A Gaussian RBFs method with regularization for the numerical solution of inverse heat conduction problems

In this paper, a recursion numerical technique is considered to solve the inverse heat conduction problems, with an unknown time-dependent heat source and the Neumann boundary conditions. The numerical solutions of the heat diffusion equations are constructed using the Gaussian radial basis functions. The details of algorithms in the one-dimensional and two-dimensional cases, involving the global or partial initial conditions, are proposed, respectively. The Tikhonov regularization method, with the generalized cross-validation criterion, is used to obtain more stable numerical results, since the linear systems are badly ill-conditioned. Moreover, we propose some results of the condition number estimates to a class of positive define matrices constructed by the Gaussian radial basis functions. Some numerical experiments are given to show that the presented schemes are favourably accurate and effective.     

A hybrid regularization method for inverse heat conduction problems

This paper presents a hybrid regularization method for solving inverse heat conduction problems. The method uses future temperatures and past fluxes to reduce the sensitivity to temperature noise. A straightforward comparison technique is suggested to find the optimal number of the future temperatures. Also, an eigenvalue reduction technique is used to further improve the accuracy of the inverse solution. The method provides a physical insight into the inverse problems under study. The insight indicates that the inverse algorithm is a general purpose algorithm and applicable to various numerical methods (although our development was based on FEM), and that the inverse solutions can be obtained by directly extending Stolz's equation in the least‐squares error (LSE) sense. Direct extension of the present method to the inverse internal heat generation problems is made. Four numerical examples are given to validate the method. The effects of the future temperatures, the past fluxes, the eigenvalue reduction, the varying number of future temperatures and local iterations for non‐linear problems are studied. Copyright © 2005 John Wiley & Sons, Ltd.     

Numerical solution to a two-dimensional inverse heat conduction problem

The two-dimensional inverse heat conduction problem is solved using the method of dynamic programming. One problem involves determining two unknown heat flux histories imposed on two faces of a slab. Several numerical experiments were performed to ascertain the effects of noise and the weighting parameters.     

A finite-difference method for solving inverse boundary-value problems of heat conduction

A finite-difference search method is described for determining the temperature and heat flux on one boundary of the body if the temperature and heat flux on the other boundary are known. The results of numerical experiments, which show that the method has proved to be efficient, are discussed.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 32, No. 3, pp. 502–507, March, 1977.     

A new space marching method for solving inverse heat conduction problems

A new space marching method is presented for solving the one-dimensional nonlinear inverse heat conduction problems. The temperature-dependent thermal properties and boundary condition on an accessible part of the boundary of the body are known. Additional temperature measurements in time are taken with a sensor located in an arbitrary position within the solid, and the objective is to determine the surface temperature and heat flux on the remaining part of the unspecified boundary. The temperature distribution throughout the solid, obtained from the inverse analysis, is then used for the computation of thermal stresses in the entire domain, including the boundary surfaces. The proposed method is appropriate for on-line monitoring of thermal stresses in pressure components. The three presented example show that the method is stable and accurate.     

Multi-model method for solving nonlinear transient inverse heat conduction problems

The multi-model inverse method for nonlinear inverse problems is established based on the multi-model control theory. First the model switching variable is chosen and several typical operating balance points in the workspace of the balance variable are selected. Then for each operating balance point the linear local model is established, and the local controller is designed for each linear local model. Finally, according to the instantaneous matching degree between the actual model and the local models, the inversion results of each local controller are weighted and synthesized to obtain the final inversion result. Numerical tests are implemented to solve the one-dimensional nonlinear inverse heat conduction problem by the multi-model inverse method associated with the dynamic matrix control (DMC) and DMC filter, respectively. Numerical results by the multi-model inverse method based on DMC demonstrate that the multi-model inverse method is a highly computationally efficient and accurate algorithm for inverse problems with complicated direct problems. Numerical results by the multi-model inverse method based on DMC filter show that the presented method can extend the applied field of the complicated linear inverse algorithms such as digital filter to the nonlinear inverse problems and it can obtain satisfactory inversion results.     

Numerical solution of inverse nonlinear problems in unsteady heat conduction

Results are presented for solution of inverse heat-conduction problems, solved by a trialand-error method using analog and digital computers (implicit scheme, mesh method).     

