A direct boundary element approach for unsteady non-linear heat transfer problems

Unsteady non-linear problems are usually solved based on the time marching scheme together with iterative methods. In this paper, we employed an example to illustrate a new direct boundary element technique, which can provide accurate solutions of unsteady non-linear problems without the need for iterations. The principle and validity of the technique is demonstrated by considering an unsteady non-linear hyperbolic heat transfer problem, together with a fully implicit difference scheme in the time domain. It illustrates that the proposed non-iterative boundary element approach is numerically rather efficient for unsteady non-linear problems. Thus, it provides an alternative to traditional iterative methods for unsteady non-linear problems.     

General boundary element method for non-linear heat transfer problems governed by hyperbolic heat conduction equation

The general Boundary Element Method (BEM) for strongly non-linear problems proposed by Liao (1995) is further applied to solve a two-dimensional unsteady non-linear heat transfer problem in the time domain, governed by the hyperbolic heat conduction equation (HHCE) with the temperature-dependent thermal conductivity coefficients which are different in the x and y directions. This paper confirms that the general BEM can be used to solve even those non-linear unsteady heat transfer problems whose governing equations do not contain any linear terms in spatial domain.     

Radial integration BEM for transient heat conduction problems

In this paper, a new boundary element analysis approach is presented for solving transient heat conduction problems based on the radial integration method. The normalized temperature is introduced to formulate integral equations, which makes the representation very simple and having no temperature gradients involved. The Green's function for the Laplace equation is adopted in deriving basic integral equations for time-dependent problems with varying heat conductivities and, as a result, domain integrals are involved in the derived integral equations. The radial integration method is employed to convert the domain integrals into equivalent boundary integrals. Based on the central finite difference technique, an implicit time marching solution scheme is developed for solving the time-dependent system of equations. Numerical examples are given to demonstrate the correctness of the presented approach.     

Analysis of static non-linear response by explicit time integration

In general, a system of linear (or non-linear) algebraic equations is solved on an analogue computer by integrating an appropriately defined system of associated first-order differential equations, the steady-state solution of which is the desired solution of algebraic system. This approach usually works very well so long as the associated dynamical system is stable. The idea of using the same approach numerically is also not altogether new. One can, ‘analogously’, construct a dynamical system associated with a given system of non-linear algebraic equations and solve it numerically by any of a number of explicit time integration schemes. In fact, similar ideas provide the basis for certain ‘dynamic relaxation’ methods, incremental loading methods and the widely acclaimed methods of invariant imbedding.     

A new family of time integration methods for heat conduction problems using numerical green’s functions

This paper is concerned with the formulation and numerical implementation of a new class of time integration schemes applied to linear heat conduction problems. The temperature field at any time level is calculated in terms of the numerical Green’s function matrix of the model problem by considering an analytical time integral equation. After spatial discretization by the finite element method, the Green’s function matrix which transfers solution from t to t + Δt is explicitly computed in nodal coordinates using efficient implicit and explicit Runge-Kutta methods. It is shown that the stability and the accuracy of the proposed method are highly improved when a sub-step procedure is used to calculate recursively the Green’s function matrix at the end of the first time step. As a result, with a suitable choice of the number of sub-steps, large time steps can be used without degenerating the numerical solution. Finally, the effectiveness of the present methodology is demonstrated by analyzing two numerical examples.     

A simple spatial integration scheme for solving Cauchy problems of non-linear evolution equations

In this study, we address a new and simple non-iterative method to solve Cauchy problems of non-linear evolution equations without initial data. To start with, these ill-posed problems are analysed by utilizing a semi-discretization numerical scheme. Then, the resulting ordinary differential equations at the discretized times are numerically integrated towards the spatial direction by the group-preserving scheme (GPS). After that, we apply a two-stage GPS to integrate the semi-discretized equations. We reveal that the accuracy and stability of the new approach is very good from several numerical experiments even under a large random noisy effect and a very large time span.     

Extremum principles for some nonlinear heat transfer problems

Summary Recent results on extremum principles for various nonlinear boundary value problems are applied to heat transfer problems involving space radiators such as fins and other parts of spacecraft. The results are illustrated by obtaining quite accurate variational solutions for such problems involving the fourth-power law of radiation.     

A meshless method for conjugate heat transfer problems

Mesh reduction methods such as boundary element methods, method of fundamental solutions, and spectral methods all lead to fully populated matrices. This poses serious challenges for large-scale three-dimensional problems due to storage requirements and iterative solution of a large set of non-symmetric equations. Researchers have developed several approaches to address this issue including the class of fast-multipole techniques, use of wavelet transforms, and matrix decomposition. In this paper, we develop a domain decomposition, or the artificial sub-sectioning technique, along with a region-by-region iteration algorithm particularly tailored for parallel computation to address the coefficient matrix issue. The meshless method we employ is based on expansions using radial-basis functions (RBFs).An efficient physically based procedure provides an effective initial guess of the temperatures along the sub-domain interfaces. The iteration process converges very efficiently, offers substantial savings in memory, and features superior computational efficiency. The meshless iterative domain decomposition technique is ideally suited for parallel computation. We discuss its implementation under MPI standards on a small Windows XP PC cluster. Numerical results reveal the domain decomposition meshless methods produce accurate temperature predictions while requiring a much-reduced effort in problem preparation in comparison to other traditional numerical methods.     

