derived for a Boussinesq fluid, over an arbitrary domain in two dimensions. Solutions by control volumes are presented for a square enclosure subjected to known temperature and concentration boundary conditions, assuming a source term depending on average vertical solute concentration. Reasonably accurate solutions are obtained at least up to Ra=10⁵ with the source models considered, for Lewis numbers ranging from 0.1 to 10. Noisy data solutions are regularized by stopping the iterations according to the discrepancy principle of Alifanov, before the high frequency components of the random noises are reproduced.     

An inverse problem in estimating the reaction functions and solute concentration simultaneously in a reversible process

An inverse algorithm basing on the Iterative Regularization Method (IRM) is applied in this study in determining the unknown time-dependent reaction functions and solute concentration in the solution, i.e. three unknown time-dependent functions, simultaneously in a reversible process by using measurements of concentration components. It is assumed that no prior information is available on the functional form of the unknown functions in the present study, it can thus be classified as function estimation for the inverse calculations. The accuracy of this inverse problem is examined by using the simulated exact and inexact concentration measurements in the numerical experiments. Results show that the estimation of the time-dependent reaction functions and solute concentration in the solution can be obtained in a very short CPU time on a HP d2000 2.66 GHz personal computer. Moreover, the sensors should be placed as close to the boundary as possible to obtain better estimations.     

Control of the boundary heat flux during the heating process of a solid material

In this paper we apply the conjugate gradient method to solve the inverse problem of determining a time-dependent boundary heat flux in order to achieve a given temperature distribution at the final time. The derivation of sensitivity and adjoint equations in conjunction with the conjugate gradient algorithm are given in detail. The zeroth-order Tikhonov regularization is introduced to stabilize the inverse solution. Solutions by finite differences are obtained for various heat flux profiles. It is found that the time-dependent heat flux may be predicted only for a non-dimensional time of the order of 0.1 while the control problem can be satisfactorily solved for an arbitrary period of time.     

Estimation of the time-dependent heat flux using the temperature distribution at a point by conjugate gradient method

In this paper, the conjugate gradient method coupled with adjoint problem is used in order to solve the inverse heat conduction problem and estimation of the time-dependent heat flux using the temperature distribution at a point. Also, the effects of noisy data and position of measured temperature on final solution are studied. The numerical solution of the governing equations is obtained by employing a finite-difference technique. For solving this problem the general coordinate method is used. We solve the inverse heat conduction problem of estimating the transient heat flux, applied on part of the boundary of an irregular region. The irregular region in the physical domain (r,z) is transformed into a rectangle in the computational domain (ξ,η). The present formulation is general and can be applied to the solution of boundary inverse heat conduction problems over any region that can be mapped into a rectangle. The obtained results for few selected examples show the good accuracy of the presented method. Also the solutions have good stability even if the input data includes noise and that the results are nearly independent of sensor position.     

Recovering A Heat Source and Initial Value by a Lie-Group Differential Algebraic Equations Method

We consider an inverse problem of a nonlinear heat conduction equation for recovering unknown space-dependent heat source and initial condition under Cauchy-type boundary conditions, which is known as a sideways heat equation. With the aid of two extra measurements of temperature and heat flux which are being polluted by noisy disturbances, we can develop a Lie-group differential algebraic equations (LGDAE) method to solve the resulting differential algebraic equations, and to quickly recover the unknown heat source and initial condition simultaneously. Also, we provide a simple LGDAE method, without needing extra measurement of heat flux, to recover the above two unknown functions. The estimated results are quite promising and robust enough against large random noise.     

