Runge–Kutta discontinuous Galerkin method using WENO limiters II: Unstructured meshes

In [J. Qiu, C.-W. Shu, Runge–Kutta discontinuous Galerkin method using WENO limiters, SIAM Journal on Scientific Computing 26 (2005) 907–929], Qiu and Shu investigated using weighted essentially non-oscillatory (WENO) finite volume methodology as limiters for the Runge–Kutta discontinuous Galerkin (RKDG) methods for solving nonlinear hyperbolic conservation law systems on structured meshes. In this continuation paper, we extend the method to solve two-dimensional problems on unstructured meshes, with the goal of obtaining a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, nonoscillatory shock transition for RKDG methods. Numerical results are provided to illustrate the behavior of this procedure.     

Local DG method using WENO type limiters for convection–diffusion problems

The local discontinuous Galerkin (LDG) method is a spatial discretization procedure for convection–diffusion equations, which employs useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers serving as numerical fluxes and limiters, which is termed as Runge–Kutta LDG (RKLDG) when TVD Runge–Kutta method is applied for time discretization. It has the advantage of flexibility in handling complicated geometry, h-p adaptivity, and efficiency of parallel implementation and has been used successfully in many applications. However, the limiters used to control spurious oscillations in the presence of strong shocks are less robust than the strategies of essentially non-oscillatory (ENO) and weighted ENO (WENO) finite volume and finite difference methods. In this paper, we investigated RKLDG methods with WENO and Hermite WENO (HWENO) limiters for solving convection–diffusion equations on unstructured meshes, with the goal of obtaining a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, non-oscillatory shock transition. Numerical results are provided to illustrate the behavior of these procedures.     

用偏压限制器控制KT－5C托卡马克等离子体边缘电场的实验研究

在托卡马克ＫＴ－５Ｃ上，我们首次采用可灵活组合的多块限制器进行了施加偏压以控制等离子体边缘电场的实验，结果表明，在本装置上限制器加正偏压优于加负偏压，正偏压改变等离子体边缘电场效果更明显。偏压限制器收集面积、纵场和偏置电压的大小对偏压效果的影响都是敏感的。改变限制器收集面积和偏压导致的边缘电场的变化最终将趋于饱和，另外发现，偏压限制器与等离子体柱的相对位置对等离子体边缘电场的影响也是重要的。     

Water diffusion in biomedical systems as related to magnetic resonance imaging

Water diffusion within the brain is studied numerically for various clinical conditions. The numerical procedure used in this work is based on the Galerkin weighted residual method of finite-element formulation. A wide range of pertinent parameters such as Lewis number, cell volume, and the buoyancy ratio are considered in the present study. Comparisons with previously published work show excellent agreement. The results show that the diffusion coefficient, cell volume, and the buoyancy ratio play significant roles on the characterization of the mass and heat transfer mechanisms within the cell. Concentration maps are developed for various clinical conditions. Pertinent results for the streamlines, isotherms and the mass and heat transfer rates in terms of the average Sherwood and Nusselt numbers are presented and discussed for different parametric values. Experimental tests are also conducted to produce an 8 Tesla image which is compared with our numerical simulation. The present study provides essential maps for brain disorders classified based on several pertinent clinical attributes.     

Biased trapping issue on weighted hierarchical networks

In this paper, we present trapping issues of weight-dependent walks on weighted hierarchical networks which are based on the classic scale-free hierarchical networks. Assuming that edge’s weight is used as local information by a random walker, we introduce a biased walk. The biased walk is that a walker, at each step, chooses one of its neighbours with a probability proportional to the weight of the edge. We focus on a particular case with the immobile trap positioned at the hub node which has the largest degree in the weighted hierarchical networks. Using a method based on generating functions, we determine explicitly the mean first-passage time (MFPT) for the trapping issue. Let parameter a (0 < a < 1) be the weight factor. We show that the efficiency of the trapping process depends on the parameter a; the smaller the value of a, the more efficient is the trapping process.     

