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Article(id=1244340203951866803, tenantId=1146029695717560320, journalId=1244295746212642849, issueId=1244340201636610993, articleNumber=null, orderNo=null, doi=10.11776/j.issn.1000-4939.2025.06.023, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1733068800000, receivedDateStr=2024-12-02, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1774603423344, onlineDateStr=2026-03-27, pubDate=1765728000000, pubDateStr=2025-12-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774603423344, onlineIssueDateStr=2026-03-27, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774603423344, creator=13701087609, updateTime=1774603423344, updator=13701087609, issue=Issue{id=1244340201636610993, tenantId=1146029695717560320, journalId=1244295746212642849, year='2025', volume='42', issue='6', pageStart='1207', pageEnd='1446', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1774603422792, creator=13701087609, updateTime=1774603751062, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1244341578555306992, tenantId=1146029695717560320, journalId=1244295746212642849, issueId=1244340201636610993, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1244341578555306993, tenantId=1146029695717560320, journalId=1244295746212642849, issueId=1244340201636610993, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=1439, endPage=1446, ext={EN=ArticleExt(id=1244340204752978878, articleId=1244340203951866803, tenantId=1146029695717560320, journalId=1244295746212642849, language=EN, title=Application of CS-FEM to finite element modelling of knee joints, columnId=1244340204681675708, journalTitle=Chinese Journal of Applied Mechanics, columnName=Biomechanics, runingTitle=null, highlight=null, articleAbstract=
Objective

To improve the quality and efficiency of the mesh in the finite element model of the knee joint by using the cell-based smoothed finite element method（CS-FEM）.

Methods

The gradient smoothing technique is introduced on the basis of the traditional finite element method, and CS-FEM is used to mesh the finite element model of the knee joint, and to compare which of the two methods（traditional finite element method and CS-FEM）is more accurate for the experimental data of the previous study.

Results

CS-FEM is more accurate and reduces the complexity of meshing than the traditional finite element method, and reduces the number of cells, making the solution process more efficient.

Conclusion

The conventional FEM methods offer high computational accuracy and stability when dealing with relatively simple geometries and linear materials due to their wide application and maturity. However, in biomechanical simulation of complex structures and nonlinear materials, CS-FEM provides higher computational efficiency and accuracy by optimizing meshing and reducing stress concentration. Therefore, CS-FEM is expected to be more widely used in the modelling of knee joints and other complex bio-logical structures in the future.

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目的

通过使用光滑有限元法（cell-based smoothed finite element method，CS-FEM）来提高膝关节有限元模型中网格的质量与效率。

方法

在传统有限元法的基础上引入梯度光滑技术，采用基于单元的光滑有限元法（CS-FEM）对膝关节有限元模型进行网格划分，对比传统有限元法与CS-FEM实验数据的精确性。

结果

CS-FEM较传统有限元法更加精确且减少了网格划分的复杂度，同时降低了单元数量，使得求解过程更加高效。

结论

传统有限元法（finite element method，FEM）方法由于其应用广泛和成熟性，在处理相对简单的几何结构和线性材料时具有较高的计算精度和稳定性。然而，在复杂结构和非线性材料的生物力学模拟中，CS-FEM通过优化网格划分和减少应力集中，提供了更高的计算效率和精度。因此，在未来膝关节和其他复杂生物结构的建模中，CS-FEM有望得到更广泛的应用。

