Multidimensional Heat Conduction Research Articles

On some linear two-point inverse problem for a multidimensional heat conduction equation with semi-nonlocal boundary conditions

On some linear two-point inverse problem for a multidimensional heat conduction equation with semi-nonlocal boundary conditions

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.

Heat Flux Identification of Aircraft Structure with Artificial Neural Network Compensation

Aerodynamic heating measurement of reusable hypersonic vehicles has always been an important aspect of hypersonic vehicle design. In this paper, a mechanistic-model-based heat flux identification method with artificial neural network (ANN) compensation is established to determine the spatially distributed heat flux of the aircraft structure. A one-dimensional heat conduction model is used to estimate heat flux by a robust and efficient algorithm integrating Tikhonov regularization with Levenberg–Marquardt method. The one-dimensional estimated heat flux has large errors for not considering multidimensional heat conduction effects. The proposed mechanistic-model-based method is then utilized to compensate the multidimensional heat conduction by ANN. The performance of the proposed method will be assessed by the determination of the heat flux of a two-dimensional plate and aircraft structure. Results show that compared with the one-dimensional inversion results, ANN compensation method can significantly improve the accuracy of estimated heat flux and is also applicable for larger levels of heat flux. The proposed compensation method is an effective technique to identify the nonuniform surface heat flux of multidimensional structures.

A Comprehensive Review on Multi-Dimensional Heat Conduction of Multi-Layer and Composite Structures: Analytical Solutions

A Comprehensive Review on Multi-Dimensional Heat Conduction of Multi-Layer and Composite Structures: Analytical Solutions

Deducing flux from single point temperature history when relative spatial variation of flux is prescribed

Surface heat transfer in convective and radiative environments is sometimes measured by recording the surface temperature history in a transient experiment and interpreting this surface temperature with the aid of a suitable model for transient conduction within the substrate. The semi-infinite one-dimensional model is often adopted, and several well-developed techniques for application of this model to surface temperature data are available. However, when a spatial variation of heat flux exists across the surface, the application of the semi-infinite one-dimensional approach may not always be a reasonable approximation because the errors are transient and typically increase with time. In this paper we introduce a method for treatment of the measured surface temperature history that is more accurate than the semi-infinite one-dimensional approximation when substrate lateral conduction is significant and the relative spatial distribution of the flux is known a priori. This new method uses the so-called Neumann heat kernel, which evolves a temperature over an insulated domain with unit energy initially deposited at a specified point. A useful impulse response function is formed by integrating this Neumann heat kernel against the spatial variation of flux over the surface of the domain. By applying the heat kernel result for the sphere to the analysis of a convective experiment using hemispherical-nosed probes, we demonstrate how the theoretical results enhance the practical analysis of transient surface temperature measurements. The current approach is superior to former methods relying on semi-empirical approximations because the multi-dimensional heat conduction within the substrate is modelled with greater fidelity using the heat kernel analysis.

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.

Effect of heat treatment on the microstructure evolution and mechanical behaviour of a selective laser melted Inconel 718 alloy

This study aims to address the fact that after the selective laser melting (SLM) of an Inconel 718 alloy, the microstructures at each fabrication position are anisotropic due to the coupling effect of multi-dimensional heat conduction caused by the forming mode, and combined with the formation laws of each phase in the Inconel 718 alloy, the microstructure evolution mechanism and corresponding mechanical behaviour laws at each fabrication position under different heat treatment regimes (including precipitation ageing (DA), δ ageing + precipitation ageing (SA) and high-temperature microstructure homogenization + ageing + precipitation ageing (HSA)) were studied. It is found that with increasing heat treatment temperature (DA→SA→HSA), the microstructure of each fabrication position gradually changes from anisotropic to isotropic, including the weakening and disappearance of the primary fabricated characteristic microstructure (“perlage” molten pools and long strip molten channels) and the primary crystal structure, and the corresponding recrystallization degree significantly increases. Among them, the nucleation and growth of recrystallized grains occurs through the process of high temperature polygonization recovery and the grain boundary migration mode, and the orientation of the recrystallized grains is determined by the direction of the grain boundary migration during recrystallization. Additionally, with the continuous dissolution of a large number of intergranular Laves phases precipitates in the primary fabricated microstructure (AB) and the further uniform diffusion of solute from intergranular to intragranular, the number of second-phase strengthening particles (γ″ phase and γʹ phase) precipitated in the crystal increases and the distribution uniformity is improved significantly. Furthermore, the intergranular δ phase precipitation will gradually change from intergranular to intragranular, and when the heat treatment temperature is close to the melting temperature of the δ phase, its shape changes from a primary short rod-like shape and needle-like shape to a bulk shape. Thus, under the influence of the precipitation-dissolution coupling between the strengthening phase in the crystal and the intragranular brittle phase, the comprehensive mechanical strength of the samples after heat treatment shows the trend of SA ＞ HSA ＞ DA ＞ AB, while the elongation shows the opposite trend (AB ＞ DA ＞ HSA ＞ SA) with the precipitation content of the δ phase, and the mechanical behaviours are all superior to those of the mechanical standards of forging and casting of Inconel 718 alloy.

