Transient Heat Conduction Problems Research Articles

An accelerated sequential function specification method based on LM gradient for transient inverse heat conduction problem

The sequential function specification method (SFSM) and the Levenberg-Marquardt method (LM) are two typical methods for inverse heat conduction problems (IHCP). However, these two methods have poor convergence and low convergence speed. Therefore, the combination of LM with SFSM is depended on to propose a modified Sequential Levenberg-Marquardt gradient method (SLM). The computation results of three shapes of heat fluxes with noise show the SLM is averagely improved over 16.3% more than SFSM in computation accuracy, but equally reduced over 39.9% less than in computation time. On other hand, when the emphasis is on the effect of the Aitken acceleration method, Tikhonov regularization and polynomial order on SLM, a neural network prediction model of the root mean square error is trained by regularization factor, polynomial order and the number of future time steps. Subsequently, the genetic algorithm optimization method is proposed for the minimum root mean square error so that it can improve the computation accuracy and efficiency of IHCP to the maximum extent. Finally, Second-order polynomial order and small order-of-magnitude regularization factor are highly recommended for parameter selection with multiple choices. The ASLM only needs to determine the optimal number of future time steps.

A hybrid shape functions based temporal finite element method to solve transient heat conduction problems

A novel temporal finite element method is presented to solve Fourier and non-Fourier heat conduction problems. Two kinds of temporal finite element models are developed using Gurtin variational principle and weighted residual technique, respectively. A kind of hybrid shape function with polynomial and trigonometric basis is stressed to give more flexible and appropriate description of temporal behavior of temperature. A recursive algorithm is developed by which the temperature solution at a specific time can be obtained only via a matrix power product with the initial condition, and a criterion of stability analysis is derived, which is can numerically be conducted when the constitution of shape function and time step size are prescribed. The proposed approach is available for Fourier and non-Fourier heat transfer problems, and can conveniently be combined with well-developed numerical algorithms for the boundary value problem, such as FE, SBFEM, etc. Various numerical examples, including those with the nonlinear thermal conductivity, radiative boundary condition, etc. are provided to illustrate the efficiency of proposed approaches, and impacts of the temporal FE model, constitution of shape functions, and step size, etc. are taken into account. Satisfactory results are achieved in comparison with the analytical and the ABAQUS based numerical solutions.

Numerical Computation of 2D Domain Integrals in Boundary Element Method by (α, β) Distance Transformation for Transient Heat Conduction Problems

When the time-dependent boundary element method, also termed the pseudo-initial condition method, is employed for solving transient heat conduction problems, the numerical evaluation of domain integrals is necessitated. Consequently, the accurate calculation of the domain integrals is of crucial importance for analyzing transient heat conduction. However, as the time step decreases progressively and approaches zero, the integrand of the domain integrals is close to singular, resulting in large errors when employing standard Gaussian quadrature directly. To solve the problem and further improve the calculation accuracy of the domain integrals, an (α, β) distance transformation is presented. Distance transformation is a simple and efficient method for eliminating near-singularity, typically applied to nearly singular integrals. Firstly, the (α, β) coordinate transformation is introduced. Then, a new distance transformation for the domain integrals is constructed by replacing the shortest distance with the time step. With the new method, the integrand of the domain integrals is substantially smoothed, and the singularity arising from small time steps in the domain integrals is effectively eliminated. Thus, more accurate results can be obtained by the (α, β) distance transformation. Different sizes of time steps, positions of source point, and shapes of integration elements are considered in numerical examples. Comparative studies of the numerical results for the domain integrals using various methods demonstrate that higher accuracy and efficiency are achieved by the proposed method.

Analytic Modelling of 2-D Transient Heat Conduction with Heat Source Under Mixed Boundary Constraints by Symplectic Superposition

Abstract Many studies have been conducted on 2-D transient heat conduction, but analytic modelling is still uncommon for the cases with complex boundary constraints due to the mathematical challenge. With an unusual symplectic superposition method, this paper reports new analytic solutions to 2-D isotropic transient heat conduction problems with heat source over a rectangular region under mixed boundary constraints at an edge. With the Laplace transform, the Hamiltonian governing equation is derived. The applicable mathematical treatments, e.g., the variable separation and the symplectic eigenvector expansion in the symplectic space, are implemented for the fundamental solutions whose superposition yields the ultimate solutions. Benchmark results obtained by the present method are tabulated, with verification by the finite element solutions. Instead of the conventional Euclidean space, the present symplectic-space solution framework has the superiority on rigorous derivations without pre-determining solution forms, which may be extended to more issues with the complexity caused by mixed boundary constraints.

