A stable least-squares finite element method for non-linear hyperbolic problems

A class of stable least-square finite element methods for non-linear hyperbolic problems is developed and some exploratory studies made. The methods are based on modifying the L²-norm of the. residual and a related approximation to the H¹-norm of the residual. The effect of the additional terms in these residual functionals is to introduce a dissipative effect proportional to the solution gradient. This acts to stabilize the solution for non-linear hyperbolic problems which generate shocks. Numerical results for a one-dimensional nozzle and shock tube problem demonstrate the accuracy and stability of the method. Results are for an implicit scheme and calculations for linear, quadratic and cubic elements are given.     

基于改进位移模式的一维有限元超收敛算法

将多尺度方法的思想与超收敛计算的解析公式结合起来,提出了改进有限元位移模式的算法。利用超收敛计算的解析公式,将高阶有限元解的位移模式用常规有限元解的位移模式表示。用常规有限元解的位移模式与高阶有限元解的位移模式之和构造新的位移模式,采用积分形式推导了单元刚度矩阵。该算法在前处理和后处理两个阶段都使用超收敛计算公式,在常规试函数的基础上,增加了高阶试函数,使得单元内平衡方程的残差减少,从而达到提高精度的目标。对于线性单元,本文结点和单元的位移、导数都达到了h4阶的超收敛精度。     

A note on the stability and accuracy of Cn finite elements in steady diffusion-convection problems

The paper is concerned with stability and accuracy of an nth order Lagrangian family of finite element steady-state solutions of the diffusion-convection equation, and furthermore is concerned with the stability and the accuracy of on mth kind Hermitian family of finite element solutions. We discuss the stability of the numerical solution based on the fact that the characteristic finite element solution can be expressed approximately as a rational function of cell Peclet number Pe_c ( = uh/k). Moreover, it is shown that by eliminating derivatives and by using the interpolation method over elements a stable solution is obtained over the domain independent of Pe_c for P^1,3, and for P^2,5 the stable solution is obtained for Pe_c less than 44.4.     

Two-level stabilized finite element method for Stokes eigenvalue problem

A two-level stabilized finite element method for the Stokes eigenvalue problem based on the local Gauss integration is considered.This method involves solving a Stokes eigenvalue problem on a coarse mesh with mesh size H and a Stokes problem on a fine mesh with mesh size h = O(H 2),which can still maintain the asymptotically optimal accuracy.It provides an approximate solution with the convergence rate of the same order as the usual stabilized finite element solution,which involves solving a Stokes eigenvalue problem on a fine mesh with mesh size h.Hence,the two-level stabilized finite element method can save a large amount of computational time.Moreover,numerical tests confirm the theoretical results of the present method.     

A weighted least squares method for first-order hyperbolic systems

The paper presents a generalization of the classical L²-norm weighted least squares method for the numerical solution of a first-order hyperbolic system. This alternative least squares method consists of the minimization of the weighted sum of the L² residuals for each equation of the system. The order of accuracy of global conservation of each equation of the system is shown to be inversely proportional to the weight associated with the equation. The optimal relative weights between the equations are then determined in order to satisfy global conservation of the energy of the physical system. As an application of the algorithm, the shallow water equations on an irregular domain are first discretized in time and then solved using Laplace modification and the proposed least squares method.     

An extended mixture model for the simultaneous treatment of small‐scale and large‐scale interfaces

The aim of this work is to present a new model based on the volume of fluid method and the algebraic slip mixture model in order to solve multiphase gas–fluid flows with different interface scales and the transition among them. The interface scale is characterized by a measure of the grid, which acts as a geometrical filter and is related with the accuracy in the solution; in this sense, the presented coupled model allows to reduce the grid requirements for a given accuracy. With this objective in mind, a generalization of the algebraic slip mixture model is proposed to solve problems involving small‐scale and large‐scale interfaces in an unified framework taking special care in preserving the conservativeness of the fluxes. This model is implemented using the OpenFOAM^® libraries to generate a tool capable of solving large problems on high‐performance computing facilities. Several examples are solved as a validation for the presented model, including new quantitative measurements to assess the advantages of the method. Copyright © 2014 John Wiley & Sons, Ltd.     

