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Article(id=1241416387193787203, tenantId=1146029695717560320, journalId=1235980550691926019, issueId=1241416382559081210, articleNumber=null, orderNo=null, doi=10.3969/j.issn.0253-6099.2025.03.007, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1734019200000, receivedDateStr=2024-12-13, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1773906331107, onlineDateStr=2026-03-19, pubDate=1748707200000, pubDateStr=2025-06-01, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1773906331107, onlineIssueDateStr=2026-03-19, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1773906331107, creator=13701087609, updateTime=1773906331107, updator=13701087609, issue=Issue{id=1241416382559081210, tenantId=1146029695717560320, journalId=1235980550691926019, year='2025', volume='45', issue='3', pageStart='1', pageEnd='223', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1773906330003, creator=13701087609, updateTime=1773908015401, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241423451685179940, tenantId=1146029695717560320, journalId=1235980550691926019, issueId=1241416382559081210, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241423451685179941, tenantId=1146029695717560320, journalId=1235980550691926019, issueId=1241416382559081210, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=44, endPage=51, ext={EN=ArticleExt(id=1241416387499971410, articleId=1241416387193787203, tenantId=1146029695717560320, journalId=1235980550691926019, language=EN, title=Application of Six-Directional Triangular Mesh with Weak Velocity Discontinuity Line Characteristics in Slope Stability Analysis, columnId=1236276106018484431, journalTitle=Mining and Metallurgical Engineering, columnName=MINING, runingTitle=null, highlight=null, articleAbstract=

To address the issue of volumetric locking encountered in the slope stability analysis with three-node triangular elements in upper bound finite element method (UBFEM), a six-directional triangular mesh (P6) with weak velocity discontinuity line characteristics was proposed based on the mechanical equivalence effect between velocity discontinuity lines and conjugate triangular elements located in the same position. The P6 is utilized in conjunction with a six-node triangular (T6) element in UBFEM for slope stability analysis. The results show that the potential slip surface of the slope obtained using the P6 combined with T6 elements is clearly defined, with a smooth transition in dissipated energy density. As the mesh density increases, the effect of weak velocity discontinuity lines strengthens, leading to an improved accuracy in the upper bound solution of the slope stability coefficient (Ns). With the combined influences of factors such as the internal friction angle and slope gradient into consideration, the upper bound solutions of Ns obtained with P6 outperform those from three-directional triangular meshes and Delaunay triangular meshes. The uniform mesh generated by the P6 UBFEM in a single computational framework yields favorable results, facilitating batch processing operations such as strength reduction upper bound analysis. Additionally, its application scope can be extended by integrating it with adaptive mesh refinement techniques.

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针对上限有限元法中三节点三角形单元在边坡稳定性分析时存在的体积锁定问题，从速度间断线与同处共轭三角形单元的力学等效效应出发，提出一种含弱速度间断线特性的六方向三角网（P6），并配合六节点三角形（T6）单元上限有限元法对边坡稳定性进行分析。结果表明：P6联合T6单元获得的边坡潜在滑动面清晰明确，耗散能密度平滑过渡，且随着网格密度增大，弱速度间断线效应加强，边坡稳定性系数Ns上限解精度提升；考虑内摩擦角及边坡坡度等综合因素的影响，P6对应的Ns上限解均优于三方向三角网及Delaunay三角网。基于P6上限有限元法在单次计算框架内形成的均匀网格计算效果良好，有利于强度折减上限分析等批处理运算方式，亦可结合网格自适应扩展其应用范围。

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杨峰（1981—），男，陕西西安人，博士，副教授，主要从事岩土工程研究。E-mail：
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余小军（1975—），男，陕西渭南人，高级工程师，主要从事矿山资源研究。E-mail：

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余小军（1975—），男，陕西渭南人，高级工程师，主要从事矿山资源研究。E-mail：

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余小军（1975—），男，陕西渭南人，高级工程师，主要从事矿山资源研究。E-mail：

