摘要
热传导问题为最普遍的工程问题之一，当前分析热传导问题的数值方法以基于体离散技术的有限体积法、有限单元法等方法为主。在这些方法中，需要将整个求解域离散成元素，其计算量巨大，特别对于复杂结构如整车模型，体网格离散本身就是一项巨大的工程。而使用边界类型的方法，则只需要对问题的边界进行离散，其计算量较以上两种方法小得多，并且表面网格划分较体网格划分容易得多。边界类型的分析方法主要以边界积分方程为理论基础，结合不同的离散技术，形成不同的方法，其中以边界单元法为代表。然而边界单元法中的分析模型并没有完整的保存实际模型的几何信息，因此在分析含有细小特征结构时容易忽略小特征。同样以边界积分方程为理论基础，边界面法利用CAD模型中B-rep数据结构，保留了实际模型的完整几何信息。本文致力于实现边界面方法的热传导分析，主要完成如下研究工作：
（1）结合双互易方法，将边界面法应用于求解含热源的稳态热传导问题。在双互易法中使用径向基函数作为其热源密度的插值函数，而径向基函数作为一种散乱数据插值方法，具有严重的数值稳定性问题。本文使用一种变参数径向基函数插值方案，有效平衡了插值精度与数值稳定之间的矛盾。在变参数径向基函数插值方法中，形状参数的变化方案对插值精度及插值稳定性影响巨大。本文以提高数值稳定性为出发点，提出一套参数变化方案，大大降低了插值矩阵的条件数，最终提高了双互易边界面法的数值稳定性。在使用双互易边界面法分析薄型结构上的热传导问题时，径向基函数的插值稳定性问题尤其突出，本文提出一套特殊的参数变化方案，将距离很近的插值点所对应的插值函数形状错开，从而减小了插值矩阵相应列的线性相关度，提高了数值稳定性。在应用双互易边界面法的过程中，需要用到插值函数的Laplace算子特解，本文将Laplace算子转化为极坐标形式，得到二阶常微分方程，通过不定积分推导径向基函数的特解，并指出推导过程中积分常数的处理需要使得特解函数满足一定的连续性。借助变参数径向基函数及其特解，DRBFM最终用于求解稳态热传导问题，借助特殊的参数变化方案，完成了对薄型结构的稳态热传导分析。
（2）结合双互易方法，将边界面法应用于求解热应力问题。在双互易边界面法中首次使用指数型径向基函数作为插值函数，并针对指数型径向基函数提出了参数变化方案，特别是在分析薄型结构上的稳态热应力问题时，调整沿厚度方向分布距离极近两个插值点上的形状参数，最终使插值变得稳定。此外，本文使用Papkovich势函数方法首次推导了指数型径向基函数在静力学问题中的特解。最后使用双互易方法分析热应力问题，得到较高的应力计算精度。
（3）结合时域卷积法，将边界面法用于求解瞬态热传导问题。在时域卷积法中，时域卷积的计算非常耗时并且占用大量的内存空间。本文通过两种方法加速了卷积计算，第一种方法通过将时间积分之后的基本解展开成级数形式，在展开式中将时间变量和空间变量分离，一次性将展开级数中前面若干项对应的边界积分计算并存储起来。后续影响矩阵则通过这些元素矩阵与时间变量相乘相加得到，避免了重复计算边界积分。第二种方法考虑基本解的衰减性质，在距离当前时间步较长时间的边界积分使用一个积分点进行积分，极大提高了计算速度。在使用第二种方法计算卷积积分的方法中，重点关注了内含管状小孔的结构，通过定义水管单元及缺角的三角形单元，极大的减少了离散网格的数量，提高了计算效率。
（4）结合拟初始条件法，求解瞬态热传导问题。在认识到后处理中需要用到体网格，并且需要用到体网格节点上的温度和热流密度之后，本文使用拟初始条件法完成了对结构的瞬态热分析。在体网格生成之后，定义了体单元，用于计算对初始条件以及热源密度的体积分。在拟初始条件法中，域内节点上的物理量并非方程组的未知数，而最终解方程时作为方程的右端项，因此其本质上并未增加方程的规模。引入体积分，极大拓宽了边界面法在瞬态热传导分析中的应用范围。然而拟初始条件法在使用小步长计算时出现数值不稳定现象，本文提出时间步长伸缩方法，有效解决了这种数值不稳定问题。在时间步长伸缩方案中，首先计算虚拟步长时间点上的温度和热流密度，再通过时间线性插值计算实际步长时间点上的温度和热流密度，通过增大影响矩阵的时间步长以提高数值稳定性。
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.
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