INVERSE FORCED CONVECTION PROBLEM OF SIMULTANEOUS ESTIMATION OF TWO BOUNDARY HEAT FLUXES IN IRREGULARLY SHAPED CHANNELS

This article deals with the use of the conjugate gradient method of function estimation for the simultaneous identification of two unknown boundary heat fluxes in channels with laminar flows. The irregularly shaped channel in the physical domain is transformed into a parallel plate channel in the computational domain by using an elliptic scheme of numerical grid generation. The direct problem, as well as the auxiliary problems and the gradient equations, required for the solution of the inverse problem with the conjugate gradient method are formulated in terms of generalized boundary-fitted coordinates. Therefore, the solution approach presented here can be readily applied to forced convection boundary inverse problems in channels of any shape. Direct and auxiliary problems are solved with finite volumes. The numerical solution for the direct problem is validated by comparing the results obtained here with benchmark solutions for smoothly expanding channels. Simulated temperature measurements containing random errors are used in the inverse analysis for strict cases involving functional forms with discontinuities and sharp corners for the unknown functions. The estimation of three different types of inverse problems are addressed in the paper: (i) time-dependent heat fluxes; (ii) spatially dependent heat fluxes; and (iii) time and spatially dependent heat fluxes.     

Two-Dimensional Green's Function for Thermal Stresses in a Semi-Layer Under a Point Heat Source

We present specific new expressions for thermal stresses as Green's functions for a plane boundary value problem of steady-state thermoelasticity for a semi-layer. We also obtain new integration formulas of Green's type, which determine the thermal stresses in the form of integrals of the products of the given distributed internal heat source, boundary temperature, and heat flux and derived kernels. Elementary functions results obtained are formulated in a theorem, which is proved using the harmonic integral representations method to derive thermal stresses Green's functions, which are written in terms of Green's functions for Poisson's equation. A new solution to particular two-dimensional boundary value problem for a semi-layer under a boundary constant temperature gradient is obtained in explicit form. Graphical presentations for thermal stresses Green's functions created by a unit heat source (line load in out-of-plane direction) and by a temperature gradient are also included.     

Three-dimensional thermal stresses Green’s functions for an unbounded parallelepiped under an inner point heat source

The aim of this study is to derive new constructive formulas and analytical expressions for Green’s functions (GFs) to 3D generalized boundary value problem (BVP) for an unbounded parallelepiped under a point heat source. These results were obtained using the developed harmonic integral representation method. On the base of derived constructive formulas it is possible to obtain analytical expressions for thermal stresses GFs to 16 BVPs for unbounded parallelepiped. An example of such kind is presented for a spatial BVP, GFs of which are presented in the form of the sum of elementary functions and double infinite series, containing products between exponential and trigonometric functions. An integration formula for thermal stresses, caused by the thermal data, distributed on the boundary strips at homogeneous locally mixed mechanical boundary conditions was also derived. The main di?culty to obtain these results was calculating an integral of the product between two GFs for Poisson’s equation. This integral taken on the base of the earlier established statement that main thermoelastic displacement Green’s functions (MTDGFs) satisfy the boundary conditions: (a) homogeneous mechanical conditions with respect to points of findings MTDGFs and (b) homogeneous thermal conditions with respect to points of the application of the heat source.     

Two-Dimensional Inverse Heat Transfer Analysis of Functionally Graded Materials in Estimating Time-Dependent Surface Heat Flux

Two-dimensional transient inverse heat conduction problem (IHCP) of functionally graded materials (FGMs) is studied herein. A combination of the finite element (FE) and differential quadrature (DQ) methods as a simple, accurate, and efficient numerical method for FGMs transient heat transfer analysis is employed for solving the direct problem. In order to estimate the unknown boundary heat flux in solving the inverse problem, conjugate gradient method (CGM) in conjunction with adjoint problem is used. The results obtained show good accuracy for the estimation of boundary heat fluxes. The effects of measurement errors on the inverse solutions are also discussed.     

Integral transform solution of integro-differential equations in conduction-radiation problems

A new approach for solving nonlinear integro-differential equations in conductive-radiative heat transfer has been developed. The method relies on eigenfunctions expansions for the unknown potentials, following the hybrid analytical-numerical framework provided by the generalized integral transform technique. The problem of conjugated conduction–radiation in a finned-tube radiator is selected for illustrating the method, and a traditional numerical solution of the problem is performed for comparing the proposed approach. A thorough error analysis demonstrates that the proposed scheme is very effective for handling integro-differential problems. Finally, a parametric analysis is provided, demonstrating the effects of the dimensionless groups in the temperature distribution.