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Article(id=1246031931238887779, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246031922707677827, articleNumber=null, orderNo=null, doi=10.19636/j.cnki.cjsm42-1250/o3.2024.007, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1708358400000, receivedDateStr=2024-02-20, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1775006762561, onlineDateStr=2026-04-01, pubDate=1724515200000, pubDateStr=2024-08-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1775006762561, onlineIssueDateStr=2026-04-01, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1775006762561, creator=13701087609, updateTime=1775006762561, updator=13701087609, issue=Issue{id=1246031922707677827, tenantId=1146029695717560320, journalId=1241755870837649424, year='2024', volume='45', issue='4', pageStart='427', pageEnd='564', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1775006760527, creator=13701087609, updateTime=1775006830786, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1246032217470779944, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246031922707677827, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1246032217470779945, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246031922707677827, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=466, endPage=476, ext={EN=ArticleExt(id=1246031931591209327, articleId=1246031931238887779, tenantId=1146029695717560320, journalId=1241755870837649424, language=EN, title=Adaptive Phantom Node Method for Three-dimensional Dynamic Stress Intensity Factor, columnId=1244229834482757770, journalTitle=Chinese Journal of Solid Mechanics, columnName=Research Paper, runingTitle=null, highlight=null, articleAbstract=

Stress intensity factor is a crucial parameter for modeling and predicting structural fracture failure. This study evaluates the dynamic stress intensity factor for solving three-dimensional dynamic fracture problems using the adaptive phantom node method. This technique combines the phantom node method with adaptive mesh refinement, automating the generation of a dense mesh around the crack. In this approach, strong discontinuities at cracks are modeled using phantom nodes without crack tip enrichment functions or extra degrees of freedom. The theoretical framework of this technique is straightforward and easy to implement based on the finite element method, but it requires a relatively dense mesh to ensure computational accuracy. Adaptive mesh refinement technology and criteria suitable for crack problems are introduced into the phantom node method, thus obviating the need for a globally dense mesh with high computational consumption while improving computational accuracy and efficiency. A concise approach, known as constrained approximation, is adopted to deal with hanging nodes presented in the locally refined mesh. It is convenient to implement numerically, does not involve special elements or complex shape functions, and retains the interpolation and numerical integration of the standard finite element method. The stress intensity factors for several three-dimensional crack problems are evaluated using the adaptive phantom node method and compared with the theoretical solutions and numerical results obtained by the standard phantom node method. It is found that the numerical results of this method are in good agreement with the theoretical solutions, and the computational accuracy is effectively improved compared to the standard phantom node method. Additionally, compared to the locally pre-refined mesh with equivalent accuracy, the adaptive refined mesh exhibits higher computational efficiency and reduced computational consumption. This holds considerable potential value for the efficient simulation and prediction of dynamic fracture failure in large-scale complex engineering structures.

, correspAuthors=Qinglin Duan, authorNote=null, correspAuthorsNote=null, copyrightStatement=null, copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, authorCompany=null, fund=null, authors=null, authorsList=Zeyang Feng, Qinglin Duan), CN=ArticleExt(id=1246031943238792025, articleId=1246031931238887779, tenantId=1146029695717560320, journalId=1241755870837649424, language=CN, title=三维动态应力强度因子的自适应虚拟节点法, columnId=1241831201896469478, journalTitle=固体力学学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

采用自适应虚拟节点法计算三维动态应力强度因子，改善了标准虚拟节点法计算精度和效率的不足. 通过无需裂尖加强函数（和相应的额外自由度）的虚拟节点技术模拟裂纹处的强间断. 在裂纹处引入自适应网格加密技术以及契合裂纹问题的加密准则，无需高计算消耗的全局稠密网格，从而提高计算精度和效率. 采用简便的约束近似技术处理局部加密网格中的悬空节点，避免特殊单元和复杂形函数，便于数值实现. 数值算例表明，相较于标准虚拟节点法，自适应虚拟节点法可有效提高计算精度，且在同样精度下具有更少的计算消耗和更高的效率.

