Nonlinear Transient Heat Conduction Problems Research Articles

A hybrid numerical method for non-linear transient heat conduction problems with temperature-dependent thermal conductivity

This paper introduces a hybrid numerical technique aimed at effectively addressing two-dimensional (2D) non-linear transient heat conduction problems, featuring temperature-dependent thermal conductivity. This scheme involves integrating the Krylov deferred correction (KDC) method in the temporal discretization and the generalized finite difference method (GFDM) in the spatial discretization. The core of the KDC method introduces a new unknown variable in the form of the first-order time derivative of temperature, leading to a non-linear equation after temporal discretization. Subsequently, the corresponding non-linear equation is solved in the spatial domain using the GFDM, supported by the Jacobian-free Newton–Krylov (JFNK) solver and fourth-order Taylor series expansion. The accuracy and stability of the hybrid approach are verified through two numerical experiments, demonstrating the strong performance of the present algorithm known as the KDC-GFDM for the simulations of the interested problems in long-time intervals.

Multi-domain physics-informed neural networks for solving transient heat conduction problems in multilayer materials

In this paper, we aim to numerically resolve linear and nonlinear transient heat conduction problems in multilayer composite materials using a deep learning method called multi-domain physics-informed neural networks (MDPINNs). For this purpose, the multilayer media are first divided into independent sub-domains based on domain decomposition technique. The single-layer deep neural networks are first established, and each sub-domain has its corresponding sub-network. Then, each two sub-networks are connected by continuity conditions on the interface to form a MDPINNs’ framework that can directly solve the transient heat conduction problem in multilayer media. The temperature distribution in the computational domain can be obtained by training the MDPINNs, including the temperature values on the interface. A series of numerical experiments are carried out to verify that the proposed framework can achieve satisfactory accuracy, including in micrometer or even nanometer structures. Compared with conventional methods, the MDPINNs have the advantage of directly solving both linear and nonlinear heat conduction problems in multilayer materials in a unified and concise form.

A cheap preconditioner based on fast diagonalization method for matrix-free weighted-quadrature isogeometric analysis applied to nonlinear transient heat transfer problems

In this paper, IsoGeometric Analysis is applied to a transient nonlinear heat conduction problem using a predictor-corrector scheme. To obtain a better approximation of the solution, we propose the k-refinement method coupled to a Matrix-Free Weighted Quadrature approach which speeds up matrix-vector operations and also reduces memory consumption. The main goal of this work is to introduce a preconditioning strategy to improve the convergence rate of the iterative solver. This method is based on Fast Diagonalization technique which exploits the tensor structure of basis functions. We show that our preconditioner is inexpensive, easy to code and as effective as other more sophisticated methods. It is also robust with respect to polynomial degree, mesh size and time-integration scheme parameters. Numerical experiments for different geometries and materials are presented, which support our theoretical results.

Numerical Analysis of Transient State Heat Transfer by Spectral Method Based on POD Reduced-Order Extrapolation Algorithm

In order to meet the requirements of high accuracy and fast algorithm for numerical heat transfer simulation, an iterative scheme of Proper Orthogonal Decomposition (for short, POD) dimension reduction based on the classical central difference Galerkin spectral method is proposed for solving two-dimensional transient heat conduction problems. The POD dimension reduction spectral method model is constructed by taking the calculation results of classical central difference Galerkin spectral method as sample data. The numerical algorithm characteristics of flow and heat transfer are studied by using a partial differential equation as a mathematical model, and the error estimation is given. Finally, different time intervals are used as parameters to simulate experiments. The results show that the POD method is applicable to transient nonlinear heat conduction problems, and the maximum average relative error of the reconstructed temperature field is 0.89675%. Moreover, the POD method not only has a high calculation accuracy, but also has an average calculation speed as high as 310.25 times that of the central difference Galerkin algorithm. It can be seen that under the condition that the error between the solution of POD dimension reduction extrapolation algorithm and the solution of classical central difference Galerkin spectrum method is small enough, the POD method can greatly reduce the calculation amount, shorten the running time, and ensure a high accuracy of the calculation results, thus verifying the effectiveness and feasibility of the algorithm.

