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Journal of Computational and Nonlinear Dynamics | Volume 5 | Issue 3 | Research Paper
Research Papers

ANCF Finite Element/Multibody System Formulation of the Ligament/Bone Insertion Site Constraints

[+] Author and Article Information
Florentina M. Gantoi

Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, 842 West Taylor Street, Chicago, IL 60607-7022fganto2@uic.edu

Michael A. Brown

Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, 842 West Taylor Street, Chicago, IL 60607-7022mabrown1@uic.edu

Ahmed A. Shabana

Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, 842 West Taylor Street, Chicago, IL 60607-7022shabana@uic.edu

J. Comput. Nonlinear Dynam 5(3), 031006 (May 14, 2010) (9 pages) doi:10.1115/1.4001373 History: Received July 09, 2009; Revised October 19, 2009; Published May 14, 2010; Online May 14, 2010
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References
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Citing Articles

The focus of this investigation is to study the mechanics of the knee joint using new ligament/bone insertion site constraint models that require the integration of multibody system and large displacement finite element algorithms. Two different sets of clamped end conditions at the ligament/bone insertion site are examined using nonlinear large displacement absolute nodal coordinate formulation (ANCF) finite elements. The first set of end conditions, called the partially clamped joint, eliminates only the translations and rotations at a point, allowing for the cross section stretch and shear at the ligament/bone connection. The second joint, called the fully clamped joint, eliminates all the translation, rotation, and deformation degrees of freedom of the cross section at the ligament/bone insertion site. In the case of the fully clamped joint, the gradient vectors do not change their length and orientation, allowing for the use of the constant strain assumptions. The partially clamped joint, on the other hand, allows for the change in length and relative orientation of the gradient vectors at the bone/ligament insertion site, leading to the cross section deformation induced by knee movements. Nanson’s formula is applied as a measure of the deformation of the cross section in the case of the partially clamped joint. In this study, the major bones in the knee joint consisting of the femur, tibia, and fibula are modeled as rigid bodies while the ligaments structures are modeled using the large displacement ANCF finite elements. Two different ANCF finite element models are developed in this investigation: the first model employs the fully parameterized three-dimensional beam element while the second model employs the three-dimensional cable element. The three-dimensional fully parameterized beam element allows for a straight forward implementation of a neo-Hookean constitutive model that can be used to accurately predict the large displacement as experienced in knee flexation and rotation. At the ligament bone insertion site, the ANCF fully parameterized beam element is used to define a fully or partially constrained joint while the ANCF cable element can only be used to define one joint type. The fully and partially clamped joint constraints are satisfied at the position, velocity, and acceleration levels using a dynamic formulation that is based on an optimum sparse matrix structure. The approach described in this investigation can be used to develop more realistic models of the knee and is applicable to future research studies on ligaments, muscles, and soft tissues. In particular, the partially clamped joint representation of the ligament/bone insertion site constraints can be used to develop improved structural mechanics models of the knee.
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References