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IEEE Access, 2024, 12: 12562-12585., articleTitle=A comprehensive survey on Delaunay triangulation: Applications, algorithms, and implementations over CPUs, GPUs, and FPGAs, refAbstract=null)], funds=[Fund(id=1241422283013673424, tenantId=1146029695717560320, journalId=1235980550691926019, articleId=1241416387193787203, awardId=2021JJ10063, language=CN, fundingSource=湖南省杰出青年基金(2021JJ10063), fundOrder=null, country=null), Fund(id=1241422283139502546, tenantId=1146029695717560320, journalId=1235980550691926019, articleId=1241416387193787203, awardId=LM（2022）-F-045, language=CN, fundingSource=伊春鹿鸣矿业有限公司科技项目(LM（2022）-F-045), fundOrder=null, country=null), Fund(id=1241422283215000020, tenantId=1146029695717560320, journalId=1235980550691926019, articleId=1241416387193787203, awardId=2024ZZTS0752, language=CN, fundingSource=中南大学中央高校基本科研业务费专项资金(2024ZZTS0752), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1241422270242017475, tenantId=1146029695717560320, journalId=1235980550691926019, articleId=1241416387193787203, xref=1., ext=[AuthorCompanyExt(id=1241422270246211779, tenantId=1146029695717560320, journalId=1235980550691926019, articleId=1241416387193787203, companyId=1241422270242017475, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1.Yichun Luming Mining Co., Ltd., Yichun 152500, Heilongjiang, China), AuthorCompanyExt(id=1241422270250406084, tenantId=1146029695717560320, journalId=1235980550691926019, articleId=1241416387193787203, companyId=1241422270242017475, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1.伊春鹿鸣矿业有限公司，黑龙江　伊春　152500)]), AuthorCompany(id=1241422270367846604, tenantId=1146029695717560320, journalId=1235980550691926019, articleId=1241416387193787203, xref=2., ext=[AuthorCompanyExt(id=1241422270376235213, tenantId=1146029695717560320, journalId=1235980550691926019, articleId=1241416387193787203, companyId=1241422270367846604, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=2.School of Civil Engineering, Central South University, Changsha 410075, Hunan, China), AuthorCompanyExt(id=1241422270384623822, tenantId=1146029695717560320, journalId=1235980550691926019, articleId=1241416387193787203, companyId=1241422270367846604, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=2.中南大学　土木工程学院，湖南　长沙　410075)])], figs=[ArticleFig(id=1241422278232166810, tenantId=1146029695717560320, journalId=1235980550691926019, articleId=1241416387193787203, language=EN, label=Fig.1, caption=Graph for equivalence relation between T3 element and velocity discontinuity line, figureFileSmall=s7O9Ir8OopuZrd+QwARFXA==, figureFileBig=N235mCYX72EnWhkwEr780Q==, tableContent=null), ArticleFig(id=1241422278303469980, tenantId=1146029695717560320, journalId=1235980550691926019, articleId=1241416387193787203, language=CN, label=图1, caption=T3单元与速度间断线等效关系图, figureFileSmall=s7O9Ir8OopuZrd+QwARFXA==, figureFileBig=N235mCYX72EnWhkwEr780Q==, tableContent=null), ArticleFig(id=1241422278408327584, tenantId=1146029695717560320, journalId=1235980550691926019, articleId=1241416387193787203, language=EN, label=Fig.2, caption=Basic structure of triangular mesh with weak velocity discontinuity line, figureFileSmall=6A9A9Rpi+jm/Hs7DIeoedQ==, figureFileBig=wrucOfvJi+n6GGb4grlQIA==, tableContent=null), ArticleFig(id=1241422278563516836, tenantId=1146029695717560320, journalId=1235980550691926019, articleId=1241416387193787203, language=CN, label=图2, caption=弱速度间断线三角网基本结构

（a）P3；（b）P6

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（a）正三角形一分为六式加密；（b）边界网格切割策略；（c）模型外部网格剔除

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（a）P6（ne=5 098）；（b）P3（ne=5 225）；（c）Delaunay三角网（ne=5 160）

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（a）P6；（b）P3；（c）Delaunay三角网

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（a）ne=5 098；（b）ne=13 183；（c）ne=47 261

