Towards an efficient numerical simulation of complex 3D knee joint motion

We present a time-dependent finite element model of the human knee joint of full 3D geometric complexity together with advanced numerical algorithms needed for its simulation. The model comprises bones, cartilage and the major ligaments, while patella and menisci are still missing. Bones are modeled by linear elastic materials, cartilage by linear viscoelastic materials, and ligaments by one-dimensional nonlinear Cosserat rods. In order to capture the dynamical contact problems correctly, we solve the full PDEs of elasticity with strict contact inequalities. The spatio-temporal discretization follows a time layers approach (first time, then space discretization). For the time discretization of the elastic and viscoelastic parts we use a new contact-stabilized Newmark method, while for the Cosserat rods we choose an energy-momentum method. For the space discretization, we use linear finite elements for the elastic and viscoelastic parts and novel geodesic finite elements for the Cosserat rods. The coupled system is solved by a Dirichlet–Neumann method. The large algebraic systems of the bone–cartilage contact problems are solved efficiently by the truncated non-smooth Newton multigrid method

A C0 interior penalty discontinuous Galerkin method for fourth‐order total variation flow I: Derivation of the method and numerical results

We consider the numerical solution of a fourth‐order total variation flow problem representing surface relaxation below the roughening temperature. Based on a regularization and scaling of the nonlinear fourth‐order parabolic equation, we perform an implicit discretization in time and a C0 Interior Penalty Discontinuous Galerkin (C0IPDG) discretization in space. The C0IPDG approximation can be derived from a mixed formulation involving numerical flux functions where an appropriate choice of the flux functions allows to eliminate the discrete dual variable. The fully discrete problem can be interpreted as a parameter dependent nonlinear system with the discrete time as a parameter. It is solved by a predictor corrector continuation strategy featuring an adaptive choice of the time step sizes. A documentation of numerical results is provided illustrating the performance of the C0IPDG method and the predictor corrector continuation strategy. The existence and uniqueness of a solution of the C0IPDG method will be shown in the second part of this paper

Spectral degeneracy and escape dynamics for intermittent maps with a hole

We study intermittent maps from the point of view of metastability. Small neighbourhoods of an intermittent fixed point and their complements form pairs of almost-invariant sets. Treating the small neighbourhood as a hole, we first show that the absolutely continuous conditional invariant measures (ACCIMs) converge to the ACIM as the length of the small neighbourhood shrinks to zero. We then quantify how the escape dynamics from these almost-invariant sets are connected with the second eigenfunctions of Perron-Frobenius (transfer) operators when a small perturbation is applied near the intermittent fixed point. In particular, we describe precisely the scaling of the second eigenvalue with the perturbation size, provide upper and lower bounds, and demonstrate $L^1$ convergence of the positive part of the second eigenfunction to the ACIM as the perturbation goes to zero. This perturbation and associated eigenvalue scalings and convergence results are all compatible with Ulam's method and provide a formal explanation for the numerical behaviour of Ulam's method in this nonuniformly hyperbolic setting. The main results of the paper are illustrated with numerical computations.Comment: 34 page

Theoretical X-Ray Absorption Debye-Waller Factors

An approach is presented for theoretical calculations of the Debye-Waller factors in x-ray absorption spectra. These factors are represented in terms of the cumulant expansion up to third order. They account respectively for the net thermal expansion $\sigma^{(1)}(T)$, the mean-square relative displacements $\sigma^2(T)$, and the asymmetry of the pair distribution function $\sigma^{(3)}(T)$. Similarly, we obtain Debye-Waller factors for x-ray and neutron scattering in terms of the mean-square vibrational amplitudes $u^2(T)$. Our method is based on density functional theory calculations of the dynamical matrix, together with an efficient Lanczos algorithm for projected phonon spectra within the quasi-harmonic approximation. Due to anharmonicity in the interatomic forces, the results are highly sensitive to variations in the equilibrium lattice constants, and hence to the choice of exchange-correlation potential. In order to treat this sensitivity, we introduce two prescriptions: one based on the local density approximation, and a second based on a modified generalized gradient approximation. Illustrative results for the leading cumulants are presented for several materials and compared with experiment and with correlated Einstein and Debye models. We also obtain Born-von Karman parameters and corrections due to perpendicular vibrations.Comment: 11 pages, 8 figure

Nonadiabatic orientation, toroidal current, and induced magnetic field in BeO molecules

It is predicted that oriented BeO molecules would give rise to unprecedentedly strong, unidirectional electric ring current and an associated magnetic field upon excitation by a right or left circularly polarized laser pulse into the first excited degenerate singlet state. The strong toroidal electric ring current of this state is dominated by the ring current of the 1π± orbital about the molecular axis. Our predictions are based on the analysis of the orbital composition of the states involved and are substantiated by high level electronic structure calculations and wavepacket simulations of the laser-driven orientation and excitation [email protected]

Magnetically responsive composites: electron beam assisted magnetic nanoparticle arrest in gelatin hydrogels for bioactuation

As emerging responsive materials, ferrogels have become highly attractive for biomedical and technical applications in terms of soft actuation, tissue engineering or controlled drug release. In the present study, bioderived ferrogels were fabricated and successfully deformed within moderate, heterogeneous magnetic fields. Synthesis was realized by arresting iron oxide nanoparticles in porcine gelatin by introduction of covalent crosslinks via treatment with energetic electrons for mesh refinement. This approach also allows for tuning thermal and mechanical stability of the gelatin matrix. Operating the bioferrogel in compression, magnetic forces on the nanoparticles are counterbalanced by the stiffness of the hydrogel matrix that is governed by a shift in thermodynamic equilibrium of swelling, as derived in the framework of osmosis. As gelatin and iron oxide nanoparticles are established as biocompatible constituents, these findings promise potential for in vivo use as contactless mechanical transducers

Dynamical effects induced by long range activation in a nonequilibrium reaction-diffusion system

We both show experimentally and numerically that the time scales separation introduced by long range activation can induce oscillations and excitability in nonequilibrium reaction-diffusion systems that would otherwise only exhibit bistability. Namely, we show that the Chlorite-Tetrathionate reaction, where autocatalytic species diffuses faster than the substrates, the spatial bistability domain in the nonequilibrium phase diagram is extended with oscillatory and excitability domains. A simple model and a more realistic model qualitatively account for the observed behavior. The latter model provides quantitative agreement with the experiments.Comment: 19 pages + 9 figure