扩展有限元法(XFEM)及其应用

扩展有限元法(extended finite element method,XFEM)是1999年提出的一种求解不连续力学问题的数值方法, 它继承了常规有限元法(CFEM)的所有优点, 在模拟界面、裂纹生长、复杂流体等不连续问题时特别有效, 短短几年间得到了快速发展与应用. XFEM与CFEM的最根本区别在于, 它所使用的网格与结构内部的几何或物理界面无关, 从而克服了在诸如裂纹尖端等高应力和变形集中区进行高密度网格剖分所带来的困难, 模拟裂纹生长时也无需对网格进行重新剖分.重点介绍XFEM的基本原理、实施步骤及应用实例等, 并进行必要的评述. 单位分解概念保证了XFEM的收敛, 基于此, XFEM通过改进单元的形状函数使之包含问题不连续性的基本成分, 从而放松对网格密度的过分要求. 水平集法是XFEM中常用的确定内部界面位置和跟踪其生长的数值技术, 任何内部界面可用它的零水平集函数表示. 第2和第3节分别简要介绍单位分解法和水平集法;第4节和第5节介绍XFEM的基本思想、详细实施步骤和若干应用实例, 同时修正了以往文献中的一些不妥之处; 最后, 初步展望了该领域尚需进一步研究的课题.      

Coupling finite difference method with finite particle method for modeling viscous incompressible flows

A hybrid approach to couple finite difference method (FDM) with finite particle method (FPM) (ie, FDM-FPM) is developed to simulate viscous incompressible flows. FDM is a grid-based method that is convenient for implementing multiple or adaptive resolutions and is computationally efficient. FPM is an improved smoothed particle hydrodynamics (SPH), which is widely used in modeling fluid flows with free surfaces and complex boundaries. The proposed FDM-FPM leverages their advantages and is appealing in modeling viscous incompressible flows to balance accuracy and efficiency. In order to exchange the interface information between FDM and FPM for achieving consistency, stability, and convergence, a transition region is created in the particle region to maintain the stability of the interface between two methods. The mass flux algorithm is defined to control the particle creation and deletion. The mass is updated by N-S equations instead of the interpolation. In order to allow information exchange, an overlapping zone is defined near the interface. The information of overlapping zone is obtained by an FPM-type interpolation. Taylor-Green vortices and lid-driven shear cavity flows are simulated to test the accuracy and the conservation of the FDM-FPM hybrid approach. The standing waves and flows around NACA airfoils are further simulated to test the ability to deal with free surfaces and complex boundaries. The results show that FDM-FPM retains not only the high efficiency of FDM with multiple resolutions but also the ability of FPM in modeling free surfaces and complex boundaries.     

Unified proof to oscillation property of discrete beam

The oscillation property （OP） is a fundamental and important qualitative property for the vibrations of single span one-dimensional continuums such as strings, bars, torsion bars, and Euler beams. Any properly discretized continuum model should keep the OP. In literatures, the OP of discrete beam models is discussed essentially by means of matrix factorization. The discussion is model-specific and boundary-condition- specific. Besides, matrix factorization is difficult in handling finite element （FE） models of beams. In this paper, according to a sufficient condition for the OP, a new approach to discuss the property is proposed. The local criteria on discrete displacements rather than global matrix factorizations are given to verify the OP. Based on the proposed approach, known results such as the OP for the 2-node FE beams via the Heilinger- Reissener principle （HR-FE beams） as well as the 5-point finite difference （FD） beams are verified. New results on the OP for the 2-node PE-FE beams and the FE Timoshenko beams with small slenderness are given. Through a simple manipulation, the qualitative property of discrete multibearing beams can also be discussed by the proposed approach.     

Dispersion error reduction for acoustic problems using the smoothed finite element method (SFEM)

The smoothed finite element method (SFEM), which was recently introduced for solving the mechanics and acoustic problems, uses the gradient smoothing technique to operate over the cell‐based smoothing domains. On the basis of the previous work, this paper reports a detailed analysis on the numerical dispersion error in solving two‐dimensional acoustic problems governed by the Helmholtz equation using the SFEM, in comparison with the standard finite element method. Owing to the proper softening effects provided naturally by the cell‐based gradient smoothing operations, the SFEM model behaves much softer than the standard finite element method model. Therefore, the SFEM can significantly reduce the dispersion error in the numerical solution. Results of both theoretical and numerical experiments will support these important findings. It is shown clearly that the SFEM suits ideally well for solving acoustic problems, because of the crucial effectiveness in reducing the dispersion error. Copyright © 2015 John Wiley & Sons, Ltd.     

