基于不同时间步长时域非结构有限体积法模拟声-弹性耦合问题

介绍一种改进的时域非结构有限体积法(FVM),并将其应用于声-弹性耦合问题。在流体与固体介质中分别求解声波动方程与弹性波方程,根据交界面上的力平衡与质点振速连续条件考虑二者的相互作用。同时考虑双线性四边形单元的线性变化项及常数项,并结合常应变三角形单元处理混合网格问题。分别对三角形单元和四边形单元进行色散分析,给出声波动方程的稳定性条件。在不同介质中采用不同时间步长,提高计算效率。求解弹性波问题、声-弹性耦合问题,结果表明,改进后的方法求解声-弹性耦合问题是有效和准确的,并且具有良好的数值稳定性。     

Free surface wave motions and interactions

This paper investigates the problems involved in the numerical simulation of free surface wave motions and surface wave effects on marine structures. Various approaches that might be taken in meeting these problems are discussed, and their relative advantages and disadvantages are considered. One particular approach combines a Lagrangian formulation of the governing equations, a triangular grid and a finite-difference solution procedure. Since this approach has some distinct advantages in the numerical calculation of fluid flows which include a free surface, it formed the basis for the development of one particular computer code, SPLISH. Sufficient progress has been made with the SPLISH code to demonstrate the attractiveness of numerical calculations for wave flow problems. Recent computational results demonstrate that realistic time-varying local flow fields, pressures and forces on and near structures such as a half-cyclinder on the ocean floor can be determined from numerical calculations for certain conditions. Good agreement is found in comparison of the numerical results from SPLISH, recent linear wave Green's function and fifth-order asymptotic solutions for wave motion over a bottom seated half-cylinder, and an experimental simulation in a wave channel.     

On the accuracy of finite difference and modal methods for computing tidal and wind wave current profiles

This paper deals with the comparative accuracy of using finite difference grids or a modal representation through the vertical in modelling tidally or wind wave induced current profiles. A point model is used in the vertical, with a no-slip condition at the sea bed. In the finite difference approach the high-shear bottom layer is resolved using either a regular grid on a logarithmic or log-linear transformed co-ordinate or an irregular grid, varying in such a manner as to retain second-order accuracy. The accuracy of these various grid schemes is considered in detail. The relative merits of using either the Crank-Nicolson or Dufort-Frankel time integration methods are considered; in the case of a fine grid in a high-viscosity region, some numerical problems are found with the Dufort-Frankel method. An alternative approach to using a finite difference grid in the vertical, namely a modal (spectral) method, is described. The form of the modes is such that they can accurately resolve the high-shear bottom boundary layer. Calculations show that the thickness of the bottom boundary layer in relation to the total water depth is important in determining the choice of grid transform and rates of convergence of solutions using finite difference or modal methods. However, for the majority of problems the modal solution is numerically attractive owing to its computational efficiency and the ease with which solution algorithms based upon it can be coded in vectorizable form suitable for the new generation of vector computers. The influence of viscosity profile, its time variation and water depth upon tidally induced or wave induced currents is considered. Calculations suggest that near-bed measurements of tidal flow in shallow water together with associated modelling would enable appropriate formulations of eddy viscosity to be determined. Similar measurements, though using a laboratory flume, would be appropriate for wind wave problems.     

