A Lagrangian surface hopping algorithm is implemented to study the two dimensional massless Dirac equation for Graphene with an electrostatic potential, in the semiclassical regime. In this problem, the crossing of the energy levels of the system at Dirac points requires a particular treatment in the algorithm in order to describe the quantum transition – characterized by the Landau-Zener probability – between different energy levels. We first derive the Landau-Zener probability for the underlying problem, then incorporate it into the surface hopping algorithm. We also show that different asymptotic models for this problem derived in [O. Morandi, F. Schurrer, J. Phys. A: Math. Theor. 44 (2011) 265301] may give different transition probabilities. We conduct numerical experiments to compare the solutions to the Dirac equation, the surface hopping algorithm, and the asymptotic models of [O. Morandi, F. Schurrer, J. Phys. A: Math. Theor. 44 (2011) 265301].

Determining the drag of a flow over a rough surface is a guiding example for the need to take geometric micro-scale effects into account when computing a macro-scale quantity. A well-known strategy to avoid a prohibitively expensive numerical resolution of micro-scale structures is to capture the micro-scale effects through some effective boundary conditions posed for a problem on a (virtually) smooth domain. The central objective of this paper is to develop a numerical scheme for accurately capturing the micro-scale effects at essentially the cost of twice solving a problem on a (piecewise) smooth domain at affordable resolution. Here and throughout the paper "smooth" means the absence of any micro-scale roughness. Our derivation is based on a "conceptual recipe" formulated first in a simplified setting of boundary value problems under the assumption of sufficient local regularity to permit asymptotic expansions in terms of the micro-scale parameter.
The proposed multiscale model relies then on an upscaling strategy similar in spirit to previous works by Achdou et al. [1], Jäger and Mikelic [29, 31], Friedmann et al. [24, 25], for incompressible fluids. Extensions to compressible fluids, although with several noteworthy distinctions regarding e.g. the "micro-scale size" relative to boundary layer thickness or the systematic treatment of different boundary conditions, are discussed in Deolmi et al. [16,17]. For proof of concept the general strategy is applied to the compressible Navier-Stokes equations to investigate steady, laminar, subsonic flow over a flat plate with partially embedded isotropic and anisotropic periodic roughness imposing adiabatic and isothermal wall conditions, respectively. The results are compared with high resolution direct simulations on a fully resolved rough domain.

Offline computation is an essential component in most multiscale model reduction techniques. However, there are multiscale problems in which offline procedure is insufficient to give accurate representations of solutions, due to the fact that offline computations are typically performed locally and global information is missing in these offline information. To tackle this difficulty, we develop an online local adaptivity technique for local multiscale model reduction problems. We design new online basis functions within Discontinuous Galerkin method based on local residuals and some optimally estimates. The resulting basis functions are able to capture the solution efficiently and accurately, and are added to the approximation iteratively. Moreover, we show that the iterative procedure is convergent with a rate independent of physical scales if the initial space is chosen carefully. Our analysis also gives a guideline on how to choose the initial space. We present some numerical examples to show the performance of the proposed method.

In this paper we consider a discontinuous Galerkin discretization of the ideal magnetohydrodynamics (MHD) equations on unstructured meshes, and the divergence free constraint (∇·B = 0) of its magnetic field B. We first present two approaches for maintaining the divergence free constraint, namely the approach of a locally divergence free projection inspired by locally divergence free elements [19], and another approach of the divergence cleaning technique given by Dedner et al. [15]. By combining these two approaches we obtain an efficient method at the almost same numerical cost. Finally, numerical experiments are performed to show the capacity and efficiency of the scheme.

In this paper, an alternative lattice Boltzmann (LB) model for incompressible flows is proposed. By modifying directly the moments of the equilibrium distribution function (EDF), the continuous expression of the EDF in tensor Hermite polynomials is derived using the moment expansion and then discretized with the discrete velocity vectors of the D2Q9 lattice. The present model as well as its counterpart, the incompressible LB model proposed by Guo, reproduces the incompressible Navier-Stokes (N-S) equations for both steady and unsteady flows. Besides, an alternative pressure formula, which represents the pressure as the diagonal part of the stress tensor, is adopted in the present model. Furthermore, in order to enhance the stability of the present LB model, an additional relaxation time pertaining to the non-hydrodynamic mode is added to the BGK collision operator. The present LB model is validated by two benchmark tests: the cavity flow with different Reynolds number (Re) and the flow past an impulsively started cylinder at Re=40 and 550.

In this paper, numerical simulations of solid particle erosion phenomena on H-type finned circular/elliptic tube surface, which is of great significance to the antiwear design of heat exchanger, are presented. The Eulerian-Lagrangian approach is applied to simulate the dilute gas-solid flow through H-type finned circular/elliptic tubes. A semi-empirical model is adopted to predict the erosion rate. The dynamics behavior of the entrained solid particles in the flow is presented. The geometry of eroded tube surface is changed with the predicted erosion which is taken into account by a UDF and the flow field is re-solved for the eroded tube surface at every time step. The influences of ten parameters (the tube bundle arrangement, particle size, particle concentration, fluid Reynolds number, fin thickness, fin pitch, fin length, fin width, slit width and the transverse tube pitch) on the maximum erosion depth of the H-type circular/elliptic finned tube surface are investigated. Using H-type finned elliptic tube surface can effectively reduce the erosion rate of tube surface compared with that using H-type finned circular tube surface. The erosion in in-line arrangement is less severe than that in staggered arrangement. With the increase of particle size, particle concentration and the fluid Reynolds number, the erosion rate of the tube surface rises. The numerically predicted effect of Reynolds number is in good agreement with previous test data. Among the six geometry parameters, the most influential parameter is the transverse tube pitch.

