The solution of the linear system that arises from the discretization of a partial differential equation (PDE) can be accomplished by direct or iterative methods. Direct methods can lead to high computational complexity and to considerable memory requirements, both of which limit the practical size of the discretization that can be used to solve the PDE. Iterative methods permit the maintenance of the sparsity pattern, reducing storage requirements while decreasing computation time as well. The primary purpose of using the preconditioner is to accelerate the convergence in the iterative phase of the computation. The chapter discusses the block structure inherent in the linear system that arises in discretizing the partial differential equation. The advantage of rowsum approach is that the operations in the iterative method with the preconditioner can be conducted as matrix-vector products rather than doing forward and backsolve steps involving the factorization.

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The energetics of ionic selectivity in the neuronal sodium channels is studied.A simple model constructed for the selectivity filter of the channel is used.The selectivity filter of this channel type contains aspartate (D), glutamate (E), lysine (K), and alanine (A) residues (the DEKA locus).We use Grand Canonical Monte Carlo simulations to compute equilibrium binding selectivity in the selectivity filter and to obtain various terms of the excess chemical potential from a particle insertion procedure based on Widom's method.We show that K + ions in competition with Na + are efficiently excluded from the selectivity filter due to entropic hard sphere exclusion.The dielectric constant of protein has no effect on this selectivity.Ca 2+ ions, on the other hand, are excluded from the filter due to a free energetic penalty which is enhanced by the low dielectric constant of protein.

DISPL is a software package for solving systems of nonlinear partial differential equations of initial value type over a two-dimensional spatial domain. This report describes an extension of the approximation procedure to include spatial domains described by polar geometry. More generally the extension includes problems for which there is periodicity in one of the spatial directions

The nature of driven motion of a vortex solid in the presence of a planar pinning defect is investigated by large-scale simulations based on the time-dependent Ginzburg-Landau equations. Three dynamic phases are identiied and characterized by their relative positional and velocity correlations. Vortices in superconductors are a well-deened system of elastic lines or points interacting by electromagnetic and hydrodynamic forces, which can be driven through a eld of pinning sites by an applied current. The dynamic response of the vortices controls most transport properties of superconductors and displays remarkable variety, including nonlinear eeects, steady-state and avalanche dynamics, and thresholds for the onset of motion. A basic understanding of these dynamics is of fundamental interest and is an essential step in controlling vortex motion in technical applications. Vortex motion may be classiied generally as elastic or plastic. In elastic motion, each vortex keeps the same neighbors, while in plastic motion the neighbors change. Plastic vortex ow has been identiied in molecular dynamics simulations 1] and in pioneering transport experiments on NbSe 2 2]. Plastic-to-elastic dynamic transitions have been predicted analytically 3] and explored in neutron scattering experiments 4,5] and transport measurements 6,7]. In this paper, we use large-scale simulations of the time-dependent Ginzburg-Landau equations to investigate the internal structure of elastic and plastic motion, following the individual motions of hundreds of vortices in the presence of a controlled driving force and a planar pinning defect. We nd two distinct plastic phases and an elastic phase, each with diierent internal symmetry. We identify and compare the characteristic features of each dynamic phase and the physical conditions favoring each. The simulations are based on the time-dependent Ginzburg-Landau equations 8,9], h 2 2m s D @ @t = L ; c 2 @A @t = L A 1 4 r r A; L = ajj 2 + b 2 jj 4 + 1 2m s h i r e s c A 2 ; where is the complex order parameter, A the vector potential, and L the Helmholtz free-energy density; the other symbols have their usual meaning. Computational details have been described in 10]. The simulated sample was a rectangular cylinder, in-nitely long and homogeneous in the eld direction. The sample had two parallel free surfaces deened by the boundary condition J s n = 0 (J s is the supercurrent density). A transport current was induced by a eld differential between the free …

Recent theoretical and experimental studies of radiation-induced segregation in alloys under irradiation with focused charged-particle beams have shown that point-defect currents generated by axial and radial displacement-rate gradients can cause significant redistribution of the alloying elements within the irradiated zone. In the case of irradiation of thin films with highly-focused electron beams, two important features have been established experimentally: (i) the diameter of the local region in which the alloy composition and phase are modified is practically equal to the beam diameter, and (ii) the time required to produce a given change in the alloy composition in the center of the irradiated zone decreases rapidly with beam diameter. Our theoretical modeling indicates that these features will also be observed in semi-infinite alloys bombarded with focused proton beams. However, in this case, the spatially-nonuniform defect production in both the axial and radial directions renders the compositional redistribution more complex. The present work shows that the ability to locally modify the alloy composition by focused electron or proton beams may offer a new method for producing local regions of controlled composition and microstructure on a submicron scale. The results of our model calculations and experimental studies will be presented to demonstrate the feasibility of this novel technique.