透射边界条件在波动谱元模拟中的实现:二维波动

将邢浩洁和李鸿晶提出的多次透射公式(multi-transmitting formula,MTF)的谱元格式应用于均匀介质中线弹性SH波动问题的谱元模拟.假定紧邻人工边界的一层谱单元为具有直线边界的四边形单元,以保证每个人工边界节点都唯一对应一条指向内域的离散网格线.人工边界节点在某时刻的位移由该离散网格线上的节点在前若干时刻的位移确定,按照MTF谱元格式进行计算.通过平面波以一定角度传播的外源问题算例和点源脉冲自由扩散的内源问题算例,验证了方法的可行性以及对实际复杂波动问题的适用性.通过不同类型初值问题算例,在时域内分析了插值多项式阶次、人工波速和透射阶次三个参数对反射误差的影响.结果表明:插值多项式阶次较高的格式会表现出更好的精度,但总体上对反射误差的影响较小;人工波速对反射误差具有显著影响,当人工波速小于介质物理波速时反射误差较大,而当人工波速等于或稍大于介质物理波速时反射误差处于较低水平;透射阶次对反射误差具有决定性影响,表现在不失稳的情形下提高透射阶次能够迅速降低反射误差,但内源问题从三阶MTF开始出现飘移失稳,外源问题从二阶MTF开始出现轻微的飘移失稳.     

Double-grid Chebyshev spectral elements for acoustic wave modeling

Highly accurate algorithms are needed for modeling wave propagation phenomena in realistic media. The spectral element methods, either based on a Chebyshev or a Legendre polynomial basis, have shown their excellent properties of high accuracy and flexibility in describing complex models outperforming other techniques. In contrast with standard grid methods, which use dense spatial meshes, spectral element methods discretize the computational domain in a very coarse mesh. With constant-property elements, this fact may in some cases reduce seriously the computational efficiency. For instance, if the medium is finely heterogeneous, it may need to be described in a much finer way than the acoustic wave field. The double-grid approach presented in this work is a viable way for overcoming this lack of the method and for handling problems where the medium changes continuously or even sharply on the small scale. The variation in the properties is taken into account by using an independent set of shape functions defined on a temporary local grid in such a way that either the small scale fluctuations are accurately handled, without the need of a global finer grid, and the macroscopic wave field propagation is solved with no loose of computational efficiency.     

非均匀介质弹性波动方程的不规则网格有限差分方法

从弹性波动方程出发，提出了一种新的空间不规则网格有限差分方法，并用于求解非均匀各向异性介质中的弹性波正演问题。这种方法简单易行，对于复杂几何结构，例如低速层、套管井和非平面界面等，在较细的不规则网格上进行离散，计算时间和占用内存更少。与多重网格差分方法相比，该方法不需要粗、细网格之间的插值，所有网格差分计算在同一次空间迭代中完成。具有复杂几何交界面的模型计算，包括地下透镜体、套管井眼等，在确定弹性常数和密度后，用不规则网格的差分方法更易实现。该方法使用了Higdon吸收边界条件解决人工边界反射问题，引入了新的稳定性条件和网格频散条件，很好地消除了非物理散射波。理论模型的效值计算表明，该方法具有良好的稳定性和计算精度，在模拟非均匀介质弹性波传播时，比相同精度的规则网格有限差分方法计算速度更快。该方法易于推广到非结构网格和三维问题中。     

A dissipation‐free numerical method to solve one‐dimensional hyperbolic flow equations

In this paper, a numerical method to capture the shock wave propagation in 1‐dimensional fluid flow problems with 0 numerical dissipation is presented. Instead of using a traditional discrete grid, the new numerical method is built on a range‐discrete grid, which is obtained by a direct subdivision of values around the shock area. The range discrete grid consists of 2 types: continuous points and shock points. Numerical solution is achieved by tracking characteristics and shocks for the movements of continuous and shock points, respectively. Shocks can be generated or eliminated when triggering entropy conditions in a marking step. The method is conservative and total variation diminishing. We apply this new method to several examples, including solving Burgers equation for aerodynamics, Buckley‐Leverett equation for fractional flow in porous media, and the classical traffic flow. The solutions were verified against analytical solutions under simple conditions. Comparisons with several other traditional methods showed that the new method achieves a higher accuracy in capturing the shock while using much less grid number. The new method can serve as a fast tool to assess the shock wave propagation in various flow problems with good accuracy.     