, correspAuthors=段庆林, authorNote=null, correspAuthorsNote=
** E-mail：.
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Comparison of the numerical performance of meshes for stationary mixed mode semi-infinite crack

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含复合型静态裂纹的半无限大板的网格数值性能对比

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Comparison of the numerical performance of meshes for stationary mode Ⅰ semi-infinite crack

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元数节点数CPU时间（秒）
初始网格32806804543.40
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含Ⅰ型静态裂纹的半无限大板的网格数值性能对比

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三维动态应力强度因子的自适应虚拟节点法
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冯泽洋 1, 2 , 段庆林 1, 2, **
固体力学学报 | 研究论文 2024,45(4): 466-476
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固体力学学报 | 研究论文 2024, 45(4): 466-476
三维动态应力强度因子的自适应虚拟节点法
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冯泽洋1, 2, 段庆林1, 2, **
作者信息
  • 1大连理工大学工业装备结构分析优化与CAE软件全国重点实验室，大连，116024
  • 2大连理工大学白俄罗斯国立大学联合学院，大连，116024

通讯作者:

Adaptive Phantom Node Method for Three-dimensional Dynamic Stress Intensity Factor
Zeyang Feng1, 2, Qinglin Duan1, 2, **
Affiliations
  • 1State Key Laboratory of Structural Analysis, Optimization and CAE Software for Industrial Equipment, Dalian University of Technology, Dalian, 116024
  • 2DUT-BSU Joint Institute, Dalian University of Technology, Dalian, 116024
出版时间: 2024-08-25 doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.007
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摘要
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采用自适应虚拟节点法计算三维动态应力强度因子，改善了标准虚拟节点法计算精度和效率的不足. 通过无需裂尖加强函数（和相应的额外自由度）的虚拟节点技术模拟裂纹处的强间断. 在裂纹处引入自适应网格加密技术以及契合裂纹问题的加密准则，无需高计算消耗的全局稠密网格，从而提高计算精度和效率. 采用简便的约束近似技术处理局部加密网格中的悬空节点，避免特殊单元和复杂形函数，便于数值实现. 数值算例表明，相较于标准虚拟节点法，自适应虚拟节点法可有效提高计算精度，且在同样精度下具有更少的计算消耗和更高的效率.

动力学裂纹  /  应力强度因子  /  自适应网格加密  /  虚拟节点法  /  扩展有限元

Stress intensity factor is a crucial parameter for modeling and predicting structural fracture failure. This study evaluates the dynamic stress intensity factor for solving three-dimensional dynamic fracture problems using the adaptive phantom node method. This technique combines the phantom node method with adaptive mesh refinement, automating the generation of a dense mesh around the crack. In this approach, strong discontinuities at cracks are modeled using phantom nodes without crack tip enrichment functions or extra degrees of freedom. The theoretical framework of this technique is straightforward and easy to implement based on the finite element method, but it requires a relatively dense mesh to ensure computational accuracy. Adaptive mesh refinement technology and criteria suitable for crack problems are introduced into the phantom node method, thus obviating the need for a globally dense mesh with high computational consumption while improving computational accuracy and efficiency. A concise approach, known as constrained approximation, is adopted to deal with hanging nodes presented in the locally refined mesh. It is convenient to implement numerically, does not involve special elements or complex shape functions, and retains the interpolation and numerical integration of the standard finite element method. The stress intensity factors for several three-dimensional crack problems are evaluated using the adaptive phantom node method and compared with the theoretical solutions and numerical results obtained by the standard phantom node method. It is found that the numerical results of this method are in good agreement with the theoretical solutions, and the computational accuracy is effectively improved compared to the standard phantom node method. Additionally, compared to the locally pre-refined mesh with equivalent accuracy, the adaptive refined mesh exhibits higher computational efficiency and reduced computational consumption. This holds considerable potential value for the efficient simulation and prediction of dynamic fracture failure in large-scale complex engineering structures.

