Multidimensional Heat Conduction Problems Research Articles

A new method for solving multidimensional heat conduction problems

In this study, we propose an efficient boundary-type method named the half boundary method (HBM) for solving multidimensional heat conduction equations. Unlike conventional numerical methods, the core idea behind the HBM is to concentrate variables along half of the boundary and employ them to represent node variables within the solution domain. This approach streamlines the solving process by dealing with smaller-order matrices, substantially reducing storage requirements. We derived the fundamental equations of the HBM for solving multidimensional heat conduction differential equations. Furthermore, by introducing the Cayley–Hamilton formula, we optimize the program to enhance computational efficiency. Additionally, we explored heat conduction problems in solution domains differing from rectangular regions. Notably, we pioneer the application of the HBM to address three-dimensional (3-D) problems for the first time, ensuring its accuracy through rigorous validation.

Development of vector algorithm using CUDA technology for three-dimensional retinal laser coagulation process modeling

For diabetic retinopathy treatment, laser coagulation is used in modern practice. During the laser surgery process, the parameters of laser exposure are selected manually by a doctor, which requires the doctor to have sufficient experience and knowledge to achieve a therapeutic effect. On the basis of mathematical modeling of the laser coagulation process, it is possible to estimate the crucial parameters without performing an operation. However, the retina has a rather complex structure, and when even low-cost numerical methods are used for modeling, it takes a long time to obtain a result. In this regard, the development of time-efficient algorithms for three-dimensional modeling is an urgent task, since the use of such algorithms will provide a compre-hensive study within a limited time. In this paper, we study the execution time of algorithms that implement various variations in the application of the splitting method and the finite difference method, adapted to the set problem of heat conduction. The study reveals the most efficient algorithm, which is then vectorized and implemented using the CUDA technology. The study was carried out using Intel Core i7-10875H and Nvidia RTX 2080 MAX Q and showed that an analog of the vector algorithm, focused on solving a multidimensional heat conduction problem, provides an acceleration of no more than 1.5 times compared to the sequential version. The developed vector-based algorithm, focused on the application of the sweep method in all directions of the three-dimensional problem, significantly reduces the time spent on copying into the memory of the video card and provides a 40-fold acceleration in comparison with the sequential three-dimensional modeling algorithm. On the basis of the same approach, a parallel algorithm of mathematical modeling was developed, which provided a 20-fold acceleration at full processor load.

The Technology of Calculating the Optimal Modes of the Disk Heating (Ball)

The paper considers the problem of optimal control of the process of thermal conductivity of a homogeneous disk (ball). An optimization problem is posed for a one-dimensional parabolic type equation with a mixed-type boundary condition. The goal of the control is to bring the temperature distribution in the disk (ball) to a given distribution in a finite time. To solve this problem, an algorithm is proposed that is based on the gradient method. The object of the study is the optimal control problem for a parabolic boundary value problem. Using the discretization of the original continuous differential problem, difference equations are obtained for which a numerical solution algorithm is proposed. Difference approximation of a differential problem is performed using an implicit scheme, which allows to increase the speed of calculations and provides the specified accuracy of calculation for a smaller number of iterations. An approximate solution of a parabolic equation is constructed using the one-dimensional sweep method. Using differentiation of the functional, an expression for the gradient of the objective functional is obtained. In this paper, it was possible to reduce the multidimensional heat conduction problem to a one-dimensional one, due to the assumption that the desired solution is symmetric. A formula is obtained for calculating the variation of a quadratic functional that characterizes the deviation of the current temperature distribution from the given one. The flowcharts and implementations of the algorithm are presented in the form of Matlab scripts, which clearly demonstrate the process of thermal conductivity and show the computation and application of optimal control in dynamics.