A Free-Lagrange method for unsteady compressible flow: simulation of a confined cylindrical blast wave

A Free-Lagrange numerical procedure for the simulation of two-dimensional inviscid compressible flow is described in detail. The unsteady Euler equations are solved on an unstructured Lagrangian grid based on a density-weighted Voronoi mesh. The flow solver is of the Godunov type, utilising either the HLLE (2 wave) approximate Riemann solver or the more recent HLLC (3 wave) variant, each adapted to the Lagrangian frame. Within each mesh cell, conserved properties are treated as piece-wise linear, and a slope limiter of the MUSCL type is used to give non-oscillatory behaviour with nominal second order accuracy in space. The solver is first order accurate in time. Modifications to the slope limiter to minimise grid and coordinate dependent effects are described. The performances of the HLLE and HLLC solvers are compared for two test problems; a one-dimensional shock tube and a two-dimensional blast wave confined within a rigid cylinder. The blast wave is initiated by impulsive heating of a gas column whose centreline is parallel to, and one half of the cylinder radius from, the axis of the cylinder. For the shock tube problem, both solvers predict shock and expansion waves in good agreement with theory. For the HLLE solver, contact resolution is poor, especially in the blast wave problem. The HLLC solver achieves near-exact contact capture in both problems. Received May 25, 1995 / Accepted September 11, 1995     

An exact multiphase Riemann solver for compressible cavitating flows

This paper presents a new one-fluid method for simulating formation and the collapse of cavitation regions in water during an isothermal process. In this method, the fluid phase changes are included into the wave pattern of an exact Riemann solver. The model behavior is assessed by comparing the numerical results with the other numerical models for several 1D Riemann problems. One-dimensional water hammer problems with vast creation and collapsing of cavitation zones are simulated as well—and the numerical results are compared to experimental results. The new model results are in very good agreement with accepted results reported in the literature. The presented results clearly show that the new model is able to capture the various behaviors of water during the phase change in the saturation dome and the vapor state, which was neglected in previous studies. Finally, the new model is adopted to an ALE method on an adaptive triangular grid to simulate an underwater explosion phenomenon inside a rigid cylinder—and the results are compared with other simulations.     

Numerical study of breaking waves by a two‐phase flow model

A two‐phase flow model, which solves the flow in the air and water simultaneously, is presented for modelling breaking waves in deep and shallow water, including wave pre‐breaking, overturning and post‐breaking processes. The model is based on the Reynolds‐averaged Navier–Stokes equations with the k ?ε turbulence model. The governing equations are solved by the finite volume method in a Cartesian staggered grid and the partial cell treatment is implemented to deal with complex geometries. The SIMPLE algorithm is utilised for the pressure‐velocity coupling and the air‐water interface is modelled by the interface capturing method via a high resolution volume of fluid scheme. The numerical model is validated by simulating overturning waves on a sloping beach and over a reef, and deep‐water breaking waves in a periodic domain, in which good agreement between numerical results and available experimental measurements for the water surface profiles during wave overturning is obtained. The overturning jet, air entrainment and splash‐up during wave breaking have been captured by the two‐phase flow model, which demonstrates the capability of the model to simulate free surface flow and wave breaking problems.Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical simulation of free‐surface flow using the level‐set method with global mass correction

A new numerical method that couples the incompressible Navier–Stokes equations with the global mass correction level‐set method for simulating fluid problems with free surfaces and interfaces is presented in this paper. The finite volume method is used to discretize Navier–Stokes equations with the two‐step projection method on a staggered Cartesian grid. The free‐surface flow problem is solved on a fixed grid in which the free surface is captured by the zero level set. Mass conservation is improved significantly by applying a global mass correction scheme, in a novel combination with third‐order essentially non‐oscillatory schemes and a five stage Runge–Kutta method, to accomplish advection and re‐distancing of the level‐set function. The coupled solver is applied to simulate interface change and flow field in four benchmark test cases: (1) shear flow; (2) dam break; (3) travelling and reflection of solitary wave and (4) solitary wave over a submerged object. The computational results are in excellent agreement with theoretical predictions, experimental data and previous numerical simulations using a RANS‐VOF method. The simulations reveal some interesting free‐surface phenomena such as the free‐surface vortices, air entrapment and wave deformation over a submerged object. Copyright © 2009 John Wiley & Sons, Ltd.     