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杨平川。E-mail:
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Computer methods in applied mechanics and engineering, 2016, 312: 130-166., articleTitle=A phasefield formulation for fracture in ductile materials: Finite deformation balance law derivation, plastic degradation, and stress triaxiality effects, refAbstract=null), Reference(id=1244340231164510956, tenantId=1146029695717560320, journalId=1244295746212642849, articleId=1244340203951866803, doi=null, pmid=null, pmcid=null, year=2015, volume=55, issue=5, pageStart=1017, pageEnd=1040, url=null, language=null, rfNumber=[39], rfOrder=48, authorNames=AMBATI M, GERASIMOV T, DE LORENZIS L, journalName=Computational mechanics, refType=null, unstructuredReference=AMBATI M, GERASIMOV T, DE LORENZIS L. Phase-field modeling of ductile fracture [J]. Computational mechanics, 2015, 55（5）: 1017-1040., articleTitle=Phase-field modeling of ductile fracture, refAbstract=null), Reference(id=1244340231319700217, tenantId=1146029695717560320, journalId=1244295746212642849, articleId=1244340203951866803, doi=null, pmid=null, pmcid=null, year=2016, volume=121, issue=null, pageStart=35, pageEnd=47, url=null, language=null, rfNumber=[40], rfOrder=49, authorNames=LIU G W, LI Q B, MSEKH M A, journalName=Computational materials science, refType=null, unstructuredReference=LIU G W, LI Q B, MSEKH M A, et al. Abaqus implementation of monolithic and staggered schemes for quasi-static and dynamic fracture phase-field model [J]. 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CT and MRI specific scanning parameters

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磁共振成像T2 TSE3DT1 TSEDWI EPI
回波时间/ms3 5005.184 900
重复时间/ms952.56119
翻转角度/（°）17010—
切片厚度/mm42.55.5
平均采集次数113
时间/s931965
), ArticleFig(id=1244340221358227786, tenantId=1146029695717560320, journalId=1244295746212642849, articleId=1244340203951866803, language=CN, label=表1, caption=

CT与MRI具体扫描参数

, figureFileSmall=null, figureFileBig=null, tableContent=
磁共振成像T2 TSE3DT1 TSEDWI EPI
回波时间/ms3 5005.184 900
重复时间/ms952.56119
翻转角度/（°）17010—
切片厚度/mm42.55.5
平均采集次数113
时间/s931965
), ArticleFig(id=1244340221525999953, tenantId=1146029695717560320, journalId=1244295746212642849, articleId=1244340203951866803, language=EN, label=Tab.2, caption=

Material and mesh parameters of the knee joint

, figureFileSmall=null, figureFileBig=null, tableContent=
划分方法结构弹性模量/MPa泊松比网格尺寸/mm接触面网格尺寸/mm六面体单元数四面体单元数
CS-FEM骨17 0000.301.01.020 64754 998
半月板590.490.50.5
关节软骨50.460.50.5
韧带215.30.41.01.0
传统FEM骨17 0000.31.01.0090 774
半月板590.490.50.5
关节软骨50.460.50.5
韧带215.30.41.01.0
), ArticleFig(id=1244340221614080343, tenantId=1146029695717560320, journalId=1244295746212642849, articleId=1244340203951866803, language=CN, label=表2, caption=

膝关节的材料与网格参数

, figureFileSmall=null, figureFileBig=null, tableContent=
划分方法结构弹性模量/MPa泊松比网格尺寸/mm接触面网格尺寸/mm六面体单元数四面体单元数
CS-FEM骨17 0000.301.01.020 64754 998
半月板590.490.50.5
关节软骨50.460.50.5
韧带215.30.41.01.0
传统FEM骨17 0000.31.01.0090 774
半月板590.490.50.5
关节软骨50.460.50.5
韧带215.30.41.01.0
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CS-FEM在膝关节有限元模型中的应用
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杨平川 , 潘红玲
应用力学学报 | 生物力学 2025,42(6): 1439-1446
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应用力学学报 | 生物力学 2025, 42(6): 1439-1446
CS-FEM在膝关节有限元模型中的应用
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杨平川 , 潘红玲
作者信息
  • 长沙理工大学体育学院，410114　长沙

通讯作者:

杨平川。E-mail:
Application of CS-FEM to finite element modelling of knee joints
Pingchuan YANG , Hongling PAN
Affiliations
  • Physical Education Institute, Changsha University of Science&Technology, 410114 Changsha, China
出版时间: 2025-12-15 doi: 10.11776/j.issn.1000-4939.2025.06.023
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目的