The Transient Performance-Evaluation Model for Desalination of Seawater Based on Matrix Algorithm – a Requirement for a Ported Arabic Smart City

Abstract Based on the transient performance model for desalination of seawater based on the matrix algorithm, the desalination device is designed according to the evaporation condensation mechanism of falling film in horizontal tube, and its structure operation and operation principle are analyzed. Then, the differential equation of the transient heat-conduction problem of the seawater desalination device is constructed. On the basis of experimental data or numerical calculation results, the temperature field of structure at some time is selected to form the snapshots matrix, and a set of optimal eigenorthogonal decomposition bases is obtained by eigenorthogonal decomposition of the snapshots matrix. Finally, the finite-element discrete scheme of the nonlinear transient heat-conduction problem is reduced by using eigenorthogonal decomposition bases of the linear problem, and the lower-order ordinary differential equations of the reduced-order model for the nonlinear transient heat-conduction problem of the seawater desalination device are obtained. The ordinary differential equations of the reduced-order model are solved by solving the ordinary differential equations of the reduced-order model. The distribution of the temperature field at each time of the structure is obtained. The performance of the model is verified by numerical examples under the conditions of two-stage temperature-difference heating and only the top-level heating. The results show that the model has good accuracy and efficiency.

A new general analytical PBEM for solving three-dimensional transient nonlinear heat conduction problems with spatially-varying heat generation

In this paper, a polygonal boundary element method (PBEM) is presented, to solve the transient nonlinear heat conduction problems with spatially variable heat generation and temperature-dependent thermal physical properties. A new general method is developed to analytically compute the domain integrals regarding arbitrary nonlinearly-varying thermal conductivity, specific heat, and density, by which the computations of numerous Gaussian points in the numerical method can be circumvented and the efficiency of the PBEM can be significantly improved. The finite difference scheme is employed to evaluate the transient term, and the Newton iterative method is used to search the exact solution of the nonlinear system equation. Four numerical examples are designed to examine the property of the proposed method. The results indicate that the proposed analytical PBEM can accurately solve the three-dimensional transient nonlinear heat conduction problems with heat generation.

Transient Nonlinear Heat Conduction in Concrete Structures: A Semi-Analytical Approach.

Thermal loading, especially in fire scenarios, challenges the safety and long-term durability of concrete structures. The resulting heat propagation within the structure is governed by the heat conduction equation, which can be difficult to solve analytically because of the nonlinearity related to the thermophysical properties of concrete. A semi-analytical approach for the transient nonlinear heat conduction problem in concrete structures was established in the present work. The nonlinearity related to the temperature-dependent thermal conductivity, mass density, and specific heat capacity of heated concrete was taken into consideration. A Taylor series approximate solution was first established within a small neighborhood, employing the Boltzmann transformation in combination with the mean value theorem. Thereafter, it was extended to the whole domain by utilizing the Bernstein polynomial. The semi-analytical approach was validated by comparing it with the numerical results of two independent Finite Element simulations of nonlinear heat conduction along concrete plates, subjected to either moderate or fierce thermal loading. Absolute values of the relative errors are smaller than 5%. The validated semi-analytical approach was further applied to prediction of the temporal evolution of the temperature field of a scaled model of a subway station, subjected to fire disaster. The nonlinearities, related to the time-dependent surface temperature and the temperature-dependent thermophysical properties of concrete, were taken into consideration. The predictions agree well with the experimental measurements. The established semi-analytical approach exhibits good accuracy and stability, providing insight into the interaction between the thermophysical properties of concrete in the heat conduction process.

The efficient calculation methods for stochastic nonlinear transient heat conduction problems

This paper proposes an efficient model reduction strategy in conjunction with non-intrusive statistical moment methods for the stochastic nonlinear transient heat conduction problem. In the proposed strategy, the deterministic reduced basis vectors (RBVs) are adopted to approximate the random temperature response, and the thermal conductivity of the random nonlinear heat conduction matrix in the RBVs space is decoupled from the geometric characteristic matrix (GCM). The computational efficiency of solving the reduced random nonlinear heat conduction models can be enhanced by implementing these two technologies. The large coefficient of variation and the small coefficient of variation stochastic heat conduction problems, respectively, are addressed by the two non-intrusive statistical methods, i.e., the quasi-Monte Carlo method (QMCM) based on number theory and the modified stochastic perturbation method (MSPM). These two methods necessitate deterministic analysis and are easily integrated into the model reduction method. In four numerical instances, the selection of RBVs, the influence of the nonlinear term on the random temperature response, and the accuracy and efficiency of the two non-intrusive statistical moment methods are discussed. The numerical results highlight that the model reduction strategy based on the decoupling of thermal conductivity and GCM, in conjunction with the QMCM and the MSPM, enables the analysis of stochastic nonlinear transient heat conduction problems with a large coefficient of variation and a small coefficient of variation, respectively.

