Transient heat conduction analysis by triple-reciprocity boundary element method

The paper deals with 2D initial-boundary value problems in the linear theory of transient heat conduction. A pure boundary element formulation is developed systematically. The time-dependent fundamental solution of the diffusion operator is employed together with higher-order polyharmonic fundamental solutions. The pseudo-initial temperature and/or heat sources density are approximated by using the triple-reciprocity formulation. All the time integrations are performed analytically in the time-marching scheme with integration within one time step and constant interpolation. The spatial discretization is reduced to boundary elements and free scattering of interior nodal points without any connectivity.     

Transient heat conduction analysis of 3D solids with fiber inclusions using the boundary element method

A boundary element formulation is presented in this work for transient heat conduction analysis of three‐dimensional (3D) fiber‐reinforced materials. The cylindrical‐shaped fibers in a 3D matrix are idealized by a system of curvilinear line elements with a prescribed diameter. The variations in the temperature and flux fields in the circumferential direction are represented in terms of a trigonometric shape function together with a linear or quadratic variation in the longitudinal direction. This approach significantly reduces the modeling effort and the computing cost. The storage requirement for the convolution integrals is eliminated by adopting an accurate integration‐based convolution algorithm for the surface of the hole and the fibers as well as a fast convolution algorithm for the outer boundary recently developed by the present authors. Numerical examples are presented to demonstrate the accuracy and applicability of the proposed method of analysis of fiber‐reinforced materials. Copyright © 2007 John Wiley & Sons, Ltd.     

Stress analysis of solids with open-ended tubular holes by BFM

Two kinds of special surface elements, curvilinear tube element and triangular element with negative parts, are proposed for modeling solids containing many slender open-ended tubular shaped holes in the framework of boundary face method. The surfaces of each tubular hole in the solid are modeled by the tube elements, and the outer surfaces that intersected and trimmed by the holes are modeled by triangular elements with negative parts. Substantial savings in both modeling effort and computational cost have been achieved. In addition, all the special surface elements are defined in the parametric space of the surface, and the exact geometry data are obtained directly from a CAD model of the solid. Therefore, automatic analysis is possible. Several numerical examples are presented to demonstrate the efficiency and accuracy of the proposed method.     

Three-dimensional transient heat conduction analysis by Laplace transformation and multiple reciprocity boundary face method

In this paper, a new multiple reciprocity formulation is developed to solve the transient heat conduction problem. The time dependence of the problem is removed temporarily from the equations by the Laplace transform. The new formulation is derived from the modified Helmholtz equation in Laplace space (LS), in which the higher order fundamental solutions of this equation are firstly derived and used in multiple reciprocity method (MRM). Using the new formulation, the domain integrals can be converted into boundary integrals and several non-integral terms. Thus the main advantage of the boundary integral equations (BIE) method, avoiding the domain discretization, is fully preserved. The convergence speed of these higher order fundamental solutions is high, thus the infinite series of boundary integrals can be truncated by a small number of terms. To get accurate results in the real space with better efficiency, the Gaver-Wynn-Rho method is employed. And to integrate the geometrical modeling and the thermal analysis into a uniform platform, our method is implemented based on the framework of the boundary face method (BFM). Numerical examples show that our method is very efficient for transient heat conduction computation. The obtained results are accurate at both internal and boundary points. Our method outperforms most existing methods, especially concerning the results at early time steps.     

Transient heat conduction in a medium with multiple spherical cavities

This paper considers a transient heat conduction problem for an infinite medium with multiple non‐overlapping spherical cavities. Suddenly applied, steady Dirichlet‐, Neumann‐ or Robin‐type boundary conditions are assumed. The approach is based on the use of the general solution to the problem of a single cavity and superposition. Application of the Laplace transform and the so‐called addition theorem results in a semi‐analytical transformed solution for the case of multiple cavities. The solution in the time domain is obtained by performing a numerical inversion of the Laplace transform. A large‐time asymptotic series for the temperature is obtained. The limiting case of infinitely large time results in the solution for the corresponding steady‐state problem. Several numerical examples that demonstrate the accuracy and the efficiency of the method are presented. Copyright © 2008 John Wiley & Sons, Ltd.     

Transient heat conduction analysis in functionally graded materials by the meshless local boundary integral equation method

Advanced computational method for transient heat conduction analysis in continuously nonhomogeneous functionally graded materials (FGM) is proposed. The method is based on the local boundary integral equations with moving least square approximation of the temperature and heat flux. The initial-boundary value problem is solved by the Laplace transform technique. Both Papoulis and Stehfest algorithms are applied for the numerical Laplace inversion to obtain the time-dependent solutions. Numerical results are presented for a finite strip and a hollow cylinder with an exponential spatial variation of material parameters.     