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（a）φ=15°；（b）φ=30°；（c）φ=45°

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（a）β=90°；（b）β=50°；（c）β=30°

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Upper bound solutions of Ns with different internal friction angles

, figureFileSmall=null, figureFileBig=null, tableContent=
φ/（°）Ns上限解
P6P3Delaunay三角网
157.227.277.29
208.368.468.50
259.839.9410.01
3011.7011.8511.97
3514.2214.4314.59
4017.7718.0918.30
4523.1223.7123.88
), ArticleFig(id=1241422282569077193, tenantId=1146029695717560320, journalId=1235980550691926019, articleId=1241416387193787203, language=CN, label=表1, caption=

不同内摩擦角对应的稳定性系数Ns上限解

, figureFileSmall=null, figureFileBig=null, tableContent=
φ/（°）Ns上限解
P6P3Delaunay三角网
157.227.277.29
208.368.468.50
259.839.9410.01
3011.7011.8511.97
3514.2214.4314.59
4017.7718.0918.30
4523.1223.7123.88
), ArticleFig(id=1241422282673934795, tenantId=1146029695717560320, journalId=1235980550691926019, articleId=1241416387193787203, language=EN, label=Table 2, caption=

Upper bound solutions of Ns with different slope angles

, figureFileSmall=null, figureFileBig=null, tableContent=
β/（°）Ns上限解
P6P3Delaunay三角网
3041.2341.9941.89
5013.7113.8713.94
708.368.468.50
905.595.635.63
), ArticleFig(id=1241422282770403789, tenantId=1146029695717560320, journalId=1235980550691926019, articleId=1241416387193787203, language=CN, label=表2, caption=

不同边坡坡度对应的稳定性系数Ns上限解

, figureFileSmall=null, figureFileBig=null, tableContent=
β/（°）Ns上限解
P6P3Delaunay三角网
3041.2341.9941.89
5013.7113.8713.94
708.368.468.50
905.595.635.63
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含弱速度间断线特性的六方向三角网在边坡稳定性分析中的应用
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余小军 1 , 秦傲韩 2 , 范佳志 1 , 杨鹰 2 , 杨峰 2 , 赵炼恒 2
矿冶工程杂志 | 采矿 2025,45(3): 44-51
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矿冶工程杂志 | 采矿 2025, 45(3): 44-51
含弱速度间断线特性的六方向三角网在边坡稳定性分析中的应用
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余小军1 , 秦傲韩2, 范佳志1, 杨鹰2, 杨峰2 , 赵炼恒2
作者信息
  • 1.伊春鹿鸣矿业有限公司，黑龙江　伊春　152500
  • 2.中南大学　土木工程学院，湖南　长沙　410075

    • 余小军（1975—），男，陕西渭南人，高级工程师，主要从事矿山资源研究。E-mail：

通讯作者:

杨峰（1981—），男，陕西西安人，博士，副教授，主要从事岩土工程研究。E-mail：
Application of Six-Directional Triangular Mesh with Weak Velocity Discontinuity Line Characteristics in Slope Stability Analysis
Xiaojun YU1 , Aohan QIN2, Jiazhi FAN1, Ying YANG2, Feng YANG2 , Lianheng ZHAO2
Affiliations
  • 1.Yichun Luming Mining Co., Ltd., Yichun 152500, Heilongjiang, China
  • 2.School of Civil Engineering, Central South University, Changsha 410075, Hunan, China
出版时间: 2025-06-01 doi: 10.3969/j.issn.0253-6099.2025.03.007
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摘要
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针对上限有限元法中三节点三角形单元在边坡稳定性分析时存在的体积锁定问题，从速度间断线与同处共轭三角形单元的力学等效效应出发，提出一种含弱速度间断线特性的六方向三角网（P6），并配合六节点三角形（T6）单元上限有限元法对边坡稳定性进行分析。结果表明：P6联合T6单元获得的边坡潜在滑动面清晰明确，耗散能密度平滑过渡，且随着网格密度增大，弱速度间断线效应加强，边坡稳定性系数Ns上限解精度提升；考虑内摩擦角及边坡坡度等综合因素的影响，P6对应的Ns上限解均优于三方向三角网及Delaunay三角网。基于P6上限有限元法在单次计算框架内形成的均匀网格计算效果良好，有利于强度折减上限分析等批处理运算方式，亦可结合网格自适应扩展其应用范围。

