An estimation of the sensitivity of numerical error norm using adjoint model

A posteriori estimation of the numerical error sensitivity to the local truncation error is addressed using adjoint model endowed with the information on the error field. The numerical error is estimated from the solution of the linear tangent model (LTM) or from a Richardson extrapolation. The local truncation error used in the LTM is obtained by the action of a high‐order finite‐difference stencil on the field computed by the main (low‐order accuracy) algorithm. Copyright © 2009 John Wiley & Sons, Ltd.     

A code-independent technique for computational verification of fluid mechanics and heat transfer problems

The goal of this paper is to present a versatile framework for solution verification of PDE＇s. We first generalize the Richardson Extrapolation technique to an optimized extrapolation solution procedure that constructs the best consistent solution from a set of two or three coarse grid solution in the discrete norm of choice. This technique generalizes the Least Square Extrapolation method introduced by one of the author and W. Shyy. We second establish the conditioning number of the problem in a reduced space that approximates the main feature of the numerical solution thanks to a sensitivity analysis. Overall our method produces an a posteriori error estimation in this reduced space of approximation. The key feature of our method is that our construction does not require an internal knowledge of the software neither the source code that produces the solution to be verified. It can be applied in principle as a postprocessing procedure to off the shelf commercial code. We demonstrate the robustness of our method with two steady problems that are separately an incompressible back step flow test case and a heat transfer problem for a battery. Our error estimate might be ultimately verified with a near by manufactured solution. While our pro- cedure is systematic and requires numerous computation of residuals, one can take advantage of distributed computing to get quickly the error estimate.     

Adaptive finite element method for an optimal control problem of Stokes flow with L2‐norm state constraint

An adaptive finite element approximation for an optimal control problem of the Stokes flow with an L²‐norm state constraint is proposed. To produce good adaptive meshes, the a posteriori error estimates are discussed. The equivalent residual‐type a posteriori error estimators of the H ¹‐error of state and L²‐error of control are given, which are suitable to carry out the adaptive multi‐mesh finite element approximation. Some numerical experiments are performed to illustrate the efficiency of the a posteriori estimators. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical approximation of generalized Newtonian fluids using Powell–Sabin–Heindl elements: I. theoretical estimates

In this paper we consider the numerical approximation of steady and unsteady generalized Newtonian fluid flows using divergence free finite elements generated by the Powell–Sabin–Heindl elements. We derive a priori and a posteriori finite element error estimates and prove convergence of the method of successive approximations for the steady flow case. A priori error estimates of unsteady flows are also considered. These results provide a theoretical foundation and supporting numerical studies are to be provided in Part II. Copyright © 2003 John Wiley & Sons, Ltd.     

Augmented mixed finite element methods for a vorticity‐based velocity–pressure–stress formulation of the Stokes problem in 2D

In this paper, we consider an augmented velocity–pressure–stress formulation of the 2D Stokes problem, in which the stress is defined in terms of the vorticity and the pressure, and then we introduce and analyze stable mixed finite element methods to solve the associated Galerkin scheme. In this way, we further extend similar procedures applied recently to linear elasticity and to other mixed formulations for incompressible fluid flows. Indeed, our approach is based on the introduction of the Galerkin least‐squares‐type terms arising from the corresponding constitutive and equilibrium equations, and from the Dirichlet boundary condition for the velocity, all of them multiplied by stabilization parameters. Then, we show that these parameters can be suitably chosen so that the resulting operator equation induces a strongly coercive bilinear form, whence the associated Galerkin scheme becomes well posed for any choice of finite element subspaces. In particular, we can use continuous piecewise linear velocities, piecewise constant pressures, and rotated Raviart–Thomas elements for the stresses. Next, we derive reliable and efficient residual‐based a posteriori error estimators for the augmented mixed finite element schemes. In addition, several numerical experiments illustrating the performance of the augmented mixed finite element methods, confirming the properties of the a posteriori estimators, and showing the behavior of the associated adaptive algorithms are reported. The present work should be considered as a first step aiming finally to derive augmented mixed finite element methods for vorticity‐based formulations of the 3D Stokes problem. Copyright © 2010 John Wiley & Sons, Ltd.     

Linear stability of swirling flows computed as solutions to the quasi-cylindrical equations of motion

The linear stability of numerical solutions to the quasi-cylindrical equations of motion for swirling flows is investigated. Initial conditions are derived from Batchelor's similarity solution for a trailing line vortex. The stability calculations are performed using a second-order-accurate finite-difference scheme on a staggered grid, with the accuracy of the computed eigenvalues enhanced through Richardson extrapolation. The streamwise development of both viscous and inviscid instability modes is presented. The possible relationship to vortex breakdown is discussed.     

