Abstract The normal depth is an essential parameter for the design, operation, and maintenance of open channels. The circular, city-gate crossing and egg-shaped sections are often used in non-pressure tunnels in hydraulic engineering, agricultural irrigation, and sewerage works. However, the equations governing the normal depth in various sections are complicated implicit transcendental equations. In order to improvements the solutions for normal depth in multiple sections of tunnels, a function model is first established for the geometric features of various sections using the mathematical transform method and while considering non-dimensional parameters. Then implement the revised PSO algorithms in MATLAB, and establish three right solution's formulas for the normal depths in different non-pressure tunnel sections through optimization. The error analysis results and project cases show that the established formula has broad applicability. The maximum relative errors of the formula for normal depths are less than 0.07%, 0.04%, and 0.07% in circular, city-gate crossing, and egg-shaped sections, respectively, which are more accurate than the existing formulas. The results of this research may be useful in design, operation, and maintenance in tunnel engineering.

中文翻译：

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多段隧道法向深度解法的改进

摘要 法向深度是明渠设计、运营和维护的重要参数。圆形、城门交叉和蛋形断面常用于水利工程、农业灌溉和污水工程中的无压隧道。然而，控制各截面法向深度的方程是复杂的隐式超越方程。为了改进隧道多段法向深度的求解方法，首先采用数学变换的方法，在考虑无量纲参数的情况下，建立了各段几何特征的函数模型。然后在MATLAB中实现改进后的PSO算法，通过优化建立了不同非承压隧道断面法向深度的三个正确解公式。误差分析结果和工程案例表明，所建立的公式具有广泛的适用性。该公式法向深度公式的最大相对误差在圆形、城门交叉和蛋形断面分别小于0.07%、0.04%和0.07%，比现有公式更准确。这项研究的结果可能对隧道工程的设计、运营和维护有用。