Harten-Lax-van Leer-contact(HLLC)approximation Riemann solver with elastic waves for one-dimensional elastic-plastic problems

A Harten-Lax-van Leer-contact （HLLC） approximate Riemann solver is built with elastic waves （HLLCE） for one-dimensional elastic-plastic flows with a hypo- elastic constitutive model and the von Mises＇ yielding criterion. Based on the HLLCE, a third-order cell-centered Lagrangian scheme is built for one-dimensional elastic-plastic problems. A number of numerical experiments are carried out. The numerical results show that the proposed third-order scheme achieves the desired order of accuracy. The third-order scheme is used to the numerical solution of the problems with elastic shock waves and elastic rarefaction waves. The numerical results are compared with a reference solution and the results obtained by other authors. The comparison shows that the pre- sented high-order scheme is convergent, stable, and essentially non-oscillatory. Moreover, the HLLCE is more efficient than the two-rarefaction Riemann solver with elastic waves （TRRSE）

A Genuinely Two-Dimensional HLL-Type Approximate Riemann Solver for Hypo-Elastic Plastic Flow

In this work,a genuinely two-dimensional HLL-type approximate Riemann solver is proposed for hypo-elastic plastic flow.To consider the effects of wave interaction from both the x-and y-directions,a corresponding 2D elastic-plastic approximate solver is constructed with elastic-plastic transition embedded.The resultant numerical flux combines one-dimensional numerical flux in the central region of the cell edge and two-dimensional flux in the cell vertex region.The stress is updated separately by using the velocity obtained with the above approximate Riemann solver.Several numerical tests,including genuinely two-dimensional examples,are presented to test the performances of the proposed method.The numerical results demonstrate the credibility of the present 2D approximate Riemann solver.

Staggered Lagrangian Discretization Based on Cell-Centered Riemann Solver and Associated Hydrodynamics Scheme

The aim of the present work is to develop a general formalism to derive staggered discretizations for Lagrangian hydrodynamics on two-dimensional unstructured grids.To this end,we make use of the compatible discretization that has been initially introduced by E.J.Caramana et al.,in J.Comput.Phys.,146(1998).Namely,momentum equation is discretized by means of subcell forces and specific internal energy equation is obtained using total energy conservation.The main contribution of this work lies in the fact that the subcell force is derived invoking Galilean invariance and thermodynamic consistency.That is,we deduce a general form of the sub-cell force so that a cell entropy inequality is satisfied.The subcell force writes as a pressure contribution plus a tensorial viscous contribution which is proportional to the difference between the nodal velocity and the cell-centered velocity.This cell-centered velocity is a supplementary degree of freedom that is solved by means of a cell-centered approximate Riemann solver.To satisfy the second law of thermodynamics,the local subcell tensor involved in the viscous part of the subcell force must be symmetric positive definite.This subcell tensor is the cornerstone of the scheme.One particular expression of this tensor is given.A high-order extension of this discretization is provided.Numerical tests are presented in order to assess the efficiency of this approach.The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of this scheme.

APPLICATION OF MFCAV RIEMANN SOLVER TO MAIRE＇S CELL-CENTERED LAGRANGIAN METHOD

In this paper, we apply arbitrary Riemann solvers, which may not satisfy the Maire＇s requirement, to the Maire＇s node-based Lagrangian scheme developed in [P. H. Maire et al., SIAM J. Sci. Comput, 29 （2007）, 1781-1824]. In particular, we apply the so-called Multi-Fluid Channel on Averaged Volume （MFCAV） Riemann solver and a Riemann solver that adaptively combines the MFCAV solver with other more dissipative Riemann solvers to the Maire＇s scheme. It is noted that neither of the two solvers satisfies the Maire＇s requirement. Numerical experiments are presented to demonstrate that the application of the two Riemann solvers is successful.

Linearized Double-Shock Approximate Riemann Solver for Augmented Linear Elastic Solid

In this work,in order to capture discontinuities correctly in linear elastic solid,augmented internal energy is defined according to the first law of thermodynamics and Hooke’s law.The non-conservative linear elastic system is then rewritten into a conservative form with the help of an augmented total energy equation.We find that the non-physical oscillations occur to the popular HLL and HLLC approximate Riemann solvers when directly applied to simulate the augmented linear elastic solid.We analyze the intrinsic reason by defining a discrepancy factor which can be used to estimate the difference of the total stress across a contact discontinuity,where it is physically required to be continuous.We discover that non-physical oscillations inevitably appear in the vicinity of the contact discontinuity if this factor is away from zero for an approximate Riemann problem solver.In order to overcome this difficulty,we propose an approximate Riemann solver based on the linearized double-shock technique.Theoretical analysis and numerical results show that in comparison to the HLL and HLLC approximate Riemann solvers,the present linearized double-shock Riemann solver can eliminate the non-physical oscillations effectively.

