An explicit algorithm for imbedding solid boundaries in Cartesian grids for the reactive Euler equations

We present a local and point-wise scheme for imposing reflective boundary conditions to stationary internal boundaries for solving the reactive Euler equations on Cartesian grids. The scheme is presented in two and three dimensions and can run efficiently on parallel machines while still maintaining the same advantages over other methods for enforcing internal boundary conditions. Level sets are used to represent internal solid regions along with a new local node sorting algorithm that decouples internal boundary nodes by establishing their connectivity to other internal boundary nodes. This approach allows us to enforce boundary conditions via a direct procedure, removing the need to solve a coupled system of equations numerically. We examine the accuracy and fidelity of our internal boundary algorithm by simulating flows past various solid boundaries in two and three dimensions, showing good agreement between our numerical results and experimental data.     

The immersed boundary method for advection–electrodiffusion with implicit timestepping and local mesh refinement

We describe an immersed boundary method for problems of fluid–solute-structure interaction. The numerical scheme employs linearly implicit timestepping, allowing for the stable use of timesteps that are substantially larger than those permitted by an explicit method, and local mesh refinement, making it feasible to resolve the steep gradients associated with the space charge layers as well as the chemical potential, which is used in our formulation to control the permeability of the membrane to the (possibly charged) solute. Low Reynolds number fluid dynamics are described by the time-dependent incompressible Stokes equations, which are solved by a cell-centered approximate projection method. The dynamics of the chemical species are governed by the advection–electrodiffusion equations, and our semi-implicit treatment of these equations results in a linear system which we solve by GMRES preconditioned via a fast adaptive composite-grid (FAC) solver. Numerical examples demonstrate the capabilities of this methodology, as well as its convergence properties.     

A ghost fluid,level set methodology for simulating multiphase electrohydrodynamic flows with application to liquid fuel injection

In this paper, we present the development of a sharp numerical scheme for multiphase electrohydrodynamic (EHD) flows for a high electric Reynolds number regime. The electric potential Poisson equation contains EHD interface boundary conditions, which are implemented using the ghost fluid method (GFM). The GFM is also used to solve the pressure Poisson equation. The methods detailed here are integrated with state-of-the-art interface transport techniques and coupled to a robust, high order fully conservative finite difference Navier–Stokes solver. Test cases with exact or approximate analytic solutions are used to assess the robustness and accuracy of the EHD numerical scheme. The method is then applied to simulate a charged liquid kerosene jet.     

An Eulerian method for multi-component problems in non-linear elasticity with sliding interfaces

This paper is devoted to developing a multi-material numerical scheme for non-linear elastic solids, with emphasis on the inclusion of interfacial boundary conditions. In particular for colliding solid objects it is desirable to allow large deformations and relative slide, whilst employing fixed grids and maintaining sharp interfaces. Existing schemes utilising interface tracking methods such as volume-of-fluid typically introduce erroneous transport of tangential momentum across material boundaries. Aside from combatting these difficulties one can also make improvements in a numerical scheme for multiple compressible solids by utilising governing models that facilitate application of high-order shock capturing methods developed for hydrodynamics. A numerical scheme that simultaneously allows for sliding boundaries and utilises such high-order shock capturing methods has not yet been demonstrated. A scheme is proposed here that directly addresses these challenges by extending a ghost cell method for gas-dynamics to solid mechanics, by using a first-order model for elastic materials in conservative form. Interface interactions are captured using the solution of a multi-material Riemann problem which is derived in detail. Several different boundary conditions are considered including solid/solid and solid/vacuum contact problems. Interfaces are tracked using level-set functions. The underlying single material numerical method includes a characteristic based Riemann solver and high-order WENO reconstruction. Numerical solutions of example multi-material problems are provided in comparison to exact solutions for the one-dimensional augmented system, and for a two-dimensional friction experiment.     

3-D Quantum Transport Solver Based on the Perfectly Matched Layer and Spectral Element Methods for the Simulation of Semiconductor Nanodevices

A 3-D quantum transport solver based on the spectral element method (SEM) and perfectly matched layer (PML) is introduced to solve the 3-D Schr?dinger equation with a tensor effective mass. In this solver, the influence of the environment is replaced with the artificial PML open boundary extended beyond the contact regions of the device. These contact regions are treated as waveguides with known incident waves from waveguide mode solutions. As the transmitted wave function is treated as a total wave, there is no need to decompose it into waveguide modes, thus significantly simplifying the problem in comparison with conventional open boundary conditions. The spectral element method leads to an exponentially improving accuracy with the increase in the polynomial order and sampling points. The PML region can be designed such that less than -100 dB outgoing waves are reflected by this artificial material. The computational efficiency of the SEM solver is demonstrated by comparing the numerical and analytical results from waveguide and plane-wave examples, and its utility is illustrated by multiple-terminal devices and semiconductor nanotube devices.     