The evolution of the implant distribution during ion implantation at elevated temperatures has been theoretically studied using a comprehensive kinetic model. In the model foreign atoms, implanted into both interstitial and substitutional sites of the host lattice, could interact with implantation-induced point defects and with extended sinks such as the bombarded surface. The synergistic effects of preferential sputtering, radiation-enhanced diffusion, and radiation-induced segregation, as well as the influence of nonuniform defect production, were taken into account. The bombarded surface was allowed to move in either direction, − x or + x , depending on ion energy, i.e., on the competition between the rates of ion deposition and sputtering. The moving surface was accounted for by means of a mathematical technique of immobilizing the boundary. The ion implantation process was cast into a system of five coupled partial differential equations, which could be solved numerically using a suitable technique. Sample calculations were performed for two systems: Si^+ and Al^+ implantations into Ni. It has been known from previous studies that in irradiated Ni, Si atoms segregate in the same direction as the defect fluxes, whereas Al solutes migrate in the opposite direction. Thus the effects of different segregation mechanisms, as well as the influence of target temperature, ion energy, and implantation rate on the evolution of implant concentrations in time and space, could be examined with the present model.

Osmotic coefficients of solutions represent experimentally observable measures of the interactions between dissolved species and are particularly important metrics for biophysical experiments in which the concentrations of solutes control biological function, as in ion-channel proteins. The computational expense associated with all-atom molecular-dynamics (MD) simulations makes it difficult to connect these data to detailed molecular models directly. It is therefore generally impractical to ensure that MD force fields are parameterized consistently with the full range of available data. Several approaches to coarse graining these systems make the estimation of osmotic coefficients more tractable. In this paper we explore one such approach, the inverse Monte Carlo (IMC) method of Lyubartsev and Laaksonen, for estimating osmotic coefficients of aqueous solutions with dissolved sodium chloride. The IMC method is used to calculate effective pairwise potential functions for the ion-ion interactions, taking as input the ion-ion radial distribution functions calculated from all-atom MD simulations. Our results show that the IMC method converges robustly and reproducibly, and we conclude that IMC and related approaches hold promise for validating force fields against experimental data, though further refinements will be important to ensure practical viability for use in parameterization. In particular, we have found that the estimates of the osmotic coefficient, which is an intensive quantity, are size dependent, but the size dependence can be corrected with a simple linear fitting procedure. Also, the Monte Carlo procedure explores the phase space extremely slowly, and it may be valuable to use accelerated sampling procedures; performance may also be improved significantly by using more sophisticated Newton–Raphson methods. It is well known that osmotic coefficients are highly sensitive to the pair potentials involved, and thus methods for their estimation must be studied and validated carefully. 2 Argonne/Math and Computer Science Reprint ANL/MCS-P1764-0610 06/09/2010

We propose an Eulerian description of the bounce-back boundary condition based on the high-order implicit time-marching schemes to improve the accuracy of lattice Boltzmann simulation in the vicinity of curved boundary. The Eulerian description requires only one grid spacing between fluid nodes when second-order accuracy in time and space is desired, although high-order accurate boundary conditions can be constructed on more grid-point support. The Eulerian description also provides an analytical framework for several different interpolation-based boundary conditions. For instance, the semi-Lagrangian, linear interpolation boundary condition is found to be a first-order upwind discretization that changes the time-marching schemes from implicit to explicit as the distance between the fluid boundary node and the solid boundary increases.

In the work of Fischer et al. (2002, “Forces on Particles in an Oscillatory Boundary Layer,” J. Fluid Mech., 468, pp. 327–347, 2005; “Influence of Wall Proximity on the Lift and Drag of a Particle in an Oscillatory Flow,” ASME J. Fluids Eng., 127, pp. 583–594) we computed the lift and drag forces on a sphere, subjected to a wall-bounded oscillatory flow. The forces were found as a function of the Reynolds number, the forcing frequency, and the gap between the particle and the ideally smooth rigid bounding wall. Here we investigate how the forces change as a function of the above parameters and its moment of inertia if the particle is allowed to freely rotate. Allowing the particle to rotate does not change appreciably the drag force, as compared to the drag experienced by the particle when it is held fixed. Lift differences between the rotating and nonrotating cases are shown to be primarily dominated in the mean by the pressure component. The lift of the rotating particle varies significantly from the fixed-particle case and depends strongly on the Reynolds number, the forcing frequency, and the gap; much less so on the moment of inertia. Of special significance is that the lift is enhanced for small Reynolds numbers and suppressed for larger ones, with a clear transition point. We also examine how the torque changes when the particle is allowed to rotate as compared to when it is held fixed. As a function of the Reynolds number the torque of the fixed sphere is monotonically decreasing in the range Re=5 to Re=400. The rotating-sphere counterpart experiences a smaller and more complex torque, synchronized with the lift transition mentioned before. As a function of the gap, the torque is significantly larger in the fixed particle case.