Hybrid weighted essentially non-oscillatory schemes with different indicators

A key idea in finite difference weighted essentially non-oscillatory (WENO) schemes is a combination of lower order fluxes to obtain a higher order approximation. The choice of the weight to each candidate stencil, which is a nonlinear function of the grid values, is crucial to the success of WENO schemes. For the system case, WENO schemes are based on local characteristic decompositions and flux splitting to avoid spurious oscillation. But the cost of computation of nonlinear weights and local characteristic decompositions is very high. In this paper, we investigate hybrid schemes of WENO schemes with high order up-wind linear schemes using different discontinuity indicators and explore the possibility in avoiding the local characteristic decompositions and the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong shocks. The idea is to identify discontinuity by an discontinuity indicator, then reconstruct numerical flux by WENO approximation in discontinuous regions and up-wind linear approximation in smooth regions. These indicators are mainly based on the troubled-cell indicators for discontinuous Galerkin (DG) method which are listed in the paper by Qiu and Shu (J. Qiu, C.-W. Shu, A comparison of troubled-cell indicators for Runge–Kutta discontinuous Galerkin methods using weighted essentially non-oscillatory limiters, SIAM Journal of Scientific Computing 27 (2005) 995–1013). The emphasis of the paper is on comparison of the performance of hybrid scheme using different indicators, with an objective of obtaining efficient and reliable indicators to obtain better performance of hybrid scheme to save computational cost. Detail numerical studies in one- and two-dimensional cases are performed, addressing the issues of efficiency (less CPU time and more accurate numerical solution), non-oscillatory property.     

MOL solution of DOM for transient radiative transfer in 3-D scattering media

A methodology based on the method of lines solution of discrete ordinates method for solution of the 3-D transient radiative transfer equation is introduced. The method is applied to the prediction of transient and steady state transmittances in a cubical enclosure containing purely scattering medium and validated against Monte Carlo solutions from the literature. The flexibility of the method for implementation of linear spatial differencing schemes, flux limiters and weighted essentially non-oscillatory methods is demonstrated. Van Leer flux limiter is found to provide stable, accurate and efficient solutions.     

A Comparison of the Performance of Limiters for Runge-Kutta Discontinuous Galerkin Methods

Discontinuities usually appear in solutions of nonlinear conservation laws even though the initial condition is smooth, which leads to great difficulty in computing these solutions numerically. The Runge-Kutta discontinuous Galerkin (RKDG) methods are efficient methods for solving nonlinear conservation laws, which are high-order accurate and highly parallelizable, and can be easily used to handle complicated geometries and boundary conditions. An important component of RKDG methods for solving nonlinear conservation laws with strong discontinuities in the solution is a nonlinear limiter, which is applied to detect discontinuities and control spurious oscillations near such discontinuities. Many such limiters have been used in the literature on RKDG methods. A limiter contains two parts, first to identify the "troubled cells", namely, those cells which might need the limiting procedure, then to replace the solution polynomials in those troubled cells by reconstructed polynomials which maintain the original cell averages (conservation). [SIAM J. Sci. Comput., 26 (2005), pp. 995-1013] focused on discussing the first part of limiters. In this paper, focused on the second part, we will systematically investigate and compare a few different reconstruction strategies with an objective of obtaining the most efficient and reliable reconstruction strategy. This work can help with the choosing of right limiters so one can resolve sharper discontinuities, get better numerical solutions and save the computational cost.     

Multi-dimensional limiting for high-order schemes including turbulence and combustion

In the present paper a fourth/fifth order upwind biased limiting strategy is presented for the simulation of turbulent flows and combustion. Because high order numerical schemes usually suffer from stability problems and TVD approaches often prevent convergence to machine accuracy the multi-dimensional limiting process (MLP) [1] is employed. MLP uses information from diagonal volumes of a discretization stencil. It interacts with the TVD limiter in such a way, that local extrema at the corner points of the volume are avoided. This stabilizes the numerical scheme and enables convergence in cases, where standard limiters fail to converge. Up to now MLP has been used for inviscid and laminar flows only. In the present paper this technique is applied to fully turbulent sub- and supersonic flows simulated with a low Reynolds-number turbulence closure. Additionally, combustion based on finite-rate chemistry is investigated. An improved MLP version (MLP^ld, low diffusion) as well as an analysis of its capabilities and limitations are given. It is demonstrated, that the scheme offers high accuracy and robustness while keeping the computational cost low. Both steady and unsteady test cases are investigated.     