Faceting transitions in crystal growth and heteroepitaxial growth in the anisotropic phase-field crystal model

We modify the anisotropic phase-field crystal model(APFC),and present a semi-implicit spectral method to numerically solve the dynamic equation of the APFC model.The process results in the acceleration of computations by orders of magnitude relative to the conventional explicit finite-difference scheme,thereby,allowing us to work on a large system and for a long time.The faceting transitions introduced by the increasing anisotropy in crystal growth are then discussed.In particular,we investigate the morphological evolution in heteroepitaxial growth of our model.A new formation mechanism of misfit dislocations caused by vacancy trapping is found.The regular array of misfit dislocations produces a small-angle grain boundary under the right conditions,and it could significantly change the growth orientation of epitaxial layers.     

Efficient thermal field computation in phase-field models

We solve the phase-field equations in two dimensions to simulate crystal growth in the low undercooling regime. The novelty is the use of a fast solver for the free space heat equation to compute the thermal field. This solver is based on the efficient direct evaluation of the integral representation of the solution to the constant coefficient, free space heat equation with a smooth source term. The computational cost and memory requirements of the new solver are reasonable and no artificial boundary conditions are needed. This allows one to solve for the thermal field in a computational domain whose size depends only on the size of the growing crystal and not on the extent of the thermal field, which can result in significant computational savings in the low undercooling regime.     

h-Multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

Being implicit in time, the space-time discontinuous Galerkin discretization of the compressible Navier–Stokes equations requires the solution of a non-linear system of algebraic equations at each time-step. The overall performance, therefore, highly depends on the efficiency of the solver. In this article, we solve the system of algebraic equations with a h-multigrid method using explicit Runge–Kutta relaxation. Two-level Fourier analysis of this method for the scalar advection–diffusion equation shows convergence factors between 0.5 and 0.75. This motivates its application to the 3D compressible Navier–Stokes equations where numerical experiments show that the computational effort is significantly reduced, up to a factor 10 w.r.t. single-grid iterations.     

h-Multigrid for space-time discontinuous Galerkin discretizations of the compressible Navier–Stokes equations

Being implicit in time, the space-time discontinuous Galerkin discretization of the compressible Navier–Stokes equations requires the solution of a non-linear system of algebraic equations at each time-step. The overall performance, therefore, highly depends on the efficiency of the solver. In this article, we solve the system of algebraic equations with a h-multigrid method using explicit Runge–Kutta relaxation. Two-level Fourier analysis of this method for the scalar advection–diffusion equation shows convergence factors between 0.5 and 0.75. This motivates its application to the 3D compressible Navier–Stokes equations where numerical experiments show that the computational effort is significantly reduced, up to a factor 10 w.r.t. single-grid iterations.     

3D quantum transport solver based on the perfectly matched layer and spectral element methods for the simulation of semiconductor nanodevices

A 3D quantum transport solver based on the spectral element method (SEM) and perfectly matched layer (PML) is introduced to solve the 3D Schrödinger equation with a tensor effective mass. In this solver, the influence of the environment is replaced with the artificial PML open boundary extended beyond the contact regions of the device. These contact regions are treated as waveguides with known incident waves from waveguide mode solutions. As the transmitted wave function is treated as a total wave, there is no need to decompose it into waveguide modes, thus significantly simplifying the problem in comparison with conventional open boundary conditions. The spectral element method leads to an exponentially improving accuracy with the increase in the polynomial order and sampling points. The PML region can be designed such that less than −100 dB outgoing waves are reflected by this artificial material. The computational efficiency of the SEM solver is demonstrated by comparing the numerical and analytical results from waveguide and plane-wave examples and its utility is illustrated by multiple-terminal devices and semiconductor nanotube devices.     

