Computational modeling of growth - A critical review, a classification of concepts and two new consistent approaches

The present contribution is dedicated to the computational modeling of growth phenomena typically encountered in modern biomechanical applications. We set the basis by critically reviewing the relevant literature and classifying the existing models. Next, we introduce a geometrically exact continuum model of growth which is not a priori restricted to applications in hard tissue biomechanics. The initial boundary value problem of biomechanics is primarily governed by the density and the deformation problem which render a nonlinear coupled system of equations in terms of the balance of mass and momentum. To ensure unconditional stability of the required time integration procedure, we apply the classical implicit Euler backward method. For the spatial discretization, we suggest two alternative strategies, a node-based and an integration point-based approach. While for the former, the discrete balance of mass and momentum are solved simultaneously on the global level, the latter is typically related to a staggered solution with the density treated as internal variable. The resulting algorithms of the alternative solution techniques are compared in terms of stability, uniqueness, efficiency and robustness. To illustrate their basic features, we elaborate two academic model problems and a typical benchmark example from the field of biomechanics.