Three-Dimensional Unsteady Heat Conduction Analysis by Triple-Reciprocity Boundary Element Method

If the initial temperature is assumed to be constant, a domain integral is not required for the solution of unsteady thermal stress problems without heat generaton using the boundary element method (BEM). However, under heat generation or nonuniform initial temperature distribution, the domain integral is necesary. In this report it is shown that the problem of unsteady heat conduction with heat generation and nonuniform initial temperature can be approximately solved without using the domain integral by the triple-reciprocity BEM. In this method, the distributions of heat generation and nonuniform initial temperature are interpolated using integral equations.

The conventional boundary element method (BEM) uses internal cells for the domain integrals when solving nonlinear problems or problems with domain effects. In the conventional BEM, however, the merit of the BEM, which is easy preparation of data, is lost. This paper presents numerical integration for a three-dimensional meshless BEM, which does not require the use of internal cells. This method uses arbitrary internal points instead of internal cells. In this paper, a three-dimensional interpolation method is proposed. Furthermore, three-dimensional numerical integration is proposed using the above-mentioned three-dimensional interpolation method. This method is based on an improved multiple-reciprocity BEM (triple-reciprocity BEM) for heat conduction analysis with heat generation. In order to investigate the efficiency of this method, several numerical examples are given.

A triple reciprocity method in Laplace transform boundary element method for three-dimensional transient heat conduction problems

In general, internal cells are required to solve elastoplastic problems using a conventional boundary element method (BEM). However, in this case, the merit of BEM, which is ease of data preparation, is lost. Triple-reciprocity BEM can be used to solve two-dimensional elastoplasticity problems with a small plastic deformation. It has been shown that three-dimensional elastoplastic problems can be solved, without the use of internal cells, by the triple-reciprocity BEM and initial strain method. In this study, an initial stress formulation is adopted and the initial stress distribution is interpolated using boundary integral equations. A new computer program was developed and applied to solving several problems.

An inverse method based on the symbolic approach is proposed to determine the boundary condition in three-dimensional inverse heat conduction problems. This method uses symbols to represent the unknown boundary condition and then executes an implicit alternating-direction finite-difference method to calculate the temperature distribution. The calculated results are expressed explicitly as functions of the symbol of the undetermined boundary condition. Then, a set of linear equations is constructed from the comparison between the output symbolic temperature and the measured numerical temperature. Thus, the set of linear equations is solved through the linear least-squares error method and leads to the solution of the unknown boundary condition. Results from examples confirm that the proposed method is applicable in solving the three-dimensional inverse heat conduction problem. In the example problems, a temporally dependent and a spatially - temporally dependent boundary condition are used to demonstrate the procedure of the proposed method. In the first example, the temporally dependent boundary conduction, the result shows that only one-point measurement is needed to estimate the temporally dependent condition whether the measurement errors are considered or not. In the second problem, the four- and eight-point measurement methods are adopted to estimate the temporally - spatially dependent boundary condition. The result shows that only the four-point measurement is needed to estimate the surface temperature when measurement errors are neglected. When measurement errors are considered, the eight-point measurement is required in order to increase the congruence of the estimated results to the exact solutions.

In general, internal cells are required to solve deformation of thin plate with arbitrary distributed load using conventional boundary element methods (BEM). However, in this case, the merit of BEM, which is the easy preparation of data, is lost. In this paper, it is shown that deformation analysis of thin plate with arbitrary distributed load can be solved without the use of internal cells by using the triple-reciprocity boundary element method. Distribution of arbitrary load is interpolated using boundary integral equations. The problem of thin plate according to Kirchhoff's theory is formulated by means of two coupled Poisson equations to be expressed in integral form using the second theorem of Green in the classical way. A new computer program was developed and applied to several example problems.

Homogeneous heat conduction analysis 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 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.

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

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

Localized method of fundamental solutions for large-scale modelling of three-dimensional anisotropic heat conduction problems – Theory and MATLAB code

Modified scaled boundary finite element analysis of 3D steady-state heat conduction in anisotropic layered media

MLS-based numerical manifold method for steady-state three-dimensional heat conduction problems of functionally graded materials

IG-DRBEM of three-dimensional transient heat conduction problems

Analytically-integrated radial integration BEM for solving three-dimensional transient heat conduction problems

In this paper, we consider a three-dimensional transient inverse heat conduction problem arising in pool boiling experiments, i.e., the reconstruction of the surface heat flux from pointwise temperature observations inside a heater. We show that the inverse problem is ill-posed and utilize Tikhonov regularization and conjugate gradient methods together with a discrepancy stopping rule for a stable solution. We investigate the proper choice of regularization terms, which not only affects stability of the reconstructions but also greatly influences the quality of reconstructions in the case of limited observations. For the numerical solution of the governing partial differential equation, a space-time finite element method is used. This allows us to compute exact gradients for the discretized Tikhonov functional, and enables the use of conjugate gradient methods for the solution of the regularized inverse problem. We discuss further aspects of an efficient implementation, including a multilevel optimization strategy, together with an implementable stopping criterion. Finally, the proposed algorithms are applied to the reconstruction of local boiling heat fluxes from experimental data.