A spacetime collocation Trefftz method for solving the inverse heat conduction problem

A collocation method with space–time radial polynomials for inverse heat conduction problems

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

This study introduces a deep neural network approach that utilizes radial basis functions (RBFs) to solve forward and inverse problems in the process of diffusion. The input layer incorporates multiquadric (MQ) RBFs, symbolizing the radial distance between the boundary points on the spacetime boundary and the source points positioned outside the spacetime boundary. The output layer is the initial and boundary data given by analytical solutions of the diffusion equation. Utilizing the concept of the spacetime coordinates, the approximations for forward and backward diffusion problems involve assigning initial data on the bottom or top spacetime boundaries, respectively. As the need for discretization of the governing equation is eliminated, our straightforward approach uses only the provided boundary data and MQ RBFs. To validate the proposed method, various diffusion scenarios, including forward, backward, and inverse problems with noise, are examined. Results indicate that the method can achieve high-precision numerical solutions for solving diffusion problems. Notably, only 1/4 of the initial and boundary conditions are known, yet the method still yields precise results.

We compute the vacuum polarization for a massless, conformally coupled scalar field on the covering space of global, four-dimensional, anti-de Sitter space-time. Since anti-de Sitter space is not globally hyperbolic, boundary conditions must be applied to the scalar field. We consider general Robin (mixed) boundary conditions for which the classical evolution of the field is well-defined and stable. The vacuum expectation value of the square of the field is not constant unless either Dirichlet or Neumann boundary conditions are applied. We also compute the thermal expectation value of the square of the field. For Dirichlet boundary conditions, both thermal and vacuum expectation values approach the same well-known limit on the space-time boundary. For all other Robin boundary conditions (including Neumann boundary conditions), the vacuum and thermal expectation values have the same limit on the space-time boundary, but this limit does not equal that in the Dirichlet case.

PurposeThis paper aims to discuss inverse problems that arise in a variety of practical thermal processes and systems. It presents some of the approaches that may be used to obtain results that lie within a small region of uncertainty. Therefore, the non-uniqueness of the solution is reduced so that the final design and boundary conditions may be determined. Optimization methods that may be used to reduce the uncertainty and to select locations for experimental data and for minimizing the error are presented. A few examples of thermal systems are given to illustrate the applicability of these methods and the challenges that must be addressed in solving inverse problems.Design/methodology/approachIn most analytical and numerical solutions, the basic equations that describe the process, as well as the relevant and appropriate boundary conditions, are known. The interest lies in obtaining a unique solution that satisfies the equations and boundary conditions. This may be termed as a direct or forward solution. However, there are many problems, particularly in practical systems, where the desired result is known but the conditions needed for achieving it are not known. These are generally known as inverse problems. In manufacturing, for instance, the temperature variation to which a component must be subjected to obtain desired characteristics is prescribed, but the means to achieve this variation are not known. An example of this circumstance is the annealing, tempering or hardening of steel. In such cases, the boundary and initial conditions are not known and must be determined by solving the inverse problem to obtain the desired temperature variation in the component. The solutions, thus, obtained are generally not unique. This is a review paper, which discusses inverse problems that arise in a variety of practical thermal processes and systems. It presents some of the approaches or strategies that may be used to obtain results that lie within a small region of uncertainty. It is important to realize that the solution is not unique, and this non-uniqueness must be reduced so that the final design and boundary conditions may be determined with acceptable accuracy and repeatability. Optimization techniques are often used for minimizing the error. This review presents several methods that may be applied to reduce the uncertainty and to select locations for experimental data for the best results. A few examples of thermal systems are given to illustrate the applicability of these methods and the challenges that must be addressed in solving inverse problems. By considering a variety of systems, the paper also shows the importance of solving inverse problems to obtain results that may be used to model and design thermal processes and systems.FindingsThe solution of inverse problems, which frequently arise in thermal processes, is discussed. Different strategies to obtain the conditions that lead to the desired result are given. The goal of these approaches is to reduce uncertainty and obtain essentially unique solutions for different circumstances. The error of the method can be checked against known conditions to see if it is acceptable for the given problem. Several examples are given to illustrate the use of these methods.Originality/valueThe basic strategies presented here for solving inverse problems that arise in thermal processes and systems, as well as the optimization techniques used to reduce the domain of uncertainty, are fairly original. They are used for certain challenging problems that have not been considered in detail earlier. Several methods are outlined for considering different types of problems.

A sequential method is proposed to estimate boundary conditions for nonlinear heat conduction problems. An inverse solution is deduced from a e nite element method, the concept of the future time, and a modie ed Newton‐ Raphson method. The undetermined boundary condition at each timestep is denoted as theunknown variablein a set of nonlinear equations, which are formulated from the measured temperature and the calculated temperature. Then, an iterative process is used to solve the set of nonlinear equations. No selected function isneeded to represent the undetermined function in advance. Two examples are used to demonstrate the characteristics of the proposed method. Thecloseagreementbetween theexactvaluesand theestimated resultscone rmsthevalidity and accuracy of the proposed method. The results show that the proposed method is an accurate, stable, and efe cient method to determine the boundary conditions in two-dimensional nonlinear inverse heat conduction problems.