Brillouin scattering was used to study the effect of high-power microwave fields on an array of permalloy particles and the results are compared with simulations. The simulations are of two types: one is based on a model in which each particle is treated as a single spin, the second model relies on generalized micromagnetic codes that include external driving fields and enable magnon–magnon coupling. Experimental results as well as simulations show clear, but sometimes different, evidence of non-linear behavior.

Spin-wave modes in a thin submicron cobalt square with a closure domain structure are obtained by using a micromagnetic equation of motion approach. In addition to modes with amplitude over the whole sample, some low-frequency modes, localized at the center, corners, and diagonals of the square, are also found. In analogy with the modes found in a circular vortex, the nonlocalized modes can be broadly classified into radial-like and azimuthal-like modes, and their frequencies can be understood qualitatively in terms of the dispersion relation of spin-wave modes of an unconfined film. Other modes that can be interpreted as the combination of radial and azimuthal modes are also observed.

A dynamical simulation approach is used in this work to study the origin of the stripe domains for an anisotropic nanostructured cobalt magnetic bar. Results show different remnant states for different field histories. It is also shown that the final domain structure at low field can be predicted from a high-field analysis of the frequencies of the standing-wave-like modes.

We theoretically study the relation between stripe domains at remanence and magnetic normal modes in a single crystal Co bar. We find different stripe patterns depending on field history and in each case the domain structure can be related to a soft mode that triggers a phase transition. The stripe domain structure when the external field is along the long axis of the bar is shown to be generated by a standing wave mode, which has the same spatial structure as the stripes. At all fields this mode has the lowest frequency of all the standing wave modes. This mode goes soft at a second-order phase transition where the stripe domains emerge. For other directions of the field, the symmetry of soft modes is found to be consistent with the change in symmetry of the ground state and that the phase transition can be first order. An analytical model relating phase transitions and soft mode behavior is also briefly discussed

We report on the lift and drag forces on a stationary sphere subjected to a wall-bounded oscillatory flow. We show how these forces depend on two parameters, namely, the distance between the particle and the bounding wall, and on the frequency of the oscillatory flow. The forces were obtained from numerical solutions of the unsteady incompressible Navier–Stokes equations. For the range of parameters considered, a spectral analysis found that the forces depended on a small number of degrees of freedom. The drag force manifested little change in character as the parameters varied. On the other hand, the lift force varied significantly: We found that the lift force can have a positive as well as a negative time-averaged value, with an intermediate range of external forcing periods in which enhanced positive lift is possible. Furthermore, we determined that this force exhibits a viscous-dominated and a pressure-dominated range of parameters.

This paper describes a technique to compute magnetic normal modes of nanosized particles. The technique is based on the Landau-Lifshitz formalism of micromagnetics and accounts fully for both the exchange and the dipolar field. It requires no more than the specification of the material parameters and the geometry of the sample; in particular, it does not require the specification of boundary conditions. It also allows the large-amplitude nonlinear regime to be probed. The technique is applied to a model of a polycrystalline iron particle, which is shown to possess a rich variety of normal modes. Some of these modes are reminiscent of standing waves, while others are more or less localized in parts of the sample. The variation of the mode frequencies with the applied field is analyzed and compared with existing approximations.

We present a method to compute the magnetic normal modes of a ferromagnetic particle. The method is a hybrid of micromagnetic simulations and a "dynamical matrix" approach similar to that used for vibrational studies. We use the method to calculate the normal modes of an Fe parallelepiped and compare the results with the modes recently extracted from a purely micromagnetic simulation. The results of the two approaches are in excellent agreement. We discuss the pros and cons of both approaches. We also present information on standing waves with wave vector perpendicular to the applied field and on a family of modes localized at the particle ends.

We have recently developed two methods to calculate the magnetic normal modes of a magnetic nano-particle. One of the methods is based on a conventional micromagnetic approach in which the time evolution of the magnetization of each cell is monitored. After filtering in frequency domain, the magnetic normal modes can be reconstructed. The second method is based on solving the same micromagneitc system in a dynamical matrix formulation. The results of the two methods, applied to a rectangular parallelepiped of Fe, will be presented and compared.