A Lagrange multiplier mixed finite element formulation for three-dimensional contact of biphasic tissues

Journal of Biomechanical Engineering

Volumn 129, Issue 3, 2007, Pages 457-471

  Yang, Taiseung ^a   Spilker, Robert L ^a  

^a RENSSELAER POLYTECHNIC INSTITUTE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

BIPHASIC TISSUES; FLUID PHASES; MIXTURE THEORY; TISSUE CONTACT ANALYSIS;

BONE; COMPRESSIVE STRESS; FINITE ELEMENT METHOD; JOINTS (ANATOMY); LAGRANGE MULTIPLIERS; THREE DIMENSIONAL;

TISSUE;

ALGORITHM; ANALYTIC METHOD; ARTICLE; CALCULATION; CARTILAGE; COMPRESSION; CONTACT ANGLE; FINITE ELEMENT ANALYSIS; FLUID FLOW; HUMERUS; HYDRATION; LAGRANGE MULTIPLIER METHOD; LIQUID; MATHEMATICAL ANALYSIS; PRESSURE; SHOULDER; SOFT TISSUE; SOLID; STRESS; TENSILE STRENGTH; VALIDATION PROCESS; VELOCITY;

ACCELERATION; ALGORITHMS; COMPUTER SIMULATION; FINITE ELEMENT ANALYSIS; HUMANS; HUMERUS; MODELS, BIOLOGICAL; PRESSURE; REPRODUCIBILITY OF RESULTS; SHOULDER JOINT; STRESS, MECHANICAL; SURFACE PROPERTIES; VISCOSITY;

EID: 34447515217     PISSN: 01480731     EISSN: None     Source Type: Journal    
DOI: 10.1115/1.2737056     Document Type: Article

References (50)

