Estimation for inner surface geometry of furnace wall using inverse process combined with grey prediction model

In this work the inner surface geometry of a cylindrical furnace wall is estimated using inverse process method combined with grey prediction model. In estimating process a virtual area extended from the inner surface of furnace wall is used for analysis. The heat conduction equation and the boundary condition are first discretized by finite difference method to form a linear matrix equation; the inverse model is then optimized by linear least-squares error method and the temperatures of virtual boundary are obtained from a few of measured temperatures in furnace wall using the linear inverse model; and finally the temperature distribution of system is got by direct process and the inner surface geometry of furnace wall can be estimated accordingly. The result shows that using inverse process combined with grey prediction model the geometry can be exactly estimated from relatively small number of measured temperatures. Moreover, the effects of measurement error, location, and number of measured points on the estimation for inner surface geometry of furnace wall are discussed in detail.     

Geometry estimation for the inner surface in a furnace wall made of functionally graded materials

The aim of this study is to solve an inverse geometry heat conduction problem (shape identification problem) to estimate the unknown geometry of the inner surface in a furnace wall which is made of functionally graded materials (FGMs). The inner surface geometry is estimated from the temperatures of measured points within the furnace wall. The inverse algorithm used in the study is based on the conjugate gradient method (CGM) and the discrepancy principle. The effect of measurement errors and measurement locations on the estimation accuracy is also investigated. Two different examples are discussed. Results show that the unknown geometry of the inner wall surface can be predicted precisely by using the present approach.     

Virtual boundary element method in conjunction with conjugate gradient algorithm for three-dimensional inverse heat conduction problems

In this article, the virtual boundary element method (VBEM) in conjunction with conjugate gradient algorithm (CGA) is employed to treat three-dimensional inverse problems of steady-state heat conduction. On the one hand, the VBEM may face numerical instability if a virtual boundary is improperly selected. The numerical accuracy is very sensitive to the choice of the virtual boundary. The condition number of the system matrix is high for the larger distance between the physical boundary and the fictitious boundary. On the contrary, it is difficult to remove the source singularity. On the other hand, the VBEM will encounter ill-conditioned problem when this method is used to analyze inverse problems. This study combines the VBEM and the CGA to model three-dimensional heat conduction inverse problem. The introduction of the CGA effectively overcomes the above shortcomings, and makes the location of the virtual boundary more free. Furthermore, the CGA, as a regularization method, successfully solves the ill-conditioned equation of three-dimensional heat conduction inverse problem. Numerical examples demonstrate the validity and accuracy of the proposed method.     

A semi-analytical boundary collocation solver for the inverse Cauchy problems in heat conduction under 3D FGMs with heat source

Abstract

In this article, the inverse Cauchy problems in heat conduction under 3D functionally graded materials (FGMs) with heat source are solved by using a semi-analytical boundary collocation solver. In the present semi-analytical solver, the combined boundary particle method and regularization technique is employed to deal with ill-pose inverse Cauchy problems. The domain mapping method and variable transformation are introduced to derive the high-order general solutions satisfying the heat conduction equation of 3D FGMs. Thanks to these derived high-order general solutions, the proposed scheme can only require the boundary discretization to recover the solutions of the heat conduction equations with a heat source. The regularization technique is used to eliminate the effect of the noisy measurement data on the accessible boundary surface of 3D FGMs. The efficiency of the proposed solver for inverse Cauchy problems is verified under several typical benchmark examples related to 3D FGM with specific spatial variations (quadratic, exponential and trigonometric functions).     

Geometry estimation of the furnace inner wall by an inverse approach

The geometry of a furnace inner wall is estimated by an inverse method in this work. Based upon the concept of a virtual area, in the analysis process the heat conduction equation with boundary conditions was first discretized by a finite difference method to form a matrix equation. And then the linear least-squares error method was applied to determine the temperature of virtual boundary by inverse process. Finally, the geometry of the furnace inner wall can be obtained by direct process. Furthermore, the effects of the measurement errors, number of measurements and position of measurements on the deviation of geometry prediction are also discussed.     