边坡稳定性  /  网格划分  /  三角网  /  三角形单元  /  上限有限元  /  滑动面  /  耗散能密度  /  速度间断线

To address the issue of volumetric locking encountered in the slope stability analysis with three-node triangular elements in upper bound finite element method (UBFEM), a six-directional triangular mesh (P6) with weak velocity discontinuity line characteristics was proposed based on the mechanical equivalence effect between velocity discontinuity lines and conjugate triangular elements located in the same position. The P6 is utilized in conjunction with a six-node triangular (T6) element in UBFEM for slope stability analysis. The results show that the potential slip surface of the slope obtained using the P6 combined with T6 elements is clearly defined, with a smooth transition in dissipated energy density. As the mesh density increases, the effect of weak velocity discontinuity lines strengthens, leading to an improved accuracy in the upper bound solution of the slope stability coefficient (Ns). With the combined influences of factors such as the internal friction angle and slope gradient into consideration, the upper bound solutions of Ns obtained with P6 outperform those from three-directional triangular meshes and Delaunay triangular meshes. The uniform mesh generated by the P6 UBFEM in a single computational framework yields favorable results, facilitating batch processing operations such as strength reduction upper bound analysis. Additionally, its application scope can be extended by integrating it with adaptive mesh refinement techniques.

slope stability  /  mesh division  /  triangular mesh  /  triangular element  /  upper bound finite element  /  slip surface  /  dissipated energy density  /  velocity discontinuity line
余小军, 秦傲韩, 范佳志, 杨鹰, 杨峰, 赵炼恒. 含弱速度间断线特性的六方向三角网在边坡稳定性分析中的应用. 矿冶工程杂志, 2025 , 45 (3) : 44 -51 . DOI: 10.3969/j.issn.0253-6099.2025.03.007
Xiaojun YU, Aohan QIN, Jiazhi FAN, Ying YANG, Feng YANG, Lianheng ZHAO. Application of Six-Directional Triangular Mesh with Weak Velocity Discontinuity Line Characteristics in Slope Stability Analysis[J]. Mining and Metallurgical Engineering, 2025 , 45 (3) : 44 -51 . DOI: 10.3969/j.issn.0253-6099.2025.03.007