The dimension coupling method for 3D transient heat conduction problem with variable coefficients

In this study, a dimension coupling method (DCM) is presented to study the variable coefficients transient heat conduction problems. The dimension splitting method (DSM) is introduced to split the 3D domain into some 2D domains, which are discretized by selecting the improved element-free Galerkin (IEFG) method; thus the corresponding discretized equation in each 2D domain is derived, and the finite element method (FEM) is employed to deal with these discretized equations in a third direction, and the time-dependent term is discretized by selecting the finite difference method (FDM), thus the final solved equation of the transient heat conduction problems is obtained. In the first example, the convergence is proved by increasing the nodes and meshes on the influence of relative errors. Some numerical results illustrate that the proposed method can substantially enhance the computational efficiency of the IEFG method. When an exact solution of the numerical example is a nonlinear function about splitting direction with natural boundary condition is given, the proposed method has shorter computing time than the hybrid element-free Galerkin (HEFG) method when dealing with this situation.

Precise integration boundary element method for solving nonlinear transient heat conduction problems with temperature dependent thermal conductivity and heat capacity

The precise integration boundary element method (PIBEM) is used to solve nonlinear transient heat conduction problems with temperature dependent thermal conductivity and heat capacity in this article. At first, a boundary-domain integral equation can be obtained by using the fundamental solution for steady linear heat conduction problems. Secondly, the domain integrals are converted into boundary integrals by using the radial integration method to get a pure boundary integral equation, which can retain the advantage of dimension reduction of the boundary element method. Then, after discretization, a first-order nonlinear differential equation system in time is obtained, and the precise integration method with predictor-corrector technique is used to solve the system of nonlinear equations. In the end, several numerical examples are presented to verify the accuracy and stability of the presented method. The results obtained by the presented method are less affected by the time step size and satisfactory results can be obtained even for a large time step size.

A domain decomposing reduced-order method for solving the multi-domain transient heat conduction problems

In this article, a domain decomposition algorithm combined with a reduced-order model (ROM) is proposed to deal with the transient heat conduction problems in multi-domain structures. Using the domain decomposition algorithm and the finite element method (FEM), the original problem is decomposed into multiple non-overlapping subdomains, which can be solved independently under appropriate boundary conditions. However, additional inter-regional connected calculation will be brought through the simple geometry decomposition. Thus, the proper orthogonal decomposition (POD), a projection-based ROM technique, is utilized to further reduce the number of unknowns and the calculation time for each subdomain. To be more specific, the local POD bases of each subdomain are obtained by decomposing a snapshot matrix that gathered according to time by the application of the virtual boundary conditions on interfaces of the adjacent subdomains and other boundaries. Furthermore, the problems under different boundary conditions can be linked by the variable condensation technique on the interfaces, which enhances the efficiency of the numerical solver dramatically. Finally, the accuracy and efficiency of the proposed method are demonstrated by comparing the results calculated using both the proposed algorithm and the direct finite element simulation.

A new analytical method for transient heat conduction in composite disks applied to thermal plug and abandonment of oil wells

A new hybrid method for transient heat conduction problems is developed and applied to simulate a Plug and Abandonment (P&A) operation of oil wells. An oil well is approximated by concentric disks in a one-dimensional configuration, which allows for use of polar coordinates. The application of the Separation of Variables Method (SVM) is used as the analytical framework for the solution of the conductive heat transfer arising from a volumetric heat source located in the center of the disks. The SVM is able to solve only time-independent boundary conditions. However, using the Duhamel's theorem, the solution determined with the SVM can be used to achieve the solution when both time-dependent boundary conditions and internal heat generation are prescribed. Lastly, for verification purposes, a commercial software that solves the transient temperature field by means of numerical procedures, providing reliability to the analytical method proposed in this work.

The thermal-mass-redistributed finite element method combined with the Houbolt method for 2D transient heat conduction problems

The thermal-mass-redistributed finite element method combined with the Houbolt method for 2D transient heat conduction problems

Fundamental Solutions of the Equations of Classical and Generalized Heat Conduction Models

This article presents the mathematical formulations of transient heat conduction problems corresponding to the models of classical heat conduction using the Fourier law and generalized heat conduction based on the Cattaneo–Vernotta–Lykov law (Maxwell–Cattaneo model), as well as the generalized Green–Nagdy type II and III models. The Fourier transforms in spatial coordinates and the Laplace transforms in time were used to obtain the fundamental solutions of the equations of the Maxwell–Cattaneo and Green–Nagdy type II and III models of classical and generalized heat conduction. The results were displayed graphically and analyzed. Differences between the considered heat conduction models were shown, and suggestions for their practical application were given.