Three-dimensional unsteady heat conduction analysis by triple-reciprocity boundary element method

The conventional boundary element method (BEM) requires a domain integral in heat conduction analysis with heat generation or an initial temperature distribution. In this paper it is shown that the three-dimensional heat conduction problem can be solved effectively using the triple-reciprocity BEM without internal cells. In this method, the distributions of heat generation and initial temperature are interpolated using integral equations and time-dependent fundamental solutions are used. A new computer program was developed and applied to solving several problems.     

Transient heat conduction analysis in a piecewise homogeneous domain by a coupled boundary and finite element method

A coupled finite element–boundary element analysis method for the solution of transient two‐dimensional heat conduction equations involving dissimilar materials and geometric discontinuities is developed. Along the interfaces between different material regions of the domain, temperature continuity and energy balance are enforced directly. Also, a special algorithm is implemented in the boundary element method (BEM) to treat the existence of corners of arbitrary angles along the boundary of the domain. Unknown interface fluxes are expressed in terms of unknown interface temperatures by using the boundary element method for each material region of the domain. Energy balance and temperature continuity are used for the solution of unknown interface temperatures leading to a complete set of boundary conditions in each region, thus allowing the solution of the remaining unknown boundary quantities. The concepts developed for the BEM formulation of a domain with dissimilar regions is employed in the finite element–boundary element coupling procedure. Along the common boundaries of FEM–BEM regions, fluxes from specific BEM regions are expressed in terms of common boundary (interface) temperatures, then integrated and lumped at the nodal points of the common FEM–BEM boundary so that they are treated as boundary conditions in the analysis of finite element method (FEM) regions along the common FEM–BEM boundary. Copyright © 2002 John Wiley & Sons, Ltd.     

Transient heat conduction by the boundary element method: D‐BEM approaches

This work is concerned with the development of different domain‐BEM (D‐BEM) approaches to the solution of two‐dimensional diffusion problems. In the first approach, the process of time marching is accomplished with a combination of the finite difference and the Houbolt methods. The second approach starts by weighting, with respect to time, the basic D‐BEM equation, under the assumption of linear and constant time variation for the temperature and for the heat flux, respectively. A constant time weighting function is adopted. The time integration reduces the order of the time derivative that appears in the domain integral; as a consequence, the initial conditions are directly taken into account. Four examples are presented to verify the applicability of the proposed approaches, and the D‐BEM results are compared with the corresponding analytical solutions.Copyright © 2011 John Wiley & Sons, Ltd.     

Transient heat conduction in a medium with multiple circular cavities and inhomogeneities

A two‐dimensional transient heat conduction problem of multiple interacting circular inhomogeneities, cavities and point sources is considered. In general, a non‐perfect contact at the matrix/inhomogeneity interfaces is assumed, with the heat flux through the interface proportional to the temperature jump. The approach is based on the use of the general solutions to the problems of a single cavity and an inhomogeneity and superposition. Application of the Laplace transform and the so‐called addition theorem results in an analytical transformed solution. The solution in the time domain is obtained by performing a numerical inversion of the Laplace transform. Several numerical examples are given to demonstrate the accuracy and the efficiency of the method. The approximation error decreases exponentially with the number of the degrees of freedom in the problem. A comparison of the companion two‐ and three‐dimensional problems demonstrates the effect of the dimensionality. Copyright © 2009 John Wiley & Sons, Ltd.     

Transient heat conduction in hollow spheres with a moving inner boundary

The finite integral transform method is used to obtain the solution of unsteady heat conduction problems for a hollow sphere with a moving internal boundary and various boundary conditions at the outer surface. For the solution of the problems of interest integral transform formulas are presented with kernels (16), (20), and (24) and the corresponding inversion formulas (18), (22), (26), (29) and characteristic equations (17), (21), (25), (28), (31), (33).Nomenclature a, thermal diffusivity and conductivity - t temperature of phase transformation - density - heat transfer coefficient - Q total quantity of heat passing through inner boundary - F latent heat of phase transformation - Fo(1,)=a/R ₁ ² , Fo(i,)=/r _i ² , Fo(i, _i)=a _i/r _i ² Fourier numbers - Bi₂=R₂/ Biot number     

Analysis of heat conduction in solids by space-time finite element method

A functional in the form of a convolution of the temperature variation in the space–time domain has been derived. It has been used as the basis for a finite element formulation of heat conduction in solids. Numerical illustrations have indicated that the space–time finite element algorithm provides a more rapid convergence to the exact solutions than the usual finite element analysis with discretization in space only.     

