修正型缓坡方程的有限元模型

与缓坡方程相比，修正型缓坡方程增加了地形曲率项和坡度平方项，从而提高了数值求解的复杂性。本文将计算域划分为内域和外域，内域为水深变化区域，使用修正型缓坡方程，其中的地形曲率项和坡度平方项可用有限单元各节点的水深信息和单元插值函数表示，外域为水深恒定区，速度势满足Helmholtz方程，通过内外域的边界匹配建立有限元方程，并用高斯消去法求解。进而分别模拟了波浪传过Homma岛和圆形浅滩的变形，其结果与相关的解析解和实验数据吻合良好，证明了本文有限元模型的正确性。同时，通过与实验数据的对比也明显看出，在地形坡度较陡的情况下，修正型缓坡方程较缓坡方程具有更高的计算精度。

FEM model of the modified mild slope equation

Compared with the mild slope equation, the bottom curvature and slope-squared terms are contained in the modified mild slope equation, which increases the complexity of solving the equation numerically. In this paper, the physical domain is divided into a finite inner region and an infinite outer region. The inner region of a variable depth is studied with the modified mild slope equation, and the outer region has a constant water depth, the velocity potential satisfying the Helmholtz equation. The bottom curvature and slope-squared terms in the equation are evaluated from the input bathymetry at each node of every element using the interpolation functions. With the boundary matching method and Gaussian elimination technique, the finite element equations are established and solved. Then, wave transformation over a Homma Island and a circular shoal is simulated respectively, and the results agree well with analytical solutions and experimental data, verifying the validity of the FEM model in this paper. Meanwhile, the comparison between the numerical and experimental results indicates the modified mild slope equation can provide more accurate predictions than the mild slope equation for wave propagation over relatively steep bathymetry.