A non‐iterative finite element method for inverse heat conduction problems

A non‐iterative, finite element‐based inverse method for estimating surface heat flux histories on thermally conducting bodies is developed. The technique, which accommodates both linear and non‐linear problems, and which sequentially minimizes the least squares error norm between corresponding sets of measured and computed temperatures, takes advantage of the linearity between computed temperatures and the instantaneous surface heat flux distribution. Explicit minimization of the instantaneous error norm thus leads to a linear system, i.e. a matrix normal equation, in the current set of nodal surface fluxes. The technique is first validated against a simple analytical quenching model. Simulated low‐noise measurements, generated using the analytical model, lead to heat transfer coefficient estimates that are within 1% of actual values. Simulated high‐noise measurements lead to h estimates that oscillate about the low‐noise solution. Extensions of the present method, designed to smooth oscillatory solutions, and based on future time steps or regularization, are briefly described. The method's ability to resolve highly transient, early‐time heat transfer is also examined; it is found that time resolution decreases linearly with distance to the nearest subsurface measurement site. Once validated, the technique is used to investigate surface heat transfer during experimental quenching of cylinders. Comparison with an earlier inverse analysis of a similar experiment shows that the present method provides solutions that are fully consistent with the earlier results. Although the technique is illustrated using a simple one‐dimensional example, the method can be readily extended to multidimensional problems. Copyright © 2003 John Wiley & Sons, Ltd.     

A meshless method for the stable solution of singular inverse problems for two-dimensional Helmholtz-type equations

We investigate a meshless method for the stable and accurate solution of inverse problems associated with two-dimensional Helmholtz-type equations in the presence of boundary singularities. The governing equation and boundary conditions are discretized by the method of fundamental solutions (MFS). The existence of boundary singularities affects adversely the accuracy and convergence of standard numerical methods. Solutions to such problems and/or their corresponding derivatives may have unbounded values in the vicinity of the singularity. Moreover, when dealing with inverse problems, the stability of solutions is a key issue and this is usually taken into account by employing a regularization method. These difficulties are overcome by combining the Tikhonov regularization method (TRM) with the subtraction from the original MFS solution of the corresponding singular solutions, without an appreciable increase in the computational effort and at the same time keeping the same MFS discretization. Three examples for both the Helmholtz and the modified Helmholtz equations are carefully investigated.     

A method of fundamental solutions for inverse heat conduction problems in an anisotropic medium

Recently, Hon and Wei proposed a method of fundamental solutions for solving isotropic inverse heat conduction problems (IHCP). It provides an efficient global approximation scheme in both spatial and time domains. In this paper, we try to extend the inherently meshless and integration-free method to solve 2D IHCP in an anisotropic medium. First, we acquire the fundamental solution of the governing equation through variables transformation. Then the truncated singular value decomposition and the L-curve criterion are applied to solve the resulting matrix equation which is highly ill-conditioned. Results for several numerical examples are presented to demonstrate the efficiency of the method proposed. The relationship between the accuracy of the numerical solutions and the value of the parameter T is also investigated.     

The reproducing kernel particle method for two-dimensional unsteady heat conduction problems

The numerical solution to two-dimensional unsteady heat conduction problem is obtained using the reproducing kernel particle method (RKPM). A variational method is employed to furnish the discrete equations, and the essential boundary conditions are enforced by the penalty method. Convergence analysis and error estimation are discussed. Compared with the numerical methods based on mesh, the RKPM needs only the scattered nodes instead of meshing the domain of the problem. The effectiveness of the RKPM for two-dimensional unsteady heat conduction problems is examined by two numerical examples.     

A new numerical method for the boundary optimal control problems of the heat conduction equation

A new numerical method is developed for the boundary optimal control problems of the heat conduction equation in the present paper. When the boundary optimal control problem is solved by minimizing the objective function employing a conjugate‐gradient method, the most crucial step is the determination of the gradient of objective function usually employing either the direct differentiation method or the adjoint variable method. The direct differentiation method is simple to implement and always yields accurate results, but consumes a large amount of computational time. Although the adjoint variable method is computationally very efficient, the adjoint variable does not have sufficient regularity at the boundary for the boundary optimal control problems. As a result, a large numerical error is incurred in the evaluation of the gradient function, resulting in premature termination of the conjugate gradient iteration. In the present investigation, a new method is developed that circumvents this difficulty with the adjoint variable method by introducing a partial differential equation that describes the temporal and spatial dynamics of the control variable at the boundary. The present method is applied to the Neumann and Dirichlet boundary optimal control problems, respectively, and is found to solve the problems efficiently with sufficient accuracy. Copyright © 2001 John Wiley & Sons, Ltd.     

Use of splines in the solution of inverse boundary problems of heat conduction

A stable algorithm is proposed for the solution of one-dimensional inverse boundary problems of heat conduction based on the solution of the Cauchy problem. The incorrectness of the problem is eliminated by the use of regularized splines.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 34, No. 2, pp. 338–343, February, 1978.     