7阶WENO-S格式的计算效率研究

WENO-S格式是一类适合于含间断问题数值模拟的加权本质无振荡格式.这类格式的光滑因子满足对单频波为常数,这使得其近似色散关系与线性基底格式一致,并且具有良好的小尺度波动模拟能力．计算效率是数值方法性能指标的一个重要方面.由于WENO-S格式的光滑因子在各子模板上的计算公式除下标不同外形式一致,在计算线性对流方程相邻数值通量时,部分光滑因子完全相同．为此提出一种消除WENO-S格式冗余光滑因子计算的方法.该方法要求一条网格线上用于重构或插值的量可以表示为一个序列.基于此要求分析其对于几种不同物理问题的可行性和使用方法.以7阶WENO-S格式为例介绍了格式性质和去除冗余光滑因子计算的方法.该方法中预先计算和存储一条网格线上的所有光滑因子,在网格点较多的情况下,光滑因子计算次数约为原7阶WENO-S格式的1/4.对一维对流问题、球面波传播问题、二维旋转问题、二维小扰动传播问题及一维和二维无黏流动问题进行了数值模拟.结果表明该格式对多种流动结构具有良好的捕捉能力,并且同时具有良好的计算效率,去除冗余计算后又降低了约20%的计算时间.     

Triangular grid method for stress-wave propagation in 2-D orthotropic materials

A triangular grid method is presented to calculate propagation problems of elastic stress waves in 2-D orthotropic materials. This method is based on the dynamic equilibrium equations of the computational cells formed among the auxiliary triangular grids. The solution is obtained by calculating alternately the nodal displacements and the central point stresses of the spatial grids. The numerical results are compared with the corresponding solutions of the finite element method. Comparisons show that the triangular grid method yields a higher calculational speed than the finite element method. The stress concentrations are investigated from wave-field analyses when the stress wave propagates within an orthotropic plate with a hole. Finally, the presented numerical method is used to study the features of wave propagation and diffraction in a square orthotropic plate with a hole when an impact load is applied to the top of the plate.This work was supported by National Natural Science Foundation of China (Nos. 10025212 and 10232040) and Natural Science Foundation of Liaoning province (No. 20021070).     

The surface marker and micro cell method

A new method is presented for the simulation of two-dimensional, incompressible, free surface fluid flow problems. The surface marker and micro cell (SMMC) method is capable of simulating transient free surface fluid flow problems that include multivalued free surfaces, impact of free surfaces with solid obstacles and converging fluid fronts (including wave breaking). New approaches are presented for the advection of the free surface, the calculation of the tentative velocity, final velocity and pressure fields and the use of multivalued velocities to treat converging fluid fronts. Simulation results are compared with experimental results for water sloshing in a tank to demonstrate the validity of the new method. Convergence of the new method is demonstrated by a grid refinement study. © 1997 John Wiley & Sons, Ltd.     

Performance study of the multiwavelet discontinuous Galerkin approach for solving the Green-Naghdi equations

This paper presents a multiresolution discontinuous Galerkin (DG) scheme for the adaptive solution of Boussinesq-type equations. The model combines multiwavelet (MW)–based grid adaptation with a DG solver based on the system of fully nonlinear and weakly dispersive Green-Naghdi (GN) equations. The key feature of the adaptation procedure is to conduct a multiresolution analysis using MWs on a hierarchy of nested grids to improve the efficiency of the reference DG scheme on a uniform grid by computing on a locally refined adapted grid. This way, the local resolution level will be determined by manipulating MW coefficients controlled by a single user-defined threshold value. The proposed adaptive multiwavelet DG solver for GN equations is assessed using several benchmark problems related to wave propagation and transformation in nearshore areas. The numerical results demonstrate that the proposed scheme retains the accuracy of the reference scheme, while significantly reducing the computational cost.     

