1.Introduction

In the past few decades,high order finite difference methods have been developed in ComputationalAeroAcoustics(CAA).Most of these methods evolve from the Dispersion-Relation-Preserving(DRP)scheme proposed by Tam and Webb,1or the compact finite difference scheme by Lele.2Though having gained great success in CAA,the high order finite difference methods cannot be applied to practical CAA multi-scale problems with complex geometry difficultly,since it requires that the flux across the grid interface has to be continuous.When facing aeroacoustics problems with complex geometries,multi-size structured grids are usually employed,which greatly improves the capability of application in some practical Engineering problems such as aircraft high lift systems and landing gears.However,how to construct the interface flux efficiently and stably is still a hot issue.

Aimed at multi-scale CAA problems,Tam and Kurbatskii3proposed low-dissipation&low-dispersion DRP scheme by optimizing the interpolation parameters with comparing multiple grid scales ratio.Besides,Tam improved the stencil DRP scheme and artificial selective damping terms which are used in the interface flux construction,in order to compute the flux accurately.

WENO scheme was initially proposed by Liu and Osher4based on Essentially Non-Oscillatory(ENO)scheme.5The weights depend on the local smoothness of the data.Smoothness measurements cause stencils that span large flow field gradients to have relative small weights; any candidate stencil containing a shock receives a nearly zero weight.In completely smooth regions,weights revert to optimal values,where optimal value is defined by maximum order of accuracy or maximum bandwidth.In the following years,Jiang and Shu6,7cast WENO scheme in to finite difference form.This scheme,which is referred to as WENO-JS hereinafter,can be capable of resolving shocks with high resolution.However,WENO-JS is too dissipative for the detailed simulation of turbulent flow.Martin et al.8,9developed two new formulations of a symmetric WENO method for the direct numerical simulation of compressible turbulence.The schemes are designed to maximize order of accuracy and bandwidth,while minimizing dissipation.The formulations and the corresponding coefficients are introduced.Numerical solutions to canonical flow problems are used to determine the dissipation and bandwidth properties of the numerical schemes.In addition,the suitability and accuracy of the bandwidth-optimized schemes for direct numerical simulations of turbulent flows are assessed in decaying isotropic turbulence and supersonic turbulent boundary layers. Wu et al.10presenteda maximum order preserving optimized WENO scheme which is a weighted average of the maximum order scheme and an optimized scheme.Hou et al.11modified the weights of WENO scheme to be smoother,which can eliminate the fluctuation within the grid variation.Lin and Hu12presented two groups of WENO schemes based on the dissipation and dissipation,which are investigated for computational aeroacoustics.

In this paper,the modified WENO scheme is deduced firstly.Then,we apply modified WENO scheme to establish the model of interface flux.On this basis,the accuracy of the model in multi-scale grid problems has been verified by several standard verification cases.The numerical simulation results with interface flux method are presented in Section 5,and the brief conclusions are given in Section 6.

2.Optimized WENO scheme

In the design of a traditional WENO scheme,the practice is to maximize the order of accuracy of the WENO scheme given the size of the difference stencil.However,high order schemes may not be the best for CAA problems.Aimed at short waves,the finite difference scheme needs to maintain its lowdissipation and low-dispersion property,as shown by Tam and Webb.To fix the idea,the initial value problem associated with the scalar wave equation is considered as follows:

7.Zhang SH,Jiang GS,Shu CW.Improvement of convergence to steady state solutions of Euler equations of Euler equations with the WENO schemes.J Sci Comput 2011;47(2):216–38.

Given a uniform grid xi=iΔx with the same grid spacing Δx,the semi-discretized form of Eq.(1)is

where i is integer number,aui+1/2and aui-1/2are numerical fluxes which depend on k(r+s+1=k,r≥0,s≥0;r and s are different integer numbers respectively)grid points including xiitself,i.e.,

Here crjcould be obtained by achieving k-th order accuracy in Taylor series truncation expansion.Considering five-point stencil WENO-JS scheme,according to Ref.6,it would be divided to 3 candidate stencils{S1，S2，S3}.Each of candidate stencils has three nodes,as shown in Fig.1.The second order polynomial approximation uk(x)=akx2+bkx+ckcould be obtained by using the function value of u(x)in each candidate stencil,where k=1，2，3.

In each candidate stencil,the formula for uk(x)could be obtained by Taylor series expansion at xjas follows:

When x=Δx/2,Eq.(4)can be approximated by

Then the numerical flux in five-point-stencil WENO scheme could be denoted with weights as

where ω is weight coefficient.