A modified sequential function specification finite element-based method for parabolic inverse heat conduction problems

A method for enhancing the stability of parabolic inverse heat conduction problems (IHCP) is presented. The investigation extends recent work on non-iterative finite element-based IHCP algorithms which, following Becks two-step approach, first derives a discretized standard form equation relating the instantaneous global temperature and surface heat flux vectors, and then formulates a least squares-based linear matrix normal equation in the unknown flux. In the present study, the non-iterative IHCP algorithm is stabilized using a modified form of Becks sequential function specification scheme in which: (i) inverse solution time steps, t, are set larger than the data sample rate, , and (ii) future temperatures are obtained at intervals equal to . These modifications, contrasting with the standard approach in which the computational, experimental, and future time intervals are all set equal, are designed respectively to allow for diffusive time lag (under the typical circumstance where is smaller than, or on the order of the characteristic thermal diffusion time scale), and to improve the temporal resolution and accuracy of the inverse solution. Based on validation tests using three benchmark problems, the principle findings of the study are as follows: (i) under dynamic surface heating conditions, the modified and standard methods provide comparable levels of early-time resolution; however, the modified technique is not subject to over-damped estimation (as characteristic of the standard scheme) and provides improved error suppression rates, (ii) the present method provides superior performance relative to the standard approach when subjected to data truncation and thermal measurement error, and (iii) in the nonlinear test problem considered, both approaches provide comparable levels of performance. Following validation, the technique is applied to a quenching experiment and estimated heat flux histories are compared against available analytical and experimental results.     

Transient two-dimensional heat conduction problems solved by the finite element method

A finite element weighted residual process has been used to solve transient linear and non-linear two-dimensional heat conduction problems. Rectangular prisms in a space-time domain were used as the finite elements. The weighting function was equal to the shape function defining the dependent variable approximation. The results are compared in tables with analytical, as well as other numerical data. The finite element method compared favourably with these results. It was found to be stable, convergent to the exact solution, easily programmed, and computationally fast. Finally, the method does not require constant parameters over the entire solution domain.     

Approximate method for the numerical solution of unsteady heat conduction problems with variable boundary conditions of the third kind

An approximate method for solving unsteady heat conduction problems with variable boundary conditions of the third order, which can be used for bodies of complex shape, is described.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 17, No. 4, pp. 688–696, September, 1969.     

An overdeterrnined two-dimensional transient inverse heat conduction problem

The method of solving the two-dimensional nonlinear inverse heat conduction problems is presented. The time- and space temperature distribution inside a solid and heat transfer coefficient distribution on the boundary of the solid is determined based on the temperature histories measured in several selected inside locations. The problem is overdetermined because the number of measuring points is higher than the number of calculated parameters (components of heat transfer coefficient).     

Simulation of processes in heat sensors based on solution of inverse problems in heat conduction

The paper deals with nonstationary problems in heat conduction, which arise in connection with the determination of the heat flux density and temperature on the surface of a model in intermittent high-enthalpy wind-tunnel facilities by the results of temperature measurements using intramodel heat sensors. The solution of inverse problem in heat conduction in a one-dimensional formulation with an arbitrary time dependence of the heat flux density is obtained by two methods, namely, by iterations and by integral transformations with finite limits. In the former method, the inverse problem is reduced to a system of two coupled integral and integro-differential equations of the Volterra type relative to the temperature and heat flux density on the external boundary. Calculations demonstrate that the numerical solution asymptotically approaches the exact solution, and the iteration method exhibits smoothing properties and is stable with respect to random errors of measurement. In the integral method, an inverse problem for the class of boundary functions satisfying the Dirichlet conditions and represented by a partial sum of the Fourier series reduces to a set of algebraic equations which has a unique solution. In the absence of measurement errors, the solution of inverse problem is exact. Examples are given of constructing solutions in the presence of random noise; it is demonstrated that, in the case of reasonable restriction of the range of frequencies to be analyzed, the errors in the solution do not exceed the mean-square level of noise.Translated from Teplofizika Vysokikh Temperatur, Vol. 43, No. 1, 2005, pp. 071–085.Original Russian Text Copyright © 2005 by E. P. Stolyarov.     

BEM approach to inverse heat conduction problems

In the paper the inverse problem of the estimation of the temperature and heat flux on the surface of a heat conducting body is considered. Since the problem belongs to the ill-posed, the method of solving the boundary probelem as well as the method of stabilizing the results of calculations are required. The boundary element method is applied to solve the boundary problem whereas combined ‘future steps’ and the regularization method is applied to obtain stable results. A numerical example is included.