dynamic crack  /  stress intensity factor  /  adaptive mesh refinement  /  phantom node method  /  extended finite element method
冯泽洋, 段庆林. 三维动态应力强度因子的自适应虚拟节点法. 固体力学学报, 2024 , 45 (4) : 466 -476 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.007
Zeyang Feng, Qinglin Duan. Adaptive Phantom Node Method for Three-dimensional Dynamic Stress Intensity Factor[J]. Chinese Journal of Solid Mechanics, 2024 , 45 (4) : 466 -476 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.007
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0　引言
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Belytschko等[1,2]发展的扩展有限元法（eX-tended Finite Element Method，XFEM）是目前模拟裂纹行为最主流的数值方法之一，并广泛应用于各类非连续问题，如疲劳裂纹[3]、裂纹反演分析[4]、裂缝渗流[5]、动力学裂纹[6]等. 该方法在有限元法（Finite Element Method，FEM）的位移插值空间中引入了捕捉奇异性的裂尖加强函数[1]、描述裂纹间断的Heaviside加强函数以及相应的额外自由度[2]. 这使得裂纹几何与计算网格相互独立，无需复杂的网格重构算法即可对任意裂纹行为进行高精度模拟. 然而引入加强函数，尤其是裂尖加强函数，不仅会产生需特殊处理的混合单元[7]，更会引发线性相关问题. 即使XFEM仅在裂纹局部进行插值加强，其总体刚度阵的条件数依然急剧增加甚至严重病态，导致准静态问题和动力学问题的隐式迭代求解过程缓慢甚至无法收敛，严重阻碍其在大规模复杂工程问题中的应用[8]. 在裂纹扩展过程中，裂尖单元会随着裂尖位置的改变而演化为完全断裂单元，与之相关的加强函数和额外自由度也不断变化. 在动力学裂纹问题中，与裂尖额外自由度相对应的物理量同样会随着裂尖的移动而被舍弃，若不施以适当的处理则将违背能量守恒定律[9].
Song等[10]在XFEM的基础上发展了简便的虚拟节点法. 该方法直接舍弃了裂尖加强函数和额外自由度，从而有效缓解了上述问题. 在对插值函数与自由度进行重新排列组合后，裂纹间断可由两个包含虚拟节点的重叠单元来描述. 虚拟节点法可通过对现有有限元程序进行少量修改来实现，既保留了标准XFEM中裂纹独立于计算网格的优势，同时也具备了标准FEM的高效性与鲁棒性. Duan等[11]将该方法拓展至三维动力学裂纹问题，并提出了描述三维裂纹面的单元局部水平集方法. Wang等[12]基于虚拟节点法提出了一种损伤传递算法，并应用于复合层压板的失效研究. Yang等[13]采用该方法成功捕捉了热循环载荷下芯片边角处的应力集中和裂纹扩展，且数值结果与实验结果吻合.
相较于标准XFEM，虚拟节点法的计算精度通常依赖于计算网格的尺寸. 薛河等[14]在不同网格尺寸下使用XFEM计算三维应力强度因子，并与公式解析解进行比较. 结果表明XFEM具有良好的计算精度，且对网格尺寸的依赖程度较低. 应力强度因子是反映裂纹尖端应力场强弱关键参量，也是判断裂纹扩展趋势以及扩展方向的重要指标. 精确计算应力强度因子是裂纹问题研究中的一项重要工作. 在舍弃了裂尖加强函数的虚拟节点法中，通常需要使用较为稠密的网格以保证其计算精度. 这对于三维裂纹问题会导致计算量的激增，降低计算效率. Feng等[15]在标准虚拟节点法的基础上发展了自适应虚拟节点法，数值结果表明该方法可有效提升应力强度因子的计算精度并控制计算量，为裂纹扩展过程的高精度模拟奠定了基础.
本文将采用自适应虚拟节点法来提高三维动态应力强度因子的计算精度与效率. 虚拟节点法可有效缓解标准XFEM的病态刚度阵和能量不守恒问题，确保数值结果的正确性与稳定性. 在裂纹周边采用自适应网格加密技术[16]，既弥补了标准虚拟节点法在计算精度上的缺陷，又可有效节省计算量，提升计算效率. 采用约束近似技术处理局部加密网格上的悬空节点，无需构造复杂形函数或特殊单元. 后文将对这些进行简要阐述，并采用数值算例验证自适应虚拟节点法在计算精度与效率上的优势.
1　自适应虚拟节点法
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1.1　虚拟节点法
在无裂尖加强函数的XFEM中，含间断的位移近似可写为：