A polygonal element differential method for solving two-dimensional transient nonlinear heat conduction problems

A novel polygonal element differential method (PEDM) is presented for solving two-dimensional nonlinear transient heat conduction problems for the first time. New shape functions as well as their derivatives with respect to isoparametric coordinates are derived to treat the polygonal elements with an internal node. System equations of the PBEM are formulated in terms of the governing equation and heat flux equilibrium condition, in which, a finite difference scheme is executed for calculating the transient term. Then, the nonlinearity system equations are dealt with by the Newton iterative method. Finally, examples with different structural complexities are designed to examine the property of the proposed method. The results show that the PEDM can effectively solve general two-dimensional transient nonlinear heat conduction problems with excellent accuracy and efficiency.

The MLS-based numerical manifold method for nonlinear transient heat conduction problems in functionally graded materials

In this study, we aim to numerically solve two-dimensional nonlinear transient heat conduction problems of functionally gradient materials (FGMs), where the governing equation is a quasilinear partial differential equation of second order. For this purpose, the weak form of the initial boundary value problem is first established. Then influence domains of nodes in the moving least squares (MLS) instead of finite elements are used as the mathematical patches to construct the mathematical cover of the numerical manifold method (NMM); while the shape functions of MLS-nodes as the weight functions subordinate to the mathematical cover, leading to the moving least squares based numerical manifold method (MLS-NMM). With the weak form, the spatial discretization is carried out using MLS-NMM, whereas temporal discretization follows the Euler backward difference. Finally, a series of numerical experiments are conducted concerning transient heat conduction problems of FGMs with heat source and heat convection, suggesting that the proposed MLS-NMM enjoys advantages of both MLS and NMM in solving nonlinear transient heat conduction of FGMs.

Space-time backward substitution method for nonlinear transient heat conduction problems in functionally graded materials

In this paper, an efficient space-time backward substitution method (STBSM) is presented for solving nonlinear transient heat conduction problems in 2D and 3D functionally graded materials (FGMs). To improve the computation efficiency of the traditional backward substitution method (BSM) for transient problems, the space-time radial basis function (STRBF) and space-time trigonometric basis function (STTBF) are introduced for interpolation in the proposed method. The time dimension is regarded as a pseudo-space dimension to generate the space-time Euler metric r T in STRBF, which can be applied to transform the d -dimensional transient governing equation into ( d + 1 ) -dimensional steady-state governing equation. Benefiting from merits of the traditional BSM, an accurate numerical solution is available with only a few of computational points. Several examples are solved to verify the accuracy and efficiency of the proposed method and its merits and flaws are discussed in the end. • A space-time method is attempted for 2D and 3D heat conduction problems in FGMs. • Space-time domain and space-time radial basis function are introduced into backward substitution method. • By introducing pseudo-space-time dimension to transform the origin transient problem into steady-state problem. • The proposed method has excellent computational efficiency.

An efficient method for transient heat conduction problems with local nonlinearities based on the quasi-superposition principle

PurposeThis paper aims to develop an efficient numerical method for nonlinear transient heat conduction problems with local radiation boundary conditions and nonlinear heat sources.Design/methodology/approachBased on the physical characteristic of the transient heat conduction and the distribution characteristic of the Green’s function, a quasi-superposition principle is presented for the transient heat conduction problems with local nonlinearities. Then, an efficient method is developed, which indicates that the solution of the original nonlinear problem can be derived by solving some nonlinear problems with small structures and a linear problem with the original structure. These problems are independent of each other and can be solved simultaneously by the parallel computing technique.FindingsWithin a small time step, the nonlinear thermal loads can only induce significant temperature responses of the regions near the positions of the nonlinear thermal loads, whereas the temperature responses of the remaining regions are very close to zero. According to the above physical characteristic, the original nonlinear problem can be transformed into some nonlinear problems with small structures and a linear problem with the original structure.Originality/valueAn efficient and accurate numerical method is presented for transient heat conduction problems with local nonlinearities, and some numerical examples demonstrate the high efficiency and accuracy of the proposed method.

