边界点法在传热问题数值分析中的应用

将一种新的数值分析方法-边界点法应用于传热问题的研究，对无内热源稳态热传导问题，通过传统边界元法将边界积分方程离散化，发现可以不直接求解影响系数矩阵，而是通过对偶关系，由域外虚源构造方程组的特解场形成边界已知和未知温度，热流密度的系数矩阵，而且域内温度和热流密度的求解将不依赖于边界未知参数的求解，对于有内热源的问题，可以将非齐次方程的解转换为齐次方程的解和某一确定解的叠加，对于非线性问题，可以通过基尔霍夫变换，将非线性问题转化为线性问题求解，这种边界点方法不但具有边界元法降维的优势，而且不须求解奇异积分，大大节约了计算时间，计算精度极高，以有内热源非线性稳态热传导问题的实例印证了这种方法的高效性。

Application of boundary point method in the numerical analysis for heat transfer problems

A novel numerical method, boundary point method(BPM), is applied to analyze heat transfer problems. For the steady heat transfer problems without inner heat sources, the boundary integral equations can be discretized by traditional boundary element method. According to the conjunctive relations, the influence coefficient matrix of equations may not be directly calculated and the relative matrix of known and unknown parameters, temperature and heat fluxes, can be formed by the particular solutions, which are constructed by utilization of the virtual unit sources out of the domain and its corresponding fundamental solutions. Particularly, the calculation of temperature and heat fluxes of any points in the domain does not depend on boundary unknowns to be solved. As the equations including inner heat sources are considered, the solutions of non\|homogeneous equations can be obtained by sum of the solutions of homogeneous equations and the corresponding special solutions. Nonlinear problems can be converted to linear problems by Kirchhoff transformation. The dimensions of problems can be reduced by using boundary point method, the same as by using boundary element method, and interpolation and numerical integration are not needed and numerical treatment for singular integration is avoided. Therefore, the efficiency of boundary point method is very high and the accuracy is excellent. A nonlinear steady heat transfer problem including inner heat source was analyzed by this method in this paper and the high efficiency of this method was verified.