被动光学限幅器的机制与研究进展

被动光学限幅器是利用介质的非线性光学性质实现对光的强度进行控制的器件，它在科研，军事和民用方面都具有广泛的应用，文章讨论了被动光学限幅器的反饱和吸收，双光子吸收，自由载流子吸收，非线性折射，光致散射和光折变机制，每种机制在光限幅器应用中的有效性及局限性，介绍了国内外研究状况及自己的工作；指出了今后被动光学限幅器研究的发展方向。     

Multi-dimensional limiting process for hyperbolic conservation laws on unstructured grids

The present paper deals with an efficient and accurate limiting strategy for the multi-dimensional hyperbolic conservation laws on unstructured grids. The multi-dimensional limiting process (MLP) which has been successfully proposed on structured grids is extended to unstructured grids. The basic idea of the proposed limiting strategy is to control the distribution of both cell-centered and cell-vertex physical properties to mimic multi-dimensional nature of flow physics, which can be formulated to satisfy so called the MLP condition. The MLP condition can guarantee high-order spatial accuracy and improved convergence without yielding spurious oscillations. Starting from the MUSCL-type reconstruction on unstructured grids followed by the efficient implementation of the MLP condition, MLP slope limiters on unstructured meshes are obtained.Thanks to its superior limiting strategy and maximum principle satisfying characteristics, the newly developed MLP on unstructured grids is quite effective in controlling numerical oscillations as well as accurate in capturing multi-dimensional flow features. Numerous test cases are presented to validate the basic features of the proposed approach.     

Volume fraction flux approximation in a two-fluid flow

The broken dam problem flow is tested to check accuracy of different procedures for gas-liquid interface resolution based on solution of the additional equation for the volume fraction of liquid phase. The study is focused on the numerical schemes used to approximate advection fluxes of this equation. In particular, the MUSCL scheme with QUICK interpolants and compressive minmod TVD limiters with the slope modification technique for the volume fraction fluxes is applied, as well as the upwind-downwind donor acceptor procedure designed in the VOF method. As the first stage, the quite simple and explicit procedure adopting the artificial compressibility method is used to solve the velocity and pressure equations. Computations are initially performed with a careful grid and time step independence studies. Importance of the wall boundary condition is also discussed. To present free surface motion, results of numerical investigation are shown in terms of contour plots for the volume fraction at successive times, as well as surge front and column height positions versus time. The work was financially supported by the President of the Russian Federation (NSh No. 454.2008.1), Russian Foundation for Basic Research (Grant No. 06-01-00724), Integration Project of SB RAS (No. 2-16), and National Scientific Council of Taiwan (NSC, R.O.C., contract NSC-92-2212-E006-102).     

Scaling of average receiving time and average weighted shortest path on weighted Koch networks

In this paper we present weighted Koch networks based on classic Koch networks. A new method is used to determine the average receiving time (ART), whose key step is to write the sum of mean first-passage times (MFPTs) for all nodes to absorption at the trap located at a hub node as a recursive relation. We show that the ART exhibits a sublinear or linear dependence on network order. Thus, the weighted Koch networks are more efficient than classic Koch networks in receiving information. Moreover, average weighted shortest path (AWSP) is calculated. In the infinite network order limit, the AWSP depends on the scaling factor. The weighted Koch network grows unbounded but with the logarithm of the network size, while the weighted shortest paths stay bounded.     

高分辨率间断有限元方法

间断有限元方法是集高分辨率有限差分方法和有限体积方法的优点发展起来的一种数值方法,在计算流体动力学问题上显示了优良的效能.利用守恒问题给出间断有限元方法的基本概念和过程,利用简单算例给出该方法的精度分析和限制器对精度的影响,并给出浅水波问题、交通流问题和波传播问题的数值模拟结果,进一步,综合评介该方法在椭圆、抛物、对流扩散、Hamilton-Jacobi方程、Navier-Stokes方程等的实际应用进展.     

电磁辐射作用于计算机主板的模拟及效应评估

 提出了一种电磁辐射（EMR）效应评估的方法：通过区域分解并且引入子域敏感因子,采用电场强度的加权平均作为EMR效应评估的参考指标。以一款普通计算机主板作为研究对象,使用时域有限差分法（FDTD）进行模拟计算,得到了EMR作用下计算机主板耦合的电磁场分布。基于该效应评估方法,比较了3~12 GHz 的平面电磁波在0°~90°入射时对计算机主板的效应。数值模拟结果表明：随着平面电磁波入射角度的增加,EMR对计算机主板的影响趋于减弱;EMR频率的变化对效应没有显著影响。当EMR入射角度为40°时,最高主板表面耦合的电场强度达到最大值;0°入射时,主板上平均电场强度达到最大值;随着入射EMR频率的升高,主板表面的最大电场强度趋于减弱。     