Hermite梯度重构核近似配点法及其在功能梯度材料板的静力分析中的应用

由于直接配点法在求解边值问题时边界上的求解精度较低,本文提出了Hermite梯度重构核近似配点法（HGCM）来改进边界求解精度。重构核近似是无网格法中一种常用的近似函数,但是其在求解高阶导数时格式复杂且非常耗时。HGCM采用梯度重构核近似构建形函数的任意高阶导数,提高了计算效率;通过Hermite配点法构建离散方程,提高了边界求解精度。这种方法在求解对应变系数四阶偏微分方程的功能梯度材料板的静力问题时精度高,计算效率高,并可进一步推广应用于高阶偏微分方程描述的边值问题。     

附面层修正应用于无网格算法的研究

研究了无网格算法中的附面层修正方法,在一种布置点云方法的基础上,发展一种曲面拟合的重构方式构造流场物理量;找出了无网格算法与网格算法之间的联系,成功将AUSM＋-up格式移植到无网格算法当中,并应用于计算欧拉方程的数值通量;计算中采用了一种改进的隐式时间推进,并引入当地时间步长和残值光顺等加速收敛措施,成功的将附面层修...     

Nodal integral method solutions for Bénard natural convection

The nodal integral method is a relatively new numerical technique that has been used recently to solve both static and dynamic multidimensional problems in heat transfer, fluid flow and neutron transport. The method offers significant advantages in terms of stability, accuracy and efficiency over conventional finite elements when the problem can be adequately modelled in Cartesian co-ordinates. This method was used to investigate bifurcation phenomena in the Bénard problem for aspect ratios in the range of one to nine. Automatic search techniques were used with a static version to find the first four critical Rayleigh values for a square cavity, to map the first two critical Rayleigh values as a function of aspect ratio, and to examine the solution types. Accuracy enhancement was obtained by factorization and extrapolation. Critical values, obtained by interpolation, were verified dynamically. Aspect ratio crossover and transition values were found for the first two critical Rayleigh numbers, with an accuracy of the order of ±3 per cent. The precision achieved in the results for Ra* and Ra** as a function of β is usually within 0.1%–0.2% except at high β (i.e. near β=9.0) and at large critical values of Ra (i.e. the first few values of Ra** near β=1). Specific results at β=1.0 are Ra*=2584±0.5, Ra**=6807, Ra³* = 19 734 and Ra⁴*=22 586.     

饱和两相介质动力固结问题时域求解的精细时程积分方法

针对u-p形式的饱和两相介质波动方程,采用精细时程积分方法计算固相位移u,采用向后差分算法求解流体压力p,建立了饱和两相介质动力固结问题时域求解的精细时程积分方法。针对标准算例,对该方法的计算精度进行了校核。开展了该方法相关算法特性的研究,对采用不同数值积分方法计算非齐次波动方程特解项计算精度的差异进行了对比研究,并对采用不同积分点数目的高斯积分法计算特解项条件下计算精度的差异进行了对比研究。研究结果表明,(1)该方法具有良好的计算精度。(2)计算非齐次波动方程特解项的数值积分方法中,梯形积分法的计算精度最差,高斯积分法、辛普生积分法和科茨积分法都具有较好的计算精度。(3)增加高斯积分点数目对于提高计算精度的作用并不显著。     

A perturbation technique for solving boundary value problems arising in the electrodynamics of conducting bodies

The electrodynamics of wave reflection from conducting media lead to difficult mathematical problems because of the matching conditions which must be met at the interface between conductors and nonconductors. Simplified boundary conditions have been proposed by Leontovitch and others which considerably simplify certain of the mathematical problems. We discuss the Leontovitch condition together with certain of its shortcomings and present a new method which overcomes some of the difficulties of the Leontovitch condition. The new method is a perturbation away from infinite conductivity which allows the solution of electrodynamics problems to be calculated to any order of accuracy in the quantity (ωε/σ)^1/2 for some important cases. An instance in which the perturbation method fails is also discussed.     

