Nonlinear Heat Conduction Problem Research Articles

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.

Exponential Euler and Backward Euler Methods for Nonlinear Heat Conduction Problems

In this paper a variant of nonlinear exponential Euler scheme is proposed for solving nonlinear heat conduction problems. The method is based on nonlinear iterations where at each iteration a linear initial-value problem has to be solved. We compare this method to the backward Euler method combined with nonlinear iterations. For both methods we show monotonicity and boundedness of the solutions and give sufficient conditions for convergence of the nonlinear iterations. Numerical tests are presented to examine performance of the two schemes. The presented exponential Euler scheme is implemented based on restarted Krylov subspace methods and, hence, is essentially explicit (involves only matrix-vector products).

Экспоненциальная и неявная схемы Эйлера для решения нелинейных задач теплопроводности

In this paper a variant of nonlinear exponential Euler scheme is proposed for solving nonlinear heat conduction problems. The method is based on nonlinear iterations where at each iteration a linear initial-value problem has to be solved. We compare this method to the backward Euler method combined with nonlinear iterations. For both methods we show monotonicity and boundedness of the solutions and give sufficient conditions for convergence of the nonlinear iterations. Numerical tests are presented to examine performance of the two schemes. The presented exponential Euler scheme is implemented based on restarted Krylov subspace methods and, hence, is essentially explicit (involves only matrix-vector products).

Choice of Regularization Methods in Experiment Processing: Solving Inverse Problems of Thermal Conductivity

This work is aimed at numerical studies of inverse problems of experiment processing (identification of unknown parameters of mathematical models from experimental data) based on the balanced identification technology. Such problems are inverse in their nature and often turn out to be ill-posed. To solve them, various regularization methods are used, which differ in regularizing additions and methods for choosing the values of the regularization parameters. Balanced identification technology uses the cross-validation root-mean-square error to select the values of the regularization parameters. Its minimization leads to an optimally balanced solution, and the obtained value is used as a quantitative criterion for the correspondence of the model and the regularization method to the data. The approach is illustrated by the problem of identifying the heat-conduction coefficient on temperature. A mixed one-dimensional nonlinear heat conduction problem was chosen as a model. The one-dimensional problem was chosen based on the convenience of the graphical presentation of the results. The experimental data are synthetic data obtained on the basis of a known exact solution with added random errors. In total, nine problems (some original) were considered, differing in data sets and criteria for choosing solutions. This is the first time such a comprehensive study with error analysis has been carried out. Various estimates of the modeling errors are given and show a good agreement with the characteristics of the synthetic data errors. The effectiveness of the technology is confirmed by comparing numerical solutions with exact ones.

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.

Estimation of the Total Heat Exchange Factor for the Reheating Furnace Based on the First-Optimize-Then-Discretize Approach and an Improved Hybrid Conjugate Gradient Algorithm

The total heat exchange factor is one of the most important thermal physical parameters in the heat transfer model for a reheating furnace machine. In this paper, a novel general strategy, which is combined with the first-optimize-then-discretize (FOTD) approach and an improved hybrid conjugate gradient (IHCG) algorithm, is proposed to identify the total heat exchange factor by solving a nonlinear inverse heat conduction problem (IHCP). Firstly, a nonlinear IHCP with the Dirichlet-type boundary condition Tm(t)=T(0,t) is built to determine the unknown total heat exchange factor w(t). Secondly, the analysis of the Fréchet gradient of the cost functional is given and the gradient is proved as Lipschitz continuous by the FOTD approach. Thirdly, based on the gradient information by FOTD, a new IHCG algorithm, whose global convergence is proved by us, is proposed for fast solving of the optimization problem. Finally, simulation experiments are given to verify the effectiveness of the proposed strategy. Compared with the first-discretize-then-optimize (FDTO) approach, the FOTD approach can reduce running time and iteration number. Compared with other CG algorithms, the proposed IHCG algorithm has better convergence performance. The experimental data by the thermocouples experiments from a reheating furnace are also given to identify the total heat exchange factor.