使用非定常无反射边界条件模拟上游尾迹/叶片排干扰问题

本文采用显式时间推进方法求解二维Euler方程计算了上游尾迹与下游叶片排相互干扰而形成的复杂流场。根据Giles提出的理论编制了无反射边界条件非定常计算程序，取得了满意的效果。进口上游尾迹的特性白尾迹模型给定．预测到了尾迹在叶栅流道内的切割，迁移及剪切等重要的非定常现象。     

云计算网络中边界节点识别方法改进研究

目前,云计算网络为人们的生产和生活提供了各种应用和服务,网络边界节点的识别问题一直较难解决。传统的网络中边界节点类型复杂,边界部署成本高,较多感知模型和静态场景难以实现。为此,提出一种改进的云计算网络中边界节点识别方法,通过制定边界部署规则确定边界节点部署数量及要求,对边界节点感知漏洞进行修补,保证边界节点对网络区域内的全覆盖识别,最后设计出云计算网络识别模型,实现了云计算网络中边界节点正确识别。仿真实验表明,提出的边界节点识别方法在稳定性、识别率和识别数量上都比传统方法有优越性,具有应用价值。     

模拟多介质界面问题的虚拟流体方法综述

虚拟流体方法为模拟具有清晰物质界面的多介质流动问题提供了一种简便途径.尤其基于多介质Riemann问题解的修正虚拟流体方法及其变体，能够真实考虑到界面附近非线性波的相互作用和物质性质的影响，可以有效解决各种界面强间断等挑战性难题，具有巨大的工程应用潜力.文章重点回顾了虚拟流体方法的发展历史，总结和对比了各种代表性版本在模拟可压缩多介质流时的界面条件定义方式和多维推广方式，并介绍了该方法的设计原则和精度分析方面的研究进展.文章还回顾了该方法在其他更广泛和更具挑战性典型科学问题中的最新应用进展，并对方法的优势和特点进行了总结.      

Implementation of the Jacobian-free Newton–Krylov method for solving the first-order ice sheet momentum balance

We have implemented the Jacobian-free Newton–Krylov (JFNK) method for solving the first-order ice sheet momentum equation in order to improve the numerical performance of the Glimmer-Community Ice Sheet Model (Glimmer-CISM), the land ice component of the Community Earth System Model (CESM). Our JFNK implementation is based on significant re-use of existing code. For example, our physics-based preconditioner uses the original Picard linear solver in Glimmer-CISM. For several test cases spanning a range of geometries and boundary conditions, our JFNK implementation is 1.8–3.6 times more efficient than the standard Picard solver in Glimmer-CISM. Importantly, this computational gain of JFNK over the Picard solver increases when refining the grid. Global convergence of the JFNK solver has been significantly improved by rescaling the equation for the basal boundary condition and through the use of an inexact Newton method. While a diverse set of test cases show that our JFNK implementation is usually robust, for some problems it may fail to converge with increasing resolution (as does the Picard solver). Globalization through parameter continuation did not remedy this problem and future work to improve robustness will explore a combination of Picard and JFNK and the use of homotopy methods.     

Two-dimensional fracture analysis of piezoelectric material based on the scaled boundary node method

A scaled boundary node method(SBNM) is developed for two-dimensional fracture analysis of piezoelectric material,which allows the stress and electric displacement intensity factors to be calculated directly and accurately. As a boundarytype meshless method, the SBNM employs the moving Kriging(MK) interpolation technique to an approximate unknown field in the circumferential direction and therefore only a set of scattered nodes are required to discretize the boundary. As the shape functions satisfy Kronecker delta property, no special techniques are required to impose the essential boundary conditions. In the radial direction, the SBNM seeks analytical solutions by making use of analytical techniques available to solve ordinary differential equations. Numerical examples are investigated and satisfactory solutions are obtained, which validates the accuracy and simplicity of the proposed approach.     

Simulation of laser-generated ultrasonic wave propagation in solid media and air with application to NDE

Ultrasonic methods are well known as powerful and reliable tool for defect detection. In the previous decades focus and interest have been directed to non-contact sensors and methods, showing many advantages over contact techniques where inspection depends on contact conditions (pressure, coupling medium, contact area). The non-contact hybrid ultrasonic method described here is of interest for many applications, requiring periodic inspection in service or after manufacturing. Despite the potential impact of laser-generated ultrasound in many areas of industry, robust tools for studying the phenomenon are lacking and thus limit the design and optimization of non-destructive testing and evaluation techniques. Here a specific numerical method is presented to efficiently and accurately solve ultrasound wave propagation problems with frequencies in the MHz range traveling in relatively large bodies and through air. This work improves a previous numerical model where propagation of the acoustic waves through air had not been considered, allowing us to simulate the presence of a non-contact transducer in reception in order to simulate numerically the complete experimental setup. It is very important to limit the amount of air to be considered in the FE analyses; otherwise the computational cost would often exceed the resources available. A way to solve the problem is to implement non-reflecting boundary conditions. A non-reflecting boundary condition allows all outgoing waves to exit the domain at the boundary where they have been imposed without reflection; thus, it is possible to model only the portion of air between the non-contact transducer and the solid under testing. Several numerical and experimental analyses were conducted on a 136 lb AREMA rail; here we study in detail two fully non-contact testing configurations for the rail head and web. The information that can be acquired is very valuable for choosing the right setup and configuration when performing non-contact hybrid ultrasonic inspection.     