一类新型自适应反扩散近似Riemann求解器及其应用

对于包含激波、剪切层等复杂结构的流动问题,为了精确模拟剪切层等精细结构,且保证激波计算的稳定性,必须采用低耗散且强鲁棒的数值通量方法。传统的HLL近似Riemann求解器的耗散性较大,Roe、HLLEM和HLLC等近似Riemann求解器在计算某些含有强激波的物理问题时会出现非物理解,容易导致不稳定。针对这一问题,本文在Riemann求解器中通过合理设计反扩散矩阵,发展了一类具有自适应反扩散的新型Riemann求解器,并将其应用到高阶加权紧致格式,实现了高阶精度求解。通过典型数值算例验证了新型方法的计算精度和稳定性,结果表明本文提出的新型自适应反扩散Riemann求解器克服了传统Riemann求解器的缺陷,既能准确识别剪切层等精细结构,又能保证激波解的稳定性。

A hybrid multidimensional Riemann solver to couple self-similar method with MULTV method for complex flows

Since proposed,the self-similarity variables based genuinely multidimensional Riemann solver is attracting more attentions due to its high resolution in multidimensional complex flows.However,it needs numerous logical operations in supersonic cases,which limit the method’s applicability in engineering problems greatly.In order to overcome this defect,a hybrid multidimensional Riemann solver,called HMTHS(Hybrid of MulTv and multidimensional HLL scheme based on Self-similar structures),is proposed.It simulates the strongly interacting zone by adopting the MHLLES(Multidimensional Harten-Lax-van Leer-Eifeldt scheme based on Self-similar structures)scheme at subsonic speeds,which is with a high resolution by considering the second moment in the similarity variables.Also,it adopts the MULTV(Multidimensional Toro and Vasquez)scheme,which is with a high resolution in capturing discontinuities,to simulate the flux at supersonic speeds.Systematic numerical experiments,including both one-dimensional cases and twodimensional cases,are conducted.One-dimensional moving contact discontinuity case and sod shock tube case suggest that HMTHS can accurately capture one-dimensional expansion waves,shock waves,and linear contact discontinuities.Two-dimensional cases,such as the double Mach reflection case,the supersonic shock/boundary layer interaction case,the hypersonic flow over the cylinder case,and the hypersonic viscous flow over the double-ellipsoid case,indicate that the HMTHS scheme is with a high resolution in simulating multidimensional complex flows.Therefore,it is promising to be widely applied in both scholar and engineering areas.

A CELL-CENTERED ALE METHOD WITH HLLC-2D RIEMANN SOLVER IN 2D CYLINDRICAL GEOMETRY

This paper presents a second-order direct arbitrary Lagrangian Eulerian(ALE)method for compressible flow in two-dimensional cylindrical geometry.This algorithm has half-face fluxes and a nodal velocity solver,which can ensure the compatibility between edge fluxes and the nodal flow intrinsically.In two-dimensional cylindrical geometry,the control volume scheme and the area-weighted scheme are used respectively,which are distinguished by the discretizations for the source term in the momentum equation.The two-dimensional second-order extensions of these schemes are constructed by employing the monotone upwind scheme of conservation law(MUSCL)on unstructured meshes.Numerical results are provided to assess the robustness and accuracy of these new schemes.

ARobust Riemann Solver for Multiple Hydro-Elastoplastic Solid Mediums

We propose a robust approximate solver for the hydro-elastoplastic solid material,a general constitutive law extensively applied in explosion and high speed impact dynamics,and provide a natural transformation between the fluid and solid in the case of phase transitions.The hydrostatic components of the solid is described by a family of general Mie-Gruneisen equation of state(EOS),while the deviatoric component includes the elastic phase,linearly hardened plastic phase and fluid phase.The approximate solver provides the interface stress and normal velocity by an iterative method.The well-posedness and convergence of our solver are proved with mild assumptions on the equations of state.The proposed solver is applied in computing the numerical flux at the phase interface for our compressible multi-medium flow simulation on Eulerian girds.Several numerical examples,including Riemann problems,shock-bubble interactions,implosions and high speed impact applications,are presented to validate the approximate solver.

