The properties of hydrogen atoms in a nano-diamond grain surrounded by an amorphous carbon shell are studied with Tight Binding computer simulations. Our samples model nano-diamond grains, of a few nanometers in size, that nucleate within an amorphous carbon matrix, as observed in deposition from a hydrocarbon rich plasma. The calculations show that the average hydrogen interstitial formation energy in the amorphous region is lower than in the nano-diamond core, therefore hydrogen interstitial sites in the in the amorphous region are more stable than in the nano-diamond core. This formation energy difference is the driving force for the diffusion of hydrogen atoms from nano-diamond grains into amorphous carbon regions. An energy well was observed on the amorphous side of the nano-diamond amorphous carbon interface: hydrogen atoms are expected to be trapped here. This scenario agrees with experimental results which show that hydrogen retention of diamond films increases with decreasing grain size, and suggest that hydrogen is bonded and trapped in nano-diamond grain boundaries and on internal grain surfaces.

Variational image segmentation based on the Mumford and Shah model [31], together with implementation by the piecewise constant level-set method (PCLSM) [26], leads to fully nonlinear Total Variation (TV)-Allen-Cahn equations. The commonly-used numerical approaches usually suffer from the difficulties not only with the non-differentiability of the TV-term, but also with directly evolving the discontinuous piecewise constant-structured solutions. In this paper, we propose efficient dual algorithms to overcome these drawbacks. The use of a splitting-penalty method results in TV-Allen-Cahn type models associated with different “double-well” potentials, which allow for the implementation of the dual algorithm of Chambolle [8]. Moreover, we present a new dual algorithm based on an edge-featured penalty of the dual variable, which only requires to solve a vectorial Allen-Cahn type equation with linear ∇(div)-diffusion rather than fully nonlinear diffusion in the Chambolle’s approach. Consequently, more efficient numerical algorithms such as time-splitting method and Fast Fourier Transform (FFT) can be implemented. Various numerical tests show that two dual algorithms are much faster and more stable than the primal gradient descent approach, and the new dual algorithm is at least as efficient as the Chambolle’s algorithm but is more accurate. We demonstrate that the new method also provides a viable alternative for image restoration.

Detecting corrosion by electrical field can be modeled by a Cauchy problem of Laplace equation in annulus domain under the assumption that the thickness of the pipe is relatively small compared with the radius of the pipe. The interior surface of the pipe is inaccessible and the nondestructive detection is solely based on measurements from the outer layer. The Cauchy problem for an elliptic equation is a typical ill-posed problem whose solution does not depend continuously on the boundary data. In this work, we assume that the measurements are available on the whole outer boundary on an annulus domain. By imposing reasonable assumptions, the theoretical goal here is to derive the stabilities of the Cauchy solutions and an energy regularization method. Relationship between the proposed energy regularization method and the Tikhonov regularization with Morozov principle is also given. A novel numerical algorithm is proposed and numerical examples are given.

This paper presents an exponential compact higher order scheme for Convection-Diffusion Equations (CDE) with variable and nonlinear convection coefficients. The scheme is for one-dimensional problems and produces a tri-diagonal system of equations which can be solved efficiently using Thomas algorithm. For two-dimensional problems, the scheme produces an accuracy over a compact nine point stencil which can be solved using any line iterative approach with alternate direction implicit procedure. The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be positive. Wave number analysis has been carried out to establish that the scheme is comparable in accuracy with spectral methods. The higher order accuracy and better rate of convergence of the developed scheme have been demonstrated by solving numerous model problems for one and two-dimensional CDE, where the solutions have the sharp gradient at the solution boundary.

The emission of scission neutrons from fissioning nuclei is of high practical interest. To study this process we have used the sudden approximation and also a more realistic approach that takes into account the scission dynamics. Numerically, this implies the solution of the bi-dimensional Schrödinger equation, both stationary and time-dependent. To describe axially symmetric extremely deformed nuclear shapes, we have used the Cassini parametrization. The Hamiltonian is discretized by using finite difference approximations of the derivatives. The main computational challenges are the solution of algebraic eigenvalue problems and of linear systems with large sparse matrices. We have employed appropriate procedures (Arnoldi and bi-conjugate gradients). The numerical solutions have been used to evaluate physical quantities, like the number of emitted neutrons per scission event, the primary fragments’ excitation energy and the distribution of the emission points.

Spectral element methods are well established in the field of wave propagation, in particular because they inherit the flexibility of finite element methods and have low numerical dispersion error. The latter is experimentally acknowledged, but has been theoretically shown only in limited cases, such as Cartesian meshes. It is well known that a finite element mesh can contain distorted elements that generate numerical errors for very large distortions. In the present work, we study the effect of element distortion on the numerical dispersion error and determine the distortion range in which an accurate solution is obtained for a given error tolerance. We also discuss a double-grid calculation of the spectral element matrices that preserves accuracy in deformed geometries.