Sequential particle filter estimation of a time-dependent heat transfer coefficient in a multidimensional nonlinear inverse heat conduction problem

In the applied mathematical modelling of heat transfer systems, the heat transfer coefficient (HTC) is one of the most important parameters. This paper proposes a combination of the Method of Fundamental Solutions (MFS) with particle filter Sequential Importance Resampling (PF-SIR) to estimate the time-dependent HTC in two-dimensional transient inverse heat conduction problems from non-standard boundary integral measurements. These measurements ensure the unique solvability of the boundary coefficient identification problem. Numerical results show high performance on several test cases with both linear and nonlinear Robin boundary conditions, supporting the synergy between the MFS and simulation-based particle filter sequential analysis methods.

Temperature field distribution and microstructure of cement-based materials under cryogenic freeze-thaw cycles

Cryogenic temperature is one of the most severe conditions for the application of cement-based materials. In this paper, the effects of different factors on the temperature field distribution in cement paste and mortar under cryogenic freeze-thaw cycles were investigated. Based on the analysis of internal pore structure and micro-morphology, the main influencing factors were determined, and the failure mechanism of cement-based materials was analyzed. The results indicated that under the condition of unsteady multi-dimensional heat conduction, the temperature field in the cement-based materials presented gradient distribution and the temperature variation in the specimen lagged, the temperature in the center of the specimen lagged most because the heat transfer in the specimen had energy consumption. Several factors such as adding fine aggregate, reducing water-to-cement ratio and low water content could improve the freeze-thaw cycles resistance of cement-based materials to a certain extent. The cryogenic freeze-thaw damage was highly related to the freezing and migration of pore water due to the cryopump effect. The pore structure was destroyed and many micro-cracks below 20 nm were produced. Under the further crystallization pressure, the larger pores were formed by destroying, expanding and connecting, thus affecting the mechanical properties and durability of cement-based materials.

Microstructure characteristics and mechanical behaviour of a selective laser melted Inconel 718 alloy

In this study, an Inconel 718 alloy was prepared by selective laser melting (SLM) with a pre-alloyed powder, and the microstructure and mechanical behaviour of the fabricated parts were studied. First, the settled layer structure, composed of several interlaced "perlage" molten pools and molten channels, was captured by low-magnification optical microscopy (OM), and the formation mechanism was revealed by the Gaussian heat source distribution characteristics of the laser source used for the SLM fabrication. Second, high-magnification scanning electron microscopy (SEM) captured the different crystal structures in different fabricated positions; meanwhile, the "cross-border growth phenomenon" along the forming direction and the "growth and distribution characteristics of the crystal" in the radial direction (perpendicular to the forming direction) were also captured. By introducing the component super-cooling theory, the growth mechanism of crystal structures in different positions under a multi-dimensional temperature gradient heat conduction coupling mode is proposed. Finally, combined with transmission electron microscopy (TEM) and a mechanical properties test, it was found that the mechanical properties were between those of parts fabricated by casting and those by forging because the γ" phase and γ' phase were distributed in the crystal, while the carbide phase and Laves phase were distributed in the intercrystal region.