Axial symmetric stationary heat conduction analysis of non-homogeneous materials by triple-reciprocity boundary element method

The heat conduction problems in homogeneous media can be easily solved by the boundary element method. The spatial variations of heat sources as well as material coefficients gives rise to domain integrals in integral formulations for solution of boundary value problems in functionally gradient materials (FGM), since the fundamental solutions are not available for partial differential equations with variable coefficients, in general. In this paper, we present the development of the triple reciprocity method for solution of axial symmetric stationary heat conduction problems in continuously non-homogeneous media with eliminating the domain integrals. In this method, the spatial variations of domain “sources” are approximated by introducing new potential fields and using higher order fundamental solutions of the Laplace operator.     

Acoustic problems analysis of 3D solid with small holes by fast multipole boundary face method

In this paper, an adaptive fast multipole boundary face method is introduced to implement acoustic problems analysis of 3D solids with open-end small tubular shaped holes. The fast multipole boundary face method is referred as FMBFM. These holes are modeled by proposed tube elements. The hole is open-end and intersected with the outer surface of the body. The discretization of the surface with circular inclusions is achieved by applying several special triangular elements or quadrilateral elements. In the FMBFM, the boundary integration and field variables approximation are both performed in the parametric space of each boundary face exactly the same as the B-rep data structure in standard solid modeling packages. Numerical examples for acoustic radiation in this paper demonstrated the accuracy, efficiency and validity of this method.     

Transient two-dimensional heat conduction problems solved by the finite element method

A finite element weighted residual process has been used to solve transient linear and non-linear two-dimensional heat conduction problems. Rectangular prisms in a space-time domain were used as the finite elements. The weighting function was equal to the shape function defining the dependent variable approximation. The results are compared in tables with analytical, as well as other numerical data. The finite element method compared favourably with these results. It was found to be stable, convergent to the exact solution, easily programmed, and computationally fast. Finally, the method does not require constant parameters over the entire solution domain.     

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.     

Nonlinear heat conduction with nonstationary boundary conditions

We study the one-dimensional nonstationary temperature field in a solid when the thermal conductivity and heat capacity depend linearly on the temperature.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 19, No. 5, pp. 880–885, November, 1970.     

Application of the boundary element method to transient heat conduction

An advanced boundary element method (BEM) is presented for the transient heat conduction analysis of engineering components. The numerical implementation necessarily includes higher-order conforming elements, self-adaptive integration and a multiregion capability. Planar, three-dimensional and axisymmetric analyses are all addressed with a consistent time-domain convolution approach, which completely eliminates the need for volume discretization for most practical analyses. The resulting general purpose algorithm establishes BEM as an attractive alternative to the more familiar finite difference and finite element methods for this class of problems. Several detailed numerical examples are included to emphasize the accuracy, stability and generality of the present BEM., Furthermore, a new efficient treatment is introduced for bodies with embedded holes. This development provides a powerful analytical tool for transient solutions of components, such as casting moulds and turbine blades, which are cumbersome to model when employing the conventional domain-based methods.     

Free vibration analysis of anisotropic solids with the boundary element method

In this paper, the boundary element method is employed for the solution of three-dimensional anisotropic free vibration problems. The formulation is based upon the use of static fundamental solutions in conjunction with the dual reciprocity method. This approach is very advantageous for the solution of free vibration problems and circumvents the problems related to the anisotropic dynamic fundamental solutions. By means of numerical examples, the influence of the internal collocation points on the representation of the mass matrix and the occurrence of complex-valued eigenfrequencies is investigated. The eigenfrequencies and mode shapes obtained with the boundary element method are compared to finite element computations and excellent agreement is observed.     

Two-dimensional steady heat conduction in functionally gradient materials by triple-reciprocity boundary element method

Homogeneous heat conduction can be easily solved by means of the boundary element method. However, domain integrals are generally necessary to solve the heat conduction problem in the functionally gradient materials. This paper shows that the two-dimensional heat conduction problem in the functionally gradient materials can be solved approximately without a domain integral by the triple-reciprocity boundary element method. In this method, the distribution of domain effects is interpolated by integral equations. A new computer program is developed and applied to several problems.