中心刚体-柔性梁刚柔耦合动力学模型降阶研究

有限单元法被广泛的采用来描述柔性体的弹性变形,然而有限元节点坐标数目庞大,将会给动力学方程求解带来巨大的计算负担.如何降低柔性体的自由度,是当前柔性多体系统动力学研究的一个重要命题.本文以中心刚体-柔性梁系统为例,采用Krylov方法和模态方法进行降价.然后分别采用有限元全模型、Krylov降阶模型和模态降阶模型,对中心刚体-柔性梁进行刚-柔耦合动力学仿真.仿真结果表明,与采用模态降阶方法相比,采用Krylov模型降阶方法只需要较低的自由度,就可以得到与采用有限元方法完全一致的结果.说明Krylov模型降阶方法能够有效的用于柔性多体系统的模型降价研究.     

Non-stationary iterative solvers on a PC cluster

In this study, we introduce cost effective strategies and algorithms for parallelizing the Krylov subspace based non-stationary iterative solvers such as Bi-CGM and Bi-CGSTAB for distributed computing on a cluster of PCs using ANULIB message passing libraries. We investigate the effectiveness of the parallel solvers on the linear systems resulting in numerical solution of some 2D and 3D nonlinear partial differential equations governing heat convection process by finite element, finite difference and wavelet based numerical schemes. Largely Bi-CGM is found to give better performance measured in terms of speedup factors.     

The communication-hiding pipelined BiCGstab method for the parallel solution of large unsymmetric linear systems

A High Performance Computing alternative to traditional Krylov subspace methods, pipelined Krylov subspace solvers offer better scalability in the strong scaling limit compared to standard Krylov subspace methods for large and sparse linear systems. The typical synchronization bottleneck is mitigated by overlapping time-consuming global communication phases with local computations in the algorithm. This paper describes a general framework for deriving the pipelined variant of any Krylov subspace algorithm. The proposed framework was implicitly used to derive the pipelined Conjugate Gradient (p-CG) method in Hiding global synchronization latency in the preconditioned Conjugate Gradient algorithm by P. Ghysels and W. Vanroose, Parallel Computing, 40(7):224–238, 2014. The pipelining framework is subsequently illustrated by formulating a pipelined version of the BiCGStab method for the solution of large unsymmetric linear systems on parallel hardware. A residual replacement strategy is proposed to account for the possible loss of attainable accuracy and robustness by the pipelined BiCGStab method. It is shown that the pipelined algorithm improves scalability on distributed memory machines, leading to significant speedups compared to standard preconditioned BiCGStab.     

Application of block Krylov subspace algorithms to the Wilson-Dirac equation with multiple right-hand sides in lattice QCD

It is well known that the block Krylov subspace solvers work efficiently for some cases of the solution of differential equations with multiple right-hand sides. In lattice QCD calculation of physical quantities on a given configuration demands us to solve the Dirac equation with multiple sources. We show that a new block Krylov subspace algorithm recently proposed by the authors reduces the computational cost significantly without losing numerical accuracy for the solution of the O(a)-improved Wilson-Dirac equation.     

Forward dynamical analysis of flexible multibody systems using system-level model reduction

Fast simulation (e.g., real-time) of flexible multibody systems is typically restricted by the presence of both differential and algebraic equations in the model equations, and the number of degrees of freedom required to accurately model flexibility. Model reduction techniques can alleviate the problem, although the classically used body-level model reduction and general-purpose system-level techniques do not eliminate the algebraic equations and do not necessarily result in optimal dimension reduction. In this research, Global Modal Parametrization, a model reduction technique for flexible multibody systems is further developed to speed up simulation of flexible multibody systems. The reduction of the model is achieved by projection on a curvilinear subspace instead of the classically used fixed vector space, requiring significantly less degrees of freedom to represent the system dynamics with the same level of accuracy. The numerical experiment in this paper illustrates previously unexposed sources of approximation error: (1) the rigid body motion is computed in a forward dynamical analysis resulting in a small divergence of the rigid body motion, and (2) the errors resulting from the transformation from the modal degrees of freedom of the reduced model back to the original degrees of freedom. The effect of the configuration space discretization coarseness on the different approximation error sources is investigated. The trade-offs to be defined by the user to control these approximation errors are explained.     

Multi-order Arnoldi-based model order reduction of second-order time-delay systems

In this paper, we discuss the Krylov subspace-based model order reduction methods of second-order systems with time delays, and present two structure-preserving methods for model order reduction of these second-order systems, which avoid to convert the second-order systems into first-order ones. One method is based on a Krylov subspace by using the Taylor series expansion, the other method is based on the Laguerre series expansion. These two methods are used in the multi-order Arnoldi algorithm to construct the projection matrices. The resulting reduced models can not only preserve the structure of the original systems, but also can match a certain number of approximate moments or Laguerre expansion coefficients. The effectiveness of the proposed methods is demonstrated by two numerical examples.     