正文
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露天开采、尾矿堆积等工程常形成存在安全风险的陡峭边坡[1-2]，因此，有必要对此类边坡进行稳定性评估，以保障矿山安全生产。边坡稳定性评估数值计算方法逐渐多元化[2-4]。其中塑性极限分析架构下的上限有限元法是开展露天矿/尾矿坝等边坡工程稳定性分析、潜在滑动面搜索的有力手段[5]，但上限有限元法存在体积锁定等缺陷。
已有研究表明，体积锁定引起的上限有限元法计算精度问题可采取以下措施进行改善：①应用高阶单元[6-8]；②增设速度间断线[9-15]；③优化网格拓扑结构等[16-18]。在单元类型方面，相比三节点三角形（T3）单元的常应变率，六节点三角形（T6）单元的应变率和塑性乘子线性变化，适用于模拟滑动面/剪切带等应变率梯度较高的破坏现象[6-8]；而速度间断线因坐标相同节点与相邻单元表征的速度矢量并不连续，为滑动面等力学特性的模拟提供了有利条件[7-10]，因此，含速度间断线三角网的上限法在岩土稳定性问题的应用相当广泛[9-16]。
除高阶单元和速度间断线外，网格拓扑结构对提升上限有限元法计算精度也有积极作用[17-19]。实际上，速度间断线可视为一对共轭单元厚度压缩至极限的特殊形式[15]，若压缩尺度有限，则压缩范围内的一组三角形单元共同具有速度间断线的部分特性。因此，不含速度间断线三角网的网格结构优化及计算精度提升问题值得深入探讨。
本文针对边坡工程稳定性问题，重点考虑三角网拓扑结构等因素，在单次计算框架内，基于均布网格前提，探索T6的最优网格布局，特别提出一种具备弱速度间断线特性的六方向三角网（P6）及其构建方法，与三方向三角网（P3）及均匀Delaunay三角网进行对比分析，揭示综合因素影响下的上限解精度演化规律，为上限有限元法进行边坡稳定性极限分析等相关研究提供新视角。
1　六方向三角网构思
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1.1　三节点三角形单元与速度间断线
一对共轭T3单元R和M，当单元厚度δ趋近于0时，与其方位相同的速度间断线具有力学等效关系[17-18]，等效关系图如图1所示。
按速度间断线局部坐标系t-n，单元R内任一点的速度可表示为：
式中：N（3）（t）为三节点三角形单元形函数[17]；uR为三节点速度矢量，uR=[utR，unR]T。
由图1可知，当δ趋于0时，单元面积AR亦趋于0，单元塑性应变率趋于无穷大，而与AR之积可表示为：
式中：和分别为沿速度间断线的切向和法向塑性正应变率；为塑性剪应变率。共轭单元M表达式同式（2）。将共轭单元R和M的内部耗散能求和：
式中和分别为单元R和单元M与塑性应变率对应的应力。可以看出，当厚度δ趋于0时，按共轭单元导出的耗散能公式恰好与同处速度间断线耗散能公式[17]完全一致，由此印证了T3单元与速度间断线之间的力学等效关系。
进一步推测，当δ值较小时，平行排列的单元组将呈现速度间断线的部分特征，称其为弱速度间断线特性。此外，式（3）源于T3单元，可推知T6单元也具有类似性质。
1.2　六方向三角网拓扑结构
对于含速度间断线三角网的上限法，文献[12]进一步提出节点可动的刚体平动运动单元计算思路，发现上限解精度依赖速度间断线的特定方位，且有效速度间断线常呈现两向贯通滑移线网的特征。
同时考虑弱速度间断线特性及贯通式网状结构需求，提出一种规则六方向三角网拓扑结构。一个正三角形单元具备三个方向，当基本结构不断缩小尺寸并大范围规则布置时，将形成三方向通路，即P3，如图2（a）所示。进一步将正三角形沿形心一分为六，即形成P6基本结构，如图2（b）所示。
2　六方向三角网生成
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P6生成流程如图3所示。
2.1　背景三角网构建
为生成适应边坡稳定性计算模型的P6，首先需构建规则背景三角网。设置正三角形区域并完全覆盖计算模型，经多次一分为四式分裂加密形成背景三角网，其中区域边长l和最小正三角形单元边长ls为主控指标，可用于调控网格密度。此外，边坡坡脚等关键点应与背景网格节点对齐。
终次加密时连接最小正三角形单元形心与顶点、各边中点，实现一次性一分为六式加密，即形成规则的P6[图3（a）]。从整体上看，该网格具备六方向通路，且随着网格密度增加，各通路内单元厚度减小，六方向上弱速度间断线特性显现。
需说明的是，若不进行终次一分为六式加密，则对应生成规则P3。
2.2　边界网格切割
边坡模型边界需切割背景网格。
