Nonlinear Transient Heat Transfer Problems Research Articles

Explicit spectral element collocation method for nonlinear transient heat transfer

In this paper, we proposed a novel collocation method to solve nonlinear transient heat transfer problems in typical intelligent materials. The new method is derived from the weighted residual method in which the weight of Jacobian is proven to be the best choice. In the paper, the physical variables and their fluxes are respectively approximated by two series of Lagrange interpolation polynomials at Chebyshev-Gauss-Lobatto nodes. In the proposed method, the mass matrices of the final linear system of equations are diagonal and the explicit time discretization method can be employed. Using the proposed method, we solved the heat transfer in intelligent materials and it is proven to be quite efficient and accurate, which benefits from that no large sparse matrix is generated and solved. By the proposed method, one can easily solve the problems with complex geometries.

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.

Full-Scale Bounds Estimation for the Nonlinear Transient Heat Transfer Problems with Interval Uncertainties

A package solution is presented for the full-scale bounds estimation of temperature in the nonlinear transient heat transfer problems with small or large uncertainties. When the interval scale is relatively small, an efficient Taylor series expansion-based bounds estimation of temperature is stressed on the acquirement of first and second-order derivatives of temperature with high fidelity. When the interval scale is relatively large, an optimization-based approach in conjunction with a dimension-adaptive sparse grid (DSG) surrogate is developed for the bounds estimation of temperature, and the heavy computational burden of repeated deterministic solutions of nonlinear transient heat transfer problems can be efficiently alleviated by the DSG surrogate. A temporally piecewise adaptive algorithm with high fidelity is employed to gain the deterministic solution of temperature, and is further developed for recursive adaptive computing of the first and second-order derivatives of temperature. Therefore, the implementation of Taylor series expansion and the construction of DSG surrogate are underpinned by a reliable numerical platform. The parallelization is utilized for the construction of DSG surrogate for further acceleration. The accuracy and efficiency of the proposed approaches are demonstrated by two numerical examples.

Solving heat conduction problems with phase-change under the heat source term approach and the element-free Galerkin formulation

The present work has been conducted in order to introduce an alternative solution for heat transfer problems with phase-change and moving boundaries, via the element-free Galerkin (EFG) method. In this novel approach, the phase-change non-linearity is addressed under the heat source term methodology. For this purpose, the global weak-form of the internal energy balance for such a non-linear transient heat transfer problem has been developed by including the latent heat as a source vector. In order to prove the reliability of this approach, the EFG formulation has been adapted to the solution of a 1-D alloy solidification benchmark problem. The suitability and potentialities of this novel approach have also been depicted in the solution of a variable domain (moving boundary) solidification problem, by considering the start-up stage of a round billet steel continuous casting process as an applied example. Results have revealed that this technique can be successfully used to deal appropriately with the marked non-linearity related to the fusion latent heat release in heat transfer problems with phase change, and its straightforward implementation leads to both accurate and stable numerical results.

Application of a Differential Transform Method to the Transient Natural Convection Problem in a Vertical Tube with Variable Fluid Properties

AbstractThe transient natural convection of a viscous fluid in a heated vertical tube is studied using the two-dimensional differential transform method (DTM). A time-dependent Dirichlet boundary condition is imposed for tube wall temperature. The partial differential equations for the velocity and temperature fields within the tube are solved by the DTM while considering temperature-dependent viscosity and thermal conductivity of the fluid. As a result, tractable solutions in double-series form are derived for the temperature and flow velocity. The transformed functions included in the solutions are obtained through a simple recursive procedure. Numerical results illustrate the effects of temperature-dependent properties on transient temperature and flow behaviour, including the Nusselt number and volumetric flow rate. The DTM gives accurate series solutions without any special functions for nonlinear transient heat transfer problems which are advantageous in finding the derivative or integral.

