Abstract
Osteoporosis is considered as a major health problem in the world. An understanding of the behavior of human bone under cyclic load requires numerical simulation of the physics. For that purpose, a large scale poroleastic solver is developed based on the mixed finite element method. This approach is free of numerical instabilities yet the discretization leads to an indefinite system that needs special attention. In this work, a comparison is made on several preconditioners that work efficiently in parallel environments.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Adachi, T.J., Kameo, Y., Hojo, M.: Trabecular bone remodelling simulation considering osteocytic response to fluid-induced shear stress. Phil. Trans. R. Soc. A 368, 2669–2682 (2010)
Aguilar, G., Gaspar, G., Lisbona, F., Rodrigo, C.: Numerical stabilization of Biot’s consolidation model by a perturbation on the flow equation. Int. J. Numer. Methods Eng. 75, 1282–1300 (2008)
Arbenz, P., van Lenthe, G.H., Mennel, U., Müller, R., Sala, M.: A scalable multi-level preconditioner for matrix-free μ-finite element analysis of human bone structures. Int. J. Numer. Methods Eng. 73(7), 927–947 (2008)
Benzi, M., Golub, G.H., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–90 (2005)
Biot, M.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941)
Bowen, R.M.: Theory of mixtures. In: Eringen, A.C. (ed.) Continuum Physics, vol. 3, pp. 1–127. Academic Press, New York (1976)
Bundschuh, J., Arriaga, M.C.S.: Introduction to the Numerical Modeling of Groundwater and Geothermal Systems. Taylor and Francis, London (2010)
Coussy, O.: Poromechanics, 2nd edn. Wiley, Chichester (2004)
Coussy, O.: Mechanics and Physics of Porous Solids. Wiley, Chichester (2010)
Cowin, S.C.: Bone poroelasticity. J. Biomech. 32, 217–238 (1999)
Detournay, E., Cheng, D.A.H.: Fundamentals of poroelasticity. In: Fairhurst, C. (ed.) Comprehensive Rock Engineering: Principles, Practice & Projects, vol. 2, pp. 113–171. Pergamon, Oxford (1993)
Ferronato, M., Castelletto, N., Gambolati, G.: A fully coupled 3-D mixed finite element model of Biot consolidation. J. Comput. Phys. 229, 4813–4830 (2010)
Flaig, C., Arbenz, P.: A scalable memory efficient multigrid solver for micro-finite element analyses based on CT images. Parallel Comput. 37(12), 846–854 (2011)
Gambolati, G., Ferronato, M., Janna, C.: Preconditioners in computational geomechanics: A survey. Int. J. Numer. Anal. Meth. Geomech. 35, 980–996 (2011)
Girault, V., Raviart, P.A.: Finite Element Approximation of the Navier–Stokes Equations. Springer, Berlin (1979)
Heroux, M.A., Bartlett, R.A., Howle, V.E., Hoekstra, R.J., Hu, J.J., Kolda, T.G., Lehoucq, R.B., Long, K.R., Pawlowski, R.P., Phipps, E.T., Salinger, A.G., Thornquist, H.K., Tuminaro, R.S., Willenbring, J.M., Williams, A., Stanley, K.S.: An overview of the Trilinos project. ACM Trans. Math. Softw. 31(3), 397–423 (2005)
International Osteoporosis Organization (2012), http://www.iofbonehealth.org
Lipnikov, K.: Numerical Methods for the Biot Model in Poroelasticity. Ph.D. thesis, University of Houston (2002)
Murphy, M.F., Golub, G.H., Wathen, A.J.: A note on preconditioning for indefinite linear systems. SIAM J. Matrix Anal. Appl. 21(6), 1969–1972 (2000)
Paige, C.C., Saunders, M.A.: Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal. 12, 617–629 (1975)
METIS-Family of Multilevel Partitioning Algorithms, http://glaros.dtc.umn.edu/gkhome/views/metis
Saad, Y.: Iterative Methods for Sparse Linear Systems, 2nd edn. SIAM, Philadelphia (2003)
Sleijpen, G.L.G., van der Vorst, H.A., Modersitzki, J.: Differences in the effects of rounding errors in Krylov solvers for symmetric indefinite linear systems. SIAM J. Matrix Anal. Appl. 22(3), 726–751 (2001)
The Trilinos Project Home Page, http://trilinos.sandia.gov/
Truty, A.: A Galerkin/least-squares finite element formulation for consolidation. Int. J. Numer. Methods Eng. 52, 763–786 (2001)
Wang, H.F.: Theory of Linear Poroelasticity with Applications to Geomechanics and Hydrogeology. Princeton University Press, Princeton (2000)
White, J.A., Borja, R.I.: Stabilized low-order finite elements for coupled solid-deformation/fluid-diffusion and their application to fault zone transients. Comput. Methods Appl. Mech. Engrg. 197, 4353–4366 (2008)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Arbenz, P., Turan, E. (2013). Preconditioning for Large Scale Micro Finite Element Analyses of 3D Poroelasticity. In: Manninen, P., Öster, P. (eds) Applied Parallel and Scientific Computing. PARA 2012. Lecture Notes in Computer Science, vol 7782. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36803-5_26
Download citation
DOI: https://doi.org/10.1007/978-3-642-36803-5_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36802-8
Online ISBN: 978-3-642-36803-5
eBook Packages: Computer ScienceComputer Science (R0)