This paper presents an approach using the method of separation of variables applied to 2D Helmholtz equations in the Cartesian coordinate. The solution is then computed by a series of solutions resulted from solving a sequence of 1D problems, in which the 1D solutions are computed using pollution free difference schemes. Moreover, non-polluted numerical integration formulae are constructed to handle the integration due to the forcing term in the inhomogeneous 1D problems. Consequently, the computed solution does not suffer the pollution effect. Another attractive feature of this approach is that a direct method can be effectively applied to solve the tridiagonal matrix resulted from numerical discretization of the 1D Helmholtz equation. The method has been tested to compute 2D Helmholtz solutions simulating electromagnetic scattering from an open large cavity and rectangular waveguide.

The 2D Maxwell eigenproblems are studied from a new point of view. An electromagnetic problem is cast from the Lagrangian system with single variable into the Hamiltonian system with dual variables. The electric and magnetic components transverse to the wave propagation direction are treated as dual variables to each other. Higher order curl-conforming and divergence-conforming vector basis functions are used to construct dual vector spectral elements. Numerical examples demonstrate some unique advantages of the proposed method.

In this paper, the numerical error of two widely used methods for remapping of discrete quantities from one computational mesh to another is investigated. We compare the intuitive, but resource intensive method utilizing intersections of computational cells with the faster and simpler swept-region-based method. Both algorithms are formally second order accurate, however, they are known to produce slightly different quantity profiles in practical applications. The second-order estimate of the error formula is constructed algebraically for both algorithms so that their local accuracy can be evaluated. This general estimate is then used to assess the dependence of the performance of both methods on parameters such as the second derivatives of the remapped distribution, mesh geometry or mesh movement. Due to the complexity of such analysis, it is performed on a set of simplified elementary mesh patterns such as cell corner expansion, rotation or shear. On selected numerical tests it is demonstrated that the swept-based method can distort a symmetric quantity distribution more substantially than the intersection-based approach when the computational mesh moves in an unsuitable direction.

In the past two decades, a rigorous solution for the shape of human red blood cell (RBC) with a negative spontaneous curvature $c_0$ has been derived with the Helfrich model under the condition that both the osmotic pressure ∆$p$ and tensile stress $λ$ are equal to zero. By fitting the experimentally observed shape of RBC, $c_0$$R_0$ has been predicted to be −1.62, the minus golden ratio, where $R_0$ is the radius of a sphere which has the same surface area as RBC. In this paper, we verify this prediction by comparing experimental data with an analytical equation describing the relation between volume and surface area. Furthermore, it is also found $ρ$_max /$ρ_B$ ≈ 1.6 with $ρ$_max the maximal radius and $ρ_B$ the characteristic radius of RBC, showing an approximate beautiful golden cross section of RBC. On the basis of a complete numerical calculation, we propose a mechanism behind the beauty of the minus golden ratio that $c_0$$R_0$ results from the balance between the minimization of the surface area and the requirement of adequate deformability of RBC to allow it passing through the spleen, the so called "physical fitness test".

A consistent dual-mesh hybrid LES/RANS framework for turbulence modeling has been proposed recently (H. Xiao, P. Jenny, A consistent dual-mesh framework for hybrid LES/RANS modeling, J. Comput. Phys. 231 (4) (2012)). To better enforce componentwise Reynolds stress consistency between the LES and the RANS simulations, in the present work the original hybrid framework is modified to better exploit the advantage of more advanced RANS turbulence models. In the new formulation, the turbulent stresses in the filtered equations in the under-resolved regions are directly corrected based on the Reynolds stresses provided by the RANS simulation. More precisely, the new strategy leads to implicit LES/RANS consistency, where the velocity consistency is achieved indirectly via imposing consistency on the Reynolds stresses. This is in contrast to the explicit consistency enforcement in the original formulation, where forcing terms are added to the filtered momentum equations to achieve directly the desired average velocity and velocity fluctuations. The new formulation keeps the averaging procedure for the filtered quantities and at the same time preserves the ability of the original formulation to conform with the physical differences between LES and RANS quantities. The modified formulation is presented, analyzed, and then evaluated for plane channel flow and flow over periodic hills. Improved predictions are obtained compared with the results obtained using the original formulation.

A method for non-rigid image registration that is suitable for large deformations is presented. Conventional registration methods embed the image in a B-spline object, and the image is evolved by deforming the B-spline object. In this work, we represent the image using B-spline and deform the image using a composition approach. We also derive a computationally efficient algorithm for calculating the B-spline coefficients and gradients of the image by adopting ideas from signal processing using image filters. We demonstrate the application of our method on several different types of 2D and 3D images and compare it with existing methods.