Accelerating iterative linear solvers using multiple graphical processing units

In this paper, we develop, study and implement iterative linear solvers and preconditioners using multiple graphical processing units (GPUs). Techniques for accelerating sparse matrix–vector (SpMV) multiplication, linear solvers and preconditioners are presented. Four Krylov subspace solvers, a Neumann polynomial preconditioner and a domain decomposition preconditioner are implemented. Our numerical tests with NVIDIA C2050 GPUs show that the SpMV kernel can be sped over 40 times faster using four GPUs. Our linear solvers and preconditioners have similar speedup.     

Discrete system reduction via impulse-response Gramians and itsrelation to q-Markov covers

A method for model reduction of linear discrete systems is proposed. It is based on the impulse-response Gramian proposed by the authors (1989) for discrete systems. This Gramian is an extension of the one proposed for linear continuous systems and contains information on the input-output behavior of the system. The rth-order reduced-order models are made to retain the first r Markov parameters and the first r×r elements of the impulse-response Gramian of the original system. The relation between this method and the q-Markov cover method is also discussed. The method is illustrated by a numerical example     

GMRES implementations and residual smoothing techniques for solving ill-posed linear systems

There are verities of useful Krylov subspace methods to solve nonsymmetric linear system of equations. GMRES is one of the best Krylov solvers with several different variants to solve large sparse linear systems. Any GMRES implementation has some advantages. As the solution of ill-posed problems are important. In this paper, some GMRES variants are discussed and applied to solve these kinds of problems. Residual smoothing techniques are efficient ways to accelerate the convergence speed of some iterative methods like CG variants. At the end of this paper, some residual smoothing techniques are applied for different GMRES methods to test the influence of these techniques on GMRES implementations.     

Sparse iterative algorithm software for large-scale MIMD machines: An initial discussion and implementation

The parallelization of sophisticated applications has dramatically increased in recent years. As machine capabilities rise, greater emphasis on modeling complex phenomena can be expected. Many of these applications require the solution of large sparse matrix equations which approximate systems of partial differential equations (PDEs). Therefore we consider parallel iterative solvers for large sparse non-symmetric systems and issues related to parallel sparse matrix software. We describe a collection of parallel iterative solvers which use a distributed sparse matrix format that facilitates the interface between specific applications and a variety of Krylov subspace techniques and multigrid methods. These methods have been used to solve a number of linear and non-linear PDE problems on a 1024-processor NCUBE 2 hypercube. Over 1 Gflop sustained computation rates are achieved with many of these solvers, demonstrating that high performance can be attained even when using sparse matrix data structures.     

基于Krylov-Schur重启技术的Arnoldi模型降阶方法

Krylov子空间模型降阶方法是模型降阶中的典型方法之一,Arnoldi模型降阶方法是这类方法中的一类基本方法。运用重正交化的Arnoldi算法得到[r]步Arnoldi分解;执行Krylov-Schur重启过程,导出基于Krylov-Schur重启技术的Arnoldi模型降阶方法。运用此方法对大规模线性时不变系统进行降阶,得到具有较高近似精度的稳定的降阶系统,从而改善了Krylov子空间降阶方法不能保持降阶系统稳定性的不足。数值算例验证了此方法是行之有效的。     

Model order reduction of nonlinear models based on decoupled multi-model via trajectory piecewise linearization

In this paper a novel model order reduction method for nonlinear models, based on decoupled multi-model, via trajectory piecewise-linearization is proposed. Through different strategies in trajectory piecewise-linear model reduction, model order reduction of a trajectory piecewise-linear model based on output weighting (TPWLOW), has been developed by authors of current work. The structure of mentioned work was founded based on Krylov subspace method, which is appropriate for high order models. Indeed the size of the Krylov subspaces may increase with the number of inputs of the system. As a result, the use of Krylov subspace method may become deficient the case for multi-input systems of order few decades. This paper aims to generalize the idea of model reduction of TPWLOW model by concentrating on balanced truncation technique which is appropriate for medium size systems. In addition, the proposed method either guarantees or provides guaranteed stability in some mentioned conditions. Finally, applicability of the reduced model, instead of high-order decoupled multi-model in weighting multi-model controllers, is investigated through the control of a nonlinear Alstom gasifier benchmark.     

Comparison of model reduction techniques for large mechanical systems

Model reduction is a necessary procedure for simulating large elastic systems, which are mostly modeled by the Finite Element Method (FEM). In order to reduce the system’s large dimension, various techniques have been developed during the last decades, many of which share some common characteristics (Guyan, Dynamic, CMS, IRS, SEREP). A fact remains that many reduction approaches do not succeed in reducing the system’s dimension without damaging the dynamical properties of the model. The mathematical field of control theory offers alternative reduction methods, which can be applied to second order Ordinary Differential Equations (ODEs), derived by the FE-discretization of large elastic Multi Body Systems (MBS), e.g., Krylov subspace method or balanced truncation. In this paper, some of these methods are applied to the elastic piston rod. The validity of the reduced models is checked by applying Modal Correlation Criteria (MCC), since only the eigenfrequency comparison is not sufficient. Diagonal Perturbation is proposed as an efficient method for iteratively solving ill-conditioned large sparse linear systems (A x=b, A: ill-conditioned) when direct methods fail due to memory capacity problems. This is the case of FE-discretized systems, when tolerance failure occurs during the discretization procedure.     