The residual formulation of the method of manufactured solutions for computationally efficient solution verification

The method of manufactured solutions (MMS) is a solution verification methodology for determining whether the implementation of a discretization method is achieving its theoretical order of accuracy. This methodology combines the advantages of grid refinement studies and comparison with exact solution, by modifying the governing equations solved within a code by adding a source term to drive the solution towards a predetermined analytic function. By solving the modified equations on a sequence of grids and comparing the differences between the converged solution and manufactured solution, the order of accuracy of the implementation can be investigated. However, in its current form, converged solutions on a sequence of grids are required which can be quite costly and difficult to obtain. In this paper, by comparing the MMS to the method for determining the theoretical order of accuracy of a discretization method, the residual formulation of the MMS is developed. This new formulation only requires that the residual of the discretized governing equations to be calculated and not the solution to the discretized equations, thus avoiding the computational cost and difficulties inherent in obtaining converged solutions. Furthermore, since only the residuals are interrogated, individual components of the flow solver can be tested, in isolation, allowing the MMS to be used more effectively in locating errors within the code. This new approach is demonstrated to yield the same order of accuracy as the original MMS using three different cases—one-dimensional porous media equation, one-dimensional St Venant equations and two-dimensional unstructured Navier–Stokes simulations.     

基于改进位移模式的一维C1有限元超收敛算法

提出了基于改进位移模式的一维C1有限元超收敛算法。利用单元内部需满足平衡方程的条件,推导了超收敛计算解析公式的显式,即将高阶有限元解的位移模式用常规有限元解的位移模式表示。用常规有限元解的位移模式与高阶有限元解的位移模式之和构造新的位移模式。采用积分形式推导了单元刚度矩阵。该算法在前处理阶段使用了超收敛计算公式,在常规试函数的基础上,增加了高阶试函数,使得单元内平衡方程的残差减少,从而达到提高精度的目标。对于Hermite单元,本文的结点和单元的位移、导数都达到了h4阶的超收敛精度。     

p-version least squares finite element formulation for two-dimensional,incompressible fluid flow

A p-version least squares finite element formulation for non-linear problems is applied to the problem of steady, two-dimensional, incompressible fluid flow. The Navier-Stokes equations are cast as a set of first-order equations involving viscous stresses as auxiliary variables. Both the primary and auxiliary variables are interpolated using equal-order C⁰ continuity, p-version hierarchical approximation functions. The least squares functional (or error functional) is constructed using the system of coupled first-order non-linear partial differential equations without linearization, approximations or assumptions. The minimization of this least squares error functional results in finding a solution vector {δ} for which the partial derivative of the error functional (integrated sum of squares of the errors resulting from individual equations for the entire discretization) with respect to the nodal degrees of freedom {δ} becomes zero. This is accomplished by using Newton's method with a line search. Numerical examples are presented to demonstrate the convergence characteristics and accuracy of the method.     

Kernel-independent fast multipole method within the framework of regularized Stokeslets

The method of regularized Stokeslets (MRS) uses a radially symmetric blob function of infinite support to smooth point forces and allows for evaluation of the resulting flow field. This is a common method to study swimmers at zero Reynolds number where the Stokeslet is the fundamental solution corresponding to the kernel of the single layer potential. Simulating the collective motion of N micro-swimmers using the MRS results in at least N² pair-wise interactions. Efficient simulation of a large number of swimmers in free space is observed with the implementation of the kernel-independent fast multipole method (FMM) for radial basis functions. We illustrate the complexity of the algorithm on a simple test case where we study regularized point forces, showing that the method is of order N. Additionally, we explore accuracy in time for the MRS where the swimmers are modeled as Kirchhoff rods and the kernel-independent FMM is compared to the direct calculation using the standard MRS. Optimal hydrodynamic efficiency is also explored for different configurations of swimmers.     