A new MIB-based time integration method for transient heat conduction analysis of discrete and continuous systems

By incorporating the matched interface and boundary (MIB) method into time integration computation, a new MIB-based time integration (MIBTI) approach is proposed for solving transient heat conduction problems in this work. Indeed, this is a truly effective time-domain MIB method that can directly discretize space and time derivatives, and the transient heat conduction problem (an initial-value problem) is transformed into a pseudo-space boundary-value problem. The initial and final time points are regarded as boundary points in the transformed space domain. Two algorithms are developed to deal with the final time point. Unlike conventional recursive schemes for time integration, all unknown variables can be determined by solving the transformed boundary-value problem in the present approach. Various discrete and continuous systems without and with internal heat sources under different boundary conditions are considered to examine the working performance of the proposed method. The applicability of the time stability criterion in the conventional recursive schemes to the proposed method is also discussed. It is found that excellent numerical stability and convergence can be achieved by using the present MIBTI method for solving transient heat conduction problems.

A new distance transformation method of estimating domain integrals directly in boundary integral equation for transient heat conduction problems

Accurate estimation of domain integrals is of crucial importance for the pseudo-initial condition method applied in transient heat conduction analysis. When small time step is used, the integrand of the domain integral changes dramatically and is close to singular. A straightforward evaluation of the domain integrals using Gaussian quadrature may produce large errors. To improve the computational accuracy of two-dimensional domain integrals in boundary integral equation, a new distance transformation method is presented in this paper. The proposed method takes the time step as the transformation parameter, implementing the new transformation in the radial direction directly. With the new distance transformation method, the integrand of the domain integral can be smoother, thus more accurate results can be obtained. Numerical examples have demonstrated the efficiency and accuracy of the proposed method.

Development of Methods for Solving Problems of Transient Heat Conduction of Plane Layered Media

Development of Methods for Solving Problems of Transient Heat Conduction of Plane Layered Media

Finite Difference-Peridynamic Differential Operator for Solving Transient Heat Conduction Problems

Finite Difference-Peridynamic Differential Operator for Solving Transient Heat Conduction Problems

An implicit thermo-mechanical coupling model for heat conduction and thermal cracking in brittle materials

In this paper, an implicit thermo-mechanical coupling model is developed to simulate heat conduction and thermal cracking in brittle materials. In the thermal analysis, an implicit particle heat conduction model (PHCM) is proposed. A linear algebraic system of heat conduction equations is derived based on the temperatures of the particle system and the heat flux among adjacent particles. An analytical expression is introduced for the involved correction coefficient, with which the macroscopic thermal conductivity coefficient can be directly adopted without calibration. Furthermore, a thermal conductivity reduction coefficient is introduced to account for the thermal resistance of the cracks. In the mechanical analysis, particle discontinuous deformation analysis (PDDA), which is an implicit variant of the discrete element method, is employed. The PHCM is first verified against analytical and finite element solutions for transient heat conduction problems. The results show that the PHCM is an effective and accurate heat conduction model that readily reflects the thermal resistance of cracks. Then, the thermo-mechanical coupling model is verified by the expansion of heated specimens under different conditions. Finally, the successful application of the coupling model in the simulation of thermal shock tests demonstrates its effectiveness for thermal cracking problems.

Detection of surface moving heat source using experimental temperature measurements on the opposite surface and inverse techniques

The identification of mobile sources is a widely researched area in multiple scientific and technical domains. Detecting mobile heat sources is an essential task in different fields, including fire safety, environmental monitoring, material science, and energy systems. In this investigation, we present an innovative solution to solve the inverse problem of detecting mobile heat sources, utilizing a three-dimensional transient inverse heat conduction problem (IHCP) formulation. The goal is to identify a moving heat source from thermographic infrared measurements of the rear face. The proposed method includes a finite volume approach with model reduction techniques to improve computational efficiency. Additionally, we use the conjugate gradient method to solve the inverse problem. We verified the proposed approach by conducting experiments on a metal plate with a moving heat source from a laser and comparing recalculated temperature of the front face with thermocouple measurements. Our results indicate that our method can detect the location and trajectory of the mobile heat source with high accuracy and efficiency. This study also highlighted the limits of source speed that can be detected by this method. This technique could be considered as an excellent option for real-time monitoring of mobile heat sources in practical applications.

