Fourth-order runge-kutta schemes for fluid mechanics applications

Multiple high-order time-integration schemes are used to solve stiff test problems related to the Navier-Stokes (NS) equations. The primary objective is to determine whether high-order schemes can displace currently used second-order schemes on stiff NS and Reynolds averaged NS (RANS) problems, for a meaningful portion of the work-precision spectrum. Implicit-Explicit (IMEX) schemes are used on separable problems that naturally partition into stiff and nonstiff components. Non-separable problems are solved with fully implicit schemes, oftentimes the implicit portion of an IMEX scheme. The convection-diffusion-reaction (CDR) equations allow a term by term stiff/nonstiff partition that is often well suited for IMEX methods. Major variables in CDR converge at near design-order rates with all formulations, including the fourth-order IMEX additive Runge-Kutta (ARK₂) schemes that are susceptible to order reduction. The semi-implicit backward differentiation formulae and IMEX ARK₂ schemes are of comparable efficiency. Laminar and turbulent aerodynamic applications require fully implicit schemes, as they are not profitably partitioned. All schemes achieve design-order convergence rates on the laminar problem. The fourth-order explicit singly diagonally implicit Runge-Kutta (ESDIRK4) scheme is more efficient than the popular second-order backward differentiation formulae (BDF2) method. The BDF2 and fourth-order modified extended backward differentiation formulae (MEBDF4) schemes are of comparable efficiency on the turbulent problem. High precision requirements slightly favor the MEBDF4 scheme (greater than three significant digits). Significant order reduction plagues the ESDIRK4 scheme in the turbulent case. The magnitude of the order reduction varies with Reynolds number. Poor performance of the high-order methods can partially be attributed to poor solver performance. Huge time steps allowed by high-order formulations challenge the capabilities of algebraic solver technology.     

Fourth-Order Runge–Kutta Schemes for Fluid Mechanics Applications

Multiple high-order time-integration schemes are used to solve stiff test problems related to the Navier–Stokes (NS) equations. The primary objective is to determine whether high-order schemes can displace currently used second-order schemes on stiff NS and Reynolds averaged NS (RANS) problems, for a meaningful portion of the work-precision spectrum. Implicit–Explicit (IMEX) schemes are used on separable problems that naturally partition into stiff and nonstiff components. Non-separable problems are solved with fully implicit schemes, oftentimes the implicit portion of an IMEX scheme. The convection–diffusion-reaction (CDR) equations allow a term by term stiff/nonstiff partition that is often well suited for IMEX methods. Major variables in CDR converge at near design-order rates with all formulations, including the fourth-order IMEX additive Runge–Kutta (ARK₂) schemes that are susceptible to order reduction. The semi-implicit backward differentiation formulae and IMEX ARK₂ schemes are of comparable efficiency. Laminar and turbulent aerodynamic applications require fully implicit schemes, as they are not profitably partitioned. All schemes achieve design-order convergence rates on the laminar problem. The fourth-order explicit singly diagonally implicit Runge–Kutta (ESDIRK4) scheme is more efficient than the popular second-order backward differentiation formulae (BDF2) method. The BDF2 and fourth-order modified extended backward differentiation formulae (MEBDF4) schemes are of comparable efficiency on the turbulent problem. High precision requirements slightly favor the MEBDF4 scheme (greater than three significant digits). Significant order reduction plagues the ESDIRK4 scheme in the turbulent case. The magnitude of the order reduction varies with Reynolds number. Poor performance of the high-order methods can partially be attributed to poor solver performance. Huge time steps allowed by high-order formulations challenge the capabilities of algebraic solver technology.     

Dispersion and Dissipation Error in High-Order Runge-Kutta Discontinuous Galerkin Discretisations of the Maxwell Equations

Different time-stepping methods for a nodal high-order discontinuous Galerkin discretisation of the Maxwell equations are discussed. A comparison between the most popular choices of Runge-Kutta (RK) methods is made from the point of view of accuracy and computational work. By choosing the strong-stability-preserving Runge-Kutta (SSP-RK) time-integration method of order consistent with the polynomial order of the spatial discretisation, better accuracy can be attained compared with fixed-order schemes. Moreover, this comes without a significant increase in the computational work. A numerical Fourier analysis is performed for this Runge-Kutta discontinuous Galerkin (RKDG) discretisation to gain insight into the dispersion and dissipation properties of the fully discrete scheme. The analysis is carried out on both the one-dimensional and the two-dimensional fully discrete schemes and, in the latter case, on uniform as well as on non-uniform meshes. It also provides practical information on the convergence of the dissipation and dispersion error up to polynomial order 10 for the one-dimensional fully discrete scheme.     