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2
Silva, J. F., and Flores, P., 2007, “Dynamic Modeling and Analysis of the Knee Joint,” "Proceedings of the National Conference on the Dynamics of Multibody Systems", P.Flores and M.Silva, eds., Actas da DSM2007: National Conference on Multibody System Dynamics, Guimaraes, Portugal, Dec. 6–7, p. 88.
3
Yamaguchi, G., 2001, "Dynamic Modeling of Musculoskeletal Motion", Kluwer Academic, Dordrecht, The Netherlands.
4
Weed, D., Maqueda, L. G., Brown, M., and Shabana, A. A., 2008, “A Multibody/Finite Element Nonlinear Formulation of a Two-Ligament Knee Joint,” Proceedings of the 2008 ASME International Mechanical Engineering Congress and Exhibition , Boston, MA, Oct. 31–Nov. 6.
5
Sugiyama, H., Escalona, J. L., and Shabana, A. A., 2003, “Formulation of Three-Dimensional Joint Constraint Using the Absolute Nodal Coordinates,” Nonlinear Dyn., 31 , pp. 167–195.  [CrossRef]
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Gerstmayr, J., and Shabana, A. A., 2006, “Analysis of Thin Beams and Cables Using the Absolute Nodal Coordinate Formulation,” Nonlinear Dyn., 45 , pp. 109–130.  [CrossRef]
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Hussein, B. A., Weed, D., and Shabana, A. A., 2009, “Clamped End Conditions and Cross Section Deformation in the Finite Element Absolute Nodal Coordinate Formulation,” Multibody Syst. Dyn., 21 (4), pp. 375–393.  [CrossRef]
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Garcia-Vallejo, D., Mayo, J., Escalona, J. L., and Dominguez, J., 2007, “Modeling Three-Dimensional Rigid-Flexible Multibody Systems by Using Absolute Coordinates,” 12th IFToMM World Congress , Besancon, Jun. 18–21.
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García-Vallejo, D., Mayo, J., and Escalona, J. L., 2008, “Three-Dimensional Formulation of Rigid-Flexible Multibody Systems With Flexible Beam Elements,” Multibody Syst. Dyn., 20 (1), pp. 1–28.  [CrossRef]
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Benjamin, M., Toumi, H., Ralphs, J. R., Bydder, G., Best, T. M., and Milz, S., 2006, “Where Tendons and Ligaments Meet Bone: Attachment Sites (‘Entheses’) in Relation to Exercise and/or Mechanical Load,” J. Anat., 208 , pp. 471–490.  [CrossRef]
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LaPrade, R. F., Engebretsen, A. H., Thuan, V. L., Johansen, S., Wentorf, F. A., and Engebretsen, L., 2007, “The Anatomy of the Medial Part of the Knee,” J. Bone Jt. Surg., 89 , pp. 2000–2010.  [CrossRef]
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Meister, B. R., Michael, S. P., Moyer, R. A., Kelly, J. D., and Schneck, C. D., 2000, “Anatomy and Kinematics of the Lateral Collateral Ligament of the Knee,” Am. J. Sports Med., 28 , pp. 869–878.
22
Pedowitz, R. A., O'Connor, J. J., and Akeson, W. H., 2003, "Daniel's Knee Injuries: Ligament and Cartilage, Structure, Function, and Repair", 2nd ed., Lippincott Williams & Willkins, Baltimore, MD.

Figures

Grahic Jump Location
+Fully parameterized ANCF beam element
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Fully parameterized ANCF beam element
Figure 1

Fully parameterized ANCF beam element

Grahic Jump Location
+Anatomy of the knee (http://www.aclsolutions.com/anatomy.php)
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Anatomy of the knee (http://www.aclsolutions.com/anatomy.php)
Figure 2

Anatomy of the knee (http://www.aclsolutions.com/anatomy.php)

Grahic Jump Location
+ANCF joint coordinate systems: (a) gradients, (b) tangent frame, and (c) cross section frame
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ANCF joint coordinate systems: (a) gradients, (b) tangent frame, and (c) cross section frame
Figure 3

ANCF joint coordinate systems: (a) gradients, (b) tangent frame, and (c) cross section frame

Grahic Jump Location
+Coordinate systems used in the joint formulation
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Coordinate systems used in the joint formulation
Figure 4

Coordinate systems used in the joint formulation

Grahic Jump Location
+Knee joint model
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Knee joint model
Figure 5

Knee joint model

Grahic Jump Location
+Normal strain ε22 at the midpoint (—◼— LCL partially clamped, —●— MCL partially clamped, —★— LCL fully clamped, —▼— MCL fully clamped)
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Normal strain ε22 at the midpoint (—◼— LCL partially clamped, —●— MCL partially clamped, —★— LCL fully clamped, —▼— MCL fully clamped)
Figure 10

Normal strain ε22 at the midpoint (—◼— LCL partially clamped, —●— MCL partially clamped, —★— LCL fully clamped, —▼— MCL fully clamped)

Grahic Jump Location
+The strain ε33 at LCL, MCL/tibia insertion site (—◼— LCL partially clamped, —●— MCL partially clamped, —★— LCL fully clamped, —▼— MCL fully clamped)
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The strain ε33 at LCL, MCL/tibia insertion site (—◼— LCL partially clamped, —●— MCL partially clamped, —★— LCL fully clamped, —▼— MCL fully clamped)
Figure 13

