An adaptive Newton-method based on a dynamical systems approach

The traditional Newton method for solving nonlinear operator equations in Banach spaces is discussed within the context of the continuous Newton method. This setting makes it possible to interpret the Newton method as a discrete dynamical system and thereby to cast it in the framework of an adaptive step size control procedure. In so doing, our goal is to reduce the chaotic behavior of the original method without losing its quadratic convergence property close to the roots. The performance of the modified scheme is illustrated with various examples from algebraic and differential equations.

Experimental validation of a nonlinear μ FE model based on cohesive-frictional plasticity for trabecular bone

Trabecular bone is a porous mineralized tissue playing a major load bearing role in the human body. Prediction of age-related and disease-related fractures and the behavior of bone implant systems needs a thorough understanding of its structure-mechanical property relationships, which can be obtained using microcomputed tomography-based finite element modeling. In this study, a nonlinear model for trabecular bone as a cohesive-frictional material was implemented in a large-scale computational framework and validated by comparison of μFE simulations with experimental tests in uniaxial tension and compression. A good correspondence of stiffness and yield points between simulations and experiments was found for a wide range of bone volume fraction and degree of anisotropy in both tension and compression using a non-calibrated, average set of material parameters. These results demonstrate the ability of the model to capture the effects leading to failure of bone for three anatomical sites and several donors, which may be used to determine the apparent behavior of trabecular bone and its evolution with age, disease, and treatment in the future.