Implementation of Geometrical Domain Decomposition Method for Solution of Axisymmetric Transient Inverse Heat Conduction Problems

The objective of this article is to study the performance of iterative parameter and function estimation techniques to solve simultaneously two unknown functions (quadratic in time, and linear in time and space) using transient inverse heat conduction method in conjunction with a geometrical domain decomposition approach, in cylindrical coordinates. For geometrical decomposition of physical domain, a multi-block method has been used. The numerical scheme for the solution of the governing partial differential equations is the finite element method. The results of the present study for a configuration composed of two joined disks with different heights are compared to those of exact heat source and temperature boundary condition using inverse analysis. Good agreement between the estimated results and exact functions has been observed for parameter estimation techniques in contrast to those of function estimation approach. In summary, the results show that the function estimation technique is sensitive to the location of measurement points, but is useful to estimate unknown functions without a priori knowledge of the functions' spatial and/or temporal distributions. However, the function estimation technique suffers from a drawback: its implementation and data extraction are less straightforward than parameter estimation method. Finally, it is shown that the use of geometrical domain decomposition offers the possibility of developing a robust inverse analysis code for general purpose heat conduction problems.     

Application of the method of fundamental solutions and radial basis functions for inverse heat source problem in case of steady-state

The paper deals with the inverse determination of heat sources in steady 2-D heat conduction problem. The problem is described by Poisson equation in which the function of the right hand side is unknown. The identification of the strength of a heat source is given by using the boundary condition and a known value of temperature in chosen points placed inside the domain. For the solution of the inverse problem of identification of the heat source the method of fundamental solution with radial basis functions is proposed. The accurate results have been obtained for five test problems where the analytical solutions were available.     

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.     

Identification of boundary conditions for non-Fourier heat conduction problems by differential transformation DRBEM and improved cuckoo search algorithm

Abstract

The differential transformation method is combined with the dual reciprocity boundary element method to solve the non-Fourier heat conduction problems in functionally gradient materials. The cuckoo search algorithm is improved by the Broyden–Fletcher–Goldfarb–Shanno algorithm to identify the boundary conditions for the heat conduction problems. The polynomial function related to coordinate and time is proposed to approximate the unknown boundary conditions. Numerical examples discuss the influences of measurement point numbers and measurement errors on inverse solutions. Numerical results demonstrate the effectiveness and accuracy of the proposed method.     

ACCURATE IMPLEMENTATION OF LINE AND DISTRIBUTED SOURCES IN HEAT CONDUCTION PROBLEMS BY THE BOUNDARY-ELEMENT METHOD

This article is concerned with the boundary-element analysis of two- and three-dimensional problems of nonlinear heat conduction. Line and distributed heat sources within the domain are implemented with no need for domain discretization. The boundary integrals developed in 2-D analysis are evaluated analytically, so no numerical integration is required. This enhances the accuracy of the solutions, especially when the geometric boundary of the domain is complex or the body is thin, or in cases where solution in the vicinity of the boundary is required. Accuracy of analysis with analytical integration in comparison with numerical integration is highlighted by an example, and the efficiency of the algorithms for implementation of various kinds of heat sources in 2-D and 3-D are explored through several other examples.     

A decentralized fuzzy inference method for solving the two-dimensional steady inverse heat conduction problem of estimating boundary condition

This paper addresses a new technique for solving the two-dimensional steady inverse heat conduction problem, which named decentralized fuzzy inference (DFI) method. First of all, a group of decentralized fuzzy inference units are designed, and the fuzzy inference for each fuzzy inference unit is conducted which bases on the difference between the measured and the computed temperature at each measuring location. The computed temperatures are obtained by solving the direct heat conduction problem with the finite difference method. And then, inference results of fuzzy inference units are weighted to yield compensation values of the unknown boundary temperatures. The unknown boundary temperatures are estimated by updating guess temperatures continuously with compensation values. Numerical experiments are carried out with different initial guesses, the number of measuring points and measurement errors. Comparing results of DFI method and Levenberg–Marquardt (L–M) method, we can conclude that DFI method is valid.     

Exact analytical solution of unsteady axi-symmetric conductive heat transfer in cylindrical orthotropic composite laminates

This study presents an exact analytical solution of transient heat conduction in cylindrical multilayer composite laminates. This solution is valid for the most generalized linear boundary conditions consisting of the conduction, convection and radiation heat transfer. Here, it is supposed that the fibers are winded around the cylinder and their direction can be changed in each lamina. Laplace transformation is applied to change the domain of the solutions from time into the frequency. An appropriate Fourier transformation has been derived using the Sturm–Liouville theorem. Here, a set of equations for Fourier coefficients are obtained based on the boundary conditions both inside and outside the cylinder, and the continuity of temperature and heat flux at boundaries between adjacent layers. The exact solution of this set of equations is obtained using Thomas algorithm and Fourier coefficients are expressed by recessive relations. Due to the difficulty of applying the inverse Laplace transformation, the Meromorphic function method is utilized to find the transient temperature distribution in laminate. Some industrial examples are presented to investigate the ability of current solution for solving the wide range of applied steady and unsteady problems.     