通过使用光滑有限元法（cell-based smoothed finite element method，CS-FEM）来提高膝关节有限元模型中网格的质量与效率。

方法

在传统有限元法的基础上引入梯度光滑技术，采用基于单元的光滑有限元法（CS-FEM）对膝关节有限元模型进行网格划分，对比传统有限元法与CS-FEM实验数据的精确性。

结果

CS-FEM较传统有限元法更加精确且减少了网格划分的复杂度，同时降低了单元数量，使得求解过程更加高效。

结论

传统有限元法（finite element method，FEM）方法由于其应用广泛和成熟性，在处理相对简单的几何结构和线性材料时具有较高的计算精度和稳定性。然而，在复杂结构和非线性材料的生物力学模拟中，CS-FEM通过优化网格划分和减少应力集中，提供了更高的计算效率和精度。因此，在未来膝关节和其他复杂生物结构的建模中，CS-FEM有望得到更广泛的应用。

膝关节  /  有限元模型  /  CS-FEM网格划分技术
Objective

To improve the quality and efficiency of the mesh in the finite element model of the knee joint by using the cell-based smoothed finite element method（CS-FEM）.

Methods

The gradient smoothing technique is introduced on the basis of the traditional finite element method, and CS-FEM is used to mesh the finite element model of the knee joint, and to compare which of the two methods（traditional finite element method and CS-FEM）is more accurate for the experimental data of the previous study.

Results

CS-FEM is more accurate and reduces the complexity of meshing than the traditional finite element method, and reduces the number of cells, making the solution process more efficient.

Conclusion

The conventional FEM methods offer high computational accuracy and stability when dealing with relatively simple geometries and linear materials due to their wide application and maturity. However, in biomechanical simulation of complex structures and nonlinear materials, CS-FEM provides higher computational efficiency and accuracy by optimizing meshing and reducing stress concentration. Therefore, CS-FEM is expected to be more widely used in the modelling of knee joints and other complex bio-logical structures in the future.