A Stokes model with cavitation for the numerical simulation of hydrodynamic lubrication

We present a cavitation model based on the Stokes equation and formulate adaptive finite element methods for its numerical solution. A posteriori error estimates and adaptive algorithms are derived, and numerical examples illustrating the theory are supplied, in particular with comparison to the simplified Reynolds model of lubrication. Copyright © 2010 John Wiley & Sons, Ltd.     

On the use of anisotropic a posteriori error estimators for the adaptative solution of 3D inviscid compressible flows

This paper describes the use of an a posteriori error estimator to control anisotropic mesh adaptation for computing inviscid compressible flows. The a posteriori error estimator and the coupling strategy with an anisotropic remesher are first introduced. The mesh adaptation is controlled by a single‐parameter tolerance (TOL) in regions where the solution is regular, whereas a condition on the minimal element size h_min is enforced across solution discontinuities. This h_min condition is justified on the basis of an asymptotic analysis. The efficiency of the approach is tested with a supersonic flow over an aircraft. The evolution of a mesh adaptation/flow solution loop is shown, together with the influence of the parameters TOL and h_min. We verify numerically that the effect of varying h_min is concordant with the conclusions of the asymptotic analysis, giving hints on the selection of h_min with respect to TOL. Finally, we check that the results obtained with the a posteriori error estimator are at least as accurate as those obtained with anisotropic a priori error estimators. All the results presented can be obtained using a standard desktop computer, showing the efficiency of these adaptative methods. Copyright © 2008 John Wiley & Sons, Ltd.     

Robust high-order semi-implicit backward difference formulas for Navier-Stokes equations

Taylor-Hood finite elements provide a robust numerical discretization of Navier-Stokes equations (NSEs) with arbitrary high order of accuracy in space. To match the accuracy of the lowest degree Taylor-Hood element, we propose a very efficient time-stepping methods for unsteady flows, which are based on high-order semi-implicit backward difference formulas (SBDF) and the inclusion of grad -div term in the NSE. To mitigate the impact on the numerical accuracy (in time) of the extrapolation of the nonlinear term in SBDF, several variants of nonlinear extrapolation formulas are investigated. The first approach is based on an extrapolation of the nonlinear advection term itself. The second formula uses the extrapolation of the velocity prior to the evaluation of the nonlinear advection term as a whole. The third variant is constructed such that it produces similar error on both velocity and pressure to that with fully implicit backward difference formulas methods at a given order of accuracy. This can be achieved by fixing one-order higher than usually done in the extrapolation formula for the nonlinear advection term, while keeping the same extrapolation formula for the time derivative. The resulting truncation errors (in time) between these formulas are identified using Taylor expansions. These truncation error formulas are shown to properly represent the error seen in numerical tests using a 2D manufactured solution. Lastly, we show the robustness of the proposed semi-implicit methods by solving test cases with high Reynolds numbers using one of the nonlinear extrapolation formulas, namely, the 2D flow past circular cylinder at Re=300 and Re = 1000 and the 2D lid-driven cavity at Re = 50 000 and Re = 100 000. Our numerical solutions are found to be in a good agreement with those obtained in the literature, both qualitatively and quantitatively.     

风力机气动特性的浸入边界法模拟1)

风力发电机的空气动力学性能是决定风力机安全与效率的最重要因素之一。但由于影响风力机气动性能的参数众多,更加高效精确地模拟风力机气动特性一直是风力机的重要发展方向。本文提出了基于浸入边界法的风力机建模,网格离散,以及数值模拟的统一性框架。利用同伦变形来生成光滑的叶片模型,并且使用仿射变换来处理叶片的渐缩与扭转问题。首先,针对二维翼型的升阻力,检验了算法的数值精度。表明此方法对于阻力的模拟具有非常严格的一阶精度,进而提出采用理查森外推法来精确高效修正升阻力模拟结果。同时,模拟研究了拱曲度以及厚度对二维翼型升阻力的影响。随后,模拟研究了单风力机(包含塔架)在不同尖速比下的功率系数,并对塔架与叶片间的相互气动作用进行了初步分析。最后,模拟研究了双风力机在风场中不同前后间隔距离下的气动干涉问题。本文主要意义在于验证建模,离散,与数值模拟的一体化框架的有效可行性,进而为后续研究(给定约束下风力机自动优化选型)提供坚实基础。     

基于建筑风场数值模拟的网格层块加密法

在建筑风场的数值模拟中,当前普遍采用的离散网格多是计算前一次性布置的固定网格,通常很难适应实际流场变量的变化要求.为提高数值模拟的精度,基于结构化同位网格系统及控制容积离散微分方程的方法,将适应性网格局部加密(AMR)的思想引入到采用压力校正迭代算法的建筑风场模拟中,提出了一种半自适应的层块网格加密方法.该方法可结合误差分析对误差较大的区域网格实行自动判别并实施逐层块状加密.算例分析表明,该方法能在较高的效益下提高数值解的精度.     