Transient heat transfer in concentric cylinders using periodic boundary condition and asymmetric heat generation applied to thermal plug and abandonment of oil wells

When a well does not fulfil its objectives, it is repurposed or permanently plugged, a shift in budget from revenue to Plug and Abandonment (P&A) expenditures occurs. New technologies being developed for P&A make use of a powerful heat emitter for melting the surrounding of the well forming a solid plug after the cool down. However, there is a gap in understanding the heat transport process in the well and much need for appropriate mathematical tools and solutions for estimating the effectiveness of Thermal P&A. The thermal analysis of the process requires eigenvalues in polar coordinates, which return only real quantities due to its implicitly dependence on the angular eigenvalues. Here, a hybrid analytical/numerical method is applied to an asymmetric transient heat conduction problem. The oil well is conceived as a 2-D multi-layer disc cast in polar coordinates, where the Separation of Variables Method (SVM) was applied to achieve a closed-form solution. Asymmetric boundary conditions of first, second and third kind can be implemented utilizing the proposed framework. A Finite Volume Method (FVM) numerical solution was produced for code verification. Lastly, research results show that temperature levels arising from the thermite reaction throughout the composite cylinder domain in radial and azimuthal ranges are enough for surpassing the melting point of the production tube steel. The results evidenced the thermal efficiency of the TP&A procedure and suggested that the production column may be destroyed by fusion, thus reducing tamponing expenses significantly.

An overset improved element-free Galerkin-finite element method for the solution of transient heat conduction problems with concentrated moving heat sources

A novel Overset Improved Element-Free Galerkin-Finite Element Method (Ov-IEFG-FEM) for solving transient heat conduction problems with concentrated moving heat sources is introduced in this communication. The method is a mesh-less/mesh-based chimera-type approach that utilises a coarse finite element mesh to discretise the problem geometry, while a separate set of overlapping nodes (patch nodes) moves with the heat source to capture the marked thermal gradients with higher accuracy using the Improved Element-Free Galerkin (IEFG) technique. Outside of the heat source area, where accuracy requirements are significantly lower, the thermal problem is solved using the Finite Element Method (FEM). The approach involves solving the problem over these two overlapping computational domains and transferring numerical information between the approximations performed on both. Such transfer of information occurs through immersed boundaries that are properly defined, enabling straightforward achievement of accurate results. The proposed Ov-IEFG-FEM is conceived to provide an enriched solution by appropriately coupling the temperature fields computed on the patch nodes and the coarse background mesh using IEFG and FEM, respectively. A comprehensive explanation concerning the appropriate coupling between the temperature fields of both the coarse background finite element mesh and the fine arrangement of moving patch nodes for the IEFG computations, is also provided in this communication. Numerical experiments demonstrate the method effectiveness in accurately and efficiently solving transient heat conduction problems with concentrated moving heat sources.

An arbitrary order numerical framework for transient heat conduction problems

This paper presents an arbitrary order numerical approach for achieving highly accurate solutions in transient heat conduction problems. In this framework, the arbitrary order Krylov deferred correction method (KDC) is first utilized to discretize the temporal direction of the problems, ensuring high-accuracy results by selecting an adequate number of Gaussian nodes. During each time iteration, a boundary value problem in terms of the inhomogeneous modified Helmholtz equation is generated by utilizing the KDC methodology. Subsequently, the arbitrary order GFDM is introduced to solve the boundary value problem, allowing for flexible selection of the Taylor series expansion order to improve spatial accuracy. Consequently, the present numerical framework is of arbitrary order of accuracy and very stable through integrating advantages of the arbitrary order KDC and the arbitrary order GFDM. The developed numerical framework is ultimately verified for its characteristic of high accuracy and stability through two numerical examples in two-dimensions and three numerical examples in three-dimensions. The results highlight the significant improvement in numerical accuracy achieved by the present arbitrary-order numerical approach for long-time simulations.

Geometric inverse estimation for the inner wall of a furnace under transient heat conduction based on dual reciprocity boundary element method

A boundary shape estimation problem for the furnace’s inner surface is solved using the dual reciprocity boundary element method (DRBEM) and the conjugate gradient method (CGM) under transient heat conduction. The DRBEM is utilized to eliminate the drawbacks of traditional numerical methods and classical boundary element method of discretizing the entire computational domain, with only boundary discretization. The inversion results are obtained by applying the CGM to minimize the objective function, in which the sensitivity coefficients are calculated with the complex variable derivation method (CVDM), making the calculation precise and independent of step size. To verify the accuracy of the DRBEM in solving the transient heat conduction problem, the influencing factors including radial-basis function, the number of internal collocation points, and time step size are investigated. The influences of measurement time interval, future time step, initial guess, measurement error, and the number and position of measurement points on the inversion results are analyzed. Meanwhile, the effectiveness of the proposed approach is tested by numerical examples, and the inversion results show that it is stable, accurate, and efficient for identifying different and complicated unknown boundary shapes of the furnace.