独立覆盖流形法的通用计算公式和通用程序设计——(三)算例验证

doi:10.11988/ckyyb.202401142

摘要/Abstract

摘要：

在前两篇提出的通用计算公式、通用程序设计方法的基础上,通过求解弹性力学运动微分方程、传导方程、波动方程(含稳态和瞬态分析),给出了一至三维包括位移场、温度场、渗流场、声场、静电场和势流场等算例进行全面验证,涵盖了任意形状和任意连接的网格、精确几何边界的模拟及本质边界条件的准确施加、高阶级数逼近、裂纹尖端附近解析级数的应用等独立覆盖流形法的特色功能。最后对全文进行总结,并提出“级数流形元(级数元)”的新名称。

关键词: 数值流形方法, 独立覆盖, 偏微分方程, 级数解, 网格剖分, 精确几何, 级数流形元<br

Abstract:

Based on the general calculation formulas and programming methods proposed in the previous two articles, we conduct a comprehensive validation across one- to three-dimensional problems. These involve the displacement field, temperature field, seepage field, sound field, electrostatic field, and potential flow field by solving the differential equations of motion in elasticity, the conduction equation, and the wave equation (covering both steady-state and transient analyses). The provided examples demonstrate the distinctive characteristics of the manifold method based on independent covers. These features include the ability to handle meshes of any shape and connection, the accurate simulation of geometric boundaries, the precise application of essential boundary conditions, high-order series approximation, and the use of analytical series near crack tips. Finally, we summarize the entire article and propose a new term “series manifold element (series element)”.

Key words: numerical manifold method, independent covers, partial differential equations, series solutions, mesh division, exact geometry, series manifold element

中图分类号:

O241.82偏微分方程的数值解法

苏海东, 宋文硕, 龚亚琦, 韩陆超, 韦玉霞. 独立覆盖流形法的通用计算公式和通用程序设计——(三)算例验证[J]. raybet体育在线院报, 2025, 42(4): 211-218.

SU Hai-dong, SONG Wen-shuo, GONG Ya-qi, HAN Lu-chao, WEI Yu-xia. General Formulas and Program Design for Manifold Method Based on Independent Covers Ⅲ: Example Verification[J]. Journal of Changjiang River Scientific Research Institute, 2025, 42(4): 211-218.

图/表 23

图1 一维稳态分析算例的数值解与理论解对比(10个覆盖,10阶多项式级数)[1]

Fig.1 Comparison between numerical solutions and theoretical solutions of one-dimensional steady state analysis examples (10 covers and 10-order polynomials)[1]

图2 一维稳态分析算例的数值解与理论解对比(60个覆盖,3阶多项式级数)[1]

Fig.2 Comparison between numerical solutions and theoretical solutions of one-dimensional steady state analysis examples (60 covers and 3-order polynomials)[1]

表1 x=0.5处不同时刻的u值对比

Table 1 Comparison of u values at different instances when x=0.5

t/s	u值
t/s	数值解	理论解
0.1	0.151 059×10⁰	0.151 059×10⁰
0.2	0.562 983×10^-1	0.562 986×10^-1
0.3	0.209 827×10^-1	0.209 829×10^-1
0.4	0.782 035×10^-2	0.782 049×10^-2
0.5	0.291 469×10^-2	0.291 475×10^-2
0.6	0.108 632×10^-2	0.108 635×10^-2
0.7	0.404 883×10^-3	0.404 892×10^-3
0.8	0.150 906×10^-3	0.150 906×10^-3
0.9	0.562 467×10^-4	0.562 440×10^-4
1.0	0.209 655×10^-4	0.209 625×10^-4

图3 不同时刻u、 ∂ u ∂ x的函数图像

Fig.3 Function images of u and ∂ u ∂ x at different instances

图4 在自重作用下的半圆柱[4]

Fig.4 Half cylinder under self weight[4]

图5 旋转45°的裂纹分析模型

Fig.5 Crack analysis model with a rotation of 45°

图6 护坡桩算例[9]

Fig.6 Example of a slope protection pile[9]

图7 圆柱绕流算例[10]

Fig.7 Example of flow around a cylinder[10]

图8 流场分析的流形法网格

Fig.8 Manifold method meshes for flow field analysis

图9 c-d边上的速度分布

Fig.9 Velocity distribution on the c-d line

图10 同轴传输线的截面示意图及1/4分析模型[11]

Fig.10 Cross section diagram and 1/4 analysis model of coaxial transmission lines[11]

图11 电势Φ计算结果对比

Fig.11 Comparison of calculation results of potential Φ

图12 水库渗流算例[12]

Fig.12 Calculation example of reservoir seepage[12]

图13 区域中点处的H随时间的变化

Fig.13 Variation of H over time at the midpoint of the region

图14 声压随时步的变化

Fig.14 Changes in sound pressure with time step

图15 三维计算模型及边界条带

Fig.15 Three-dimensional computational model and boundary strips

图16 3#坝段的装配块体分布[14]

Fig.16 Distribution of assembly blocks for 3# dam block[14]

图17 流形法网格各方向视图[14]

Fig.17 Views of manifold method meshes in several directions[14]

图18 上游面应力[14]

Fig.18 Stresses on the upstream surface[14]

图19 参考解的有限元网格[13]

Fig.19 Finite element meshes of the reference solution[13]

图20 中心点温度历程

Fig.20 Temperature history at center point

图21 拱坝网格[3]

Fig.21 Arch dam meshes[3]

图22 z向位移时程[3]

Fig.22 Time history of z-direction displacement[3]

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