As for CAA problems,it is very important to simulate the short waves with a limited stencil scheme as much as possible.We may equate Eq.(6)and Eq.(1)to yield

Fig.1 Five-point-stencil WENO-JS scheme.

Fig.2 Integrated error E of different ω0values.

Table 1 Different free parameter ω0and λ.

Case name λ ω0WENO_opt1 0.85 0.1405944 WENO_opt2 0.7 0.1205971 WENO_opt3 0.55 0.0988999 WENO_opt4 0.4 0.752431 WENO_opt5 0.25 0.0492365 WENO_opt6 0.1 0.0207083 WENO_JS 0.1

where

2.Lele SK.Compact finite difference schemes with spectral-like resolution.J Comput Phys 1992;103(1):16–42.

InEq.(8),wechooseω0as a free para meter.According to Ref.12,the original definition for the integrated error is in the form ofThe integrated error minimizes the distances both betweenand αΔx and betweenand zero in the form of the L2norm.However,the distance betweenand a Gaussian function is minimized instead of that betweenand zerosince the former distance is almost zero when the value of αΔx is far from π.Therefore,ω0is chosen to optimize the integrated error Edefined as(σ =0.2675π)

whereis the real part ofandis the imaginary part ofFor a given λ,ω0is determined by minimizing the integrated error E.In Fig.2,the optimization results with variation of ω0values are presented.In Table 1,the values of optimized parameter ω0are given.WENO_opt1–6 are different optimized WENO schemes.

Fig.3 shows the detailed comparison of the dispersion&dissipation property of various WENO schemes.WENO_opt1 scheme could achieve the best sound wave resolution with the least number of grid nodes and the least phase error.The amplitude error of WENO_opt6 scheme is the smallest.If the tolerance ofis set to be 0.01,then the point per wavenumber of the WENO_opt1scheme is4.58,the point per wavenumber of the WENO_opt2 scheme is 5.15,the point per wavenumber of the WENO_opt3 scheme is 5.71,the point per wavenumber of the WENO_opt4 scheme is 6.68,the point per wavenumber of the WENO_opt5 scheme is 7.05,the point per wavenumber of the WENO_opt6 scheme is 7.48,the point per wavenumber of the WENO_JS scheme is 5.61,and the point per wavenumber of the DRP scheme is 4.83.

Fig.3 Phase and amplitude errors of partial derivative ∂u/∂x.

Fig.4 Comparison of numerical simulation and exact solution.

Fig.5 Illustration of interface flux reconstruction method.

When function u(x)is discontinuous,smoothness factor has to be defined to change the redistribution of weights,which could guarantee that the node value approximation is essentially non-oscillatory around the discontinuity.Due to the design method proposed by Jiang and Shu6,in this fivepoint-stencil WENO scheme,the smoothness factor βmcould be denoted as

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In order to avoid the denominator being zero,ε=10-6,m=0，1，2.

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Fig.6 Schematic diagrams of computational domain and different patch meshes.

By calculating the non-linear weighting parameters,the flux term of Eq.(6)could be denoted as

For numerical stability,modification has been made to the flux term of WENO scheme by the Lax-Friedrichs flux-vector splitting method,and the flux term of Eq.(1)could be denoted as

where0.5(Aui+1/2- β(max|λ(A)|)Δu).When β is set to be different

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3.Modified WENO scheme verification

In this section,the comparison of dispersion and dissipation between the modified WENO scheme and the DRP scheme has been conducted with 1D convection equation,in which multiple initial conditions and different weights have been taken into account.

(3)其他因素产生的矿井充水通道。如为封闭或者封闭不良的钻孔，当井下巷道或采煤工作面揭露或者接近钻孔时，地表水甚至地下水都通过钻孔涌入巷道，引起水患事故。本区所有钻孔均按设计要求进行了封闭，封闭结论为合格。但未启封检查。故在钻孔附近开采时应适时监测钻孔的导水性。

Consider 1D linear convection equation

13.Allampalli V,Hixon R,Nallasamy M.High-accuracy large-step explicit Runge-Kutta(HALE-RK)schemes for computational aeroacoustics.J Comput Phys 2009;228(18):3837–50.

values,the dissipation of the scheme would be different.The scheme acts as central when β=0,and acts as upwind when β=1.According to the β definition in Ref.12,the definition of β within[s1，s2]has been modified as shown below:

where,andin which

where G(x，η，z)=e-β(x-z)2,a=0.5,z=-0.7,γ=10,δ=0.005 and η=lg2/36δ2.