knee joint  /  finite element modelling  /  CS-FEM meshing technique
杨平川, 潘红玲. CS-FEM在膝关节有限元模型中的应用. 应用力学学报, 2025 , 42 (6) : 1439 -1446 . DOI: 10.11776/j.issn.1000-4939.2025.06.023
Pingchuan YANG, Hongling PAN. Application of CS-FEM to finite element modelling of knee joints[J]. Chinese Journal of Applied Mechanics, 2025 , 42 (6) : 1439 -1446 . DOI: 10.11776/j.issn.1000-4939.2025.06.023
正文
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在人体膝关节生物力学分析中，由于膝关节解剖结构复杂，且构成膝关节的骨、软骨、韧带、肌肉和肌腱实体表面往往是不规则的自由曲面，故如何建立合理有效的膝关节有限元模型一直是一项非常重要和复杂的工作[1]。网格单元是将一个复杂的连续体（如膝关节）分割成小的离散单元的基本单元，网格单元质量的好坏将直接决定有限元的计算结果[2]，不合理的网格划分会导致结果的不稳定性和收敛性问题。因此，合适的网格划分对于获得准确结果至关重要[3]。
传统膝关节有限元模型的网格划分方法主要有四面体网格自动划分法和六面体映射网格划分法[4-6]。例如:鲍春雨等[4]通过Geomagic Studio软件将人体膝关节有限元模型划分为六面体（Solid185）。姜亚琼等[5]选择四节点线性四面体单元，网格大小3 mm，对膝关节各部件进行网格划分。张楠等[6]通过HyperMesh软件将排球运动员膝关节有限元模型划分为六面体（Solid185）。传统的网格划分方法具有一定的局限性:①四面体单元计算精度较低[7]；②四面体单元对于复杂的接触条件常伴随计算不收敛，无法满足复杂载荷条件下的力学分析[8]；③六面体单元耗时耗力且对于曲率变化复杂的畸形表面难以实现[9]。
LIU等[10]提出的基于单元的光滑有限元法（cell-based smoothed finite element method，CS-FEM）有效解决了传统网格划分方法的局限性。与传统的有限元方法相比，CS-FEM网格划分方法通过采用切割单元的技术在计算中提供了更大的灵活性和精度，尤其是在处理复杂或不规则域的情况下。使用类似于传统有限元法（finite element method，FEM）的单元网格直接计算形状函数值，并执行光滑操作以获得光滑应变，而无需对形状函数进行微分[11]。其中单元被进一步划分以形成平滑单元，然后可以通过沿平滑单元边界的边界积分沿着进行平滑应变计算[12-13]。在CS-FEM中，光滑Galerkin弱形式与光滑应变场一起使用，因此通常被称为弱化弱列式。从理论上讲，CS-FEM生成的刚度矩阵总是比使用相同底层网格的FEM生成的刚度矩阵“更软”[11，14]。CS-FEM的这一特殊功能引入了许多基本特征[15]，能建立一个上限解[16]甚至接近范数[17]的精确解的模型。由于软化效应，CS-FEM可以处理高扭曲网格和n边形单元。
本研究采用CS-FEM对人体膝关节有限元模型进行网格划分，通过比较两种方法（传统有限元法与CS-FEM）得出哪种方法对于之前研究的实验数据更加精确。
通过采用CS-FEM方法，有限元分析能够更加高效、精确地应对实际工程中的复杂问题，尤其是在模拟不规则几何、动态变化和多物理场耦合时具有明显优势。这使得CS-FEM在许多领域，如生物力学、结构力学、流体力学等，成为了非常重要的工具。
1　CS-FEM算法
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1.1　梯度光滑技术
如图1所示，考虑域Ω，将域Ω分为N个边界为Γp(p=1，2，…N )的不重合光滑域Ωp。每个光滑域Ωp对应一个物质点Xp[18]。
光滑域Ωp内的场变量ui的光滑梯度定义如下
式中，为场变量的梯度（二维问题，i，j＝1，2；三维问题，i，j＝1，2，3），；为光滑后的场变量梯度，；W(X；X-Xp )为光滑函数，。
在光滑域内，光滑函数可表示为
式中，Ap为光滑域的面积（三维问题为光滑域的体积）。
应用散度定理至式（2）可得
式中，n为光滑域边界上Γp的单位外法向量的大小。
将式（3）代入式（4）得到光滑梯度的计算式
梯度光滑的求解方式从域积分转变为边界积分，并且梯度光滑计算只需要场变量本身和光滑域边界的单位外法向量。
1.2　单元刚度矩阵的构造