Multi-domain physics-informed neural network for solving forward and inverse problems of steady-state heat conduction in multilayer media

In this paper, a novel deep learning technique, called multi-domain physics-informed neural network (M-PINN), is presented to solve forward and inverse problems of steady-state heat conduction in multilayer media. By adopting the domain decomposition technique, the multilayer media is first divided into several sub-domains. Then, the fully connected neural network is employed to approximate the temperature field on each sub-domain. Finally, a large total network framework is formed by combining subnetworks of all the mediums and using continuity conditions on interfaces. By training the total network, we can obtain the temperature distribution over the whole computational domain, including the interface between every two mediums. In the proposed method, the boundary conditions are introduced into the loss function, and the governing equation is used as a constrain item, which ensures the accuracy and stability of numerical approximation. As a meshless collocation technology, the M-PINN does not require tedious procedures such as meshing and numerical integration, and can freely address forward and inverse problems of thin body and coating structure. Several numerical examples are given to illustrate the efficiency and performance of the new method. Results indicate that the Swish and the Sigmoid functions are two better activation functions for such problems. As the number of nodes increases, the number of hidden layers does not need to be increased. Even for the thin film at nanoscale, the M-PINN still obtains accurate results. Moreover, the proposed scheme shows better performance than the traditional boundary element method in solving nonlinear heat conduction problems.

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.

Automatic selection of regularization parameter in inverse heat conduction problems

This paper presents a fast, robust and automatic approach for selecting the regularization parameter in Tikhonov regularization. The methodology is based on the Generalized Cross-Validation (GCV) method, where the minimum of the GCV curve is sought via a numerical optimization algorithm. Typical convergence issues near the global minimum are addressed by changing the variables in the optimization problem in an appropriate fashion. Evidences for the effectiveness of the proposed approach are presented by performing three numerical experiments, consisting of linear and nonlinear inverse heat conduction problems. The obtained estimates are in good agreement with the reference values, showing that the obtained solutions in both problems were appropriately regularized. In this methodology, inverse problem needs not be solved with multiple regularization parameter candidates. Instead, the inverse problem is solved only once, using the optimal parameter. This feature is even more pressing in nonlinear problems, for the optimal regularization parameter may change during the iterative solution of the least squares problem, thus being adjusted at each iteration. Therefore, manual selection of the regularization parameter is not necessary, thus yielding an automatic selection methodology. Finally, as for the “robust” statement, it was made based on the numerical experiments shown on the paper, where the optimal regularization parameters for each problem vary in orders of magnitude.

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.

LaSDI: Parametric Latent Space Dynamics Identification

Enabling fast and accurate physical simulations with data has become an important area of computational physics to aid in inverse problems, design-optimization, uncertainty quantification, and other various decision-making applications. This paper presents a data-driven framework for parametric latent space dynamics identification procedure that enables fast and accurate simulations. The parametric model is achieved by building a set of local latent space model and designing an interaction among them. An individual local latent space dynamics model achieves accurate solution in a trust region. By letting the set of trust region to cover the whole parameter space, our model shows an increase in accuracy with an increase in training data. We introduce two different types of interaction mechanisms, i.e., point-wise and region-based approach. Both linear and nonlinear data compression techniques are used. We illustrate the framework of Latent Space Dynamics Identification (LaSDI) enable a fast and accurate solution process on various partial differential equations, i.e., Burgers’ equations, radial advection problem, and nonlinear heat conduction problem, achieving O(100)x speed-up and O(1)% relative error with respect to the corresponding full order models.

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.

Analysis of Nonlinear Heat Conduction Problems with Temperature-Dependent Conductivity Using Peridynamic Differential Operator

In this paper, a nonlocal approach for handling nonlinear heat conduction problems is developed using the peridynamic differential operator (PDDO). The boundary conditions and heat conduction equations are transformed from local differential form to nonlocal integral form by introducing peridynamic functions. The algebraic equations are established by the Lagrange multiplier method and variational analysis, which is solved with Newton–Raphson iterative method. Nonlinearities resulting from the temperature-dependence of material properties have been taken into account in irregular boundaries and cracked plates. Numerical examples demonstrate the effectiveness and accuracy of the PDDO for solving nonlinear heat conduction problems with temperature-dependent conductivity.

Surface transfer coefficients estimation for heat conduction problem using the Bayesian framework

This work deals with an inverse two-dimensional nonlinear heat conduction problem to determine the top and lateral surface transfer coefficients. For this, the Bayesian framework with the Markov Chain Monte Carlo algorithm is used to determine the posterior distribution of unknown parameters. To handle the computational burden, a lumped one-dimensional model is proposed. The lumped model approximations are considered within the parameter estimation procedure thanks to the Approximation Error Model. The experiments are carried out for several configurations of chamber ventilator speed. Experimental observations are obtained through a complete measurement uncertainty propagation. By solving the inverse problem, accurate probability distributions are determined. Additional investigations are performed to demonstrate the reliability of the lumped model, in terms of accuracy and computational gains.