The grid scale has been set as Δx=0.004,with time step Δt=0.0001 and the number of iterations being 10000.The time marching method used is six-level four-order HALERK6 scheme.13,14Fig.4 shows the comparison of different spatial discretion schemes with analytical solution with grid scale as Δx=0.004.It is also shown in these two figures that:(A)the DRP scheme could induce serious oscillations.Though the oscillation could be attenuated by adding artificial dissipation,the oscillation near the discontinuity is still very severe.(B)All the WENO scheme solutions match the analytical solutions very well,with the accuracy getting better when spatial grid scale is smaller.(C)As for the same WENO scheme,the computational result varies very little while using different flux splitting methods.

Fig.7 Distribution of sound pressure at time t=100Δt.

Fig.8 Sound pressure curves at y=0 line.

4.Interface flux reconstruction

In order to further prove the simulation capability of the interface flux reconstruction method applied to complex con figuration problems,the periodical point sound source propagation across triple 2D cylinder test case from the NASA 4th CAA workshop18–20has been selected for verification.

WENO schemes are applied at the interface of blocks,which well keep the numerical stability without damaging the overall computational accuracy.Besides,in order to further improve the computational stability,WENO_opt6 is applied at node C,while WENO_opt4 is applied at node B and WENO_opt1 at node A.

体视显微镜的光源数目为一个或两个。除了调节光线的强弱，趋光角度不同，镜下标本的成像也会有很大不同。正确的选择上下光源，通过结合体视显微镜上方的变倍旋钮，让我们即可清晰的观察标本的结构形态等细节特点，又可以宏观对比不同标本的形态特征，观察活体标本的行动轨迹[5]。

5.Numerical simulation

In order to verify the feasibility of the interface flux reconstruction in multi-block patch/non-patch interface,two study cases have been conducted.

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5.1.Gauss impulse propagation

One Gauss impulse sound source has been set at the center of the computational domain,as shown in Fig.6,with the background Mach number as zero.At the initial time t=0,the initial value of the Gauss sound source is set as

where u′and v′are component of fluctuation velocity,ρ′is fluctuation density,p′is fluctuation pressure.

OLED显示采用4线串行SPI，使用4线制串行模式的接口信号线。RST引脚用来硬件复位OLED，让其恢复默认状态；DC引脚是命令/数据标志，用于读与写；SDIN引脚是数据线，不同模式下对应的信号线不同，SCLK引脚是时钟线，只有在串行模式下，D0时钟线为信号线，D1为数据线。

In order to avoid the numerical contamination from the sound wave reflection,Perfect Match Layer(PML)nonreflection boundary condition17has been applied at the farfield.The unified unitless time step dt=0.5 has been used in sound field computation.Three different grid topologies have been used,which are named as Case 1(Δd1=1，Δd2=1),Case 2 (Δd1=1，Δd2=1/2), and Case 3 (Δd1=1/2，Δd2=1).As shown in the right part of Fig.6,it is the grid topology of Case 2 con figuration.

Figs.7 and 8 includes the sound pressure distribution contour of the Gauss impulse at t=100Δt and the sound pressure curve at the location of y=0 station.It is shown in the sound propagation contour that sound wave could propagate uniformly to the space with constant speed in different kinds of patched grids.From the comparison of the computational results and the analytical solutions,it could be concluded that the interface flux reconstruction method could be effectively applied in sound wave propagation.Fig.9 is comparison of sound pressure by using three grid topologies.Compared to Case 1,the calculation results of Case 2 and Case 3 are more closer to analytical solution.

5.2.A Harmonic energy source scattering from multiple cylinders

DRP spatial discretion scheme coupled with artificial dissipation is usually applied to the computation of flux at the interface.With the function test results in Part 3,when the magnitude of discontinuity at the interface is unknown,the amount of the artificial dissipation could not be determined with this method.The flux dissipation would be too much when the artificial dissipation is excessive,while it could result in oscillations when the dissipation is inadequate.In this paper,modi fied WENO scheme is applied at the interface,with hybrid Lax-Friedirchs flux-vector splitting method15,16used for flux splitting,and high order DRP scheme has been used inside the grid elements,as shown in Fig.5.F is flux term.The flux at grid node A of the interface of Block 1 could be discreetly solved.The grid node variable of Block 2 could be interpolated from the neighbor grid nodes.

