Multiquadric method for the numerical solution of a biphasic mixture model. (English) Zbl 0910.76059

Summary: A computational algorithm based on the multiquadric method has been devised to solve the biphasic mixture model. The model includes a set of constitutive equations for the fluid flows through the solid phase, a set of momentum equations for stress-strain equilibrium, and continuity equations for the solid phase and the fluid phase. The numerical method does not require the generation of mesh as in the finite element method and hence gives high flexibility in applying the method to irregular geometry. Numerical examples are made to compute the solution of the confined compression problem which approximates the nonlinear response of soft hydrated tissues under external loadings.


76M25 Other numerical methods (fluid mechanics) (MSC2010)
76T99 Multiphase and multicomponent flows
76Z05 Physiological flows
Full Text: DOI


[1] Spilker, R. L.; Suh, J. K., Formulation and evaluation of a finite element model for the biphasic model of hydrated soft tissues, Comp. Struct., 35, 425-439 (1990) · Zbl 0727.73061
[2] Suh, J. K.; Spilker, R. L.; Holmes, M. H., A penalty finite element analysis for nonlinear mechanics of biphasic hydrated soft tissues under large deformation, Int. J. Num. Meth., 32, 1411-1439 (1991), in Eng. · Zbl 0763.73057
[3] Torzilli, P. A.; Mow, V. C., On the fundamental fluid transport mechanisms through normal and pathological articular cartilage, Biomech., 9, 541-552 (1976)
[4] Mow, V. C., The role of lubrication in biomechanical joints, ASME J. Lubr. Technol. Trans., 91, 320-329 (1969)
[5] Simon, B. R.; Wu, J. S.S.; Carlton, M. W.; Evans, J. H.; Kazarian, L. E., Structural models for human spinal motion segments based on a poroelastic view of the intervertebral disk, ASME J. Biomech. Eng., 107, 327-335 (1985)
[6] Salzetein, R. A.; Pollack, S. R.; Mak, A. F.T.; Petron, N.; Brankov, G., Electromechanical potentials in cortical bone, Part 1: A continuum approach, Biomech., 20, 681-692 (1987)
[7] Taber, L. A.; Yang, M.; Keller, B. B.; Clark, E. B., A poroelastic model for the trabecular embryonic heart, (Vanderby, R., Advances in Bioengineering (1992), ASME), 623
[8] Kenyon, D. E., A mathematical model of water flux through aortic tissue, Bull. Math. Biol., 41, 79-90 (1979)
[9] Oomens, C. W.J.; VanCampen, D. H.; Grootenboer, H. J., A mixture approach to the mechanics of skin, Biomech., 20, 877-885 (1982)
[10] Mak, A. F.T.; Huang, L.; Wang, Q., A biphasic poroelastic analysis of the flow dependent subcutaneous tissue pressure and compaction due to epidermal loadings: issues in pressure core, ASME J. Biomech. Eng., 116, 421-429 (1994)
[11] Wayne, J. S.; Woo, S. L.Y.; Kwan, M., Finite element analysis of repaired articular surfaces, Eng. Medicine, 205, 155-162 (1992)
[12] Hon, Y. C.; Huang, D. T.; Lu, M. W.; Mak, A. F.T.; Xue, W. M., A new penalty formulation for the numerical solution of biphasic mixture model, J. Neural, Parallel & Scientific Computations, 4, 465-474 (1996)
[13] Hardy, R. L., Multiquadric equations of topography and other irregular surfaces, Geophys. Res., 176, 1905-1915 (1971)
[14] Franke, R., Scattered data interpolation: test of some methods, Math. Comput., 38, 181-200 (1982) · Zbl 0476.65005
[15] Kansa, E. J., Multiquadrics—a scattered data approximation scheme with applications to computational fluid dynamics—II. Solution to parabolic, hyperbolic and elliptic partial differential equations, Computers Math. Applic., 19, 147-161 (1990) · Zbl 0850.76048
[16] Mow, V. C.; Holmes, M. H.; Lai, W. M., Fluid transport and mechanical properties of articular cartilage: A review, Biomech., 17, 377-394 (1984)
[17] Tarwater, A. E., A parameter study of Hardy’s multiquadric method for scattered data interpolation, UCRL-54670 (Sept. 1985)
[18] Madych, W. R., Miscellaneous error bounds for multiquadric and related interpolators, Computers Math. Applic., 24, 121-138 (1992) · Zbl 0766.41003
[19] Golberg, M. A.; Chen, C. S., On a method of Atkinson for evaluating domain integrals in the boundary element method, Appl. Math. Comput., 60, 125-138 (1994) · Zbl 0797.65090
[20] Bogomonly, A., Fundamental solutions method for elliptic boundary value problems, SIAM J. Numer. Anal., 22, 644-669 (1985)
[21] Cheng, R. S.C., Delta-trigonomeric and spline methods using the single-layer potential representation, (Ph.D. Thesis (1987), Univ. of Maryland)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.