This study shows a new way to implement terrain-following σ-coordinate in a numerical model, which does not lead to the well-known “pressure gradient force (PGF)” problem. First, the causes of the PGF problem are analyzed with existing methods that are categorized into two different types based on the causes. Then, the new method that bypasses the PGF problem all together is proposed. By comparing these three methods and analyzing the expression of the scalar gradient in a curvilinear coordinate system, this study finds out that only when using the covariant scalar equations of σ-coordinate will the PGF computational form have one term in each momentum component equation, thereby avoiding the PGF problem completely. A convenient way of implementing the covariant scalar equations of σ-coordinate in a numerical atmospheric model is illustrated, which is to set corresponding parameters in the scalar equations of the Cartesian coordinate. Finally, two idealized experiments manifest that the PGF calculated with the new method is more accurate than using the classic one. This method can be used for oceanic models as well, and needs to be tested in both the atmospheric and oceanic models.

Dislocations are line defects in crystalline materials. The Peierls-Nabarro models are hybrid models that incorporate atomic structure of dislocation core into continuum framework. In this paper, we present a numerical method for a generalized Peierls-Nabarro model for curved dislocations, based on the fast multipole method and the iterative grid redistribution. The fast multipole method enables the calculation of the long-range elastic interaction within operations that scale linearly with the total number of grid points. The iterative grid redistribution places more mesh nodes in the regions around the dislocations than in the rest of the domain, thus increases the accuracy and efficiency. This numerical scheme improves the available numerical methods in the literature in which the long-range elastic interactions are calculated directly from summations in the physical domains; and is more flexible to handle problems with general boundary conditions compared with the previous FFT based method which applies only under periodic boundary conditions. Numerical examples using this method on the core structures of dislocations in Al and Cu and in epitaxial thin films are presented.

Numerical atomic orbitals have been successfully used in molecular simulations as a basis set, which provides a nature, physical description of the electronic states and is suitable for 𝒪(N) calculations based on the strictly localized property. This paper presents a numerical analysis for some simplified atomic orbitals, with polynomial-type and confined Hydrogen-like radial basis functions respectively. We give some a priori error estimates to understand why numerical atomic orbitals are computationally efficient in electronic structure calculations.

A new solution methodology is proposed for solving efficiently Helmholtz problems. The proposed method falls in the category of the discontinuous Galerkin methods. However, unlike the existing solution methodologies, this method requires solving (a) well-posed local problems to determine the primal variable, and (b) a global positive semi-definite Hermitian system to evaluate the Lagrange multiplier needed to restore the continuity across the element edges. Illustrative numerical results obtained for two-dimensional interior Helmholtz problems are presented to assess the accuracy and the stability of the proposed solution methodology.

In this paper, hp-adaptive finite element methods are studied for time-harmonic Maxwell’s equations. We propose the parallel hp-adaptive algorithms on conforming unstructured tetrahedral meshes based on residual-based a posteriori error estimates. Extensive numerical experiments are reported to investigate the efficiency of the hp-adaptive methods for point singularities, edge singularities, and an engineering benchmark problem of Maxwell’s equations. The hp-adaptive methods show much better performance than the h-adaptive method.

A multi-hierarchy simulation model aiming magnetic reconnection studies is developed and improved in which macroscopic and microscopic physics are computed consistently and simultaneously. Macroscopic physics is solved by mag-netohydrodynamics (MHD) algorithm, while microscopic dynamics is expressed by particle-in-cell (PIC) algorithm. The multi-hierarchy model relies on the domain decomposition method, and macro- and micro-hierarchies are interlocked smoothly by hand-shake scheme. As examination, plasma flow injection is simulated in the multi-hierarchy model. It is observed that plasmas flow from a macro-hierarchy to a micro-hierarchy across the magnetic field smoothly and continuously.

The classical discrete element approach (DEM) based on Newtonian dynamics can be divided into two major groups, event-driven methods (EDM) and time-driven methods (TDM). Generally speaking, TDM simulations are suited for cases with high volume fractions where there are collisions between multiple objects. EDM simulations are suited for cases with low volume fractions from the viewpoint of CPU time. A method combining EDM and TDM called Hybrid Algorithm of event-driven and time-driven methods (HAET) is presented in this paper. The HAET method employs TDM for the areas with high volume fractions and EDM for the remaining areas with low volume fractions. It can decrease the CPU time for simulating granular flows with strongly non-uniform volume fractions. In addition, a modified EDM algorithm using a constant time as the lower time step limit is presented. Finally, an example is presented to demonstrate the hybrid algorithm.

