An efficient and accurate numerical method is presented for the solution of highly oscillatory differential equations in one spatial dimension. While standard methods would require a very fine grid to resolve the oscillations, the presented approach uses first an analytic WKB-type transformation, which filters out the dominant oscillations. The resulting ODE-system is much smoother and can hence be discretized on a much coarser grid, with significantly reduced numerical costs. Here we are concerned with stationary two-band Schrodinger equations employed in quantum transport applications. 

We focus on the Kane–model and the two band 𝑘 ⋅ 𝑝–model. The accuracy of the presented method is illustrated on a numerical example.

We consider electrons in a circular nanoring of zero width, in magnetic field, and with Rashba spin-orbit interaction. We include the Coulomb interaction in the the ”exact diagonalization” manner. The Coulomb interaction has strong effects on the spin polarization which may be totally different than for noninteracting electrons. Our current results include up to four electrons, but this number can easily be increased.

Polymer nanocomposites based on polyolefins and layered clay minerals (smectites) are multi-layered structure. Even unfilled polyolefins (polyethylene, polypropylene, etc.) are structurally heterogeneous environment consisting of amorphous and crystalline phases. Adding filler further complicates the structure of the material. The results of computer simulation of interaction of silicate inclusions, crystalline supermolecular formations in the matrix (spherulites) and its amorphous part. Composite is modeled as a square periodicity cell with inclusion and spherulites. Inclusion was a pack of silica plates separated by thin layers of polymer. Spherulite modeled as “snowflakes” of crystallized lamellae (band areas with an ordered arrangement of polymer molecules). Problem was solved by finite element method in nonlinear-elastic formulation (plane strain).

This article examines the theoretical description of the mechanical behavior of elastomeric nanocomposites based on butadiene styrene polymer, and several species with different filler volume fraction in 30phr and 50phr. To construct the determining equations we use the scheme, whose points are connected by elastic, viscous, plastic and transmission elements. To describe the properties of each element used well-known equations of nonlinear elasticity theory, the theory of nonlinear viscous fluids, the theory of plastic flow of material in the finite deformation of the medium. To obtain the constants of the model used stepwise algorithm. Used in the experiments (cyclic loading, relaxation and creep) can get more information about the viscoelastic properties of rubber.

Last years, fluid flows in nano-sized domains are intensively studied [1–4] due to nontriviality of observed effects and practical importance of this part of hydrodynamics. At present, there are no general equations of nanohydrodynamics. Usually, the molecular dynamics is used for computations. As for analytical approaches, the simplest one involves introducing the slip condition at the boundary [5] together with classical hydrodynamics equations. The small scale of nanochannels gives us the possibility to use, in some cases, the Stokes approximation for motion equations [6].

In this work we apply the planar Stokes model [7] with slip boundary conditions for describing nano-flows. We have developed a method of flow calculation, which is based on the expansion of pressure in a complete system of harmonic functions. Using the pressure distribution, we calculate the velocity field and stress on the boundary. This method can be used for description of various problems of nanofluidics: hydrodynamics of nanochannels, flows along superhydrophobic surfaces, etc

In the paper the method of molecular dynamics is used to investigate the features of structure transformation, which are taking place during the process of surface treatment. The force field of a cylindrical shape was used to imitate the motion with constant velocity of hard indenter. The following parameters of tasks were varied: the radius of indenter, initial immersion depths, surface roughness, number of passes and the initial position of the indenter. Calculations were performed for the pure crystallites of copper and iron. According to the modeling the treatment of the surface layer can leads to formation of numerous structural defects, which can provide effect of nano-fragmentation of material near the surface. A comparison of surface topography before and after treatment was analyzed also. Results of our study are in good agreement with experimental data.

Stability of 2D triangular lattice under finite arbitrary strain is investigated. The lattice is considered infinite and consisting of particles which interact by pair force central potential. Dynamic stability criterion is used: frequency of elastic waves is required to be real for any real wave vector. Two stability regions corresponding to horizontal and vertical orientations of the lattice are obtained. It means that a structural transition, which is equal to the change of lattice orientation, is possible.

Essentially nonlinear model of a crystalline bi-atomic lattice described by coupled nonlinear equations, is considered. Its nonlinear wave solutions account for dynamic variations in an internal structure of the lattice due to an influence of a dynamic loading. Numerical simulations are performed to study evolution of a kink-shaped dynamic variations in an internal structure of the lattice. Special attention is paid on the transition from kink-shaped to bell-shaped variations. It is shown how predictions of the known exact traveling wave solutions may help in understanding and explanation of evolution of localized waves of permanent shape and velocity in numerical solutions.