ANALITYCAL SOLUTION OF THE ONE DIMENSIONAL NONLINEAR TRANSIENT HEAT CONDUCTION PROBLEM USING GREEN’S FUNCTIONS

Analytical solutions showed to be an important and strong tool for understand thermal problems using mathematic tools. In this work we propose an approach about one dimensional analytical solution for a nonlinear transient heat conduction problem, were used mathematical elements such as Kirchhoff transformation, Green’s functions and the combination of them. The combination of this two methods showed that was possible to determinate an analytical solution for the nonlinear thermal problem, and showed a good approximation when compared with results from numerical methods.

AN UNCONDITIONALLY STABLE METHOD FOR TRANSIENT HEAT CONDUCTION

Time-dependent transient heat conduction problems are widely encountered in aerospace, civil engineering, metallurgical engineering, etc., and for such problems, accurate and fast numerical approaches have always attracted attention in the past decades. To achieve this goal, this paper proposes an unconditionally stable single-step time integration method for general transient heat conduction systems. In the proposed method, the temperature vector and its time derivative are formulated independently by the Langrage interpolation function, and then the relation between the temperature vector and its time derivative is defined with the weighted residual method. Theoretical analysis, including convergence rate and amplification factor, illustrates that the proposed method is strictly second-order accurate for the temperature vector and its time derivative, and it has the strong algorithmic dissipation (L-dissipation), meaning that it can quickly filter out the unwanted numerical oscillations in the high-frequency range. At present, most existing time integration methods, such as the Crank-Nicolson method and the Galerkin method, are unconditionally stable for linear transient heat conduction systems, but they are conditionally stable for nonlinear ones. To this end, this work improved the stability analysis theory for nonlinear transient heat conduction systems proposed by Hughes and used the improved stability analysis theory to design the free parameters of the proposed method. Because of this reason, the proposed method is unconditionally stable for both linear and nonlinear transient heat conduction problems. Due to the desirable algorithmic stability, the proposed method can still provide accurate and stable predictions for nonlinear transient heat conduction problems where the excellent Crank-Nicolson method fails. Some linear and nonlinear transient heat conduction problems are solved in this paper, and the results of these problems show that compared to the currently popular time integration methods, such as the Crank-Nicolson method and the backward difference formula, the proposed method enjoys noticeable advantages in accuracy, dissipation and stability.

Radial point interpolation method and high-order continuation for solving nonlinear transient heat conduction problems

Radial point interpolation method and high-order continuation for solving nonlinear transient heat conduction problems

Topology optimization of transient nonlinear heat conduction using an adaptive parameterized level-set method

ABSTRACT In this article, topology optimization of transient nonlinear heat conduction problems is solved by a parameterized level-set method. Nonlinear density-based and continuum shape sensitivity analyses are conducted for topology generation and shape optimization. Meanwhile, the backward time transient adjoint structures are derived. To suppress dependence of the initial guess for the parameterized level-set-based topology optimization, the solid isotropic material with penalization method is adopted to generate the initial topology design. Then, the parameterized level-set method is further performed to obtain the optimal shape. The material volume is strictly controlled by restricting the coefficients of radial basis functions within a dynamic bound, which can avoid incorrect topological results due to topological changes. Finally, three numerical examples are solved with the parameterized level-set method, which demonstrates that the proposed approach can obtain complex topological details with smooth boundaries for transient nonlinear heat conduction problems.