In this paper, we study the effects of inserted needle on the subcutaneous interstitial flow. A goal is to describe the physical stress affecting cells during acupuncture treatment. The model consists of the convective Brinkman equations to describe the flow through a fibrous medium. Numerical studies in FreeFem++ are performed to illustrate the acute physical stress developed by the implantation of a needle that triggers the physiological reactions of acupuncture. We emphasize the importance of numerical experiments for advancing in modeling in acupuncture.

In this study long bubble rising in a narrow channel was investigated using multiphase lattice Boltzmann method. The problem is known as a Bretherton or Taylor bubble flow [2] and is used here to verify the performance of the scheme proposed by [13]. The scheme is modified by incorporation of multiple relaxation time (MRT) collision scheme according to the original suggestion of the author. The purpose is to improve the stability of the method. The numerical simulation results show a good agreement with analytic solution provided by [2]. Moreover the convergence study demonstrates that the method achieves more than the first order of convergence. The paper investigates also the influence of simulation parameters on the interface resolution and shape.

A unified process for the construction of hierarchical conforming bases on a range of element types is proposed based on an ab initio preservation of the underlying cohomology. This process supports not only the most common simplicial element types, as are now well known, but is generalized to squares, hexahedra, prisms and importantly pyramids. Whilst these latter cases have received (to varying degrees) attention in the literature, their foundation is less well developed than for the simplicial case. The generalization discussed in this paper is effected by recourse to basic ideas from algebraic topology (differential forms, homology, cohomology, etc) and as such extends the fundamental theoretical framework established by the work of Hiptmair and Arnold et al. for simplices. The process of forming hierarchical bases involves a recursive orthogonalization and it is shown that the resulting finite element mass, quasi-stiffness and composite matrices exhibit exponential or better growth in condition number.

The present paper follows our previous work [Yang et al., Phys. Rev. E, 90 (2014), 063011] in which the bending modes of a symmetric flexible fiber in viscous flows were studied by using a coupling approach of smoothed particle hydrodynamics (SPH) and element bending group (EBG). It was shown that a symmetric flexible fiber can undergo four different bending modes including stable U-shape, slight swing, violent flapping and stable closure modes. For an asymmetric flexible fiber, the bending modes can be different. This paper numerically studies the fiber shape, flow field and fluid drag of an asymmetric flexible fiber immersed in a viscous fluid flow by using the SPH-EBG coupling method. An asymmetric number is defined to describe the asymmetry of a flexible fiber. The effects of the asymmetric number on the fiber shape, flow field and fluid drag are investigated.

Plasma waves with frequencies close to the particular gyrofrequencies of the charged particles in the plasma lose energy due to cyclotron damping. We briefly discuss the gyro-resonance of low frequency plasma waves and ions particularly with regard to particle-in-cell (PiC) simulations. A setup is outlined which uses artificially excited waves in the damped regime of the wave mode's dispersion relation to track the damping of the wave's electromagnetic fields. Extracting the damping rate directly fromthe field data in real or Fourier space is an intricate and non-trivial task. We therefore present a simple method of obtaining the damping rate Γ from the simulation data. This method is described in detail, focusing on a step-by-step explanation of the course of actions. In a first application to a test simulation we find that the damping rates obtained from this simulation generally are in good agreement with theoretical predictions. We then compare the results of one-, two- and three-dimensional simulation setups and simulations with different physical parameter sets.

A finite difference method which is second-order accurate in time and in space is proposed for two-dimensional fractional percolation equations. Using the Fourier transform, a general approximation for the mixed fractional derivatives is analyzed. An approach based on the classical Crank-Nicolson scheme combined with the Richardson extrapolation is used to obtain temporally and spatially second-order accurate numerical estimates. Consistency, stability and convergence of the method are established. Numerical experiments illustrating the effectiveness of the theoretical analysis are provided.

This paper presents a numerical and experimental study on hydrodynamic behavior of thin liquid films in rectangular domains. Three-dimensional computer simulations were performed using the lattice Boltzmann equation method (LBM). The liquid films laying on solid and liquid substrates are considered. The rupture of liquid films in computations is initiated via the thermocapillary (Marangoni) effect by applying an initial spatially localized temperature perturbation. The rupture scenario is found to depend on the shape of the temperature distribution and on the wettability of the solid substrate. For a wettable solid substrate, complete rupture does not occur: a residual thin liquid film remains at the substrate in the region of pseudo-rupture. For a non-wettable solid substrate, a sharp-peaked axisymmetric temperature distribution induces the rupture at the center of symmetry where the temperature is maximal. Axisymmetric temperature distribution with a flat-peaked temperature profile initiates rupture of the liquid film along a circle at some distance from the center of symmetry. The outer boundary of the rupture expands, while the inner liquid disk transforms into a toroidal figure and ultimately into an oscillating droplet.