Assessment of a discontinuous Galerkin method for the simulation of vortical flows at high Reynolds number

This paper focuses on the assessment of a discontinuous Galerkin method for the simulation of vortical flows at high Reynolds number. The Taylor–Green vortex at Re = 1600 is considered. The results are compared with those obtained using a pseudo‐spectral solver, converged on a 512³ grid and taken as the reference. The temporal evolution of the dissipation rate, visualisations of the vortical structures and the kinetic energy spectrum at the instant of maximal dissipation are compared to assess the results. At an effective resolution of 288³, the fourth‐order accurate discontinuous Galerkin method (DGM) solution (p = 3) is already very close to the pseudo‐spectral reference; the error on the dissipation rate is then essentially less than a percent, and the vorticity contours at times around the dissipation peak overlap everywhere. At a resolution of 384³, the solutions are indistinguishable. Then, an order convergence study is performed on the slightly under‐resolved grid (resolution of 192³). From the fourth order, the decrease of the error is no longer significant when going to a higher order. The fourth‐order DGM is also compared with an energy conserving fourth‐order finite difference method (FD4). The results show that, for the same number of DOF and the same order of accuracy, the errors of the DGM computation are significantly smaller. In particular, it takes 768³ DOF to converge the FD4 solution. Finally, the method is also successfully applied on unstructured high quality meshes. It is found that the dissipation rate captured is not significantly impacted by the element type. However, the element type impacts the energy spectrum in the large wavenumber range and thus the small vortical structures. In particular, at the same resolution, the results obtained using a tetrahedral mesh are much noisier than those obtained using a hexahedral mesh. Those obtained using a prismatic mesh are already much better, yet still slightly noisier. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical simulation of turbulent free-surface flow in curved channel

This paper presents first results of numerical simulation of turbulent free-surface flow. Simple implementation of surface capturing method is based on the variable density approach. The flow is treated as if there is only one fluid, but with variable material properties (density, viscosity). The switch in these values is done by a function resulting from the mass conservation principle. This approach simplifies the implementation of turbulence model. In this case the SST k−ω model was chosen in modification given by Hellsten.Numerical solution was carried out by finite-volume method with explicit Runge-Kutta time-integration. The artificial compressibility method was used for time-marching search for steady state solution. The whole model was tested on horizontally placed square-sectioned 90^∘ bend, which was partially filled by the water. The main goal of this study was to demonstrate the applicability of this model and solution method for capturing the water-air interface as well as for predicting the turbulent effects in both fluids.     

A high‐order mass‐lumping procedure for B‐spline collocation method with application to incompressible flow simulations

This paper presents new developments of the staggered spline collocation method for cost‐effective solution to the incompressible Navier–Stokes equations. Maximal decoupling of the velocity and the pressure is obtained by using the fractional step method of Gresho and Chan, allowing the solution to sparse elliptic problems only. In order to preserve the high‐accuracy of the B‐spline method, this fractional step scheme is used in association with a sparse approximation to the inverse of the consistent mass matrix. Such an approximation is constructed from local spline interpolation method, and represents a high‐order generalization of the mass‐lumping technique of the finite‐element method. A numerical investigation of the accuracy and the computational efficiency of the resulting semi‐consistent spline collocation schemes is presented. These schemes generate a stable and accurate unsteady Navier–Stokes solver, as assessed by benchmark computations. Copyright © 2003 John Wiley & Sons, Ltd.     

渐进多尺度展开方法的精度和物理意义

多尺度渐进展开方法(MsAEM)是分析周期复合材料结构力学行为的代表性方法,可以通过加权残量等方法实现,作者曾针对MsAEM的精度和力学含义进行研究。本文对作者的工作进行了总结,进一步明确了一维周期结构的单元阶次、摄动阶次和精确解的关系,揭示了不同阶次虚拟载荷和影响函数的物理意义,从物理角度强调了二阶展开项是不可缺少的,并对未来工作进行了展望。     

Analysis of a discontinuous Galerkin approximation of the Stokes problem based on an artificial compressibility flux

In this work, we propose and analyse a discontinuous Galerkin (DG) method for the Stokes problem based on an artificial compressibility numerical flux. A crucial step in the definition of a DG method is the choice of the numerical fluxes, which affect both the accuracy and the order of convergence of the method. We propose here to treat the viscous and the inviscid terms separately. The former is discretized using the well‐known BRMPS method. For the latter, the problem is locally modified by adding an artificial compressibility term of the form (1/c²)(?p/?t) for the sole purpose of interface flux computation. The flux is obtained as the exact solution of a local Riemann problem. The analysis of the method extends the well‐established strategies for the DG discretization of the Laplacian to the resulting partially coercive problem. Copyright © 2007 John Wiley & Sons, Ltd.