Bipolar-power-transistor-based limiter for high frequency ultrasound imaging systems

High performance limiters are described in this paper for applications in high frequency ultrasound imaging systems. Limiters protect the ultrasound receiver from the high voltage (HV) spikes produced by the transmitter. We present a new bipolar power transistor (BPT) configuration and compare its design and performance to a diode limiter used in traditional ultrasound research and one commercially available limiter. Limiter performance depends greatly on the insertion loss (IL), total harmonic distortion (THD) and response time (RT), each of which will be evaluated in all the limiters. The results indicated that, compared with commercial limiter, BPT-based limiter had less IL (−7.7 dB), THD (−74.6 dB) and lower RT (43 ns) at 100 MHz. To evaluate the capability of these limiters, they were connected to a 100 MHz single element transducer and a two-way pulse-echo test was performed. It was found that the −6 dB bandwidth and sensitivity of the transducer using BPT-based limiter were better than those of the commercial limiter by 22% and 140%, respectively. Compared to the commercial limiter, BPT-based limiter is shown to be capable of minimizing signal attenuation, RT and THD at high frequencies and is thus suited for high frequency ultrasound applications.     

Non-linear coupled slosh dynamics of liquid-filled laminated composite containers: a two dimensional finite element approach

This paper brings into focus some of the interesting effects arising from the non-linear motion of the liquid free surface, due to sloshing, in a partially filled laminated composite container along with the associated coupling due to fluid-structure interaction effects. The finite element method based on two-dimensional fluid and structural elements is used for the numerical simulation of the problem. A numerical scheme is developed on the basis of a mixed Eulerian-Lagrangian approach, with velocity potential as the unknown nodal variable in the fluid domain and displacements as the unknowns in the structure domain. The FE formulation based on Galerkin weighted residual method along with an iterative solution procedure are explained in detail followed by a few numerical examples. Numerical results obtained by the present investigation for the rigid containers are first compared with the existing solutions to validate the code for non-linear sloshing without fluid-structure coupling. Thereafter the computational procedures are advanced to obtain the coupled interaction effect of non-linear sloshing in laminated composite containers.     

Investigation of nonlinear characteristics of intensity limiters of high-power laser radiation

The problem of determining nonlinear-optical characteristics of working substances of limiters of high-power laser radiation is analyzed. A new method of simultaneous calculation of the beam-waist radius and nonlinear absorption coefficient from the Z-scan data is proposed. The value of nonlinear optical characteristics of ZnSe, porphyrin-graphene (Graphene-TPP), fullerene-graphene (Graphene-60), polyethylene oxide containing multilayer carbon nanotubes (PEO/MWNT), along with polymethine dyes PD-792 and PD-7098 and dicyanomethylene-pyran dyes DCM-627 and DCM-684, are obtained. Different working substances are compared for the first time under the same conditions by plotting output characteristics of laser-radiation limiters. The change of the laser-pulse shape after propagation through the limiters with different working substances is demonstrated.     

饱和铁心型超导限流器

饱和铁心型超导限流器具有限流功能可靠、输电稳态损耗小、限流恢复时间短等优异特性。根据其工作原理,一方面,用于直流励磁的超导绕组自身不承受短路电流冲击,这避免了其它类型超导限流器中由于超导绕组失超所带来的诸多技术难题;另一方面,其主体电抗器可以借鉴成熟的铁心电抗器技术。因此,饱和铁心型超导限流器的以其优异的性能特性和成熟的工程实现技术,成为迄今为止人们研究的各类超导限流器中的最具实用价值的一种。在电网容量不断扩大,智能化水平不断提高的今天,饱和铁心型超导限流器有非常广阔的应用前景。文中将叙述饱和铁心型超导限流器的基本原理、结构形式、功能特性,并对已在云南普吉变电站挂网运行的35kV/90MVA超导限流器进行简要介绍。     

超导电力技术新进展及其未来发展的思考

近年来高温超导材料研究取得很大进展,它在电力领域的应用研究已受到广泛关注,一些示范样机,诸如高温超导输电电缆、变压器、故障电流限制器、电机和储能装置已经研制成功并投入示范性试验.超导技术是21世纪具有战略经济意义的高新技术,文章将介绍高温超导电力应用研究的新进展及其未来发展的思考。