Keywords: biological soft tissues, FETI, parallel computing, transverse isotropy, polyconvex, domain decomposition.

This paper shows the modeling of biological soft tissues as they appear in arterial walls and the three-dimensional simulation based on the FETI-DP algorithm using a transversely isotropic material model. Such biological soft tissues are characterized by a nearly incompressible, anisotropic hyperelastic material behavior in the physiological range of deformations. For the representation of such materials we apply a polyconvex strain energy function, cf. [1], in order to ensure the existence of minimizers [2] and in order to satisfy the Legendre-Hadamard condition automatically. To account for the anisotropy the concept of structural tensors and representation theorems for anisotropic tensor functions are used and the energy is formulated in terms of the basic and mixed invariants of the deformation and structural tensor.

The three-dimensional discretization results in large systems of equations, therefore a parallel algorithm is applied to solve the equilibrium problem. Domain decomposition methods like the FETI-DP (Dual-Primal Finite Element Tearing and Interconnecting [4]) method are designed to solve large linear equation systems that arise from the discretization of partial differential equations on parallel computers. Their numerical and parallel scalabilty, as well as their robustness, also in the incompressible limit, has been shown theoretically e.g. [5] and in numerical simulation, e.g. [6]. At the end of this paper we show some results of preliminary numerical simulations using the dual-primal FETI method. There, we apply our anisotropic material model to run unaxial tension tests of soft biological tissues.

1

Balzani, D., Neff, P., Schröder, J., Holzapfel, G.A., "A Polyconvex Framework for Soft Biological Tissues. Adjustment to Experimental Data", Int. J. Solids Struct, 43/20, 6052-6070, 2006. doi:10.1016/j.ijsolstr.2005.07.048

2

Schröder, J., Neff, P., "Invariant Formulation of Hyperelastic Transverse Isotropy Based on Polyconvex Free Energy Functions", Int. J. Solids Struct, 40, 401-445, 2003. doi:10.1016/S0020-7683(02)00458-4

3

Balzani, D., "Polyconvex Anisotropic Energies and Modeling of Damage Applied to Arterial Walls", PhD-Thesis, Scientific Report of the Institute of Mechanics, University Duisburg-Essen, Verlag Glückauf Essen 2006.

4

Farhat, C., Lesoinne, M., LeTallec, P., Pierson, K.H., Rixen, D., "FETI-DP: A dual-primal unified FETI method - Part I: A faster alternative to the two-level FETI method", Internat. J. Numer. Methods Engrg, 50, 1523-1544, 2001. doi:10.1002/nme.76

5

Klawonn, A., Widlund, O.B., "Dual-Primal FETI Methods for Linear Elasticity", Comm. Pure Appl. Math., 59, 1523-1572, 2006. doi:10.1002/cpa.20156

6

Klawonn, A., Rheinbach, O., "A parallel implementation of Dual-Primal FETI methods for three dimensional linear elasticity using a transformation of basis", SIAM J. Sci. Comput., 28/5, 1886-1906, 2006. doi:10.1137/050624364

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