其中NI为节点有限元形函数矩阵，uI和qI分别为节点自由度和额外自由度，f（x）为水平集值，其分布如图1所示. H（x）是Heaviside加强函数，可表示为：
虚拟节点法忽略了裂尖位于单元内部的情况，将单元分为完整单元和（完全）断裂单元. Song等[10]通过对式（1）中的各项重新组合，将断裂单元替换为两个含有虚拟节点的重叠单元，其位移近似可表示为：
其中E1和E2为图1中由真实节点和虚拟节点组成的重叠单元，和为相应的节点自由度，表示为：
在虚拟节点法的框架下，含裂纹三维弹性体Ω中的动力学求解方程可离散化为：
其中B为应变矩阵，D为弹性矩阵，N为有限元形函数矩阵，b为体力，为边界上施加的牵引力，ρ为材料密度. VE为断裂单元E的体积，和分别为重叠单元内的有效体积（即图1中的蓝色区域），且和nE为单元E内的裂纹面及其法向量，τc为裂纹面上的内聚牵引力. 由式（6）-（8）可以看出，仅需计算重叠单元中有效体积的比值，即可获得断裂单元的数值积分结果. 因此虚拟节点法可在有限元程序的基础上轻易实现，并具备了与FEM同等的高效性与鲁棒性.
1.2　自适应网格加密
在加密过程中，通常使用单元等级来表示单元的加密次数，单元等级为0则表示该单元是初始单元. 加密后的网格及单元等级的变化过程如图2所示. 裂纹是一种局部现象，可使用距离函数作为自适应加密范围的判断准则. 单元等级i对应的加密半径定义为：
其中h为初始单元的平均棱长，Lmax为本次加密过程中设置的最大单元等级. m是人工参数，控制等级为Lmax的单元层数. 通常根据所选取的断裂准则来确定m的值. 对于最大主应力准则，m取4～6即可，而对于以应力强度因子为基础的判断准则，则需要根据应力强度因子的提取域范围来确定. 当两个等级不同的单元相邻时，会产生需要特殊处理的悬空节点. 为简化处理，本文限制相邻单元等级的最大差值为1，即每个棱或面上最多包含一个悬空节点且位于中心点，并采用约束近似法来处理悬空节点. 值得注意的是，若一个单元中同时存在虚拟节点和悬空节点，则会增加节点处理的复杂性. 因此需控制m的取值以避免此现象的发生.
在图3所示的局部加密网格中，单元G含有棱悬空节点6和10以及面悬空节点9，它们的位移可通过所关联的真实节点线性插值得到：
单元G内的位移近似可表示为：
将式（10）代入上式后可得到：
其中为单元G的新有限元形函数矩阵，可由NI表示为：
由上式可知，约束近似法无需构造复杂形函数，只需将悬空节点上的有限元形函数均匀分配给关联的真实节点. 在组装总体刚度阵、质量阵和节点载荷向量时，同样将悬空节点的对应项分配给关联的真实节点. 约束近似法保留了标准有限元法的插值格式和数值积分，同时具备有限元法的高效性与鲁棒性，对于进行大规模复杂裂纹的数值模拟至关重要.
2　动态应力强度因子
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本文采用相互作用积分来计算裂纹前沿任意点的动态应力强度因子[16]. 如图4所示，任取一点s建立裂尖局部坐标系及相互作用积分的计算域. 其中X2和X3分别为s点处的裂纹面单位法向量和裂纹前沿单位切向量，且X1=X2×X3. 引入辅助场，则相互作用积分在该局部坐标系内定义为[9]：
其中上角标“aux”表示该变量为辅助场变量，σij，ui，和分别为应力，位移，速度和加速度的分量. q是一个连续的试探函数，在本文中被定义为：
其中RV和2HV分别为圆柱体提取域的底面半径和高度. 在数值实现的过程中，选取当前计算网格中与该圆柱体相交的单元进行数值积分，避免在提区域内重新构造计算网格. 记等级为Lmax的单元的平均棱长为，通常取，. 其中n为人工参数，且需满足RV<RLmax，以保证提取域内的单元等级均为Lmax.
相互作用积分也可由真实场和辅助场的应力强度因子表示为[9,17]：
其中KⅠ，KⅡ和KⅢ为应力强度因子. f1，f2和f3可分别表示为：
其中λ和μ为拉梅常数，为裂纹扩展速度，cd和cs分别为纵波和横波速度. C为提取域内的裂纹前沿，即图4中的曲线ab. 为简化计算，通常认为ab为直线段，则可得到：
将式（21）代入式（16），并取，和，可得：
则可计算KⅠ为：
同理可得到KⅡ和KⅢ
3　数值算例
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3.1　含复合型静态裂纹的半无限大板
该算例是验证动态应力强度因子计算精度的经典算例[9,17,18]，Lee等[19]于1990年给出解析解. 几何尺寸和边界条件如图5所示. 作平面应变假设，即约束前后两个边界面在厚度方向上的自由度. 材料参数为杨氏模量E=200 GPa，泊松比ν=0.25，密度ρ=7833 kg/m3. 数值模型为有限平板，解析解仅在平板右侧反射的应力波达到裂尖之前是有效的，因此模拟时间为t≤3tc=3（a/c）d=542×10-6 s. 分别使用tc和）对模拟时间和动态应力强度因子进行无量纲化处理.