A new alternating iteration strategy based on the proper orthogonal decomposition for solving large-scaled transient nonlinear heat conduction problems

In this paper, the application of Free Element Method (FREM) is extended to the transient nonlinear heat conduction problems, and the characteristics of anisotropic, heterogeneous, temperature-dependent thermophysical properties and heat generation are also involved. In order to improve the efficiency during the nonlinear iterations, an alternating iteration strategy, FREM-ROM, is proposed, in which the proper orthogonal decomposition (POD) technique is used to establish a sectionalized extrapolation algorithm with lower dimensions and sufficiently high accuracy. This iteration strategy reduces the computational scale by using the Reduced-Order Model (ROM) technique and can guarantee the accuracy by continually correcting the database and POD bases every few time steps during the process of iterations. It also has the potential to be applied to other numerical methods for improved the efficiency. The accuracy and efficiency are examined by two numerical examples, in which the 2D example illustrates the relationship between truncated basis and accuracy in detail, and the heat transfer performance of a large-scale 3D cross-fin heat sink(CFHS) is investigated to highlight the efficiency advantages of solving large-scaled practical problems.

A new approach to solve multi-medium nonlinear transient heat conduction problems using interface integration BEM

In order to solve multi-medium nonlinear transient heat conduction problems, based on interface integration boundary element method, a new approach, which can convert the multi-domain integrals to the boundary integrals with high precision, is firstly proposed in this paper. The boundary element method with the internal and exterior source is used to establish the interface integral equation of the multi-medium heat conduction problems, and then there are some new multi-domain integrals that should be converted to the boundary integrals. However, different from the domain integral in the single medium problems, the conventional transformation technique cannot be directly applied to convert multi-domain integrals. To maintain the advantage of only boundary discretization for the BEM and take the discontinuity of integrands in multi-domain integrals into consideration, a uniformly continuous function over additional topology domains is defined. Therefore, an improved method is presented to transform the multi-domain integrals to the boundary and interface integrals based on the conventional radial integration method (RIM). What's more, the idea of the proposed approach can theoretically prove the correctness of applying RIM to the complex models with multi-connected domain, which has been only confirmed from some numerical examples. Finally, two numerical examples of multi-medium nonlinear heat conduction problems are shown to demonstrate the correctness and feasibility of the new proposed method.

An efficient and accurate hybrid weak-form meshless method for transient nonlinear heterogeneous heat conduction problems

Our purpose is to establish a numerical method meeting the requirements of efficiency, stability, accuracy and easy-using. Faced with these demands, in this paper, a hybrid method combining the advantages of 2 weak-form meshless methods is proposed for solving steady and transient heat conduction problems with temperature-dependent thermophysical properties in heterogeneous media. In the process of the calculation for an arbitrary model discretized by the hybrid method, two different weak-form methods are used for the nodes categorized as: (1) boundary nodes, (2) near boundary interior nodes and (3) interior nodes. A Galerkin free element method (GFREM) is proposed for the interior nodes of the Cartesian grid; and the local radial point interpolation method (LRPIM) is used for the other two types of nodes. The GFREM using the Cartesian grid can greatly improve the accuracy and efficiency of the hybrid scheme. Special decomposition technology to determine the irregular integral domain for LRPIM can increase the flexibility for complex models. In addition, the full implicit finite difference scheme and Newton–Raphson method are used for solving transient nonlinear problems. And non-crossing interface integral domain and support domain are employed for heterogeneous media. Compared with other methods, numerical results show better accuracy, stability and efficiency from the numerical experiments.

A fast reduced-order model for radial integral boundary element method based on proper orthogonal decomposition in nonlinear transient heat conduction problems

In order to efficiently solve nonlinear transient heat conduction problem, a method combining the radial integration boundary element method (RIBEM) and the proper orthogonal decomposition (POD) is proposed to establish the nonlinear reduced-order model, and the implementation of the reduced-order model makes fast numerical simulation possible in engineering field. Transient heat conduction problems with temperature-dependent and temperature-independent thermal conductivity are firstly solved by the RIBEM respectively, and two types of snapshots are composed of these solved transient temperature fields. Subsequently, the POD is applied in the analysis of the snapshots, and then a set of optimal orthogonal basis can be obtained. In the procedure of establishing the reduced-order model, we need to redistribute the nonlinear equations. The strategy is to separate the boundary nodes defined as Dirichlet condition, and then the orthogonal basis is used to derive a reduced-order expression for the remaining unknowns. Therefore, the degree of freedom that needs to be solved can be greatly reduced. Numerical examples show that different reduced-order models based on nonlinear basis and linear basis are all consistent with the full-order model. What is more, the computational efficiency is greatly improved due to the introduction of the reduced-order model.