As shown in Fig.10,three cylinders of various diameters have been laid out in the computational domain.The left cylinder is at(x=-4，y=0)with diameter being 1,the upper right and lower right cylinder are at (x=3，y=4) and(x=3，y=-4)respectively,with both of the diameters being 0.75.At the center of the computational domain,(x=0，y=0)is a periodical point sound source,the formula of which is as follows:

Fig.9 Sound pressure curves of different cases.

Fig.10 Topology of grid blocks.

where S is sound source term.Slide wall boundary has been applied in order to simulate the interference,diffraction and other complex phenomenon of the sound waves,while the PML non-reflection boundary condition17has been used at the far- field.Seven-point four-order DRP scheme is used for spatial discretion,seven-point stencil dissipation scheme is used inside the grid block,the flux reconstruction scheme with modified WENO scheme is applied at the interface,and time advancing method used is the 6-level 4-order Runge-Kutta scheme.21The size of the complete computational domain is(x ∈ [-9，9]，y ∈ [-9，9]),the topology of the grid domain is shown in Fig.10.Two different grid distribution methods have been used in this case.At the interface near the three cylinders,1–1 grid node matching and 1–2 patch are both used,with the grid details at the cylinder as shown in Fig.11.Grid scale Δx=0.025 has been used for non-patch grid sound field simulation,with unified unitless time step Δt=0.002 and the number of iterations as 105.

Figs.12–14 are the sound pressure distribution contour of the periodical sound source propagation procedure at different moments.From the comparison,it could be shown that(A)interference always occur between the sound waves regardless of any grid subdivision methodology,so does the diffraction;(B)the computational results by 1–2 patch grid could obtain higher definition of sound wave compared to the 1–1 node matching results.

Fig.11 Enlarged part of grid around one cylinder.

Fig.12 Sound pressure field at t=200.

Fig.13 Sound pressure field at t=1000.

Fig.14 Enlarged graph of sound pressure field at t=1000.

Fig.15 Comparison of numerical results with analytical results for Root Mean Square(RMS)of sound pressure at center line.

Fig.16 Comparison of numerical results with analytical results for RMS of sound pressure at surface of cylinders.

In order to quantitatively analyze the capability of interface flux reconstruction in the program,the RMS of the sound pressure distribution on the centroid line y=0 together with that on the cylinder surface is presented,as shown in Figs.15 and 16.From the comparison,it is shown that the peak value of the sound wave computed in this paper is slightly less than that of exact solution.In Fig.16,the sound wave peak value on the right cylinder surface is also slightly less than that of exact solution.The possible reason that causes the discrepancy might be that interface of block had more large dissipation.However,the total trend has shown that the current results well match that of the references,and the patch grid could even obtain better results.

6.Conclusions

For multi-scale grid problems,interface flux reconstruction method based on modi fied WENO scheme has been introduced.Based on the methodology of DRP scheme,the modified WENO scheme has been deduced.

(1)For different weights of the phase errors and amplitude errors,the influence of this scheme on the grid definition and dissipation has been analyzed.Then a hybrid flux vector splitting method with treatment to grid discontinuity has been developed and verified using sine wave function and hybrid wave function.The results have shown that modified WENO scheme could effectively simulate non-discontinuous wave and continuous wave problems with enough grid definition.

(2)According to the characteristic of the dissipation of the modified WENO scheme, flux reconstruction method has been developed using amplitude error accumulation at the discrete grid node on the interface.The feasibility and accuracy of the developed flux reconstruction method applied in multi-scale grid problem have been verified by comparing the numerical result of Gauss impulse sound source radiation with the analytical solutions.

(3)The accuracy of the developed method based on multiscale grid applied to complex con figuration problems has been verified by analysis of triple cylinder interference case.

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Selection of drainage system based on analytic hierarchy process and fuzzy comprehensive evaluation

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总而言之，作为一个理学家，董玘虽未能开辟新境，但也无凌虚蹈空之论，而力求平实妥帖，不为异说以惑世。因此，综上所述，笔者认为董玘是一个虽然有点固执木讷但却是坚持原则的纯粹而不驳杂的儒者。

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将两组Gesell测试各能区所得的发育商（DQ）分别进行配对样本的t检验，结果显示：病例组在“大运动”能区的发育商低于对照组，两者差异有统计学意义，而另外四个能区，包括“适应性”、“精细动作”、“语言”和“个人－社会”能区病例组和对照组差异均无统计学意义。见表2。

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皮下注射索马鲁肽治疗2型糖尿病有效性和安全性的Meta分析 …………………………………………… 杨 婷等（20）：2856

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