A Splitting Scheme for Diffusion and Heat Conduction Problems

The problem of mathematical modeling and optimization of nonstationary diffusion and heat conduction processes is considered. An approach that uses the idea of splitting and computation of the obtained difference schemes using explicit schemes of point to point computing is proposed for numerical solution of multidimensional diffusion and heat conduction initial–boundary-value problems. Construction of difference splitting schemes, approximation and stability on initial data are investigated. Differential properties of the quality functional are analyzed for the numerical solution of the optimal control problem for a parabolic equation. An iterative algorithm for finding the optimal control is proposed.

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.

A backward-forward Lie-group shooting method for nonhomogeneous multi-dimensional backward heat conduction problems under a long time span

Based on the idea of the Lie-group shooting method and the backward group preserving scheme, a novel backward-forward algorithm is developed to solve non-linear, nonhomogeneous, multi-dimensional backward heat conduction problems under long time spans. For a nonhomogeneous, multi-dimensional, backward heat conduction problem, it is very difficult to integrate towards the time direction, even when using a high-order numerical scheme. To avoid time integration and increase the computational efficiency, a novel backward-forward Lie-group scheme is proposed. According to the quadratic equation of the Lie-group shooting method, a solution is applied to obtain the initial condition and to examine the final condition. Using the reciprocal relationship between the solutions of forward and backward schemes, the proposed algorithm can avoid the time integration of the numerical scheme and numerical divergence. To illustrate the effectiveness and accuracy of the proposed algorithm, several benchmarks in multi-dimensions are tested. The numerical results indicate that the proposed algorithm can achieve an efficient and accurate solution, even with noisy measurement data by comparing the estimation results with the existing literatures.

A modified Lie-group shooting method for multi-dimensional backward heat conduction problems under long time span

This paper proposes a modified Lie-group shooting method to solve multi-dimensional backward heat conduction problems under long time spans. The backward heat conduction problem is renowned for being ill posed because the solutions are generally unstable and highly dependent on the given data. For dealing with those problems, the Lie-group shooting method is one of the most powerful tools to find the unknown initial condition for the backward heat conduction problems in the time domain. In previous studies, the Lie-group shooting method uses the time and spatial semi-discretization technique to change the integration direction of numerical schemes and then increase the time span. However, the conversional Lie-group shooting method cannot get to the core of divergence problems for the backward heat conduction problems, especially the increased computational time. The main reason is that a real single-parameter Lie-group element occurs at zero, and a generalized midpoint Lie-group element is not equivalent to the single-parameter Lie-group element in the Minkowski space. Hence, to overcome the above problems, the relationship of the initial condition, the final condition and a real single-parameter r is assessed. According to the constraint condition of the initial and final condition, a real single-parameter r depends on the time span to maintain the numerical convergence. Again, in order to preserve the same Lie-group property in the time direction, the high-order Lie-group scheme based on the generalized linear group in Euclidean space is introduced, which concurrently satisfies the constraint of the cone structure, the Lie-group and the Lie algebra at each time step. The accuracy and efficiency are validated, even under noisy measurement data, by comparing the estimation results with existing literature.

Numerical solution of multi-dimensional transient nonlinear heat conduction problems with heat sources by an extended element differential method

In this paper, the element differential method is extended to solve a transient nonlinear heat conduction problem with a heat source and temperature-dependent thermophysical properties for the first time. The transient term is discretized by employing a finite difference scheme. An iterative methodology is developed to deal with the nonlinearity caused by temperature-dependent thermophysical properties. Examples of two-dimensional (2D) and three-dimensional (3D) problems are given to validate the present method for solving multi-dimensional transient nonlinear heat conduction problems. The results show that the present EDM provides a promising way that is effective and with high accuracy for solving multi-dimensional transient nonlinear heat conduction problems.

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%.