Simultaneous estimation of heat transfer coefficient and reference temperature from impinging flame jets

Heat transfer from impinging flame jets to a flat plate has been assumed to be one-dimensional in most of the investigations and without radiation loss treatment. In the present work, the exact nature of diffusion of heat in the plate is investigated via solution to multidimensional heat conduction problem. Two procedures have been employed – Duhamel theorem and three dimensional transient analytical IHCP. The Duhamel theorem which is analytical model for transient one dimensional heat conduction is applied but its application failed the check of linearity requirement of the convection rate equation. From the solution by analytical IHCP for transient, three-dimensional heat conduction, the distribution of wall heat flux and the wall temperature is perfectly linear. This check confirmed that three dimensional approach has to be used. Experimental data is then analyzed by the three dimensional analytical IHCP for short and larger time intervals. It is found that for short time data, heat transfer coefficient and the reference temperature have oscillatory distribution along the radial direction on the impingement plate and for larger time data the oscillations die out. However, at larger time, radiation loss from the impingement plate becomes significant. The effect of variations in thermal conductivity of the impingement plate with the temperature on heat transfer coefficient and reference temperature is discussed. A novel method is developed to correct the heat transfer coefficient and reference temperature to incorporate radiation losses. The deviation in heat transfer coefficient and reference temperature estimated without considering variable thermal conductivity and radiation loss for large time interval is upto 50%.

A novel numerical method for solving heat conduction problems

In this work, an efficient numerical method with a high accuracy is proposed for solving the heat conduction problems. In this method, the governing equation of heat conduction in the partial differential equation form is firstly integrated over the small volume around each node point. In the resulting integrals the spatial derivatives of the unknown temperature and heat flux disappear. Then the numerical quadrature is employed to discretize the integrals. Numerical results show that when the same amount of the computer memory and CPU-time is consumed the proposed method can achieve a high accuracy in comparison with the finite volume method (FVM). Furthermore, the proposed method is more accurate than the finite element method (FEM) and boundary element method (BEM) for multi-dimensional heat conduction problems.

INVERSE VOF MESHLESS METHOD FOR EFFICIENT NONDESTRUCTIVE THERMOGRAPHIC EVALUATION

A novel computational tool based on the Localized Radial-basis Function (RBF) Collocation (LRC) Meshless method coupled with a Volume-of- Fluid (VoF) scheme capable of accurately and efficiently solving transient multi-dimensional heat conduction problems in composite and heterogeneous media is formulated and implemented. While the LRC Meshless method lends its inherent advantages of spectral convergence and ease of automation, the VoF scheme allows to effectively and efficiently simulate the location, size, and shape of cavities, voids, inclusions, defects, or de-attachments in the conducting media without the need to regenerate point distributions, boundaries, or interpolation matrices. To this end, the Inverse Geometric problem of Cavity Detection can be formulated as an optimization problem that minimizes an objective function that computes the deviation of measured temperatures at accessible locations to those generated by the LRC-VoF Meshless method. The LRC-VoF Meshless algorithms will be driven by an optimization code based on the Simplex Linear Programming algorithm which can efficiently search for the optimal set of design parameters (location, size, shape, etc.) within a predefined design space. Initial guesses to the search algorithm will be provided by the classical 1D semi-infinite composite analytical solution which can predict the approximate location of the cavity. The LRC-VoF formulation is tested and validated through a series of controlled numerical experiments. The proposed approach will allow solving the onerous computational inverse geometric problem in a very efficient and robust manner while affording its implementation in modest computational platforms, thereby realizing the disruptive potential of the proposed multi- dimensional high-fidelity non-destructive evaluation (NDE) method.

An Efficient Analytical Solution Strategy for Multi-dimensional Heat Conduction Problems

An Efficient Analytical Solution Strategy for Multi-dimensional Heat Conduction Problems

A low-order meshless model for multidimensional heat conduction problems

In this paper, a perturbation method is used to solve a two-dimensional unsteady heat conduction problem. Low-order transfer functions are defined. Step responses are obtained and compared to the complete numerical solutions given by a meshless method. The analytical results are found to be in good agreement with numerical solutions which reveals the effectiveness and convenience of the used method.

A Numerical Method on Inverse Determination of Heat Transfer Coefficient Based on Thermographic Temperature Measurement

The heat transfer coefficient in a multidimensional heat conduction problem is obtained from the solution of the inverse heat conduction problem based on the thermographic temperature measurement. The modified one-dimensional correction method (MODCM), along with the finite volume method, is employed for both twoand three-dimensional inverse problems. A series of numerical experiments are conducted in order to verify the effectiveness of the method. In addition, the effect of the temperature measurement error, the ending criterion of the iteration, etc. on the result of the inverse problem is investigated. It is proved that the method is a simple, stable and accurate one that can solve successfully the inverse heat conduction problem.