SummaryIn this paper, we address shortcomings of the method of exponential basis functions by extending it to general linear and non‐linear problems. In linear problems, the solution is approximated using a linear combination of exponential functions. The coefficients are calculated such that the homogenous form of equation is satisfied on some grid. To solve non‐linear problems, they are converted to into a succession of linear ones using a Newton–Kantorovich approach. The generalized exponential basis functions (GEBF) method developed can be implemented with greater ease compared with exponential basis functions, as all calculations can be performed using real numbers and no characteristic equation is needed. The details of an optimized implementation are described. We compare GEBF on some benchmark problems with methods in the literature, such as variants of the boundary element method, where GEBF shows a good performance. Also, in a 3D problem, we report the run time of the proposed method compared with that of Kratos, a parallel, highly optimized finite element code. The results show that in this example, to obtain the same level of error, much less computational effort is needed in the proposed method. Practical limitations might be encountered, however, for large problems because of dense matrix operations involved. Copyright © 2015 John Wiley & Sons, Ltd.

Exponential basis functions in space and time: A meshless method for 2D time dependent problems

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-

A simultaneous estimation of three boundary conditions of heat conduction problem is proposed by numerical approach. A finite-difference method is used to discretize the governing equations. The aim is to estimate the evolution of the distributions of the unknown heat fluxes from the transient temperature histories taken with several sensors sensors inside a two-dimensional sheet. The temperatures are known at three lines in the finite body. The estimation algorithm of this inverse heat conduction problem is based on the iterative regularization method and on the conjugate gradient method. The alternative direction implicit method is used to solve the direct, the adjoin and the variation problems. For each boundary condition, a descent parameter is computed. An optimal choice of the vector of the descent parameters is used in this study and shows an increase in the convergence rate. All numerical simulations are performed for two-dimensional linear heat conduction problem. The temperatures are given with measurement errors and the iterative process is stopped in accordance with the residua criterion. Numerical results are presented in this study. The temperature and heat fluxes graphs are smoothed. The accuracy and efficiency of the inverse analysis for simultaneously estimating the heat fluxes and the temperature is examined by several cases. Finally, the effects of sensor position and the magnitude of measurement error on the inverse solutions are discussed. Numerical results for some representative cases prove that heat fluxes and temperature can be predicted well by this method. In this paper, simulation results are presented and numerical performance of the method will be discussed.

In this paper, we develop a non‐iterative way to solve biharmonic boundary value problems by using a radial basis meshless method. This is an original application of meshless method to solving inverse problems without any iteration, since traditional numerical methods for inverse boundary value problems mainly are iterative and hence very time‐consuming. Numerical examples are presented for inverse biharmonic boundary value problems and corresponding direct problems, since solving direct problems is a preliminary step for inverse problems. All our examples of direct and inverse problems are solved within seconds in CPU time on a standard PC, which makes our proposed technique a great potential candidate for wide‐spread applications to other inverse problems. Copyright © 2004 John Wiley & Sons, Ltd.

The modern state of approximate integral methods used in applications, where the processes of heat conduction and heat and mass transfer are of first importance, is considered. Integral methods have found a wide utility in different fields of knowledge: problems of heat conduction with different heat-exchange conditions, simulation of thermal protection, Stefantype problems, microwave heating of a substance, problems on a boundary layer, simulation of a fluid flow in a channel, thermal explosion, laser and plasma treatment of materials, simulation of the formation and melting of ice, inverse heat problems, temperature and thermal definition of nanoparticles and nanoliquids, and others. Moreover, polynomial solutions are of interest because the determination of a temperature (concentration) field is an intermediate stage in the mathematical description of any other process. The following main methods were investigated on the basis of the error norms: the Tsoi and Postol’nik methods, the method of integral relations, the Gudman integral method of heat balance, the improved Volkov integral method, the matched integral method, the modified Hristov method, the Mayer integral method, the Kudinov method of additional boundary conditions, the Fedorov boundary method, the method of weighted temperature function, the integral method of boundary characteristics. It was established that the two last-mentioned methods are characterized by high convergence and frequently give solutions whose accuracy is not worse that the accuracy of numerical solutions.

This paper presents an efficient inverse analysis technique based on a sensitivity coefficient algorithm to estimate the unknown boundary conditions of multidimensional steady and transient heat conduction problems. Sensitivity coefficients were used to represent the temperature response of a system under unit loading conditions. The proposed method, coupled with the sensitivity analysis in the finite element formulation, is capable of estimating both the unknown temperature and heat flux on the surface provided that temperature data are given at discrete points in the interior of a solid body. Inverse heat conduction problems are referred to as ill-posed because minor inaccuracy or error in temperature measurements cause a drastic effect on the predicted surface temperature and heat flux. To verify the accuracy and validity of the new method, two-dimensional steady and transient problems are considered. Their surface temperature and heat flux are evaluated. From a comparison with the exact solution, the effects of measurement accuracy, number and location of measuring points, a time step, and regularization terms are discussed. © 1998 Scripta Technica. Heat Trans Jpn Res, 26(6): 345–359, 1997

The primary cause of gun barrel erosion is the heat generated by the shell as its travels along the barrel. Therefore, calculating the heat flux input to the gun bore is very important when investigating wear problems in the gun barrel and examining its thermomechanical properties. This paper employs the continuous-time analog Hopfield neural network (CHNN) to compute the temperature distribution in various forward heat conduction problems. An efficient technique is then proposed for the solution of inverse heat conduction problems using a three-layered backpropagation neural network (BPN). The weak generalization capacity of BPN networks when applied to the solution of nonlinear function approximations is improved by employing the Bayesian regularization algorithm. The CHNN scheme is used to calculate the temperature in a 155mm gun barrel and the trained BPN is then used to estimate the heat flux of the inner surface of the barrel. The results show that the proposed neural network analysis method successfully solves forward heat conduction problems and is capable of predicting the unknown parameters in inverse problems with an acceptable error.

Inverse identification of boundary conditions in a scramjet combustor with a regenerative cooling system