1. Mow, V. C., Holmes, M. H., and Lai, W. M., 1984, "Fluid Transport and Mechanical Properties of Articular Cartilage: A Review," J. Biomech., 17(5), pp. 377-394.
2. Woo, S. L.-Y., Mow, V. C., and Lai, W. M., 1987, "Biomechanical Properties of Articular Cartilage," Handbook of Bioengineering, Skalak, R., and Chien, S., eds., McGraw-Hill, New York, pp. 4.1-4.44.
3. Soslowsky, L. J., Ateshian, G. A., and Mow, V. C., 1992, "Quantitation of In Situ Contact Areas at the Glenohumeral Joint: A Biomechanical Study," J. Orthop. Res., 10, pp. 524-534.
4. Mow, V. C., Kuei, S. C., Lai, W. M., and Armstrong, C. G., 1980, "Biphasic Creep and Stress Relaxation of Articular Cartilage in Compression: Theory and Experiments," ASME J. Biomech. Eng., 102, pp. 73-84.
5. Truesdell, C., and Toupin, R. A., 1960, "The Classical Field Theories," Handbuck der physik, Flügge, S., editor, Springer-Verlag, Berlin.
6. Armstrong, C. G., Lai, W. M., and Mow, V. C., 1984, "An Analysis of the Unconfined Compression of Articular Cartilage," ASME J. Biomech. Eng., 106, pp. 165-173.
7. Mow, V. C., Kwan, M. K., Lai, W. M., and Holmes, M. H., 1986, "A Finite Deformation Theory for Nonlinearly Permeable Cartilage and Other Soft Hydrated Connective Tissues," Frontiers in Biomechanics, Woo, S. L.-Y., Schmid-Schonbein, G., and Zweifach, B., eds., Springer-Verlag, Berlin, pp. 153-179.
8. Kwan, M. K., Lai, W. M., and Mow, V. C., 1990, "A Finite Deformation Theory for Cartilage and Other Soft Hydrated Connective Tissues," J. Biomech., 23, pp. 145-155.
9. Holmes, M. H., 1985, "A Theoretical Analysis for Determining the Nonlinear Hydraulic Permeability of a Soft Tissue From a Permeation Experiment," Bull. Math. Biol., 47(5), pp. 669-683.
10. Lai, W. M., Mow, V. C., and Roth, V., 1981, "Effects of Nonlinear Strain-Dependent Permeability and Rate of Compression on the Stress Behavior of Articular Cartilage," ASME J. Biomech. Eng., 103, pp. 61-66.
11. Mak, A. F., 1986, "The Apparent Viscoelastic Behavior of Articular Cartilage - The Contributions From the Intrinsic Matrix Viscoelasticity and Interstitial Fluid Flows," ASME J. Biomech. Eng., 108, pp. 123-130.
12. Cohen, B., 1992, Anisotropic Hydrated Soft Tissues in Finite Deformation and the Biomechanics of the Growth Plates in Mechanical Engineering, Columbia University, New York.
13. Lai, W. M., Hou, J. S., and Mow, V. C., 1991, "A Triphasic Theory for the Swelling and Deformational Behaviors of Articular Cartilage," ASME J. Biomech. Eng., 113(3), pp. 245-258.
14. Huyghe, J. M., and Janssen, J. D., 1997, "Quadriphasic Mechanics of Swelling Incompressible Porous Media," Int. J. Eng. Sci., 35(8), pp. 793-802.
15. Eberhardt, A. W., Lewis, J. L., and Keer, L. M., 1991, "Normal Contact of Elastic Spheres With Two Elastic Layers as a Model of Joint Articulation," ASME J. Biomech. Eng., 113, pp. 410-417.
16. Ateshian, G. A., Lai, W. M., Zhu, W. B., and Mow, V. C., 1994, "An Asymptotic Solution for the Contact of Two Biphasic Cartilage Layers," J. Biomech., 27(11), pp. 1347-1360.
17. Hou, J. S., Holmes, M. H., Lai, W. M., and Mow, V. C., 1989, "Boundary Conditions at the Cartilage-Synovial Fluid Interface for Joint Lubrication and Theoretical Verifications," ASME J. Biomech. Eng., 111(1), pp. 78-87.
18. Donzelli, P. S., 1995, A Mixed-Penalty Contact Finite Element Formulation for Biphasic Soft Tissues in Mechanical Engineering, Aeronautical Engineering and Mechanics, Rensselaer Polytechnic Institute, Troy, NY.
19. Almeida, E. S., and Spilker, R. L., 1997, "Mixed and Penalty Finite Element Models for the Nonlinear Behavior of Biphasic Soft Tissues in Finite Deformation - Part I: Alternate Formulations," Comput. Methods Biomech. Biomed. Eng., 1, pp. 25-46.
20. Almeida, E. S., and Spilker, R. L., 1998, "Mixed and Penalty Finite Element Models for the Nonlinear Behavior of Biphasic Soft Tissues in Finite Deformation - Part II. Nonlinear Examples," Comput. Methods Biomech. Biomed. Eng., 1, pp. 151-170.
21. Spilker, R. L., and Suh, J.-K., 1990, "Formulation and Evaluation of a Finite Element Model of Soft Hydrated Tissue," Comput. Struct., 35(4), pp. 425-439.
22. Suh, J.-K., Spilker, R. L., and Holmes, M. H., 1991, "A Penalty Finite Element Analysis for Nonlinear Mechanics of Biphasic Hydrated Soft Tissue Under Large Deformation," Int. J. Numer. Methods Eng., 32, pp. 1411-1439.
23. Spilker, R. L., and Maxian, T. A., 1990, "A Mixed-Penalty Finite Element Formulation of the Linear Biphasic Theory for Soft Tissues," Int. J. Numer. Methods Eng., 30, pp. 1063-1082.
24. Oomens, C. W. J., Van Campen, D. H., and Grootenboer, H. J., 1987, "A Mixture Approach to the Mechanics of Skin," J. Biomech., 20(9), pp. 877-885.
25. Wayne, J. S., Woo, S. L.-Y., and Kwan, M. K., 1991, "Application of the u-p Finite Element Method to the Study of Articular Cartilage," ASME J. Biomech. Eng., 113(4), pp. 397-403.