A Riemann-Solver Free Spacetime Discontinuous Galerkin Method for General Conservation Laws

This paper summarizes a Riemann-solver-free spacetime discontinuous Galerkin method developed for general conservation laws. The method integrates the best features of the spacetime Conservation Element/Solution Element (CE/SE) method and the discontinuous Galerkin (DG) method. The core idea is to construct a staggered spacetime mesh through alternate cell-centered CEs and vertex-centered CEs within each time step. Inside each SE, the solution is approximated using high-order spacetime DG basis polynomials. The spacetime flux conservation is enforced inside each CE using the DG concept. The unknowns are stored at both vertices and cell centroids of the spatial mesh. However, the solutions at vertices and cell centroids are updated at different time levels within each time step in an alternate fashion. Thanks to the staggered spacetime formulation, there are no left and right states for the solution at the spacetime interface. Instead, the solution available to evaluate the flux is continuous across the interface. Therefore, no (approximate) Riemann solvers are needed to provide a unique numerical flux. The current method can be used to solve arbitrary conservation laws including the compressible Euler equations, shallow water equations and magnetohydrodynamics (MHD) equations without the need of any form of Riemann solvers. A set of benchmark problems of various conservation laws are presented to demonstrate the accuracy of the method.

适用于非静力大气模式的近似黎曼求解器应用研究

基于多矩非静力大气模式,开展了3类垂向近似黎曼求解器应用研究。多矩非静力大气模式具有高精度与数值守恒特性,其垂向采用守恒的有限差分格式进行数值离散,而网格单元边界通量计算是通过求解黎曼问题来实现的,因此采用合适的近似黎曼求解器对准确模拟非静力大气垂直运动显得十分关键。LLF(Local Lax-Friedrich)、LMARS(Low Mach Approximate Riemann Solver)和HLLC(Harten-Lax-van Leer Contact)为计算流体力学(CFD)中常用的3种近似黎曼求解器,它们的计算代价和复杂程度逐渐增加。一维标准数值试验表明:LLF计算最为经济,但具有较强的耗散;LMARS具有适用于大气流动的假设,对于数值粘性的控制较好且计算量不大;HLLC建立的三波模型可以避免对中间特征场的过度数值耗散。基于LLF近似黎曼求解器计算经济的特点,通过优化LLF近似黎曼求解器各特征波动的粘性系数,能够实现与LMARS、HLLC近似黎曼求解器相同的性能,且计算代价最小。二维非静力数值试验表明,优化的LLF近似黎曼求解器能够规避常规LLF近似黎曼求解器的数值耗散过大问题,正确模拟小尺度非静力垂直运动,达到更复杂的LMARS、HLLC近似黎曼求解器模拟效果且并未增加计算量,这为非静力大气数值模式提供了良好的参考价值。

基于精确Riemann求解器的明满流过渡过程模拟

Preissmann窄缝法模拟明满流过渡过程方法简单,但存在明显的非物理振荡,抑制非物理振荡是该方法应用的关键。基于Godunov格式和精确Riemann求解器对明满流过渡过程进行模拟,针对Riemann问题代数恒等式在明满流交界处不光滑问题,提出了三阶收敛方法与二分法结合的迭代求解方法,保证迭代收敛至真实解;针对由于变量空间重构方法不能准确表达变量在空间中真实物理状态而导致的非物理振荡,提出了基于精确Riemann解的变量空间重构方法,准确表达激波间断在单元内的空间分布状态,从机理上抑制了非物理振荡。实例研究表明,数值计算结果与解析解或实测值吻合良好,研究成果为明满流过渡过程的高精度数值模拟提供了新的方法。

基于Riemann解的二维流体力学Lagrange有限点无网格方法

在高维流体力学计算中,对于多介质大变形等一类问题,采用有网格方法常遇到较大的困难.针对二维问题,研究了一种无网格方法———Lagrange有限点方法:在求解区域上设置适当的离散点集,视其中每一点为流体力学Lagrange点;对于点集的任一点,确定邻点集合,并基于该点同邻点集合的联系,应用Godunov方法将流体力学Lagrange方程进行离散;考虑到算法的稳健性,方法中可设置较多邻点并采用最小二乘法.将该方法应用于典型的数值算例,取得了良好效果.

MFCAV近似Riemann解在新型拉氏方法中的应用

Maire等提出了一种新型的有限体积中心型拉氏方法,该方法大大地改善了一直困扰着一般中心型拉氏方法的虚假网格变形.然而在计算数值流和移动网格时,该方法只应用了数值黏性较大的弱波近似(weak waveapproximated method,WWAM)Riemann解,而且方法的设计表明其他类型的近似Riemann解不能简单直接地应用上去.将体平均多流管(multi fluid channel on averaged volume,MFCAV)近似Riemann解视为对WWAM的修正,成功将其应用于新型方法中,数值实验表明应用了MFCAV的新方法是有效的.研究为将其他更为有效的近似Riemann解应用于该新型方法中开辟了一条道路.

