Robust Inverse Approach for Two-Dimensional Transient Nonlinear Heat Conduction Problems

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

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.

This article develops an isogeometric independent coefficients (IGA-IC) reduced order method for transient nonlinear heat conduction analysis. Herein, we first exactly represent the geometric model via isogeometric analysis (IGA), and therein provide an accurate solution for the semi-discretized equations. Next, our proposed GSSSS-1 time-stepping framework is employed to solve the transient nonlinear temperature in space and time domains. We advance our independent coefficients (IC) reduced order method to efficiently solve IGA-based transient nonlinear heat conduction problems. We extend the IC method to significantly reduce the original full IGA-discretized formulations and calculate the reduced equilibrium formulations in each Newton–Raphson iteration. Thereby, hugely improving the efficiency and guaranteeing the accuracy simultaneously. Illustrative numerical examples validate this proposed IGA-IC method is reliable, accurate, and efficient; especially, the larger the scale of the problem, the more advantages the proposed IGA-IC will inherit.

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.

On the basis of reproducing kernel particle method(RKPM), the complex variable reproducing kernel particle method (CVRKPM) is discussed. The advantage of the CVRKPM is that the correction function of a 2-D problem is formed with 1-D basis function when the shape function is obtained. Then, we apply the complex variable method to two-dimensional transient heat conduction problems. In combination with the Galerkin weak form of transient heat conduction problems, the penalty method is employed to enforce the essential boundary conditions, the CVRKPM for transient heat conduction problems is investigated and the corresponding formulae are obtained. Compared with the conventional RKPM, the CVRKPM introduced in this paper has a higher precision and a lower computation cost. Some examples given in this paper verify the effectivity of the proposed method.

A note on the determination of the thermal properties of a material in a transient nonlinear heat conduction problem

A sequential method is proposed to estimate boundary conditions for nonlinear heat conduction problems. An inverse solution is deduced from a e nite element method, the concept of the future time, and a modie ed Newton‐ Raphson method. The undetermined boundary condition at each timestep is denoted as theunknown variablein a set of nonlinear equations, which are formulated from the measured temperature and the calculated temperature. Then, an iterative process is used to solve the set of nonlinear equations. No selected function isneeded to represent the undetermined function in advance. Two examples are used to demonstrate the characteristics of the proposed method. Thecloseagreementbetween theexactvaluesand theestimated resultscone rmsthevalidity and accuracy of the proposed method. The results show that the proposed method is an accurate, stable, and efe cient method to determine the boundary conditions in two-dimensional nonlinear inverse heat conduction problems.

The numerical manifold method for 2D transient heat conduction problems in functionally graded materials

The scaled boundary finite element method based on the hybrid quadtree mesh for solving transient heat conduction problems

A transient heat pipe was modeled by a quasi steady state one-dimensional vapor flow and a transient two-dimensional wall and wick heat conduction. The assumption of quasi steady state vapor was based on the fact that the vapor dynamics have a much smaller timescale than the wall conduction. A boundary element method was proposed to solve the transient two-dimensional heat conduction problem. The vapor flow dynamics was solved by a numerical approach, including two iterative “estimate-correction” processes. Comparisons with previous experimented and numerical studies verified the model and the solution methods. This work provides an effective means of transient heat pipe analysis.

Analytical techniques for the prediction of cooling curves by quenching of steel are reported. The nonlinear transient heat conduction problem including heat of phase transformation and transient boiling of water at the surface of a cylindrical specimen is analyzed numerically. In this report, the authors' former reports are extended to calculate cooling curves inside of a specimen up to 300mm. Calculated cooling curves clearly show dependency on latent heat of phase transformation and correspondence to reference data by the continuous-cooling-transformation diagram (CCT diagram).

異方性体の非線形非定常熱伝導問題に対する境界要素法 : 第3報(OS1-3 境界要素法)

Zonal free element method for solving nonlinear transient heat conduction problems

Application of the Laplace transform to nonlinear transient problems

In the present paper, a method to reduce the computational cost associated with solving a nonlinear transient heat conduction problem is presented. The proposed method combines the ideas of two level discretization and the multilevel time integration schemes with the proper orthogonal decomposition model order reduction technique. The accuracy and the computational efficiency of the proposed methods is discussed. Several numerical examples are presented for validation of the approach. Compared to the full finite element model, the proposed method significantly reduces the computational time while maintaining an acceptable level of accuracy.