Lightning strike thermal damage model for glass fiber reinforced polymer matrix composites and its application to wind turbine blades

A physics-based model describing the thermal interaction between a lightning channel and a composite structure has been developed. The model includes: (i) spatial and temporal evolution of the lightning channel as a function of the electric current waveform; (ii) temporary and spatially non-uniform heat flux generated at the composite structure, where the heat flux is an explicit function of the electric current waveform and the instant lightning channel radius; (iii) nonlinear transient heat transfer problem formulation for layered anisotropic composites that accounts for temperature-dependent material properties, a moving boundary of the expanding lightning channel, and phase transition moving boundary associated with instantaneous material removal due to sublimation. The model is applied for evaluation of thermal damage of the tip glass fiber reinforced polymer matrix composite panel of the Sandia 100-meter All-glass Baseline Wind Turbine Blade (SNL 100-00) subjected to lightning strike.

Application of multilevel scheme and two level discretization for POD based model order reduction of nonlinear transient heat transfer 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.

Green's function approach to the nonlinear transient heat transfer analysis of functionally graded materials

This paper presents a Green's function approach to the solution of nonlinear transient heat transfer problem in a functionally graded material (FGM) object for the purpose of both the numerical analysis accuracy and computational efficiency. The Green's functions of transient heat transfer problem for a short hollow cylinder with power law functionally graded materials have been obtained analytically. The generalized expressions of the effective material properties in FGM are simplified based on micromechanical models (Voigt model and Mori–Tanaka model). The nonlinearity resulting from the temperature-dependent properties is resolved using the artificial parameter method. Two examples have been given to compare the effectiveness of the proposed Green's function approach with the finite element method (FEM), and the accuracy is satisfactory. This approach can become a cost-effective and accurate method for FGM structure design and risk assessment.

Nonlinear Transient Heat Transfer Problems using the Meshfree ES-PIM

Article Nonlinear Transient Heat Transfer Problems using the Meshfree ES-PIM was published on December 1, 2010 in the journal International Journal of Nonlinear Sciences and Numerical Simulation (volume 11, issue 12).

The flux-correct Green element formulation for linear, nonlinear heat transport in heterogeneous media

The singular integral equation, obtained from applying Green's identity to the stationary part of the linear diffusion equation operator (the Laplace operator), is implemented in an element-by-element fashion for the solution of nonlinear heat conduction in heterogeneous media so that both the temperature and its normal directional distribution (the flux) are calculated directly at every nodal point. This formulation, referred to as the flux Green element method (flux GEM), is used to solve linear and nonlinear transient heat transfer problems in two-dimensional heterogeneous domains. The flux-based Green element formulation takes advantage of the ability of the boundary element theory to correctly calculate the fluxes so that the solution in each element is complete in the sense that any solution (temperature or its directional derivative) required at any other point other than the grid points is achieved by implementing the integrations only in the element in which the point is located. One apparent drawback of the formulation is increased number of degrees of freedom (unknowns) that are generated, but this is compensated by the fact that high accuracy is achieved with coarse discretization. Furthermore, this paper presents a novel way of addressing the closure problem associated with having more unknowns than discretized equations at internal nodes. Here, an additional compatibility equation is presented for the internal normal directional derivatives of the temperature. This equation has a universal appeal in the sense that it applies not only to this heat equation but to any situation where such directional derivatives exist at a point. For the nonlinear transient heat conduction, the Picard and Newton–Raphson algorithms are applied in linearizing the discrete equations. In all examples examined in this paper, the Newton–Raphson algorithm shows superior convergence characteristics. The flux GEM is applied to four examples, and in all cases, the flux GEM achieves comparable accuracy with between 20% and 33% of the grid of the earlier Green element formulation which differences the internal fluxes.

The Al-powder forging process: its finite element analysis

This paper addresses the finite element formulation and simulation of the powder forging process for cup-shaped aluminum products. This study is the preliminary analysis for the powder forging process of Al 6061 engine pistons of automobiles. The mechanical behavior of the powder is assumed to obey the isotropic compressible viscoplasticity, while the thermal behavior of powders is effectively modeled by a non-linear transient heat-transfer problem. A generalized Crank–Nicolson–Galerkin scheme is employed for the temporal and spatial approximations, and non-linear numerical equations are solved by the Newton–Raphson method. Through the numerical simulation, the distributions of density, temperature, stress, and punch load are investigated.