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.      

The coupling method for torsion problem of the square cross-section bar with cracks

By coupling natural boundary element method (NBEM) with FEM based on domain decomposition, the torsion problem of the square cross-sections bar with cracks have been studied, the stresses of the nodes of the cross-sections and the stress intensity factors have been calculated, and some distribution pictures of the stresses have been drawn. During computing, the effect of the relaxed factors to the convergence speed of the iterative method has been discussed. The results of the computation have confirmed the advantages of the NBEM and its coupling with the FEM. Foundation item: the State Key Laboratory of Science and Engineering Computation Biography: ZHAO Hui-ming (1971-)     

Merging phononic crystals and acoustic black holes

Phononic crystals(PCs) have recently been developed as effective components for vibration suppression and sound absorption. As a typical design of PCs, wave attenuation occurs in the so-called stop-band. However, the structural response is still significantly large in the pass-band. In this paper, we combine PCs and acoustic black holes(ABHs) in a unique device, achieving a versatile device that can attenuate vibration in the stop-band, while suppress vibration in the pass-band. This approach provides a versatile platform for controlling vibration in a multiband with a simple design.     

Spectral strip-element method for beams treated with active constrained layer damping

For vibration analysis of beams fully treated with active constrained layer damping (ACLD), a new approach called spectral strip-element method (SSEM) based on the spectral finite element method (SFEM) is proposed. It can avoid difficulties in solving the characteristic equation with higher orders and unknown parameters for wave numbers when using the SFEM; simultaneously, advantages of a very few elements and high accuracy of the SFEM are kept. A numerical example shows that the proposed method is very effective and reliable, compared with the exact solutions resulted from the spectral transfer matrix method (STMM).     

铁路道床有砟-无砟过渡段动力特性的DEM-FEM耦合分析

针对铁路道床有砟-无砟过渡段的结构特点,采用离散元-有限元耦合模型分析散体道砟和无砟道床间过渡段的动力特性。散体道砟道床和无砟道床分别采用离散元方法 DEM和有限元方法 FEM模拟,而在过渡段将道砟颗粒嵌入无砟道床以增加道砟颗粒与无砟道床间的咬合力,并在离散元和有限元耦合区域实现了力学参数的传递。采用以上DEM-FEM耦合方法对有砟-无砟道床及其过渡段在列车荷载作用下的沉降过程进行了数值分析。计算结果表明,离散元方法中道砟颗粒间的力链呈现非对称梯形分布,其与有限元方法中的应力分布趋势一致;采用嵌入式道砟颗粒的方法可以增加有砟-无砟过渡段道砟间的咬合力,有效约束道砟颗粒的位移,减少有砟-无砟道床间的沉降差异。本文计算模型可以合理地分析有砟道床的力链分布以及无砟道床的应力分布,确定列车荷载下道床有砟-无砟过渡段的动力学行为。     

MPS-FEM数值分析带自由面的流固耦合问题

随着计算科学的发展,研究人员为探索流固耦合问题的物理机理而提出了众多的数值方法。其中,耦合的移动粒子半隐式方法 MPS(Moving Particle Semi-Implicit method)和有限单元法FEM(Finite Element method)为流固耦合问题的数值仿真工作提供了新的途径。本文所有流场的数值模拟工作均采用课题组自主开发的无网格法求解器MLParticle-SJTU来完成。该求解器在原始的MPS法基础上,对核函数、压力梯度模型、压力泊松方程的求解和自由面判断方式等方面进行了改进。此外,在该求解器框架内,基于FEM法拓展了针对结构场进行求解的功能。首先,对MPS和FEM方法的理论模型及其耦合策略进行了介绍。然后,采用该自研MPS-FEM耦合求解器,数值模拟了溃坝流动对弹性结构的冲击及其相互作用的标准问题。通过将结构变形及自由面波型变化等结果与已发表结果进行对比,验证了该求解器在处理带自由面剧烈变化的粘性流体和柔性变形结构的耦合作用问题上的可行性。     

