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

Inverse heat conduction analysis provides an efficient approach for estimating the thermophysical properties of materials, the boundary conditions, or the initial conditions. In this paper, two-dimensional transient nonlinear inverse heat conduction problems are investigated, for estimating time- and space-dependent boundary heat flux, as well as the temperature-dependent thermal conductivities. Modifications are carried out to extend the previous one-dimensional inversion algorithm to solve the two-dimensional transient nonlinear heat conduction problem, to overcome the frequently occurring divergence issues, and to improve the stability of the inversion algorithm. Boundary-only measurements are used as additional information, and a dimensionless objective function is adopted. In the direct problem, formulations for solutions to the two-dimensional transient nonlinear heat conduction problem are derived and validated. Numerical examples show that the inversion algorithm is effective, efficient, accurate, and robust, for recovering multiple parameters, with and without a functional form.

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

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.

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

Shape reconstruction in transient heat conduction problems based on radial integration boundary element method

This paper presents an efficient inverse analysis technique based on a sensitivity coefficient algorithm to estimate the unknown boundary conditions of multidimensional steady and transient heat conduction problems. Sensitivity coefficients were used to represent the temperature response of a system under unit loading conditions. The proposed method, coupled with the sensitivity analysis in the finite element formulation, is capable of estimating both the unknown temperature and heat flux on the surface provided that temperature data are given at discrete points in the interior of a solid body. Inverse heat conduction problems are referred to as ill-posed because minor inaccuracy or error in temperature measurements cause a drastic effect on the predicted surface temperature and heat flux. To verify the accuracy and validity of the new method, two-dimensional steady and transient problems are considered. Their surface temperature and heat flux are evaluated. From a comparison with the exact solution, the effects of measurement accuracy, number and location of measuring points, a time step, and regularization terms are discussed. © 1998 Scripta Technica. Heat Trans Jpn Res, 26(6): 345–359, 1997

Application of the Laplace transform to nonlinear transient problems

Three-dimensional transient heat conduction analysis by boundary knot method

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.

: This study discusses the analytical and implementational difficulties associated with transient scalar boundary element methods. The fundamental solutions to steady and transient heat conduction problems are derived from first principles. Their implementation in the solutions to seven representative heat conduction problems is detailed. Keywords: boundary value problems; heat conduction problems; boundary element; first principles; Poisson Equation; Helmholtz Equation.

Parametric integral equations systems in 2D transient heat conduction analysis

The precise integration method for semi-discretized equation in the dual reciprocity method to solve three-dimensional transient heat conduction problems

A new method for identifying temperature-dependent thermal conductivity in transient heat conduction problems based on element differential method

This paper proposes a noniterative analytical thermal model for 3-D integrated circuits (3-D-ICs) based on the solution of the governing energy equations using Green’s function, separation of variables, and superposition methods. The model is applicable to both transient and steady-state heat conduction problems. Any finite number of dies equally sized in plane can be taken into consideration in current analysis. In addition, the analytical solution presented here is very general, so that it takes into account the different interface boundary conditions between the dies. Importantly, the thermal properties (i.e., $k$ , $\rho $ , $c$ ) and thickness of different dies can be different, and the spatial and time-varying volumetric heat generation can be applied in each of the different dies. The results compared with the finite-element simulations show that the temperature fields predicted by the proposed model can offer an accurate and efficient solution to compute the temperature fields in the representative 3-D-ICs, with a maximum deviation between the two being around 2.0% in the transient heat conduction. In the steady-state case, the maximum deviation can reach even less than 1.0%.