On the use of immersed boundary methods for shock/obstacle interactions

This paper describes the implementation of immersed boundary method using the direct-forcing concept to investigate complex shock–obstacle interactions. An interpolation algorithm is developed for more stable boundary conditions with easier implementation procedure. The values of the fluid variables at the embedded ghost-cells are obtained using a local quadratic scheme which involves the neighboring fluid nodes. Detailed discussions of the method are presented on the interpolation of flow variables, direct-forcing of ghost cells, resolution of immersed-boundary points and internal treatment. The method is then applied to a high-order WENO scheme to simulate the complex fluid–solid interactions. The developed solver is first validated against the theoretical solutions of supersonic flow past triangular prism and circular cylinder. Simulated results for test cases with moving shocks are further compared with the previous experimental results of literature in terms of triple-point trajectory and vortex evolution. Excellent agreement is obtained showing the accuracy and the capability of the proposed method for solving complex strong-shock/obstacle interactions for both stationary and moving shock waves.     

An MPI parallel level-set algorithm for propagating front curvature dependent detonation shock fronts in complex geometries

We present a parallel, two-dimensional, grid-based algorithm for solving a level-set function PDE that arises in Detonation Shock Dynamics (DSD). In the DSD limit, the detonation shock propagates at a speed that is a function of the curvature of the shock surface, subject to a set of boundary conditions applied along the boundaries of the detonating explosive. Our method solves for the full level-set function field, φ(x, y, t), that locates the detonation shock with a modified level-set function PDE that continuously renormalises the level-set function to a distance function based off of the locus of the shock surface, φ(x, y, t)=0. The boundary conditions are applied with ghost nodes that are sorted according to their connectivity to the interior explosive nodes. This allows the boundary conditions to be applied via a local, direct evaluation procedure. We give an extension of this boundary condition application method to three dimensions. Our parallel algorithm is based on a domain-decomposition model which uses the Message-Passing Interface (MPI) paradigm. The computational order of the full level-set algorithm, which is O(N ⁴), where N is the number of grid points along a coordinate line, makes an MPI-based algorithm an attractive alternative. This parallel model partitions the overall explosive domain into smaller sub-domains which in turn get mapped onto processors that are topologically arranged into a two-dimensional rectangular grid. A comparison of our numerical solution with an exact solution to the problem of a detonation rate stick shows that our numerical solution converges at better than first-order accuracy as measured by an L1-norm. This represents an improvement over the convergence properties of narrow-band level-set function solvers, whose convergence is limited to a floor set by the width of the narrow band. The efficiency of the narrow-band method is recovered by using our parallel model.     

Boundary symmetries in linear differential and integral equation problems applied to the self-consistent Green’s function formalism of acoustic and electromagnetic scattering

Explicit symmetry relations for the Green’s function subject to homogeneous boundary conditions are derived for arbitrary linear differential or integral equation problems in which the boundary surface has a set of symmetry elements. For corresponding homogeneous problems subject to inhomogeneous boundary conditions implicit symmetry relations involving the Green’s function are obtained. The usefulness of these symmetry relations is illustrated by means of a recently developed self-consistent Green’s function formalism of electromagnetic and acoustic scattering problems applied to the exterior scattering problem. One obtains explicit symmetry relations for the volume Green’s function, the surface Green’s function, and the interaction operator, and the respective symmetry relations are shown to be equivalent. This allows us to treat boundary symmetries of volume-integral equation methods, boundary-integral equation methods, and the T matrix formulation of acoustic and electromagnetic scattering under a common theoretical framework. By specifying a specific expansion basis the coordinate-free symmetry relations of, e.g., the surface Green’s function can be brought into the form of explicit symmetry relations of its expansion coefficient matrix. For the specific choice of radiating spherical wave functions the approach is illustrated by deriving unitary reducible representations of non-cubic finite point groups in this basis, and by deriving the corresponding explicit symmetry relations of the coefficient matrix. The reducible representations can be reduced by group-theoretical techniques, thus bringing the coefficient matrix into block-diagonal form, which can greatly reduce ill-conditioning problems in numerical applications.