一种基于TV分裂的真正多维Riemann解法器

给出了一种真正多维的HLL Riemann解法器.采用TV(Toro-Vázquez)分裂将通量分裂成对流通量和压力通量,其中对流通量的计算采用类似于AUSM格式的迎风方法,压力通量的计算采用波速基于压力系统特征值的HLL格式,并将HLL格式耗散项中的密度差用压力差代替,来克服传统的HLL格式不能分辨接触间断的缺点.为了实现数值格式真正多维的特性,分别计算网格界面中点和角点上的数值通量,并且采用Simpson公式加权中点和角点上的数值通量来得到网格界面上的数值通量.采用基于SDWLS(solution dependent weighted least squares)梯度的线性重构来获得空间的二阶精度,时间离散采用二阶Runge-Kutta格式.数值实验表明,相比于传统的一维HLL格式,该文的真正多维HLL格式具有能够分辨接触间断,消除慢行激波波后振荡以及更大的时间步长等优点.并且,与其他能够分辨接触间断的格式(例如HLLC格式)不同的是,真正多维的HLL格式在计算二维问题时不会出现数值激波不稳定现象.

用鲁棒Riemann求解器和运动重叠网格计算旋翼粘性绕流

发展了一种基于鲁棒Riemann求解器和运动重叠网格技术计算直升机悬停旋翼流场的方法。基于惯性坐标系,悬停旋翼流场是非定常流场,控制方程为可压缩Reynolds平均Navier-Stoke方程,其对流项采用Roe近似Reimann求解器离散,使用改进的五阶加权基本无振荡格式进行高阶重构,非定常时间推进采用含牛顿型LUSGS子迭代的全隐式双时间步方法。为实施旋转运动和便于捕捉尾迹,计算采用运动重叠网格技术。计算得到的桨叶表面压力分布及桨尖涡涡核位置都与实验结果吻合较好。数值结果表明:所发展方法对桨尖涡具有较高的分辨率,对激波具有较好的捕捉能力,该方法可进一步推广到前飞旋翼粘性绕流的计算。

基于HLLC Riemann求解器和重叠网格的三维可压缩粘性流场的计算

寻找准确捕捉激波和涡尾迹的高分辨率问题一直是计算流体力学领域具有挑战性的问题。本文基于HLLC近似Riemann求解器,发展了一种计算三维可压缩粘性跨音速流场的计算方法。采用HLLC Riemann求解器离散Reynolds平均Navier-Stoke方程的对流项,为提高精度,使用改进的五阶加权基本无振荡格式进行高阶重构。首先对三维固定翼绕流进行计算,然后结合静态重叠网格将方法成功推广到悬停旋翼绕流,并实现洞边界和人工外边界处高阶插值的直接进行。数值结果表明所发展方法能有效地捕捉激波,对桨尖涡也具有较高的分辨率。

基于HLLC近似Riemann求解器的天然河道水流运动模拟

天然河道断面几何形状快速变化条件下的一维复杂水流运动高精度数值模拟仍面临较大困难。基于Godunov格式,提出了基于守恒型圣维南方程的HLLC近似Riemann求解器的通量计算方法,将该求解器由浅水方程拓展至守恒型圣维南方程;提出了针对天然河道复杂断面几何形状下的变量空间重构方法:依据过流断面面积和静力矩等效原则将河道断面概化成矩形,通过线性插值构造单元界面处断面几何形状,根据水位重构结果计算界面两侧过流断面面积和静力矩的重构值,保证计算格式守恒。实例研究表明,天然河道断面几何形状快速变化条件下,该方法的计算结果与实测值吻合良好,同时对于混合水流运动具有较高模拟精度。研究成果为天然河道水动力及环境水力学高精度数值模拟提供了新的方法。

改进的PPM格式及其在Riemann问题中的应用

在Collela和Skora提出的PPM格式基础之上,为消除舍入误差带来的影响,发展了一种改进的PPM格式。将改进的PPM格式结合Riemann近似解算子应用于求解Riemann问题。选取双膨胀波和Rayleigh-Taylor不稳定性问题作为算例进行了数值验证,并将改进的PPM格式与原始PPM格式的求解结果进行了对比分析。结果表明:改进的PPM格式相较于原始PPM格式的求解精度有了明显提升;计算结果更加合理,气泡发展曲线与解析解更为接近。

基于黎曼求解器的SPH算法及其应用

基于传统光滑粒子流体动力学方法(SPH)进行数值模拟时,在边界处存在压力和速度震荡,甚至导致求解失败。为准确求解SPH方程式中的粒子的物理粘性,减小数值耗散,将黎曼求解器纳入SPH格式中,通过重整化过程对连续性方程中的梯度算子进行了校正。通过一系列经典算例的数值模拟结果表明,基于黎曼求解器的SPH算法能够准确的模拟液体的流动,且具有良好的准确性和鲁棒性。