Non‐hydrostatic 3D free surface layer‐structured finite volume model for short wave propagation

In this paper a layer‐structured finite volume model for non‐hydrostatic 3D environmental free surface flow is presented and applied to several test cases, which involve the computation of gravity waves. The 3D unsteady momentum and mass conservation equations are solved in a collocated grid made of polyhedrons, which are built from a 2D horizontal unstructured mesh, by just adding several horizontal layers. The mesh built in such a way is unstructured in the horizontal plane, but structured in the vertical direction. This procedure simplifies the mesh generation and at the same time it produces a well‐oriented mesh for stratified flows, which are common in environmental problems. The model reduces to a 2D depth‐averaged shallow water model when one single layer is defined in the mesh. Pressure–velocity coupling is achieved by the Semi‐Implicit Method for Pressure‐Linked Equations algorithm, using Rhie–Chow interpolation to stabilize the pressure field. An attractive property of the model proposed is the ability to compute the propagation of short waves with a rather coarse vertical discretization. Several test cases are solved in order to show the capabilities and numerical stability of the model, including a rectangular free oscillating basin, a radially symmetric wave, short wave propagation over a 1D bar, solitary wave runup on a vertical wall, and short wave refraction over a 2D shoal. In all the cases the numerical results are compared either with analytical or with experimental data. Copyright © 2008 John Wiley & Sons, Ltd.     

Application of local grid refinement to vortex motion due to a solitary wave passing over a submerged body

A numerical method based on the streamfunction–vorticity formulation is applied to simulate the two‐dimensional, transient, viscous flow with a free surface. This method successfully uses the locally refined grid in an inviscid–viscous model to explore the processes of vortex formation due to a solitary wave passing over a submerged bluff body. The two particular bodies considered here are a blunt rectangular block and a semicircular cylinder. Flow visualization to track dyelines is carried out in the laboratory in order to confirm the validity of the numerical results. Numerical results examined by different grid configurations ensure the locally refined grid to be useful in practical application. Flow phenomena, including the vortex motion and wave patterns during non‐linear wave–structure interaction, are also discussed. Copyright © 2002 John Wiley & Sons, Ltd.     

爆轰波强间断问题的伪弧长算法及其人为解验证

针对该问题开展了伪弧长数值算法研究,通过引入弧长参数,使网格按照一定的形式自适应移动,达到在强间断区域自动加密的效果,从而提高网格分辨率。基于伪弧长算法编写了二维程序,并对程序进行人为解方法验证。将伪弧长算法和直接有限体积法的数值结果进行对比,通过误差分析,显示出伪弧长算法能有效提高计算精度。最后将伪弧长算法应用于气相爆轰波在二维管道中的传播问题,研究了波阵面的捕捉效果和爆轰波胞格结构的形成过程。     

含孔正交各向异性复合材料板的动应力集中

运用一种改进的非结构化四边形格子法,对含孔正交各向异性板条受面内冲击拉伸时弹性应力波的传播过程和孔边的动应力集中进行了研究．非结构化格子法采用与有限元类似的网格剖分方法,并基于围绕每个节点的积分平衡方程,并自然满足复杂边界的自由边界条件．计算中不需存储刚度矩阵,因而计算速度快、效率高、节省内存,在解决应力波传播问题中具有显著的优越性．通过对多种工况进行数值模拟,分析了材料的各向异性性质、纤维方向、孔径比、加载脉冲周期等参数对孔边动应力的影响,得到了一些规律性的结果．并与现有实验结果进行对比,验证了该方法的有效性．     

波动数值模拟的一种显式方法——二维波动

将一维波动时域数值模拟的一种显式方法推广到二维,导出了二维非规则网格的节点递推公式.针对均匀正方形网格详细论述了时空精度皆为2M阶(M为正整数)的稳定递推公式的构建方法,并以构建二阶(M = 1)和四阶(M = 2)公式为例予以说明. 最后,通过算例检验了本文研究结果,特别是说明了高阶公式对提高计算效率的价值.