We also apply the LBM to simulations of an evolution of one or two holes in liquid films for two-layer systems of immiscible fluids in a rectangular cell. The computed patterns are successfully compared against the results of experimental visualizations. Both the experiments and the simulations demonstrate that the initially circular holes evolved in the rectangular cell undergoing drastic changes of their shape under the effects of the surface tension and gravity. In the case of two interacting holes, the disruption of the liquid bridge separating two holes is experimentally observed and numerically simulated.

A novel mesh deformation technique is developed based on the Delaunay graph mapping method and the inverse distance weighting (IDW) interpolation. The algorithm maintains the advantages of the efficiency of Delaunay graph mapping mesh deformation while it also possesses the ability of better controlling the near surface mesh quality. The Delaunay graph is used to divide the mesh domain into a number of sub-domains. On each sub-domain, the inverse distance weighting interpolation is applied, resulting in a similar efficiency as compared to the fast Delaunay graph mapping method. The paper will show how the near-wall mesh quality is controlled and improved by the new method

Nanosecond (ns) pulsed dielectric barrier discharge (DBD) actuator in a laminar flat plate boundary layer is investigated numerically in an attempt to gain some new insights into the understanding of ns DBD actuation mechanism. Special emphasis is put on the examination, separation and comparison of behaviors of discharge induced micro shock wave and residual heat as well as on the investigation of response of external flow to the two effects. The shock wave is found to introduce highly transient, localized perturbation to the flow and be able to significantly alter the flow pattern shortly after its initiation. The main flow tends to quickly recover to close to its undisturbed state due to the transient nature of perturbation. However, with the shock decay and final disappearance, another perturbation source in the vicinity of discharge region, which contains contribution from both residual heat and shock, becomes increasingly pronounced and eventually develops into a perturbation wave train in the boundary layer. The perturbation is relatively weak and may not be a Tollmien-Schlichting (TS) wave and not trigger the laminar-turbulent transition of boundary layer. Instead, it is more likely to manipulate the flow stability to achieve the strong control authority of this kind of actuation in the case of flow separation control. In addition, a parametric study over the different electrical and hydrodynamic parameters is also conducted.

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.

A new approach for reducing error of the volume penalization method is proposed. The mask function is modified by shifting the interface between solid and fluid by toward the fluid region, where v and η are the viscosity and the permeability, respectively. The shift length is derived from the analytical solution of the one-dimensional diffusion equation with a penalization term. The effect of the error reduction is verified numerically for the one-dimensional diffusion equation, Burgers’ equation, and the two-dimensional Navier-Stokes equations. The results show that the numerical error is reduced except in the vicinity of the interface showing overall second-order accuracy, while it converges to a non-zero constant value as the number of grid points increases for the original mask function. However, the new approach is effectivewhen the grid resolution is sufficiently high so that the boundary layer,whose width is proportional to , is resolved. Hence, the approach should be used when an appropriate combination of ν and η is chosen with a given numerical grid.

The superconvergent property of the Discontinuous Galerkin (DG) method for linear hyperbolic systems of partial differential equations in one dimension is explained by relating the DG method to a particular continuous method, whose accuracy depends in part on a local analysis, and in part on information transferred from upwind elements.

A hybrid grid based second-order finite volume algorithm has been developed for Detached-Eddy Simulation (DES) of turbulent flows. To alleviate the effect caused by the numerical dissipation of the commonly used second order upwind schemes in implementing DES with unstructured computational fluid dynamics (CFD) algorithms, an improved second-order hybrid scheme is established through modifying the dissipation term of the standard Roe's flux-difference splitting scheme and the numerical dissipation of the scheme can be self-adapted according to the DES flow field information. By Fourier analysis, the dissipative and dispersive features of the new scheme are discussed. To validate the numerical method, DES formulations based on the two most popular background turbulence models, namely, the one equation Spalart-Allmaras (SA) turbulence model and the two equation k–ω Shear Stress Transport model (SST), have been calibrated and tested with three typical numerical examples (decay of isotropic turbulence, NACA0021 airfoil at 60° incidence and 65° swept delta wing). Computational results indicate that the issue of numerical dissipation in implementing DES can be alleviated with the hybrid scheme, the resolution for turbulence structures is significantly improved and the corresponding solutions match the experimental data better. The results demonstrate the potentiality of the present DES solver for complex geometries.