The Damage Spreading (DS) method allows the investigation of the effect caused by tiny perturbations, in the initial conditions of physical systems, on their final stationary or equilibrium states. The damage (D(t)) is determined during the dynamic evolution of a physical system and measures the time dependence of the difference between a reference (unperturbed) configuration and an initially perturbed one. In this paper we first give a brief overview of Monte Carlo simulation results obtained by applying the DS method. Different model systems under study often exhibit a transition between a state where the damage becomes healed (the frozen phase) and a regime where the damage spreads arriving at a finite (stationary) value (the damaged phase), when a control parameter is finely tuned. These kinds of transitions are actually true irreversible phase transitions themselves, and the issue of their universality class is also discussed. Subsequently, the attention is focused on the propagation of damage in magnetic systems placed in confined geometries. The influence of interfaces between magnetic domains of different orientation on the spreading of the perturbation is also discussed, showing that the presence of interfaces enhances the propagation of the damage. Furthermore, the critical transition between propagation and nonpropagation of the damage is discussed. In all cases, the determined critical exponents suggest that the DS transition does not belong to the universality class of Directed Percolation, unlike many other systems exhibiting irreversible phase transitions. This result reflects the dramatic influence of interfaces on the propagation of perturbations in magnetic systems.

We deal with the numerical solution of the Navier-Stokes equations describing a motion of viscous compressible fluids. In order to obtain a sufficiently stable higher order scheme with respect to the time and space coordinates, we develop a combination of the discontinuous Galerkin finite element (DGFE) method for the space discretization and the backward difference formulae (BDF) for the time discretization. Since the resulting discrete problem leads to a system of nonlinear algebraic equations at each time step, we employ suitable linearizations of inviscid as well as viscous fluxes which give a linear algebraic problem at each time step. Finally, the resulting BDF-DGFE scheme is applied to steady as well as unsteady flows and achieved results are compared with reference data.

In this paper, we present a generalized Peierls-Nabarro model for curved dislocations using the discrete Fourier transform. In our model, the total energy is expressed in terms of the disregistry at the discrete lattice sites on the slip plane, and the elastic energy is obtained efficiently within the continuum framework using the discrete Fourier transform. Our model directly incorporates into the total energy both the Peierls energy for the motion of straight dislocations and the second Peierls energy for kink migration. The discreteness in both the elastic energy and the misfit energy, the full long-range elastic interaction for curved dislocations, and the changes of core and kink profiles with respect to the location of the dislocation or the kink are all included in our model. The model is presented for crystals with simple cubic lattice. Simulation results on the dislocation structure, Peierls energies and Peierls stresses of both straight and kinked dislocations are reported. These results qualitatively agree with those from experiments and atomistic simulations. 

The Chan-Vese method of active contours without edges [11] has been used successfully for segmentation of images. As a variational formulation, it involves the solution of a fully nonlinear partial differential equation which is usually solved by using time marching methods with semi-implicit schemes for a parabolic equation; the recent method of additive operator splitting [19,36] provides an effective acceleration of such schemes for images of moderate size. However to process images of large size, urgent need exists in developing fast multilevel methods. Here we present a multigrid method to solve the Chan-Vese nonlinear elliptic partial differential equation, and demonstrate the fast convergence. We also analyze the smoothing rates of the associated smoothers. Based on our numerical tests, a surprising observation is that our multigrid method is more likely to converge to the global minimizer of the particular non-convex problem than previously unilevel methods which may get stuck at local minimizers. Numerical examples are given to show the expected gain in CPU time and the added advantage of global solutions.

Several lumped parameter, or zero-dimensional (0-D), models of the microcirculation are coupled in the time domain to the nonlinear, one-dimensional (1-D) equations of blood flow in large arteries. A linear analysis of the coupled system, together with in vivo observations, shows that: (i) an inflow resistance that matches the characteristic impedance of the terminal arteries is required to avoid non-physiological wave reflections; (ii) periodic mean pressures and flow distributions in large arteries depend on arterial and peripheral resistances, but not on the compliances and inertias of the system, which only affect instantaneous pressure and flow waveforms; (iii) peripheral inertias have a minor effect on pulse waveforms under normal conditions; and (iv) the time constant of the diastolic pressure decay is the same in any 1-D model artery, if viscous dissipation can be neglected in these arteries, and it depends on all the peripheral compliances and resistances of the system. Following this analysis, we propose an algorithm to accurately estimate peripheral resistances and compliances from in vivo data. This algorithm is verified against numerical data simulated using a 1-D model network of the 55 largest human arteries, in which the parameters of the peripheral windkessel outflow models are known a priori. Pressure and flow waveforms in the aorta and the first generation of bifurcations are reproduced with relative root-mean-square errors smaller than 3%.

A hybrid lattice-Boltzmann finite-difference method is presented to simulate incompressible, resistive magnetohydrodynamic (MHD) flows. The lattice Boltzmann equation (LBE) with the Lorentz force term is solved to update the flow field while the magnetic induction equation is solved using the finite difference method to calculate the magnetic field. This approach is methodologically intuitive because the governing equations for MHD are solved in their respective original forms. In addition, the extension to 3-D is straightforward. For validation purposes, this approach was applied to simulate the Hartmann flow, the Orszag-Tang vortex system (2-D and 3-D) and the magnetic reconnection driven by doubly periodic coalescence instability. The obtained results agree well with analytical solutions and simulation results available in the literature.

A finite volume (FV) method for simulating 3D Fluid-Structure Interaction (FSI) is presented in this paper. The fluid flow is simulated using a parallel unstructured multigrid preconditioned implicit compressible solver, whist a 3D matrix-free implicit unstructured multigrid finite volume solver is employed for the structural dynamics. The two modules are then coupled using a so-called immersed membrane method (IMM). Large-Eddy Simulation (LES) is employed to predict turbulence. Results from several moving boundary and FSI problems are presented to validate proposed methods and demonstrate their efficiency. 

The purpose of this paper is to solve some of the trouble spots of the classical SPH method by proposing an alternative approach. First, we focus on the problem of the stability for two different SPH schemes, one is based on the approach of Vila [25] and the other is proposed in this article which mimics the classical 1D Lax Wendroff scheme. In both approaches the classical SPH artificial viscosity term is removed preserving nevertheless the linear stability of the methods, demonstrated via the von Neumann stability analysis. Moreover, the issue of the consistency for the equations of gas dynamics is analyzed. An alternative approach is proposed that consists of using Godunov-type SPH schemes in Lagrangian coordinates. This not only provides an improvement in accuracy of the numerical solutions, but also assures that the consistency condition on the gradient of the kernel function is satisfied using an equidistant distribution of particles in Lagrangian mass coordinates. Three different Riemann solvers are implemented for the first-order Godunov type SPH schemes in Lagrangian coordinates, namely the Godunov flux based on the exact Riemann solver, the Rusanov flux and a new modified Roe flux, following the work of Munz [17]. Some well-known numerical 1D shock tube test cases [22] are solved, comparing the numerical solutions of the Godunov-type SPH schemes in Lagrangian coordinates with the first-order Godunov finite volume method in Eulerian coordinates and the standard SPH scheme with Monaghan's viscosity term.

In this paper, we study splitting numerical methods for the three-dimensional Maxwell equations in the time domain. We propose a new kind of splitting finite-difference time-domain schemes on a staggered grid, which consists of only two stages for each time step. It is proved by the energy method that the splitting scheme is unconditionally stable and convergent for problems with perfectly conducting boundary conditions. Both numerical dispersion analysis and numerical experiments are also presented to illustrate the efficiency of the proposed schemes.