The strain ε33 at LCL, MCL/tibia insertion site (—◼— LCL partially clamped, —●— MCL partially clamped, —★— LCL fully clamped, —▼— MCL fully clamped)

Grahic Jump Location
+Length in the case of the fully clamped joint (—◼— LCL beam elements, —●— MCL beam elements, —★— LCL cable elements, —▼— MCL cable elements)
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Length in the case of the fully clamped joint (—◼— LCL beam elements, —●— MCL beam elements, —★— LCL cable elements, —▼— MCL cable elements)
Figure 6

Length in the case of the fully clamped joint (—◼— LCL beam elements, —●— MCL beam elements, —★— LCL cable elements, —▼— MCL cable elements)

Grahic Jump Location
+Length in the case of the partially clamped joint (—◼— LCL beam elements, —●— MCL beam elements, —★— LCL cable elements, —▼— MCL cable elements)
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Length in the case of the partially clamped joint (—◼— LCL beam elements, —●— MCL beam elements, —★— LCL cable elements, —▼— MCL cable elements)
Figure 7

Length in the case of the partially clamped joint (—◼— LCL beam elements, —●— MCL beam elements, —★— LCL cable elements, —▼— MCL cable elements)

Grahic Jump Location
+Axial strain of the midpoint in the case of the fully clamped joint (—◼— LCL beam elements, —●— MCL beam elements, —★— LCL cable elements, —▼— MCL cable elements)
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Axial strain of the midpoint in the case of the fully clamped joint (—◼— LCL beam elements, —●— MCL beam elements, —★— LCL cable elements, —▼— MCL cable elements)
Figure 8

Axial strain of the midpoint in the case of the fully clamped joint (—◼— LCL beam elements, —●— MCL beam elements, —★— LCL cable elements, —▼— MCL cable elements)

Grahic Jump Location
+Axial strain of the midpoint in the case of the partially clamped joint (—◼— LCL beam elements, —●— MCL beam elements, —★— LCL cable elements, —▼— MCL cable elements)
View Large  |  Save Figure  |  Download Slide (.ppt)
Axial strain of the midpoint in the case of the partially clamped joint (—◼— LCL beam elements, —●— MCL beam elements, —★— LCL cable elements, —▼— MCL cable elements)
Figure 9

Axial strain of the midpoint in the case of the partially clamped joint (—◼— LCL beam elements, —●— MCL beam elements, —★— LCL cable elements, —▼— MCL cable elements)

Grahic Jump Location
+The strain ε22 at LCL, MCL/tibia insertion site (—◼— LCL partially clamped, —●— MCL partially clamped, —★— LCL fully clamped, —▼— MCL fully clamped)
View Large  |  Save Figure  |  Download Slide (.ppt)
The strain ε22 at LCL, MCL/tibia insertion site (—◼— LCL partially clamped, —●— MCL partially clamped, —★— LCL fully clamped, —▼— MCL fully clamped)
Figure 11

The strain ε22 at LCL, MCL/tibia insertion site (—◼— LCL partially clamped, —●— MCL partially clamped, —★— LCL fully clamped, —▼— MCL fully clamped)

Grahic Jump Location
+Normal strain ε33 at the midpoint (—◼— LCL partially clamped, —●— MCL partially clamped, —★— LCL fully clamped, —▼— MCL fully clamped)
View Large  |  Save Figure  |  Download Slide (.ppt)
Normal strain ε33 at the midpoint (—◼— LCL partially clamped, —●— MCL partially clamped, —★— LCL fully clamped, —▼— MCL fully clamped)
Figure 12

Normal strain ε33 at the midpoint (—◼— LCL partially clamped, —●— MCL partially clamped, —★— LCL fully clamped, —▼— MCL fully clamped)

Grahic Jump Location
+Deformation of the cross section area at the midpoint (—◼— LCL partially clamped, —●— MCL partially clamped, —★— LCL fully clamped, —▼— MCL fully clamped)
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Deformation of the cross section area at the midpoint (—◼— LCL partially clamped, —●— MCL partially clamped, —★— LCL fully clamped, —▼— MCL fully clamped)
Figure 14

Deformation of the cross section area at the midpoint (—◼— LCL partially clamped, —●— MCL partially clamped, —★— LCL fully clamped, —▼— MCL fully clamped)

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