1）边界特征点单元加密：两条及以上边界交点即为特征点，连接特征点与所处单元顶点[图3（b）中案例1]。
2）过单元顶点切割：当跨界单元存在1个顶点位于特定边界时，则连接该顶点与边界交点[图3（b）中案例2]。
3）不过单元顶点切割：若跨界单元无顶点位于边界上，连接既有顶点与两交点，将原有单元分割为三部分[图3（b）中案例3]。筛选两种情况下单元面积差异较小的一种。
网格切割完成后，剔除边坡模型外围无效单元[图3（c）]，形成适应模型边界的P6。
3　六节点三角形单元上限有限元分析
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1）上限定理。依据塑性极限分析中的上限定理，在所有满足运动许可的速度场u中，极限荷载对应速度场的耗散能最小。
式中：Dp为计算域内的总塑性耗散能；dp（u）为塑性耗散能函数；若边坡稳定性考虑自重作用，Wext（u）为岩土体自重虚功率，不含其他荷载项。
2）屈服准则与相关联流动法则。满足屈服条件的应力张量σ集合为：
式中：F为屈服函数；f（σ）为屈服面。若岩土体服从Mohr-Coulomb屈服准则，相关联流动约束可表示为：
式中：为塑性乘子，；ω=[cosα+sinφ，sinφ-cosα，2sinα]T。
为形成计算效率更高的二阶锥规划模型，定义辅助变量，，则二者应满足如下约束[20]：
3）T6单元。图4为T6单元各节点上赋存的物理场量。与T3单元不同，T6单元塑性应变率及塑性乘子在单元内部线性变化[8]，但该单元对P3和P6等三角网计算精度的耦合影响仍有待深入探讨。
模型总塑性耗散能为：
式中：c和φ分别为黏聚力和内摩擦角；ne为单元总数；Ak为第k个单元的面积；为第k个单元第i个顶节点的塑性乘子。
单元k内部耗散能密度为：
4）上限有限元二阶锥规划模型。基于T6单元的边坡稳定性上限有限元分析涉及的优化变量为六节点速度v和顶点处塑性乘子，以及与对应的辅助变量ρ1，ρ2。除相关联流动约束外，整个计算模型还应施加内部及边界速度约束条件，最终形成二阶锥规划问题，见式（10），具体细节可见文献[20]。
式中：B和C为等式约束系数矩阵；s为未知变量向量；b为等式约束右侧向量。
4　算例分析与讨论
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4.1　边坡稳定性分析模型
假定岩土体为均质各向同性材料，建立边坡稳定性分析模型如图5所示[8,20]。为方便建模，坐标原点选为坡脚左侧边界点处。岩土体内摩擦角φ=15°～45°，黏聚力c=1 kPa，边坡坡度β=30°～90°，坡高H=10 m，临界容重γcr作为目标函数通过优化得到，待求解的稳定性系数Ns=γcrH/c，采用T6单元上限有限元法求解Ns及潜在滑动面。
除P6和P3外，引入均匀Delaunay三角网作为对比，“均匀”特指模型中的单元尺度基本一致。Delaunay三角剖分是目前主流网格剖分方法，旨在评价单个三角形形状质量，以内切圆与外接圆半径比（rin/rout）量化，其中正三角形（rin/rout=0.5）被视为最优形态[21]，缺点是无法满足特定单元集合的平行排列或特定方向贯通性排布等需求。
4.2　网格结构的影响
为清晰展示边坡稳定性模型的P6、P3和均匀Delaunay三角网结构特征，绘制稀疏网格条件下3种三角网格如图6所示，单元总数ne≈5 000。
P6与P3于边坡模型内部形成规则的多方向通路，且得益于网格切割，边界位置亦维持背景网格结构。
选取密集网格（ne≈23 000），采用上限有限元法计算获得P6、P3及Delaunay三角网对应的边坡稳定性系数Ns上限解及潜在滑动面，如图7所示。图中采用归一化塑性耗散能密度表征边坡潜在滑动面，即（取值范围0～1），其中为模型耗散能密度最大值。考虑T6单元数值线性变化特征，云图以精准插值的方式显示。
由图7可知，P6、P3与Delaunay三角网Ns上限解分别为14.22、14.43和14.59，依上限定理，Ns数值越小解答越优[18]。因此，计算精度排序为：P6>P3>Delaunay三角网。
3种网格的耗散能密度较大区域均集中于边坡坡脚位置；Delaunay三角网的归一化耗散能密度分布存在光滑性、连续性差等特点，而P6与P3对应的耗散能密度于潜在滑动面范围内呈平滑过渡特征；此外，相比P3，P6耗散能密度较大值的分布范围更窄，说明P6模拟边坡滑动面的效果更佳。
4.3　单元总数ne的影响
单元总数ne也是影响上限有限元法计算精度的关键因素。选取内摩擦角φ=35°、边坡坡度β=70°，设定单元总数ne范围为3 000～50 000，3种网格对应的Ns上限解曲线如图8所示。由图8可知，随着单元总数ne（或网格密度）增大，P6、P3与Delaunay三角网对应的Ns均不断减小。相比P3、Delaunay三角网，同等网格密度条件下P6的Ns值更小，表明P6计算精度更高。ne>30 000后，随着ne持续增大，P6的Ns值下降幅度大于P3、Delaunay三角网的Ns值下降幅度。
