  Fiber-reinforced soft biological tissues are typically modeled as
hyperelastic, anisotropic, and nearly incompressible materials. To enforce
incompressibility a multiplicative split of the deformation gradient into a
volumetric and an isochoric part is a very common approach. However, due to the
high stiffness of anisotropic materials in the preferred directions, the finite
element analysis of such problems often suffers from severe locking effects and
numerical instabilities. In this paper, we present novel methods to overcome
locking phenomena for anisotropic materials using stabilized P1-P1 elements. We
introduce different stabilization techniques and demonstrate the high
robustness and computational efficiency of the chosen methods. In several
benchmark problems we compare the approach to standard linear elements and show
the accuracy and versatility of the methods to simulate anisotropic, nearly and
fully incompressible materials. We are convinced that this numerical framework
offers the possibility to accelerate accurate simulations of biological
tissues, enabling patient-specfic parameterization studies, which require
numerous forward simulations.