A new family of explicit time integration methods for linear and non-linear structural dynamics

A new family of explicit single-step time integration methods with controllable high-frequency dissipation is presented for linear and non-linear structural dynamic analyses. The proposed methods are second-order accurate and completely explicit with a diagonal mass matrix, even when the damping matrix is not diagonal in the linear structural dynamics or the internal force vector is a function of velocities in the non-linear structural dynamics. Stability and accuracy of the new explicit methods are analysed for the linear undamped/damped cases. Furthermore, the new methods are compared with other explicit methods.     

Homogenization of transient heat transfer problems for some composite materials

The article presented is devoted to the homogenization of transient heat transfer problems in some composite materials. The mathematical model used in the FEM computation is based on the effective modules method introduced for periodic composites. The effective heat conductivity is calculated in the closed form; effective heat capacity and mass density for the composite are obtained by simple spatial averaging. Such a homogenization scheme makes it possible to significantly simplify the numerical analysis of transient heat phenomena in various types of composites. Computational experiments performed using symbolic mathematics show the variability of effective heat conductivity for 2D and 3D composites as a function of the reinforcement volume ratio, of composite components conductivity coefficients as well as of the probabilistic moments of material properties versus volume ratio. Finally, using the Finite Element Method program, the comparison of transient heat transfer problem for the real and homogenized composites models is carried out.     

An efficient semi-analytic time integration method with application to non-linear rotordynamic systems

This paper presents a simple and efficient time-integration method for non-symmetric and non-linear equations of motion occurring in the analysis of rotating machines. The algorithm is based on a semi-analytic formulation combining powerful methods of linear structural dynamics applied to non-linear dynamic problems. To that purpose, the total solution is separated into a linear and a non-linear part, and a further partitioning into quasi-static and dynamic parts is performed. Modal analysis is applied to the undamped equations of the dynamic parts. The quasi-static parts contain all degrees of freedom, while a cost-saving modal reduction may be easily performed for the dynamic parts. Duhamel's integral is utilized for the modal equations. The time-evolution of the unknown modal excitations due to the dissipative, non-conservative, gyroscopic and non-linear effects entering Duhamel's integral is approximated during each time-step. The resulting time-stepping procedure is performed in an implicit manner, and the method is examined in some detail, in view of stability and accuracy characteristics. A rotordynamic system serves as a benchmark problem in order to demonstrate the computational advantages of the present method with respect to various other time-integration algorithms.     

Simulation of crashworthiness problems with improved contact algorithms for implicit time integration

When studying crashworthiness problems, contact simulation can be the source of a number of problems. A first one is the discontinuities in the normal evolution for a boundary discretized by finite elements. Another problem is the treatment of the contact forces that can introduce numerical energy in the system. In this paper, we propose to combine a method of discontinuity smoothing with the energy–momentum consistent scheme that recently appeared in the literature.     

On nonlinear problems of heat and mass transfer

A system of nonlinear equations for steady one-dimensional heat and mass transfer problems is considered. Analytical solutions are obtained for certain special cases.     

Examinations for terminal condition of Lagrange multiplier for heat transfer control problems

This paper presents a method to express the terminal condition of Lagrange multiplier by the solution of the steady adjoint equation. In the control problem based on the adjoint equation method, the terminal condition of Lagrange multiplier has been set zero to satisfy the stationary condition of the extended performance function. Therefore, there is a possibility that the control value near the terminal time cannot be appropriately expressed. To solve this problem, a method to express the terminal condition of Lagrange multiplier by the solution of the steady adjoint equation is proposed and examined. Finally, to confirm the propriety of the proposed method, this method is applied to the sequential control problem, and some remarks are made. Copyright © 2007 John Wiley & Sons, Ltd.     

A comparative study on various time integration schemes for heat wave simulation

The performance of several explicit and implicit time advancement schemes of first-order ODEs are examined for heat wave simulation with different boundary conditions. It is found that the boundary conditions have a considerable influence on the stiffness property of the hyperbolic heat conduction equation, due to the occurrence of thermal shock waves, and hence, according to the type of the enforced boundary conditions, a specific time integration scheme has to be performed in order to obtain an accurate and efficient solution. The results of the considered time integration schemes are also compared with analytical solution and based on the obtained results, some recommendations regarding the numerical simulation of hyperbolic heat conduction are presented.     

Experimental research on heat transfer coefficients for cryogenic cross-counter-flow coiled finned-tube heat exchangers

The aim of present experimental research is to find out the suitable correlations for designing the coiled finned-tube heat exchangers used in cryogenic applications. In order to conduct above experimental study, cross-counter-flow coiled finned-tube heat exchangers were developed in our lab and used in actual refrigeration cycle. The experiments were conducted in the range of effective Reynolds number 500–1900. The effect of diametrical clearance on the prediction of overall heat transfer coefficient is also investigated experimentally. The results from present study were compared in the form of overall heat transfer coefficient. Results of present experimental research indicate that different correlations selected in the study can be used with reasonable accuracy for designing the coiled finned-tube heat exchangers, if they are applied with suitable method of calculation of free-flow cross-sectional area. A more accurate new correlation has also been proposed that fitted experimental data within ±10% error band.     

Integrated study of scientific and applied problems on heat transfer enhancement in tubular heat transfer apparatuses

A highly efficient method of enhancing heat transfer in annular swirl-augmented tubes, with gases and liquids flowing in them, under drop and film condensation is developed and investigated.Moscow Aviation Institute, Russian. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 65, No. 1, pp. 25–31, July, 1993.