将问题域Ω离散为N个六面体单元，问题域Ω中某个单元Ωe沿单元边的中点被分割为j个光滑域ΩCj（1≤j≤q），ΩCj⊂Ωe⊂Ω，e为问题域内单元数。Ωe＝ΩC1∪ΩC2∪…∪ΩCq，ΩC1∪ΩC2∪…∪ΩCq＝0。六面体单元划分光滑域示意图如图2所示。
FEM中单元应变ε与节点位移矢量u之间的关系为
式中：BC为单元应变-位移矩阵；x为积分点。
CS-FEM要将式（6）中的BC修正为，包括8个光滑域，其形式为
在三维问题域Ω中，每一个BCi（1≤i≤8）都是6×3的矩阵，可以表示为
式中：Ni(x)为单元形函数；V为光滑域体积；nCx、nCy、nCz为光滑域积分点上的单位外法向量。
将光滑域上一个积分点代入式（8）得
式中：xC为光滑域表面的积分点；b为光滑域上积分点个数；AC为积分点所在光滑域表面面积。
光滑域表面积分点及单位外法向量如图3所示。
将式（8）和式（9）代入式（7）得到BC的表达式。单元刚度矩阵为每个光滑域刚度矩阵求和，即
式中，D为材料矩阵。
2　网格模型验证
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2.1　模型建立
使用Light Speed VCT（GE公司，美国）与MRI Signa 3.0 T（GE公司，美国）电子计算机断层扫描（computerized tomography，CT）和磁共振成像（magnetic resonance imaging，MRI）具体扫描参数（表1）。收集一名受试者右侧膝关节的CT和MRI数据，继而在Mimics软件中，重建出膝关节三维几何模型。将得到的膝关节三维几何模型导入SolidWorks软件，进行逆向建模[4]。通过HyperMesh对膝关节三维模型进行网格划分[6]。
2.2　材料属性与边界载荷
材料定义根据前人研究得出相应膝关节各个结构的材料参数（表2）[4]和约束载荷，完整膝关节的约束和载荷条件:ACL后交叉韧带、内侧副韧带、外侧副韧带在止点处与股骨和胫骨完全固定；胫骨与腓骨相对固定；内侧副韧带与股骨和胫骨内侧光滑接触；股骨近端和中端完全固定[19]；同时有限元模型的网格尺寸（表2）。
2.3　模型验证和对比
对模型进行应变和应力测试，验证模型的有效性。具体的方法为:①完全固定股骨、膝关节处中立位，对胫骨上端施加134 N向前推力[4，19]，观察ACL等效应力。②通过在股骨顶端施加纵向750 N荷载及10 N•m内翻力矩，模拟膝内翻；观察半月板、股骨软骨、胫骨软骨等效应力与最大剪切应力[20]。
CS-FEM网格划分方法模型验证结果显示，方法一:CS-FEM模型验证结果发现胫骨平台平移为4.75 mm（图4），同样的载荷条件下，与鲍春雨等[4]的膝关节模型（4.30 mm）基本一致。且与PEŃA等（4.75 mm）[21]、刘晓敏等（4.83 mm）[22]、GABRIEL等（5.0 mm）[23]、SONG等（4.6 mm）[24]提出的膝关节有限元模型验证结果更加接近。ACL等效应力为18.04~20.29 MPa，与文献[4，19]结果基本一致。方法二:模拟膝内翻时半月板、股骨软骨、胫骨软骨等效应力为:14.93、2.22、2.50 MPa；最大剪切应力为:7.75、1.18、1.41 MPa（图4）。本研究模型与杨骏良模型[20]基本一致。
传统FEM网格划分方法模型验证结果显示，方法一:传统FEM模型验证结果发现胫骨平台平移为5.30 mm（图4），同样的载荷条件下，与CS-FEM模型相差较大。与鲍春雨等（4.30 mm）[4]、PEŃA等（4.75 mm）[21]、刘晓敏等（4.83 mm）[22]、SONG等（4.6 mm）[24]提出的膝关节有限元模型验证结果相差更大。而与GABRIEL等（5.0 mm）[23]相差较小。方法二:模拟膝内翻时半月板、股骨软骨、胫骨软骨等效应力为:12.14、1.78、2.60 MPa；最大剪切应力为:6.94、0.94、1.46 MPa（图4）。本研究模型与杨骏良模型[20]存在较大差异。
因此传统FEM划分的网格质量与CS-FEM有较大差距，从应力、应变结果看CS-FEM与前人研究具有高度相似。
3　讨论
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FEM和CS-FEM是两种在工程和生物力学中广泛应用的数值模拟技术。二者均用于建立复杂结构的仿真模型，如膝关节模型，用于分析生物组织在不同载荷条件下的应力和变形。本研究将从网格划分、计算效率、精度和适用性4个方面对这两种方法进行讨论。
3.1　网格划分