A new multi-objective optimization approach for process parameters optimization during numerical simulation of quenching steel parts

The paper presents the numerical simulation of quenching cylindrical steel sample immersed in three different quenchants: water, 5 % aquatensid solution, and isorapid oil. The quenching process starts from the initial temperature of the cylinder at 850 °C and moves through the air until it reaches the quenching bath. The quenchant is held at constant temperature of 40 °C. The cylinder is made of carefully selected steel which does not change its structure during quenching and heating. Cylindrical samples were manufactured in three different dimensions (R, H), (mm): (12.5 × 100), (25 × 150) and (37.5 × 225), so that four measuring points were installed in each sample. Each measuring point consists of thermocouple installed beneath the cylinder surface, capable of measuring the temperature every half second. Based on the experiment, the numerical simulation is recognized as transient and nonlinear two-dimensional heat conduction problem consisting of the two main tasks: direct heat transfer problem (DHTP) and inverse heat transfer problem (IHTP). The paper proposes a new multi-objective optimization approach for the estimation of heat transfer coefficients during the numerical simulation of quenching cylindrical steel sample. The proposed approach gained better results and less convergence time compared to the results from the literature. The paper includes methods, algorithms and the source code for the calculation of the temperature fields in time and heat transfer coefficient estimation of the IHTP. The simulation software has been implemented in C# programming language and can be found at http://github.com/bhrnjica/quenching_simulation.

Application of the B-Spline Method to Solve Nonlinear Problem of Heat Conduction With Radiation Boundary Conditions Using Kirchhoff Transformation

Abstract This paper concerns the joint application of the B-spline method and the Kirchhoff transformation to solve the nonlinear problem of thermal conduction with radiation type boundary conditions. The proposed method requires few iterations, sometimes none, for solids subjected to prescribed temperature boundary conditions. This method can be deployed by other numerical approaches (boundary element method, finite element method, finite element method, etc.) for the resolution of the heat conduction equation (linear or nonlinear), in terms of the Kirchhoff transformation θ. For numerical implementation, the steady-state finite element method is considered. The numerical validation was performed for a hollow aluminum cylinder whose outer surface is subjected to radiation. Three types of thermal conductivities are considered: (i) constant, (ii) linear, and (iii) nonlinear. As an application, we studied the thermal response of an aluminum reactor, in the form of an annular disk with cooling tubes, exposed to thermal radiation.

A modified sequential gradient-based method for the inverse estimation of transient heat transfer coefficients in non-linear one-dimensional heat conduction problems

In this communication, we introduce an online sequential implementation of a gradient-based method for reconstructing transient heat transfer coefficients in the context of non-linear one-dimensional heat conduction problems. Such a method employs a quasi-Newton updating strategy for computing the descent direction, in contrast with the traditional approach based on the conjugate gradient method. We denote the resulting procedure as the sequential quasi-Newton method (SQNM). The performance of the proposed algorithm was tested in the reconstruction of triangle-, sine-, and square-wave functions that models different transient heat transfer coefficients and compared with the results obtained using a standard sequential function specification method. The SQNM was capable of properly reconstruct the aforementioned exact functions independently of the location of the temperature sensors within the body. The proposed strategy is fast, robust, and reliable, which demonstrates the suitability of employing the sequential gradient-based implementation, together with the quasi-Newton updating strategy, for reconstructing transient heat transfer coefficients in the context of one- and two-dimensional non-linear heat conduction problems. Thus, the proposed method is a novel alternative strategy to other online inverse estimation procedures.

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.

Multidimensional thermal structures in the singularly perturbed stationary models of heat and mass transfer with a nonlinear thermal diffusion coefficient

A new approach to the study of multidimensional singularly perturbed problems of nonlinear heat conduction is proposed, based on the further development and use of asymptotic analysis methods. We study the question of the existence of classical Lyapunov stable stationary solutions with boundary and internal transition layers (stationary thermal structures) of the nonlinear heat transfer equation. We suggest the efficient algorithm for constructing an asymptotic approximation to the localization surface of the transition layer. To justify the constructed formal asymptotics, we use the principle of comparison.We consider the application of the results of asymptotic analysis to solving inverse problem of reconstructing the temperature dependence of the thermal conductivity coefficient from a known position of the internal layer of thermal structure.