一种新的非重复性包标记IP追踪方案

大多数概率包标记(PPM)因为重复标记和固定的标记概率而存在最弱链问题,从而导致重构路径的弱收敛性。文章提出了一种新的非重复性包标记的IP追踪方案,通过重载部分偏移域来获得更多的标记空间。具体以分片标记(FMS)方案为例,给出了标记算法和编码方式,分析了新方案在追踪范围、收敛时间、分片重组、误报和计算量等方面的性能。通过比较,证明新方案是优于FMS的。     

Time-varying PSO - convergence analysis, convergence-related parameterization and new parameter adjustment schemes

In this paper, a formal convergence analysis of the conventional PSO algorithms with time-varying parameters is presented. Based on this analysis, a new convergence-related parametric model for the conventional PSO is introduced. Finally, several new schemes for parameter adjustment, providing significant performance benefits, are introduced. Performance of these schemes is empirically compared to conventional PSO algorithms on a set of selected benchmarks. The tests prove effectiveness of the newly introduced schemes, especially regarding their ability to efficiently explore the search space.     

两种级联空时格码方案的EXIT分析

级联空时格码方案可以有效地提高空时格码系统的性能。研究了两种级联空时格码方案:PC-STTC和ST-Turbo-TC。由于两种级联方案都使用了迭代译码方法,而外信息转移(EXIT)图是分析迭代译码性能的有利工具,主要分析比较了PC-STTC和ST-Turbo-TC的EXIT性能,比较了不同信噪比和编码多项式对于译码外信息转移特性的影响。研究结果显示,PC-STTC方案比ST-Turbo-TC方案有着更好的迭代收敛性能,同时给出了BER仿真图证明了EXIT图的分析结论。     

Parameter convergence in adaptive feedback linearizing control of limit cycling systems with input saturation

Adaptive feedback linearizing control schemes are used to suppress limit cycle oscillations in nonlinear systems where the system parameters are either unknown or uncertain. Parameter convergence is desirable in these schemes as it provides a measure of robustness of the scheme and also permits the unknown/uncertain system parameters to be estimated. In recent work, we have shown how using a persistently exciting forcing it is possible to achieve parameter convergence in nonlinear limit cycling systems. In practice, however, limits on the control input to the plant due to saturation must be considered, and the main goal of this work is to analyze the effect of input saturation on parameter convergence in an adaptive feedback linearization framework. In particular, a technique known as control hedging is incorporated and the effectiveness of this method for very severe saturation constraints has been evaluated. Results are presented for a single degree-of-freedom wing rock dynamics model and a multi degree-of-freedom combustion acoustics model showing successful parameter convergence even in the presence of input saturation.     

Experimental evaluation of iterative learning control algorithms for non-minimum phase plants

The purpose of this paper is two-fold, firstly it describes the development and modelling of an experimental test facility as a platform on which to assess the performance of Iterative Learning Control (ILC) schemes. This facility includes a non-minimum phase component. Secondly, P-Type, D-Type and phase-lead types of the algorithm have been implemented on the test-bed, results are presented for each method and their performance is compared. Although all the ILC strategies tested experience eventual divergence when applied to a non-minimum phase system, it is found that there is an optimum phase-lead ILC design that maximizes convergence and minimizes error. A general method of arriving at this phase-lead from knowledge of the plant model is described. A variety of filters have been applied and assessed in order to improve the overall performance of the algorithm.     