A one-dimensional moving grid solution for the coupled non-linear equations governing multiphase flow in porous media. 2: Example simulations and sensitivity analysis

This paper presents numerical examples for the moving grid finite element algorithm derived in Part Ito solve the non-linear coupled set of PDEs governing immiscible multiphase flow in porous media in one dimension. Examples include single- and double-front simulations for two- and three-phase flow regimes and incorporating a mass sink. The modelling approach is shown to achieve significant savings in computation time and memory allocation when compared with fixed grid solutions of equivalent accuracy. This work includes sensitivity analyses for the parameters which are incorporated in the grid adaptation method, including the curvature weights, artificial viscosity and artificial repulsive force. It is found that the curvature weights are exponential functions of the negative ratio of the square root of the domain length to the number of discrete nodes. These weighting parameters are also shown to depend upon the shape of the front. On the basis of the examined simulations, it is recommended that artificial viscosity be neglected in the solution of the coupled non-linear set of PDEs governing multiphase flow in porous media. Similarly, use of a repulsive force is found to be unnecessary in simulations involving the migration of two liquid phases. For multiphase flows incorporating a gas phase it is recommended to use a non-zero value for the repulslive force to avoid development of an ill-conditioned nodal distribution matrix. An equation to evaluate the repulsive force under these circumstances is suggested.     

A convergence accelerator of a linear system of equations based upon the power method

This paper considers the convergence rate of an iterative numerical scheme as a method for accelerating at the post‐processor stage. The methodology adapted here is: (1) residual eigenmodes included in the origin of the convex hull are eliminated; (2) remaining residual terms are smoothed away by the main convergence algorithm. For this purpose, the polynomial matrix approach is employed for deriving the characteristic equation by two different methods. The first method is based on vector scaling and the second is based on the normal equations approach. The input for both methods is the solution difference between two consecutive iteration/cycle levels obtained from the main program. The singular value decomposition was employed for both methods due to the ill‐conditioned structure of the matrices. The use of the explicit form of the Richardson extrapolation in the present work overrules the need to employ the Richardson iteration with a Leja ordering. The performance of these methods was compared with the GMRES algorithm for three representative problems: two‐dimensional boundary value problem using the Laplace equation, three‐dimensional multi‐grid, potential solution over a sphere and the one‐dimensional steady state Burger equation. In all three examples both methods have the same rate of convergence, or better, as that of the GMRES method in terms of computer operational count. However, in terms of storage requirements, the method based upon vector scaling has a significant advantage over the normal equations approach as well as the GMRES method, in which only one vector of the N grid‐points is required. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical perturbation method for approximate solution of poisson's equation on a moderately deforming grid

In problems such as the computation of incompressible flows with moving boundaries, it may be necessary to solve Poisson's equation on a large sequence of related grids. In this paper the LU decomposition of the matrix A ₀ representing Poisson's equation discretized on one grid is used to efficiently obtain an approximate solution on a perturbation of that grid. Instead of doing an LU decomposition of the new matrix A , the RHS is perturbed by a Taylor expansion of A ^?1 about A ₀. Each term in the resulting series requires one ‘backsolve’ using the original LU . Tests using Laplace's equation on a square/rectangle deformation look promising; three and seven correction terms for deformations of 20% and 40% respectively yielded better than 1% accuracy. As another test, Poisson's equation was solved in an ellipse (fully developed flow in a duct) of aspect ratio 2/3 by perturbing about a circle; one correction term yielded better than 1% accuracy. Envisioned applications other than the computation of unsteady incompressible flow include: three-dimensional parabolic problems in tubes of varying cross-section, use of ‘elimination’ techniques other than LU decomposition, and the solution of PDEs other than Poisson's equation.     

Mechanical Quadrature Methods and Extrapolation Algorithms for Boundary Integral Equations with Linear Boundary Conditions in Elasticity

By potential theory, elastic problems with linear boundary conditions are converted into boundary integral equations (BIEs) with logarithmic and Cauchy singularity. In this paper, a mechanical quadrature method (MQMs) is presented to deal with the logarithmic and the Cauchy singularity simultaneously for solving the boundary integral equations. The convergence and stability are proved based on Anselone??s collective compact and asymptotical compact theory. Furthermore, an asymptotic expansion with odd powers of errors is presented, which possesses high accuracy order O(h ³). Using h ³?Richardson extrapolation algorithms (EAs), the accuracy order of the approximation can be greatly improved to O(h ⁵), and an a posteriori error estimate can be obtained for constructing a self-adaptive algorithm. The efficiency of the algorithm is illustrated by examples.     