FEM和CS-FEM的核心区别之一在于网格的划分方式和单元类型的选择[25]。传统的FEM通常使用四面体和六面体单元来划分几何结构[7-8]。六面体单元能够提供较高的计算精度，但生成高质量的六面体网格非常复杂[9]。而四面体单元虽然容易生成，但需要更多数量才能达到同等精度[8]。相比之下，CS-FEM通过在传统FEM网格的基础上加入“平滑”技术，允许使用多面体单元进行网格划分。这种方法可以减少所需单元的数量，并且更容易适应几何复杂的区域，从而显著降低计算资源的消耗[18]。本研究的膝关节有限元模型采用CS-FEM通过优化四面体和六面体网格的组合（四面体为54 998个，六面体为20 647个。见表2），使其数量分别控制在合理范围内。采用传统的FEM对膝关节有限元模型进行划分，将其全部划分为四面体（90 774个）。两种方法的单元总数量基本一致，但是传统FEM的精度较CS-FEM差非常多。CS-FEM可以通过优化四面体和六面体网格的组合，使其数量分别控制在合理范围内，而传统的FEM需要更高的单元数才能达到相似的精度。
3.2　计算效率
FEM和CS-FEM在计算效率上的差异也非常明显。传统FEM模型中，采用细密的四面体或六面体单元划分，计算时间和内存消耗会大幅增加。这在生物力学建模中尤其表现突出，如在膝关节的三维动态有限元分析中，模型通常需要数万个单元来描述膝关节在不同角度下的受力情况[26]。CS-FEM则通过平滑技术减少了网格划分的复杂度，同时降低了单元数量，使得求解过程更加高效[27-30]。在一个膝关节有限元模型中，若传统FEM需要2倍或以上的单元来达到CS-FEM相同的精度，则计算时间和内存需求也会呈线性增加。这使得CS-FEM在处理大型生物力学模型时表现出明显的优势[28]。
3.3　计算精度
在计算精度方面，传统FEM依赖于网格的精细度来提高计算结果的准确性，但这种方法存在局限性。即使是高质量的六面体单元，在某些复杂几何形状和非线性材料特性下，其应力和变形计算结果也可能出现误差[31]。CS-FEM通过在有限元单元内部引入平滑函数，将应力和应变的计算更加精确化。这种方法不仅减少了网格依赖性，还可以更好地捕捉膝关节软组织如软骨层和韧带的非线性响应，从而提高整体模型的准确性。特别是在模拟高应力集中区域，如膝关节弯曲运动中的胫骨和股骨接触面，CS-FEM能有效地减少局部应力集中现象，提供更可靠的应力分布图[32-35]。
3.4　适用性
两种方法的适用性也存在差异。传统FEM广泛应用于结构工程和生物力学建模，是研究和应用最成熟的方法之一。它在简单几何结构和线性材料建模中效果尤为突出，如钢板固定系统的有限元仿真[36]。
然而，在涉及复杂几何和生物材料（如软骨、韧带等）的建模中，CS-FEM具有优势。由于其对网格单元类型和数量的优化，CS-FEM能够在复杂的生物结构中更好地表现[37-40]。例如，在髌股关节的受力研究中，CS-FEM可以更精确地模拟髌骨在膝关节屈曲角度变化下的受力情况。此外，在预测膝关节置换前后不同结构受力分布的研究中，CS-FEM也被证明是一种高效且精确的模拟方法。
4　结束语
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FEM和CS-FEM在膝关节建模中的应用各有优劣。传统FEM方法由于其应用的广泛性和成熟性，在处理相对简单的几何结构和线性材料时具有较高的计算精度和稳定性。然而，在复杂结构和非线性材料的生物力学模拟中，CS-FEM通过优化网格划分和减少应力集中，提供了更高的计算效率和精度。因此，在未来膝关节和其他复杂生物结构的建模中，CS-FEM有望得到更广泛的应用。
基金
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  • 国家社科基金资助项目(23BTY121)
文献
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2025年第42卷第6期
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doi: 10.11776/j.issn.1000-4939.2025.06.023
  • 接收时间：2024-12-02
  • 首发时间：2026-03-27
  • 出版时间：2025-12-15
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  • 收稿日期：2024-12-02
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国家社科基金资助项目(23BTY121)
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    长沙理工大学体育学院，410114　长沙

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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
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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