The previously developed LB-DF/FD method derived from the lattice Boltzmann method and Direct Forcing/Fictitious Domain method is extended to deal with 3D particle’s Brownian motion. In the model the thermal fluctuations are introduced as random forces and torques acting on the Brownian particle. The hydrodynamic interaction is introduced by directly resolving the fluid motions. A sphere fluctuating in a cubic box with the periodic boundary is considered to validate the present model. By examining the velocity autocorrelation function (VCF) and rotational velocity autocorrelation function (RVCF), it has been found that in addition to the two relaxation times, the mass density ratio should be taken into consideration to check the accuracy and effectiveness of the present model. Furthermore, the fluctuation-dissipation theorem and equipartition theorem have been investigated for a single spherical particle. Finally, a Brownian particle trapped in a harmonic potential has been simulated to further demonstrate the ability of the LB-DF/FD model.

In this paper, we investigate the effects of kinematic transports on the nematic liquid crystal system numerically and theoretically. The model we used is a “1+2” elastic continuum model simplified from the Ericksen-Leslie system. The numerical experiments are carried out by using a Legendre-Galerkin spectral method which can preserve the energy law in the discrete form. Based on this highly accurate numerical approach we find some interesting and important relationships between the kinematic transports and the characteristics of the flow. We make some analysis to explain these results. Several significant scaling properties are also verified by our simulations.

Within the projection schemes for the incompressible Navier-Stokes equations (namely “pressure-correction” method), we consider the simplest method (of order one in time) which takes into account the pressure in both steps of the splitting scheme. For this scheme, we construct, analyze and implement a new high order compact spatial approximation on nonstaggered grids. This approach yields a fourth order accuracy in space with an optimal treatment of the boundary conditions (without error on the velocity) which could be extended to more general splitting. We prove the unconditional stability of the associated Cauchy problem via von Neumann analysis. Then we carry out a normal mode analysis so as to obtain more precise results about the behavior of the numerical solutions. Finally we present detailed numerical tests for the Stokes and the Navier-Stokes equations (including the driven cavity benchmark) to illustrate the theoretical results.

We perform a comparison in terms of accuracy and CPU time between second order BDF semi-Lagrangian and Lagrange-Galerkin schemes in combination with high order finite element method. The numerical results show that for polynomials of degree 2 semi-Lagrangian schemes are faster than Lagrange-Galerkin schemes for the same number of degrees of freedom, however, for the same level of accuracy both methods are about the same in terms of CPU time. For polynomials of degree larger than 2, Lagrange-Galerkin schemes behave better than semi-Lagrangian schemes in terms of both accuracy and CPU time; specially, for polynomials of degree 8 or larger. Also, we have performed tests on the parallelization of these schemes and the speedup obtained is quasi-optimal even with more than 100 processors.

In this paper we proposed a modified Baer-Nunziato model for compressible multi-fluid flows, with main attention on the energy exchange between the two fluids. The proposed model consists of eleven PDEs; however, the use of the particular phase evolving variables may reduce the model to have only six PDEs. The main advantage of the model is that the Abgrall's UPV criterion on mixture velocity and pressure is satisfied without affecting either its hyperbolicity or its conservations of the two individual masses, momentum or total energy. An Lax-Friedrichs scheme is built for a particular case of the proposedmodel. When the two fluids in the fluid mixture are both of the linear Mie-Gruneisen type, the scheme satisfies the Abgrall's UPV criterion on mixture velocity and pressure. Numerical experiments with polytropic, barotropic, stiffened and van der Waals fluids show that the scheme is efficient and able to treat fluids characterized with quite different thermodynamics.

A multiple-image method is proposed to approximate the reaction-field potential of a source charge inside a finite length cylinder due to the electric polarization of the surrounding membrane and bulk water. When applied to a hybrid ion-channel model, this method allows a fast and accurate treatment of the electrostatic interactions of protein with membrane and solvent. To treat the channel/membrane interface boundary conditions of the electric potential, an optimization approach is used to derive image charges by fitting the reaction-field potential expressed in terms of cylindric harmonics. Meanwhile, additional image charges are introduced to satisfy the boundary conditions at the planar membrane interfaces. In the end, we convert the electrostatic interaction problem in a complex inhomogeneous system of ion channel/membrane/water into one in a homogeneous free space embedded with discrete charges (the source charge and image charges). The accuracy of this method is then validated numerically in calculating the solvation self-energy of a point charge.

In this short note we present a derivation of the Spectral Difference Scheme from a Discontinuous Galerkin (DG) discretization of a nonlinear conservation law. This allows interpretation of the Spectral Difference Scheme as a particular discretization under the quadrature-free nodal DG paradigm. Moreover, it enables identification of the key differences between the Spectral Difference Scheme and standard nodal DG schemes.