进一步绘制不同网格密度条件下P6的云图，如图9所示。随着单元总数ne不断增大，平行排列的网格间距不断减小，六方向弱速度间断线效应更加显著；以归一化耗散能密度表征的边坡潜在滑动面宽度更窄，对应的Ns值从14.82减小至14.08，说明网格密度增加后，P6对T6上限有限元具有良好的精度优势。
4.4　内摩擦角的影响
为探讨岩土体内摩擦角对P6、P3及Delaunay三角网计算精度的影响，设定内摩擦角φ为15°～45°，单元总数ne为23 000，边坡坡度β为70°，不同内摩擦角对应的稳定性系数Ns上限解见表1。
由表1可知，随着内摩擦角增大，P6、P3与Delaunay三角网对应的Ns均增大，且同等条件下P6对应的Ns值更小，说明不同内摩擦角条件下，P6计算精度仍具有优势。
进一步绘制φ=15°、30°、45°对应的P6的云图，如图10所示。随着内摩擦角增大，边坡潜在滑动面逐渐向坡面方向移动，说明P6可较好地适应内摩擦角变化，耗散能密度平顺过渡。
4.5　边坡坡度的影响
为探讨边坡坡度对P6、P3及Delaunay三角网计算精度的影响，设定β为30°、50°、90°，φ为20°，ne为23 000，不同边坡坡度对应的稳定性系数Ns上限解见表2。由表2可知，随着边坡坡度逐渐增大，P6、P3与Delaunay三角网对应的Ns值均减小，且P6的Ns值更小，说明不同边坡坡度条件下，P6计算精度仍具有优势。
进一步绘制不同边坡坡度条件下P6对应的云图如图11所示。由图11可以看出，不同边坡坡度条件下，P6对应的以耗散能密度表征的潜在滑动面均呈现出形态光滑、宽度狭窄等较优特性。
5　结论
收起
1）针对边坡稳定性分析问题，提出一种具备弱速度间断线效应的六方向三角网（P6）构思及生成方法，P6可在单次计算框架以及均布网格前提下达到较高计算精度，有助于批处理运算。
2）P6联合T6单元，经上限有限元法计算获得的潜在滑动面清晰明确，且随着网格密度增大，弱速度间断线效应加强，边坡稳定性系数Ns上限解精度提升。
3）进一步考虑内摩擦角与边坡坡度等参数，发现综合因素影响下，相比P3和Delaunay三角网，同等网格密度条件下P6对应的Ns上限解均具备较高的计算精度，印证了P6应用于边坡稳定性分析的计算优势。
4）P6在计算域内均匀布置，存在一定程度的过度细化问题，可探索以耗散能密度或应变率梯度等指标进行网格自适应加密或稀疏，以此作为进一步提升的方向。
基金
收起
  • 湖南省杰出青年基金(2021JJ10063)
  • 伊春鹿鸣矿业有限公司科技项目(LM（2022）-F-045)
  • 中南大学中央高校基本科研业务费专项资金(2024ZZTS0752)
文献
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2025年第45卷第3期
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doi: 10.3969/j.issn.0253-6099.2025.03.007
  • 接收时间：2024-12-13
  • 首发时间：2026-03-19
  • 出版时间：2025-06-01
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  • 收稿日期：2024-12-13
基金
湖南省杰出青年基金(2021JJ10063)
伊春鹿鸣矿业有限公司科技项目(LM（2022）-F-045)
中南大学中央高校基本科研业务费专项资金(2024ZZTS0752)
作者信息
    1.伊春鹿鸣矿业有限公司，黑龙江　伊春　152500
    2.中南大学　土木工程学院，湖南　长沙　410075

通讯作者:

杨峰（1981—），男，陕西西安人，博士，副教授，主要从事岩土工程研究。E-mail：
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2种不同金属材料的力学参数
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多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
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小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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