A numerical method for determining thermal conductivity distribution of the interlayer of a sandwich plate based on thermographic temperature measurement

The thermal conductivity distribution of the interlayer of a sandwich plate in a multidimensional heat conduction problem is obtained from the solution of the inverse heat conduction problem (IHCP) based on the thermographic temperature measurement. The modified one-dimensional correction method along with the finite volume method is employed for both two-dimensional and three-dimensional inverse problems. A series of numerical experiments are conducted in order to verify the effectiveness of the method. In addition, the effect of the temperature measurement error, the stopping criterion of the iteration, etc on the result of the problem is investigated. It is proved by numerical experiments that the method is a simple, stable and accurate one for this IHCP.

Dynamic observers based on Green's functions applied to 3D inverse thermal models

This work develops a new procedure for the use of dynamic observers in inverse heat conduction problem (IHCP). The dynamic observer state technique, used here, is developed to solve not only a one-dimensional but also a three-dimensional heat transfer problem. The IHCP is represented by a classical inverse definition, that is, an unknown heat flux heat is imposed at a front surface of a sample. The heat flux is then estimated by using the dynamic observer techniques and temperature data from a sensor located at the sample far from the heat source. The novelty is the new procedure to obtain the heat transfer function of the system. In this work, this function is derived using Green's function concept instead of taking Laplace transform of a spatially discretized system, as is usual. This procedure allows a great flexibility to the observer technique and represents an easy way to solve multidimensional heat conduction problem. Tests have shown excellent results.

Inverse method for temperature and stress monitoring in complex-shaped bodies

The purpose of this work is to formulate a space marching method, which can be used to solve inverse multidimensional heat conduction problems. The method is designed to reconstruct the transient temperature distribution in a whole construction element based on measured temperatures taken at selected points on the outer surface of the construction element. Next, the Finite Element Method is used to calculate thermal stresses and stresses caused by other loads such as, for instance, internal pressure. The developed method for solving temperature and total stress distribution is tested using the measured temperatures generated from a direct solution. Transient temperature and total stress distributions obtained from the method presented below are compared with the values obtained from the direct solution. Finally, the presented method is experimentally verified during the cooling of a thick-walled cylindrical element. The model of a pressure vessel was preheated at 300 °C and then cooled by cold water injection. The comparison of results obtained from the inverse method with experimental data shows the high accuracy of the developed method. The presented method allows to optimize the power block’s start-up and shut-down operations, contributes to the reduction of heat loss during these operations and to the extension of power block’s life. The fatigue and creep usage factor can be computed in an on-line mode. The presented method herein can be applied to monitoring systems that work in conventional as well as in nuclear power plants.

Numerical method for the solution of non-linear two-dimensional inverse heat conduction problem using unstructured meshes

A new method of solving multidimensional heat conduction problems is formulated. The developed space marching method allows to determine quickly and exactly unsteady temperature distributions in the construction elements of irregular geometry. The method which is based on temperature measurements at the outer surface, is especially appropriate for determining transient temperature distribution in thick-wall pressure components. Two examples are included to demonstrate the capabilities of the new approach. Copyright © 2000 John Wiley & Sons, Ltd.

A space marching method for multidimensional transient inverse heat conduction problems

A new method for the solution of multidimensional heat conduction problems is formulated. The developed space marching method allows to determine quickly and exactly unsteady temperature distributions in the construction elements of irregular geometry. The method is especially appropriate for determining transient temperature distribution in thick-wall pressure components based on temperature measurements at the outer surface. Two examples are included to demonstrate the capabilities of the new approach.

A Finite Element Analysis for Inverse Heat Conduction Problems.

We present an efficient inverse analysis method based on a sensitivity coefficient algorithm to estimate unknown boundary conditions of multidimensional heat conduction problems. Sensitivity coefficients were used to represent the temperature response of a system under unit loading conditions. The proposed method, coupled with sensitivity analysis in the finite element formulation, is capable of estimating unknown both 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 ones because minor inaccuracy or error in temperature measurement causes a drastic effect on the predicted surface temperature and heat flux. To verify the accuracy and validity of the new method, two-dimensional steady-state and transient problems are considered and their surface temperature and heat flux are evaluated. From a comparison with exact solutions, the effects of measurement accuracy, number and location of measuring points, a time step, and regularization terms are discussed.