Convergence acceleration for the steady-state Euler equations

We consider the iterative solution of systems of equations arising from discretizations of the non-linear Euler equations governing compressible flow. The differential equation is discretized on a structured grid, and the steady-state solution is computed by a time-marching method.A convergence acceleration technique based on semicirculant approximations of the difference operator or the Jacobian is used. Implementation issues and variants of the scheme allowing for a reduction of the arithmetic complexity and memory requirement are discussed. The technique can be combined with a variety of iterative solvers, but we focus on non-linear explicit Runge-Kutta time-integration schemes. The results show that the single-stage forward Euler method can be used, and that the time step is not limited by a CFL-criterion. This results in that the arithmetic work required for computing the solution is equivalent to the work required for a fixed number of residual evaluations.     

多细胞基因表达式编程的函数优化算法

针对处理复杂的函数优化问题时传统演化算法易出现收敛性能不佳、搜索冗长和精度不高等问题,提出了一种基于多细胞基因表达式编程的函数优化新算法.该算法引入了同源基因和细胞系统思想,设计了相应新的个体编码方案、种群生成和遗传操作策略.通过对8个Benchmarks函数的对比实验,验证了该算法具有很强的全局寻优能力、较佳的收敛性能和更高的解精度.     

Anderson accelerated fixed-stress splitting schemes for consolidation of unsaturated porous media

In this paper, we study the robust linearization of nonlinear poromechanics of unsaturated materials. The model of interest couples the Richards equation with linear elasticity equations, generalizing the classical Biot equations. In practice a monolithic solver is not always available, defining the requirement for a linearization scheme to allow the use of separate simulators. It is not met by the classical Newton method. We propose three different linearization schemes incorporating the fixed-stress splitting scheme, coupled with an L-scheme, Modified Picard and Newton linearization of the flow equations. All schemes allow the efficient and robust decoupling of mechanics and flow equations. In particular, the simplest scheme, the Fixed-Stress-L-scheme, employs solely constant diagonal stabilization, has low cost per iteration, and is very robust. Under mild, physical assumptions, it is theoretically shown to be a contraction. Due to possible break-down or slow convergence of all considered splitting schemes, Anderson acceleration is applied as post-processing. Based on a special case, we justify theoretically the general ability of the Anderson acceleration to effectively accelerate convergence and stabilize the underlying scheme, allowing even non-contractive fixed-point iterations to converge. To our knowledge, this is the first theoretical indication of this kind. Theoretical findings are confirmed by numerical results. In particular, Anderson acceleration has been demonstrated to be very effective for the considered Picard-type methods. Finally, the Fixed-Stress-Newton scheme combined with Anderson acceleration shows the best performance among the splitting schemes.     

Energy conserving/decaying implicit time-stepping scheme for nonlinear dynamics of three-dimensional beams undergoing finite rotations

In this work we present an extension of time-integration energy conserving scheme which introduces desirable properties of controllable energy decay, as well as numerical dissipation of high-frequency contribution to total response. Finite element implementation details are given for the chosen model problem of geometrically exact beam undergoing finite rotations. Several numerical simulations illustrate a very satisfying performance of the proposed time-stepping scheme.     

基于八叉树建模的人工蜂群动态路径规划算法

针对群体动画路径规划过程中存在收敛速度慢、与环境的交互性差等问题,提出一种基于八叉树建模的人工蜂群路径规划算法.将环境以八叉树模型进行分解并存储,引入群体自适应机制,通过粒子适应度和环境编码信息寻找目标点,采用分层方式实现路径的动态规划.仿真实验结果表明,该算法能进行群体路径动态规划,在寻优性和收敛性上均有较好的性能.     

Concepts and Application of Time-Limiters to High Resolution Schemes

A new class of implicit high-order non-oscillatory time integration schemes is introduced in a method-of-lines framework. These schemes can be used in conjunction with an appropriate spatial discretization scheme for the numerical solution of time dependent conservation equations. The main concept behind these schemes is that the order of accuracy in time is dropped locally in regions where the time evolution of the solution is not smooth. By doing this, an attempt is made at locally satisfying monotonicity conditions, while maintaining a high order of accuracy in most of the solution domain. When a linear high order time integration scheme is used along with a high order spatial discretization, enforcement of monotonicity imposes severe time-step restrictions. We propose to apply limiters to these time-integration schemes, thus making them non-linear. When these new schemes are used with high order spatial discretizations, solutions remain non-oscillatory for much larger time-steps as compared to linear time integration schemes. Numerical results obtained on scalar conservation equations and systems of conservation equations are highly promising.     

