A Novel Localized Meshless Collocation Method for Numerical Simulation of Flume Dynamic Characteristics
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摘要:
局部边界节点法是一种基于非奇异半解析基函数和移动最小二乘原理的新型无网格配点技术，该方法把每个节点处的未知变量表示为该点对应的局部子域内节点处物理量的线性组合，该文基于局部边界节点法对数值波浪水槽进行了研究。首先，通过基准算例确定了Laplace算子非奇异半解析基函数的合理形状参数值。进一步，基于合理的参数选取，用较少的离散节点即可成功模拟波浪传播行为，将得到的数值结果与其他文献数值结果比较，可以发现局部边界节点法用更少的局部点即可得到较好的数值结果。最后，以保护近海岸建筑物为目标，模拟了水下防波堤对波浪传播的影响。结果表明，当波浪与梯形防波堤发生作用后，波峰变得比较陡峭，而波谷变得相对比较平坦，为近海岸防波堤的相关研究和设计提供了数值参考。
Abstract:The localized boundary knot method (LBKM) is a novel meshless collocation technology based on the non-singular semi-analytical basis functions and the moving least squares theory, and expresses the unknown variable at each knot as a linear combination of physical quantities at nodes inside its corresponding local subdomain. The LBKM was used to study the numerical wave flume. Firstly, the appropriate shape parameters for the non-singular semi-analytical basis functions of the Laplace operator were derived by the benchmark example. Further, the numerical results obtained with fewer nodes and appropriate parameters were in good agreement with the referential results. Finally, the effects of the underwater breakwater on wave propagation were investigated to protect coastal buildings. The results show that, when the wave interacts with the trapezoidal breakwater, the wave crest will become steeper, and the wave trough will become relatively flatter, which provides a numerical reference for the research and design of the coastal breakwater.
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图 2 在
$x = 4$ 处自由液面高程演化图：(a)不同总节点数；(b)不同时间间隔；(c)不同邻近点数Figure 2. Evolution of free-surface elevation at
$x = 4 :$ (a) different total numbers of nodes; (b) different time increments; (c) different numbers of nearest nodes
图 3 自由表面在4个不同时刻的轮廓
Figure 3. Profiles of free surface along the flume at 4 specific moments
图 4 梯形防波堤数值波浪水槽的示意图
Figure 4. The schematic diagram of the numerical wave flume with a trapezoidal submerged obstacle
图 5 梯形防波堤数值波浪水槽的布点图
Figure 5. Distributions of nodes of the numerical wave flume with a trapezoidal submerged obstacle
图 7 自由液面从
$x = 6$ 到$x = 17$ 在特定时刻的表面轮廓Figure 7. Free surface profiles from
$x = 6$ to$x = 17$ at specific moments表 1 不同形状参数下二阶Stokes波的均方根误差
Table 1. The RMSEs for 2nd-order Stokes waves with different shape parameters
$c$ 0.01 0.1 0.5 1 1.5 2 2.5 3 3.2 εRMSE1 \ $5.23 \times {10^{ - 3}}$ $6.13 \times {10^{ - 4}}$ $5.81 \times {10^{ - 4}}$ $6.16 \times {10^{ - 4}}$ $6.50 \times {10^{ - 4}}$ $6.84 \times {10^{ - 4}}$ $7.19 \times {10^{ - 4}}$ \ 注：“\”表示无法计算，后同。
Note: “\” means unable to calculate, the same below.
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表 2 不同总点数下LBKM与GFDM的均方根误差
Table 2. The RMSE1s of LBKM and GFDM under different total numbers of nodes
$N$ 889 3379 13159 20449 LBKM $7.28 \times {10^{ - 4}}$ $6.18 \times {10^{ - 4}}$ $5.81 \times {10^{ - 4}}$ $5.78 \times {10^{ - 4}}$ GFDM $5.98 \times {10^{ - 3}}$ $1.28 \times {10^{ - 3}}$ $4.71 \times {10^{ - 4}}$ $4.63 \times {10^{ - 4}}$
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表 3 不同邻近点数下LBKM与GFDM的均方根误差
Table 3. The RMSE1s of LBKM and GFDM under different numbers of nearest nodes
$m$ 6 10 14 16 18 20 25 30 LBKM $5.04 \times {10^{ - 3}}$ $6.13 \times {10^{ - 4}}$ $5.91 \times {10^{ - 4}}$ $5.85 \times {10^{ - 4}}$ $5.82 \times {10^{ - 4}}$ $5.81 \times {10^{ - 4}}$ $6.53 \times {10^{ - 4}}$ $2.56 \times {10^{ - 3}}$ GFDM \ \ $4.78 \times {10^{ - 4}}$ $4.77 \times {10^{ - 4}}$ $4.75 \times {10^{ - 4}}$ $4.71 \times {10^{ - 4}}$ $2.34 \times {10^{ - 3}}$ $3.58 \times {10^{ - 3}}$
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表 4 不同总点数下
$\phi $ ，${{\partial \phi } / {\partial x}}$ 和${{{\partial ^2}\phi } / {\partial {x^2}}}$ 的均方根误差Table 4. The RMSE2s of
$\phi $ ,${{\partial \phi } / {\partial x}}$ and${{{\partial ^2}\phi } / {\partial {x^2}}}$ under different total numbers of nodes$N$ 889 3379 13159 20449 $\phi $ $4.63 \times {10^{ - 3}}$ $3.85 \times {10^{ - 3}}$ $2.42 \times {10^{ - 3}}$ $2.35 \times {10^{ - 3}}$ ${{\partial \phi }/{\partial x}}$ $1.29 \times {10^{ - 2}}$ $6.65 \times {10^{ - 3}}$ $6.52 \times {10^{ - 3}}$ $5.26 \times {10^{ - 3}}$ ${{{\partial ^2}\phi } / {\partial {x^2}}}$ $3.96 \times {10^{ - 2}}$ $2.86 \times {10^{ - 2}}$ $2.58 \times {10^{ - 2}}$ $1.26 \times {10^{ - 2}}$
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