Modeling Spatial Motion of 3D Deformable Multibody Systems with Nonlinearities

A computational strategy for modeling spatial motion of systems of flexible spatial bodies is presented. A new integral formulation of constraints is used in the context of the floating frame of reference approach. We discuss techniques to linearize the equations of motion both with respect to the kinematical coupling between the deformation and rigid body degrees of freedom and with respect to the geometrical nonlinearities (inclusion of stiffening terms). The plastic behavior of bodies is treated by means of plastic multipliers found as the result of fixed-point type iterations within a time step. The time integration is based on implicit Runge Kutta schemes with arbitrary order and of the RadauIIA type. The numerical results show efficiency of the developed techniques.     

Preconditioned Krylov solvers on GPUs

In this paper, we study the effect of enhancing GPU-accelerated Krylov solvers with preconditioners. We consider the BiCGSTAB, CGS, QMR, and IDR(s) Krylov solvers. For a large set of test matrices, we assess the impact of Jacobi and incomplete factorization preconditioning on the solvers’ numerical stability and time-to-solution performance. We also analyze how the use of a preconditioner impacts the choice of the fastest solver.     

On the use of absolute interface coordinates in the floating frame of reference formulation for flexible multibody dynamics

In this work a new formulation for flexible multibody systems is presented based on the floating frame formulation. In this method, the absolute interface coordinates are used as degrees of freedom. To this end, a coordinate transformation is established from the absolute floating frame coordinates and the local interface coordinates to the absolute interface coordinates. This is done by assuming linear theory of elasticity for a body’s local elastic deformation and by using the Craig–Bampton interface modes as local shape functions. In order to put this new method into perspective, relevant relations between inertial frame, corotational frame and floating frame formulations are explained. As such, this work provides a clear overview of how these three well-known and apparently different flexible multibody methods are related. An advantage of the method presented in this work is that the resulting equations of motion are of the differential rather than the differential-algebraic type. At the same time, it is possible to use well-developed model order reduction techniques on the flexible bodies locally. Hence, the method can be employed to construct superelements from arbitrarily shaped three dimensional elastic bodies, which can be used in a flexible multibody dynamics simulation. The method is validated by simulating the static and dynamic behavior of a number of flexible systems.     

应用预处理AWE技术快速计算导体目标宽带RCS

应用渐近波形估计技术计算目标宽带雷达散射截面（RCS）,可有效提高计算效率。然而当目标为电大尺寸时,阻抗矩阵求逆运算将十分耗时,甚至无法计算。提出使用Krylov子空间迭代法取代矩阵逆来求解大型矩阵方程,应用双门槛不完全LU分解预处理技术降低迭代求解所需的迭代次数。数值计算表明,该方法结果与矩量法逐点求解结果吻合良好,并且计算效率大大提高。     

An efficient preconditioned iterative solver for solving a coupled fluid structure interaction problem

Modelling the interaction of an acoustic field in a fluid and an elastic structure submerged in the fluid leads to a system of complex linear equations with a complicated sparsity structure and, for higher wavenumbers and adequate modelling, the systems are very large. Direct methods are not practical. Preconditioned iterative methods, which are suitable for single operator equations, are not immediately applicable to the coupled case. This article proposes a block diagonal preconditioner of the sparse approximate inverse (SPAI) type that can accelerate the convergence of Krylov iterative solvers for the coupled system. Moreover, the proposed preconditioner can properly and implicitly scale the coupled matrix. Some numerical results are presented to demonstrate the effectiveness of the new method.     

On the scalability of inexact balancing domain decomposition by constraints with overlapped coarse/fine corrections

In this work, we analyze the scalability of inexact two-level balancing domain decomposition by constraints (BDDC) preconditioners for Krylov subspace iterative solvers, when using a highly scalable asynchronous parallel implementation where fine and coarse correction computations are overlapped in time. This way, the coarse-grid problem can be fully overlapped by fine-grid computations (which are embarrassingly parallel) in a wide range of cases. Further, we consider inexact solvers to reduce the computational cost/complexity and memory consumption of coarse and local problems and boost the scalability of the solver. Out of our numerical experimentation, we conclude that the BDDC preconditioner is quite insensitive to inexact solvers. In particular, one cycle of algebraic multigrid (AMG) is enough to attain algorithmic scalability. Further, the clear reduction of computing time and memory requirements of inexact solvers compared to sparse direct ones makes possible to scale far beyond state-of-the-art BDDC implementations. Excellent weak scalability results have been obtained with the proposed inexact/overlapped implementation of the two-level BDDC preconditioner, up to 93,312 cores and 20 billion unknowns on JUQUEEN. Further, we have also applied the proposed setting to unstructured meshes and partitions for the pressure Poisson solver in the backward-facing step benchmark domain.