Integrating rotation from angular velocity

The integration of the rotation from a given angular velocity is often required in practice. The present paper explores how the choice of the parametrization of rotation, when employed in conjuction with different numerical time-integration schemes, effects the accuracy and the computational efficiency. Three rotation parametrizations - the rotational vector, the Argyris tangential vector and the rotational quaternion - are combined with three different numerical time-integration schemes, including classical explicit Runge-Kutta method and the novel midpoint rule proposed here. The key result of the study is the assessment of the integration errors of various parametrization-integration method combinations. In order to assess the errors, we choose a time-dependent function corresponding to a rotational vector, and derive the related exact time-dependent angular velocity. This is then employed in the numerical solution as the data. The resulting numerically integrated approximate rotations are compared with the analytical solution. A novel global solution error norm for discrete solutions given by a set of values at chosen time-points is employed. Several characteristic angular velocity functions, resulting in small, finite and fast oscillating rotations are studied.     

以操作系统为中心的存储一致性模型--线程一致性模型

分布共享存储系统为保证程序的正确执行，必须通过存储一致性模型对共享存储访问顺序加以限制，而现有模型在可扩展性和操作系统级实现方面存在不足。结合多线程的特点，提出了一种以操作系统为中心的线程一致性模型，通过并行程序执行过程中线程状态的变化来观察和限制存储访问事件的正确顺序，有利于系统的可扩展性、一致性维护信息获取的方便性和完备性以及操作系统本身的设计和实现。分别从模型的定义、正确性证明、实现方案和性能分析等几个方面展开了论述。     

A unified approach for displacement,equilibrium and hybrid finite element models in elasto-plasticity

Finite element models for elasto-plastic incremental analysis are derived from a three-field variational principle. The Newton-Raphson method is applied to solve the nonlinear system of equations which is obtained from the stationarity condition of this principle. The iterative schemes are discussed in detail for pure displacement and for pure equilibrium models from which iterative schemes for hybrid models follow directly. In the displacement model, the compatibility of the strains and the plasticity criterium are satisfied during the whole iterative process, while the equilibrium of the stresses is restored only in the mean after convergence. In the equilibrium model, the plasticity criterium and the compatibility of the strains are verified in the mean during the iterative process; when convergence is achieved, the stresses are locally in equilibrium with the applied external loads. In both cases, a tangential stiffness matrix can be constructed, even for perfectly plastic materials and it allows one to obtain always very good convergence properties. Examples are shown for plane stress and axisymmetric cases.     

Parameter convergence of a new class of adaptive controllers

A new class of adaptive control schemes for minimum-phase linear time invariant (LTI) systems has recently been developed using nonlinear design techniques which guarantee improved transient performance in addition to closed-loop stability and asymptotic tracking. In this paper we establish the parameter convergence properties of this new class of schemes in the presence of persistently exciting signals and compare them with the properties of the traditional adaptive controllers. We show that the new class of adaptive controllers has stronger parameter convergence properties in the presence of overparameterization     

两种基于Narendra方案的混合自适应修正方案

本文在Ｎａｒｅｎｄｒａ等人所提出的适用于理想对象的混合自适应控制方案的基础上提出了两种混合自适应方案。本文首先将Ｎａｒｅｎｄｒａ等人提出的仅适合于理想对象的控制器方案利用规范化方法推广到对象具有未建模动态情况，提出了一种修正方案，然后又针对修正方案在某些情况下收敛速度缓慢这一缺点进行分析，发现是Ｎａｒｅｎｄｒａ方案的固有缺点，于是又提出了一种变换误差模型方案，采用两种不同的敏感函数和误差测度模型…     

多变量非线性系统的变阶采样迭代学习控制

针对存在初态误差的情形, 提出多变量非线性系统的变阶采样迭代学习控制方法. 相对固定阶迭代学习算法, 变阶算法可有效降低跟踪误差. 对变阶采样迭代学习算法进行了收敛性分析, 推导出收敛充分条件. 给出了变阶学习的两种实现策略-DD (Direct division)和DIP (Division in phases)策略. 数值仿真表明, 基于DIP策略的变阶采样迭代学习算法在获得较高的控制精度的同时, 具有较快的收敛速度.