Finite cell method compared to h-version finite element method for elasto-plastic problems

The finite cell method （FCM） combines the high-order finite element method （FEM） with the fictitious domain approach for the purpose of simple meshing. In the present study, the FCM is used to the Prandtl-Reuss flow theory of plasticity, and the results are compared with the h-version finite element method （h-FEM）. The numerical results show that the FCM is more efficient compared to the h-FEM for elasto-plastic problems, although the mesh does not conform to the boundary. It is also demonstrated that the FCM performs well for elasto-plastic loading and unloading.     

Approximate analytical solution in slow-fast system based on modified multi-scale method

A simple, yet accurate modified multi-scale method (MMSM) for an approximately analytical solution in nonlinear oscillators with two time scales under forced harmonic excitation is proposed. This method depends on the classical multi-scale method (MSM) and the method of variation of parameters. Assuming that the forced excitation is a constant, one could easily obtain the approximate analytical solution of the simplified system based on the traditional MSM. Then, this solution for the oscillator under forced harmonic excitation could be established after replacing the harmonic excitation by the constant excitation. To certify the correctness and precision of the proposed analytical method, the van der Pol system with two scales subject to slowly periodic excitation is investigated; this system presents rich dynamical phenomena such as spiking (SP), spiking-quiescence (SP-QS), and quiescence (QS) responses. The approximate analytical expressions of the three types of responses are given by the MMSM, and it can be found that the precision of the new analytical method is higher than that of the classical MSM and better than that of the harmonic balance method (HBM). The results obtained by the present method are considerably better than those obtained by traditional methods, quantitatively and qualitatively, particularly when the excitation frequency is far less than the natural frequency of the system.     

二维弹性新型快速多极虚边界元的最小二乘法

在虚边界元最小二乘法的方程求解中采用新型的快速多极展开和广义极小残值法,提出了一种二维弹性新型快速多极虚边界元最小二乘法的求解思想。基于二维弹性问题原有的快速多极虚边界元最小二乘法的展开格式,通过引入对角化的概念,以更新展开传递格式;相对于原有快速多极算法,该方法可进一步提高计算效率且仍能保证具有较高的计算精度。数值算例说明了该方法的可行性、计算效率、计算精度均较高。     

可靠性优化问题中遗传算法适应值函数的建立

应用遗传算法进行优化,约束的处理成为建立适应值函数和算法进行的关键.可靠性优化是以系统可拿性指标作为优化问题的约束条件.首先结合外罚函数法建立数学模型,处理约束的惩罚因子时根据种群情况白适应取值,构造适应值函数的映射公式.随后采用拉格朗日秉子法建立了新的约束与目标函数向适应值函数的映射公式,该公式可以避免因罚函数病态所导致的搜索终止,收敛更加快速,使遗传算法得以成功应用于可靠性优化问题中.分析计算结果表明乘子法具有更好地收敛效果,两个公式构造合理.     

快速多极虚边界元法对含圆孔薄板有效弹性模量的模拟分析

针对虚边界元法,引入快速多极展开和广义极小残值法(GMRES)的思想,以形成快速多极虚边界元法的求解思想,并将此方法用于含圆孔薄板有效弹性模量的模拟分析.由于本文方法采用了"源点"多极展开和"场点"局部展开的组合处理方案,从而使得原问题方程组求解的计算耗时量和储存量降至与所求问题的计算自由度数成线性比例.本文工作的研究目的在于:提高虚边界元法在普通台式机上的运算能力和拓宽虚边界元法对大规模复杂问题的求解(或数值模拟).文中给出了均布圆孔的正方形薄板和之字形分布圆孔薄板二个算例,以验证该方法的可行性,计算精度和计算效率.     