图6给出了四种不同的网格. 动态应力强度因子计算结果与解析解的对比如图7所示. 显然随着裂纹周边单元尺寸的减小，计算结果与解析解更为吻合. 表1详细对比了不同网格. 尽管自适应加密在提高计算精度的同时会增加计算量，但与局部预加密网格相比，该方法在计算效率上仍具有显著优势.
图8展示了不同动态应力强度因子提取域尺寸对计算结果的影响，其中为等级是Lmax的单元的平均棱长. 增加提取域范围可有效提高计算精度，但一旦达到上限，进一步增加范围只能带来较小的提升.
3.2　含Ⅰ型静态裂纹的半无限大板
含Ⅰ型静态裂纹的半无限大板算例被广泛应用于验证动态应力强度因子[17,18,20]. 假设为平面应变问题，前后两个边界面在厚度方向上被约束. 几何尺寸和边界条件如图9所示. 材料参数为杨氏模量E=210 GPa，泊松比ν=0.3，密度ρ=8000 kg/m3. 该算例的解析解为[21]：
在该算例中，应力波从上边界面传递到下边界面，随后经反射传递到裂纹. 解析解仅在这段时间内有效，则模拟时间为t≤3tc=3（H/c）d=1.009×10-3 s. 动态应力强度因子和模拟时间分别使用和tc进行无量纲化处理. 图10对比了不同计算网格的动态应力强度因子，更加详细的数据比较如表2所示. 对比结果验证了自适应虚拟节点法对计算精度和效率的提升，且数值结果的振荡明显减弱. 图11展示了应力强度因子提取域对计算精度的影响.
3.3　含圆片型裂纹的长方体
含圆片型裂纹的长方体算例通常用于评估数值方法在计算含弯曲前沿的三维裂纹动态应力强度因子时的精度[22-24]. 几何尺寸和边界条件如图12（a）所示. 考虑其对称性质，建立如图12（b）所示的四分之一数值模型并划分六面体网格. 材料参数为杨氏模量E=10 GPa，泊松比ν=0.2，密度ρ=1000 kg/m3，上、下表面的载荷为σ0=1000 Pa.
模拟时间t≤4tc=4（h/cd）=7.2×10-4 s，并进行无量纲化处理t/tc. 图13对比了图12（b）中A点处的Z方向位移，并使用σ0h/E对其进行无量纲化处理. 所得到的数值结果与Wen等[23]的已发表结果一致，且几乎不受自适应加密网格的影响. 动态应力强度因子KⅠ同样使用进行无量纲化处理，与Wen等[22,23]的数值结果对比如图14所示，所提出方法的计算精度随着网格自适应加密得到了提升.
4　结论
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本文采用自适应虚拟节点法计算三维动态应力强度因子. 该方法无需构造特殊单元或复杂形函数，在标准有限元程序的基础上即可实现，同时具备有限元法的鲁棒性和高效性. 数值结果表明，自适应虚拟节点法获得的动态应力强度因子与解析解相吻合，相较于标准虚拟节点法具有更高的计算精度和效率. 增加动态应力强度因子的提取域范围同样可以提高计算精度，但存在上限. 基于本文的工作，可将该方法拓展至动力学裂纹扩展以及多裂纹扩展等复杂裂纹问题，对于大规模复杂工程结构的断裂模拟分析具有可观的应用价值.
基金
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  • 中央高校基本科研业务费项目(DUT21GF304)
  • 科学挑战专题(TZ2018002)
  • 中白国际合作基金项目(ICR2203)
文献
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参考文献 引证文献
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2024年第45卷第4期
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doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.007
  • 接收时间：2024-02-20
  • 首发时间：2026-04-01
  • 出版时间：2024-08-25
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  • 收稿日期：2024-02-20
基金
中央高校基本科研业务费项目(DUT21GF304)
科学挑战专题(TZ2018002)
中白国际合作基金项目(ICR2203)
作者信息
    1大连理工大学工业装备结构分析优化与CAE软件全国重点实验室，大连，116024
    2大连理工大学白俄罗斯国立大学联合学院，大连，116024

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Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
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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