A Boundary Element Method for Steady-State Heat Transfer Problems Based on a Novel Type of Interpolation Elements
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摘要: 为了提高边界元法在求解稳态热问题时的计算精度,通过使用一种新型单元插值方法(称为扩展单元插值法)，实现对稳态传热问题的求解。扩展单元是在传统不连续单元的边界配置虚拟节点，把原非连续单元变成高阶的连续单元，并将其作为新型的插值单元。利用虚拟节点和内部源节点构造出的插值函数，可以精确插值边界上的连续和不连续物理场，插值精度要比原始不连续单元高两阶。另外，边界积分方程只在传统的不连续单元的内部节点处建立，只包含内部源节点的自由度，而虚拟节点的自由度可通过与内部源节点之间的关系消除掉，因此最终系统方程的求解规模不会增加。这种新型的插值单元继承了传统连续和不连续单元的优点，克服了它们的缺点。数值结果表明，此种单元插值方法用于求解稳态传热问题时可获得较高的计算精度和收敛性。Abstract: To improve the computational accuracy of the boundary element method (BEM) for solving the steady-state heat problems, a new method was studied with a new element interpolation method (called expansion element interpolation method). The novel expansion interpolation element was obtained through configuration of virtual nodes at the boundary of the traditional discontinuous element, to transform the original discontinuous element into a higher-order continuous element. The interpolation function constructed with the virtual node and the internal source node can accurately work in the continuous and discontinuous physical fields on the boundary, and the interpolation accuracy increases by 2 orders compared with that of the original discontinuous element. In addition, the boundary integral equation is only valid at the internal source nodes of the traditional discontinuous element and contains only the degrees of freedom of the source nodes, while the degrees of freedom of the virtual node can be eliminated through the relationship between the virtual node and the internal source node, so the size of the final linear system equations will not increase. This new interpolation element inherits the advantages of the traditional continuous and discontinuous elements and overcomes their disadvantages. The numerical results show that, the proposed method helps solve the steady-state heat transfer problem with high computational accuracy and convergence.
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图 2 新型单元用于插值已知边界变量
Figure 2. The novel elements used for interpolation of the known boundary variables
图 3 新型单元用于插值未知边界变量
Figure 3. The novel elements used for interpolation of the unknown boundary variables
图 4 新型单元用于插值非连续边界变量
Figure 4. The novel elements used for interpolation of the discontinuous boundary variables
图 5 带孔平板的几何模型（单位：mm）
Figure 5. The geometric model for the plate with a hole (unit: mm)
图 7 内孔边界上热流量q的数值结果
Figure 7. The numerical results of thermal flux on the boundary of the internal hole
图 8 底边和侧边界上热流量q的相对误差和收敛性图
Figure 8. The relative errors and convergence rates of q on the bottom and side edges
图 10 复合支架所施加的混合边界条件
Figure 10. The mixed boundary conditions applied for the composite bracket
图 11 左侧边界和圆弧边界（R＝0.5 m）上温度的计算结果
Figure 11. The computational results of temperature on left and arc (R＝0.5 m) boundaries
图 12 本文方法计算得到的温度云图
注 为了解释图中的颜色，读者可以参考本文的电子网页版本。
Figure 12. The temperature nephogram obtained with the proposed method
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