A posteriori error of a discontinuous Galerkin scheme for compressible miscible displacement problems with molecular diffusion and dispersion

A kind of compressible miscible displacement problems including molecular diffusion and dispersion in porous media is studied. A symmetric interior penalty discontinuous Galerkin method is applied to the coupled system of flow and transport. Explicit a posteriori error estimates are obtained based on the duality argument, approximation properties and residual notations. The resulting error bound is shown to be reliable and worthwhile for guiding dynamic mesh adaptivity. Copyright © 2009 John Wiley & Sons, Ltd.     

On the averaging principle for one-frequency systems. An application to satellite motions

This paper is related to our previous works (Morosi and Pizzocchero in J. Phys. A, Math. Gen. 39:3673–3702, 2006; Nonlinear Dyn., 2008), on the error estimate of the averaging technique for systems with one fast angular variable. In the cited references, a general method (of mixed analytical and numerical type) has been introduced to obtain precise, fully quantitative estimates on the averaging error. Here, this procedure is applied to the motion of a satellite in a polar orbit around an oblate planet, retaining only the J ₂ term in the multipole expansion of the gravitational potential. To exemplify the method, the averaging errors are estimated for the data corresponding to two Earth satellites; for a very large number of orbits, computation of our estimators is much less expensive than the direct numerical solution of the equations of motion.     

二维弹性力学问题的光滑无网格伽辽金法

计算效率低的问题长期阻碍着无网格伽辽金法(element-free Galerkin method, EFGM) 的深入发展. 为了提高EFGM 的计算速度, 本文提出一种求解二维弹性力学问题的光滑无网格伽辽金法. 该方法在问题域内采用滑动最小二乘法(moving least square, MLS)近似、在域边界上采用线性插值建立位移场函数; 基于广义梯度光滑算子得到两层嵌套光滑三角形背景网格上的光滑应变, 根据广义光滑伽辽金弱形式建立系统离散方程. 两层嵌套光滑三角形网格是由三角形背景网格本身以及四个等面积三角形子网格组成. 为了提高方法的精度, 由Richardson外推法确定两层光滑网格上的最优光滑应变. 几个数值算例验证了该方法的精度和计算效率. 数值结果表明, 随着光滑积分网格数目的增加, 光滑无网格伽辽金法的计算精度逐步接近EFGM 的, 但计算效率要远远高于EFGM的. 另外, 光滑无网格伽辽金法的边界条件可以像有限元那样直接施加. 从计算精度和效率综合考虑, 光滑无网格伽辽金法比EFGM具有更好的数值表现, 具有十分广阔的发展空间.      

Influence of the spatial resolution on fine-scale features in DNS of turbulence generated by a single square grid

We focus in this paper on the effect of the resolution of direct numerical simulations (DNS) on the spatio-temporal development of the turbulence downstream of a single square grid. The aims of this study are to validate our numerical approach by comparing experimental and numerical one-point statistics downstream of a single square grid and then investigate how the resolution is impacting the dynamics of the flow. In particular, using the Q–R diagram, we focus on the interaction between the strain-rate and rotation tensors, the symmetric and skew-symmetric parts of the velocity gradient tensor, respectively. We first show good agreement between our simulations and hot-wire experiment for one-point statistics on the centreline of the single square grid. Then, by analysing the shape of the Q–R diagram for various streamwise locations, we evaluate the ability of under-resolved DNS to capture the main features of the turbulence downstream of the single square grid.     

Efficient computation of mean drag for the subcritical flow past a circular cylinder using general Galerkin G2

General Galerkin (G2) is a new computational method for turbulent flow, where a stabilized Galerkin finite element method is used to compute approximate weak solutions to the Navier–Stokes equations directly, without any filtering of the equations as in a standard approach to turbulence simulation, such as large eddy simulation, and thus no Reynolds stresses are introduced, which need modelling. In this paper, G2 is used to compute the drag coefficient c_D for the flow past a circular cylinder at Reynolds number Re=3900, for which the flow is turbulent. It is found that it is possible to approximate c_D to an accuracy of a few percent, corresponding to the accuracy in experimental results for this problem, using less than 10⁵ mesh points, which makes the simulations possible using a standard PC. The mesh is adaptively refined until a stopping criterion is reached with respect to the error in a chosen output of interest, which in this paper is c_D. Both the stopping criterion and the mesh‐refinement strategy are based on a posteriori error estimates, in the form of a space–time integral of residuals times derivatives of the solution of a dual problem, linearized at the approximate solution, and with data coupling to the output of interest. Copyright © 2008 John Wiley & Sons, Ltd.