Efficient automatic discrete adjoint sensitivity computation for topology optimization – heat conduction applications

PurposeTopology optimization is a method used for developing optimized geometric designs by distributing material pixels in a given design space that maximizes a chosen quantity of interest (QoI) subject to constraints. The purpose of this study is to develop a problem-agnostic automatic differentiation (AD) framework to compute sensitivities of the QoI required for density distribution-based topology optimization in an unstructured co-located cell-centered finite volume framework. Using this AD framework, the authors develop and demonstrate the topology optimization procedure for multi-dimensional steady-state heat conduction problems.Design/methodology/approachTopology optimization is performed using the well-established solid isotropic material with penalization approach. The method of moving asymptotes, a gradient-based optimization algorithm, is used to perform the optimization. The sensitivities of the QoI with respect to design variables, required for optimization algorithm, are computed using a discrete adjoint method with a novel AD library named residual automatic partial differentiator (Rapid).FindingsTopologies that maximize or minimize relevant quantities of interest in heat conduction applications are presented. The efficacy of the technique is demonstrated using a variety of realistic heat transfer applications in both two and three dimensions, in conjugate heat transfer problems with finite conductivity ratios and in non-rectangular/non-cuboidal domains.Originality/valueIn contrast to most published work which has either used finite element methods or Cartesian finite volume methods for transport applications, the topology optimization procedure is developed in a general unstructured finite volume framework. This permits topology optimization for flow and heat transfer applications in complex design domains such as those encountered in industry. In addition, the Rapid library is designed to provide a problem-agnostic pathway to automatically compute all required derivatives to machine accuracy. This obviates the necessity to write new code for finding sensitivities when new physics are added or new cost functions are considered and permits general-purpose implementations of topology optimization for complex industrial applications.

A highly accurate backward-forward algorithm for multi-dimensional backward heat conduction problems in fictitious time domains

This paper proposes highly accurate one-step backward-forward algorithms for solving multi-dimensional backward heat conduction problems (BHCPs). The BHCP is renowned for being ill-posed because the solutions are generally unstable and highly dependent on the given data. In this paper, the present algorithm combines algebraic equations with a high-order Lie-group scheme to construct one-step algorithms called the backward fictitious integrate method (BFTIM) and the forward fictitious integrate method (FFTIM). First, the original parabolic equation is transformed into a new parabolic equation of an evolution type by introducing a fictitious time variable. Then, the numerical integration of the discretized algebraic equations must satisfy the constraints of the cone structure, Lie-group and Lie algebra at each fictitious time step. Finally, the algorithms with the minimum fictitious time steps along the manifold of the Lie-group scheme approach the true solution with one step when given an initial guess. In addition, this paper provides a strategy to determine the initial guess, which is the reciprocal relationship of the initial condition (IC) and the final condition (FC). More importantly, the IC and FC can be recovered by the BFTIM and FFTIM according to the relation between the IC and FC, even under large noisy measurement data. Five numerical examples of the BHCP are tested and numerical results demonstrate that the present schemes are more effective and stable. In general, the numerical implementations of the BFTIM and FFTIM are simple and have one-step convergence speeds.

The quasi‐reversibility method for a final value problem of the time‐fractional diffusion equation with inhomogeneous source

This paper is devoted to discuss a multidimensional backward heat conduction problem for time‐fractional diffusion equation with inhomogeneous source. This problem is ill‐posed. We use quasi‐reversibility regularization method to solve this inverse problem. Moreover, the convergence estimates between regularization solution and the exact solution are obtained under the a priori and the a posteriori choice rules. Finally, the numerical examples for one‐dimensional and two‐dimensional cases are presented to show that our method is feasible and effective.

Uncertainty analysis of in-vessel retention in a high power reactor during severe accident

The in-vessel retention by external reactor vessel cooling (IVR-ERVC) is one of the severe accident management strategies for mitigation or termination of the accident by holding the radioactive material in the reactor vessel. We made a lumped parameter based computer code to evaluate the thermal load on the reactor vessel, and applied the code for a probabilistic analysis for a high power (>1000MWe) reactor assuming the APR1400 condition. The present analysis showed the focusing effect in most of cases. The heat transfer models, especially the critical heat flux correlation, showed a significant impact on the result. Consideration of the multidimensional heat conduction effect in the vessel wall can reduce the excessive conservatism in the vessel failure criterion by assuming the same heat fluxes at inside and outside of the vessel wall. The influence of the probability distribution profiles for the uncertainty parameters was also investigated.