26. Almeida, E. S., 1995, Finite Element Formulations for Biological Soft Hydrated Tissues Under Finite Deformation, in Mechanical Engng, Aeronautical Engng and Mechanics, Rensselaer Polytechnic Institute, Troy, NY.
27. Vermilyea, M. E., and Spilker, R. L., 1992, "A Hybrid Finite Element Formulation of the Linear Biphasic Equations for Soft Hydrated Tissues," Int. J. Numer. Methods Eng., 33, pp. 567-594.
28. Mish, K. D., Herrmann, L. R., and Muraleetharan, K., 1992, "A Comparison of Biot Formulation Finite Element Models for Two- and Three-Dimensional Transient Soil Problems," Symposium on Computational Mechanics of Porous Materials and Their Thermal Decomposition, Salamon, N. J., and Sullivan, R. M., eds., ASME, New York, pp. 69-79.
29. Simon, B. R., Wu, J. S. S., Zienkiewicz, O. C., and Paul, D. K., 1986, "Evaluation of U-W and U-P Finite Element Methods for the Dynamic Response of Saturated Porous Media Using One-Dimensional Models," Int. J. Numer. Analyt. Meth. Geomech., 10, pp. 483-499.
30. Luenberger, D. G., 1984, Linear and Nonlinear Programming, 2nd ed., Addison-Wesley, Reading, MA.
31. Lipshitz, H., and Glimcher, M. J., 1979, "In Vitro Studies of the Wear of Articular Cartilage - II. Characteristics of the Wear of Articular Cartilage When Worn Against Stainless Steel Plates Having Characterized Surfaces," Wear, 52, pp. 297-339.
32. Lai, W. M., and Mow, V. C., 1980, "Drag-Induced Compression of Articular Cartilage During a Permeation Experiment," Biorheology, 17, pp. 111-123.
33. Yang, T. S., 2003, A Three-Dimensional Biphasic Finite Element Contact Formulation for Hydrated Soft Tissue, in Mechanical, Aeronautical and Nuclear Engineering, Rensselaer Polytechnic Institute, Troy, NY, p. 164.
34. Donzelli, P. S., and Spilker, R. L., 1998, "A Contact Finite Element Formulation for Biological Soft Hydrated Tissues," Comput. Methods Appl. Mech. Eng., 153, pp. 63-79.
35. Yang, T. S., and Spilker, R. L., 2006, "A Patch Test for a Mixed Finite Element Approach for Three-Dimensional Contact of Biphasic Tissues," ASME J. Biomech. Eng. (in review).
36. Taylor, R. L., and Papadopoulos, P., 1991, "On a Patch Test for Contact Problems in Two Dimensions," Computational Methods in Nonlinear Mechanics, Wriggers, P., and Wagner, W., eds., Springer, New York, pp. 690-702.
37. Hughes, T. J. R., 1987, The Finite Element Method: Linear Static and Dynamic Analysis, Prentice-Hall, Englewood Cliffs, p. 803.
38. Zienkiewicz, O. C., and Taylor, R. L., 1989, The Finite Element Method, 4th ed., McGraw-Hill, New York, Vol. I, p. 648.
39. Heinstein, M. W., Attaway, S. W., Swegle, J. W., and Mello, F. J., 1993, "A General-Purpose Contact Detection Algorithm for Nonlinear Structural Analysis Codes," Sandia National Laboratories: Albuquerque.
40. Oldenburg, M., and Nilsson, L., 1994, "The Position Code Algorithm for Contact Searching," Int. J. Numer. Methods Eng., 37, pp. 359-386.
41. Yang, T. S., and Spilker, R. L., 2007, "A Study of Preconditioned Krylov Subspace Methods With Reordering for Linear Systems From a Biphasic υ-p Finite Element Formulation," Comput. Methods Biomech. Biomed. Eng., 10(1), pp. 13-24.
42. Mak, A. F., Lai, W. M., and Mow, V. C., 1987, "Biphasic Indentation of Articular Cartilage - Part 1: Theoretical Analysis," J. Biomech., 20, pp. 703-714.
43. Spilker, R. L., Suh, J. K., and Mow, V. C., 1992, "A Finite Element Analysis of the Indentation Stress-Relaxation Response of Linear Biphasic Articular Cartilage," ASME J. Biomech. Eng., 114, pp. 191-201.
44. Suh, J.-K., and Spilker, R. L., 1994, "Indentation Analysis of Biphasic Articular Cartilage: Nonlinear Phenomena Under Finite Deformation," ASME J. Biomech. Eng., 116, pp. 1-9.
45. Kelkar, R., and Ateshian, G. A., 1995, "Contact Creep Response Between a Rigid Impermeable Cylinder and a Biphasic Cartilage Layer Using Integral Transforms, 1995 Bioengineering Conference," Hochmuth, R. M., Langrana, N. A., and Hefzy, M. S., eds., ASME, New York.
46. Kelkar, R., and Ateshian, G. A., 1999, "Contact Creep of Biphasic Cartilage Layers," ASME J. Biomech. Eng., 66, pp. 137-145.
47. Soslowsky, L. J., Ateshian, G. A., and Mow, V. C., 2001, "Stereophotogrammetric Determination of Joint Anatomy and Contact Areas," Biomechanics of Diarthrodial Joints, Mow, V. C., Ratcliffe, T. A., and Woo, S. L.-Y., eds., Springer-Verlag, Berlin, pp. 243-268.
48. Dunbar, W. L., Un, K., Donzelli, P. S., and Spilker, R. L., 2001, "An Evaluation of Three Dimensional Diarthrodial Joint Contact Using Penetration Data and the Finite Element Method," ASME J. Biomech. Eng., 123, pp. 333-340.
49. Un, K., and Spilker, R. L., 2006, "A Penetration-Based Finite Element Method for Hyperelastic 3-D Biphasic Tissues in Contact - Part I: Derivation of Contact Boundary Conditions," ASME J. Biomech. Eng., 128, pp. 124-130.
50. On, K., and Spilker, R. L., 2006, "A Patch Test for a Mixed Finite Element Approach for Three-Dimensional Contact of Biphasic Tissues," ASME J. Biomech. Eng. (in review).