The Boundary Face Method For Heat Conduction Problem

The heat conduction widely appears in engineering problems. The prevailing numerical tools for that problem include the finite volume method (FVM) and the finite element method (FEM). Those two methods are both based on the body discretization technology. In those methods, the considered domain is discretized into elements. A large number of computational nodes are involved and thus consumes a large quantity of time. Furthermore, the discretization is very difficult for some very complex structures such as the model of entire car. At least several monthes will be spent on modeling the entire car for a single engineer. The methods of boundary type, however, can save a lot of analysis time since only boundary discretization is required in these methods. Most of these methods are developed based on the boundary integral equation (BIE). Different dicretization methods usually lead to different methods. The boundary element method (BEM) is the most widely used method of boundary type. The analysis model in the BEM, however, keeps little geometric data from the real geometric model which usually comes from a CAD package. In the boundary face method (BFM) which is also theoretically based on the BIE, a parametric surface discretization scheme, which makes full use of the B-rep data structure in the CAD package, is developed. In this method, geometric data from the CAD model is entirely kept. The BFM is implemented for heat conduction problem in this paper. And the main contributions of the paper are listed as follows:(1) By combining with the dual reciprocity method (DRM), the BFM is applied for steady-state heat conduction problem in which the heat source is considered. In the DRM, the radial basis function (RBF) is employed as the interpolation function for the heat source. The RBF is widely used in scattered data approximation and it suffers from an ill-conditioning problem. A variable shaped RBF is proposed in this paper to balance the contradiction between the accuracy and the stability. A variation scheme for the variable shaped RBF is proposed in this paper in order to improve the stability of the interpolation. By employing that variation scheme, the condition number of the interpolation matrix is reduced, thus the stability of the DRBFM is improved. In the application of DRBFM for heat conduction on thin-shell like structures, the condition number of the RBF interpolation matrix is especially large. A special variation scheme for this type of structure is proposed in this paper. With the proposed scheme, the shapes of the RBF on points that are very close to each other are significantly distincted. The linear dependence between the corresponding lines of the interpolation matrix is largely reduced. Thus the stability of the interpolation is improved. In the application of DRBFM, the particular solution of the RBF to the heat conduction problem is necessary. To deduce the particular solution, the Laplace operator is transformed into polar coordinates form. An ordinary differential equation of the second order is obtained. The particular solution is deduced through some indefinite integration schemes. In the deduction, the integral constant is emphasized. The integral constant is set to keep the particular solution smooth. With the help of variable shaped RBF and its particular solution, the DRBFM is implemented to solve heat conduction problem. By employing the specially proposed variation scheme, a heat conduction analysis on thin-shell like structures is performed.(2) Coupling with the DRM, the BFM is extended to solve thermo-elasticity problem. In this application, an exponential RBF is introduced in the DRBFM for the first time. Variation schemes for variable shaped exponential RBF are proposed. Applications on thin-shell like structures are especially concerned. In the analysis of those structures, the shapes of the RBF on two points, which are very close to each other, are distincted through a specially proposed variation scheme. Thus a stable interpolation is obtained. Furthermore, the particular solution of the exponential RBF to the elasticity problem is deduced by using a Papkovich potential function method. With the variable shaped exponential RBF and the corresponding particular solution, the DRBFM is applied to solve thermo-elasticity problem. Accurate stress result is achieved.(3) By coupling with the time convolution method, the BFM is applied to solve transient heat conduction problem. In the traditional time convolution method, much time is consumed and huge memory is required during the computation of the time convolution. Two schemes are employed in this paper to accelerate the convolution. One is the fundamental solution expansion method. In this expansion method, the fundamental solution is expanded into series. The temporal variable and the spatial variable are separated in all terms of the series. The integrals of the spatial variables over the boundary of the considered domain is calculated and saved. The influence matrix that appears in the BIE is computed through some matrix-vector multiplications between the element matrix and vectors at arbitrary time. Thus a large quantity of time is saved for integral calculation. The other one considers the decay monotonicity of the fundamental solution with respect to temporal variables and spatial variables. In this scheme, only one integral point is used for calculating the integral over the time that is far away from the considered time and the integral over the boundary part that is far away from the source point. By reducing the integral points, the convolution is accelerated. In the application of the second scheme in the BFM, structures which contain pipe-shaped cavities are concerned. With employing a tube element and a triangular element with negative part, the number of discretization mesh is reduced and the efficiency of the method is improved. Finally, the transient heat conduction analysis of those structures is performed by the BFM.(4) By combining with the quasi-initial condition method, another application of the BFM for transient heat conduction is implemented. Since the mesh over the domain and temperature at domain nodes are required in the post-process of the BFM methods, a quasi-initial condition method, which is developed based on the domain dicretization, is implemented to perform the transient heat conduction analysis in this paper. Based on the domain mesh, several domain elements are introduced to approximate the variation of the quasi-initial temperature and heat source throughout the domain. In the quasi-initial condition method, physical variables on domain nodes is known and taken as part of the right hand vector of the system. Thus the scale of the final system of the BFM depends on the number of boundary nodes rather than domain nodes. With considering the domain integral, the BFM becomes more powerful. When small time steps are involved, however, the quasi-initial condition method becomes numerically unstable. To circumvent this unstable problem, a time step amplification method has been proposed. In the time step amplification method, the temperature and the flux at the amplified time step are computed at first. The temperature and the flux at the considering time step are computed through a linear interpolation along the time interval. Thus the actual time step employed in the computational of the influence matrix is large enough to keep the method stable.