In this paper, the meshless boundary integral quadrature method (MBIQM) is proposed to determine the conduction shape factor of heat exchanger tubes containing slits. The MBIQM is a meshless method of quadrature form by introducing the adaptive exact solution and Gaussian quadrature. In this way, the singular integral can be technically calculated free of the sense of Cauchy principal value in numerical implementation. When dealing with the boundary value problem containing a slit or so-called degenerate boundary, a rank-deficient influence matrix due to a degenerate boundary may occur. To overcome the rank-deficiency problem, we introduce the dual boundary integral equation (BIE) with the hypersingular BIE to obtain independent equations for collocation points on the slit. A feasible adaptive exact solution is also required for the problem with a degenerate boundary. Since the jump behavior cannot be described by the ordinary form of adaptive exact solution for the corresponding collocation point on the slit, we adopt the harmonic basis function in the elliptical coordinates to construct the new adaptive exact solution. This is the main novelty of this paper. In addition, the numerical instability due to the degenerate scale of an outer boundary is also observed. To avoid the appearance of numerical instability due to a degenerate scale, regularized techniques are employed. Accurate conduction shape factors for any size are obtained by using the proposed approach with regularized techniques.

中文翻译：

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计算含缝隙换热管传导形状因子的无网格边界积分求积法


本文提出了无网格边界积分求积法（MBIQM）来确定含有缝隙的换热器管的传导形状因子。 MBIQM是一种引入自适应精确解和高斯求积的求积形式的无网格方法。这样，奇异积分的计算在技术上就可以脱离数值实现中柯西主值的意义。当处理包含狭缝或所谓简并边界的边值问题时，可能会出现由于简并边界而导致的秩亏影响矩阵。为了克服秩不足问题，我们引入了对偶边界积分方程（BIE）和超奇异 BIE，以获得狭缝上搭配点的独立方程。对于具有退化边界的问题，还需要一个可行的自适应精确解。由于跳跃行为不能用狭缝上对应配置点的自适应精确解的普通形式来描述，因此我们采用椭圆坐标系中的调和基函数来构造新的自适应精确解。这是本文的主要创新点。此外，还观察到由于外边界简并尺度导致的数值不稳定。为了避免由于简并尺度而出现数值不稳定，采用正则化技术。通过使用所提出的方法和正则化技术，可以获得任何尺寸的精确传导形状因子。