Non-iteration estimation of thermal conductivity using finite volume method

The finite volume approach is developed for the inverse estimation of thermal conductivity in one-dimensional domain. The differential governing equation of heat conduction is converted to a system of linear equations in matrix form using the temperature data and heat generation at the discrete grid points as well as surface heat flux. The unknown thermal conductivities are obtained by solving the system equations directly. The features of the present method are that no prior information about the functional form of the thermal conductivity is required and no iterations in the calculation process are needed. The accuracy and robust of the present method are verified by comparing examples of inverse estimation of spatially and temperature-dependent thermal conductivities with the exact solutions.     

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.     

Weighted Least-Squares Collocation Method (WLSCM) for 2-D and 3-D Heat Conduction Problems in Irregular Domains

In this article, the weighted least-squares collocation method (WLSCM) is adopted to deal with two- and three-dimensional heat conduction problems in irregular domains. A radial basis function (RBF) is selected to construct the approximation function. To improve the accuracy and stability, some auxiliary points are increased within the domain of interest. Only inner nodes are used to construct the approximation function, and the equilibrium equations are satisfied not only at collocation points but also at auxiliary points, so the equations should be solved in a least-square sense. A 2-D case that has an analytical solution is simulated by the proposed method and the outcome verifies that the present method can obtain desired accuracy and efficiency. Then the current method is adopted to compute one 2-D and two 3-D cases of engineering heat conduction problems in irregular complex domains. The results show that the present method can deal effectively with the heat conduction problems of both 2-D and 3-D irregular domains.     

Identification of Geometric Boundary Configurations in Heat Transfer Problems Via Element-Free Galerkin and Level-Set Methods

This article presents a numerical model to solve inverse geometry heat transfer problems to determine an unknown boundary shape. The evolution of unknown shapes is described by the level-set method (LSM) and is controlled by a Hamilton–Jacobi equation which is solved by a finite-difference (FD) scheme. The element-free Galerkin method (EFGM) is employed to determine the temperature field in the process of boundary evolution via a slight adjustment of the position and number of nodes. The proposed numerical model is verified via an identification of a curvilinear boundary, and the effects of initial guess, number of probing points, measurement error, and density of EFGM nodes and the LSM FD grid are taken into account.     

Analysis of boundary inverse heat conduction problems using space marching with Savitzky-Gollay digital filter

This paper presents a method by which boundary inverse heat conduction problems can be analyzed. A space marching algorithm is used for formulating and solving parabolic and hyperbolic inverse heat conduction problems. The solution of numerical examples shows that a combination of the digital filter with the hyperbolic approximation of inverse heat conduction problem increases the stability of the results without loss of resolution. The validity of numerical solution for the inverse problem is examined by comparing the obtained results with the direct solution of the problem.     

The determination temperature-dependent thermal conductivity as inverse steady heat conduction problem

The paper deals with the non-iterative inverse determination of the temperature-dependent thermal conductivity in 2-D steady-state heat conduction problem. The thermal conductivity is modeled as a polynomial function of temperature with the unknown coefficients. The identification of the thermal conductivity is obtained by using the boundary data and additionally from the knowledge of temperature inside the domain. The method of fundamental solutions is used to solve the 2-D heat conduction problem. The golden section search is used to find the optimal place for pseudo-boundary on which are placed the singularities in the frame of method of fundamental solutions.     

A shape design problem in determining the interfacial surface of two bodies based on the desired system heat flux

A shape design problem (or inverse geometry problem) in determining the geometry of interfacial surface between two conductive bodies in a three-dimensional multiple region domains, based on the desired system heat flux and domain volume, is examined in this study. The design algorithm utilized the Levenberg–Marquardt method (LMM), B-spline surface generation and the commercial software CFD-ACE+. The validity of this shape design analysis is examined using the numerical experiments. Different desired system heat fluxes are considered in the numerical test cases to justify the validity of the present algorithm in solving the three-dimensional shape design problems. Finally, the results show that for the two different cases considered in this work, the maximum increasing in the system heat flux is obtained as 11.3% and 14.1%, respectively. It is also concluded that when the boundary control points of interfacial surface are free to move, maximum system heat flux can be obtained by the present algorithm since it has more degree of freedom in describing the interfacial surface.