基于大涡模拟的移动粒子半隐式法研究

将大涡模拟(LES)和无网格的移动粒子半隐式法(MPS)相结合, 以求解湍流中的自由表面问题. 对N-S方程进行滤波计算可得到大涡模拟的控制方程, 大涡模拟的控制方程相对于以往的移动粒子半隐式法而言仅多出雷诺应力项, 通过亚粒子应力(sub-particle-scale,SPS)模型并引入Smagorinsky涡黏模型将雷诺应力模型化, 可实现移动粒子半隐式法的大涡模拟. 将MPS-LES应用至具有大变形自由表面的共振晃荡中, 其模拟结果同实验及其他数值模拟结果都相当接近.      

Random responses of a two-degree-of-freedom structure with strongly nonlinear coupling and parametric interaction

This paper investigates the random responses of a TDOF structure with strongly nonlinear coupling and parametric vibration. With the nonlinear coupling of inertia in the equations of motion of the system being removed by successive elimination, the non-Gaussian moment equation method (NGM) is applied and 69 moment equations are integrated with central cumulative truncation technique. The stochastic central difference-cum-statistical linearization method(SCD-SL) and the digital simulation method(DSM) are also used. A comparison of results by different methods are given and the SCD-SL method is the most efficient method. The project supported by National Natural Science Foundation of China     

Multi-scale HPC system for multi-scale discrete simulation—Development and application of a supercomputer with 1 Petaflops peak performance in single precision

A supercomputer with 1.0 Petaflops peak performance in single precision, designed and established by Institute of Process Engineering, Chinese Academy of Sciences, is introduced in this brief communication. A designing philosophy utilizing the similarity between hardware, software and the problems to be solved is embodied, based on the multi-scale method and discrete simulation approaches developed at Institute of Process Engineering (IPE) and implemented in a graphic processing unit (GPU)-based hybrid computing mode. The preliminary applications of this machine in areas of multi-phase flow, molecular dynamics and so on are reported, demonstrating the supercomputer as a paradigm of green computation in new architecture.     

海冰动力学数值模拟中改进的PIC方法

为了准确地模拟海冰的动力过程,需要建立精确有效的数值方法。本文结合质点网格法(PIC)和光滑质点流体动力学方法(SPH)发展了一种改进的PIC方法。该方法在欧拉坐标下对海冰动量方程进行差分计算,在拉格朗日坐标下进行海冰质点位移、厚度和密集度计算,并采用Gauss函数进行欧拉网格点与拉格朗日质点间海冰参数的交互插值。采用改进的PIC方法对规则区域内的海冰堆积过程进行了数值试验,对渤海海冰的动力过程进行了72小时数值模拟。计算结果均表明改进的PIC方法具有计算量小,计算结果平稳精确的优点,可很好地适用于海冰动力作用过程的数值模拟。     

Developing a physical influence upwind scheme for pressure-based cell-centered finite volume methods

In the framework of a cell-centered finite volume method (FVM), the advection scheme plays the most important role in developing FVMs to solve complicated fluid flow problems for a wide range of Reynolds numbers. Advection schemes have been widely developed for FVMs employing pressure-velocity coupling methodology in the incompressible flow limit. In this regard, the physical influence upwind scheme (PIS) is developed for a cell-centered finite volume coupled solver (FVCS) using a pressure-weighted interpolation method for linking the pressure and velocity fields. The well-known exponential differencing scheme and skew upwind differencing scheme are also deployed in the current FVCS and their numerical results are presented. The accuracy and convergence of the present PIS are evaluated solving flow in a lid-driven square cavity, a lid-driven skewed cavity, and over a backward-facing step (BFS). The flow within the lid-driven square cavity is numerically solved at Reynolds numbers from 400 to 10 000 on a relatively coarse mesh with respect to other reported solutions. The lid-driven skewed cavity test case at Reynolds number of 1000 demonstrates the numerical performance of the present PIS on nonorthogonal grids. The flow over a BFS at Reynolds number of 800 is numerically solved to examine capabilities of current FVCS employing the current PIS in inlet-outlet flow computations. The numerical results obtained by the current PIS are in excellent agreement with those of benchmark solutions of corresponding test cases. Incorporating implicit role of pressure terms in a pressure-weighted interpolation method and development of PIS provides satisfactory solution convergence alongside the numerical accuracy for the current FVCS. A particular numerical verification is performed for the V velocity calculation within the BFS flow field, which confirms the reliability of present PIS.