基于重构核的最小二乘配点法求解封闭声腔响应

基于重构核思想,应用无网格配点法构造近似函数,并利用最小二乘方法的原理解决边界问题,离散控制微分方程,建立求解的代数方程.并将此方法应用于封闭声腔响应的求解,即对亥姆霍兹方程进行离散,建立其最小二乘无网格配点格式.该方法的系数矩阵是对称正定的,因而保证了解的稳定性.通过数值算例分别验证了配点均匀分布与随机分布时此方法的...     

多联通封闭空间声场响应的基于核重构的最小二乘无网格解法

针对多通域封闭空间声场响应的亥姆霍兹方程的求解问题,本文基于核重构思想,应用无网格配点法构造近似函数,并利用最小二乘方法的原理解决边界问题,离散控制微分方程,建立求解的代数方程。边界问题以及稳定性问题一直是无网格法的难点,该方法的系数矩阵是对称正定的,因此结果具有较好的稳定性。通过数值算例分析多联通域二维问题中配点均匀分布与随机分布时此方法的精确性以及稳定性,利用典型算例对比无网格方法数值解与解析解,结果证明此方法不需要进行网格划分,节点可随机分布,精度较高且具有良好的收敛性。     

非定常Stokes方程一种基于完全重叠型区域分解的有限元并行算法

基于完全重叠型区域分解技巧,本文提出了一种求解非定常Stokes方程的有限元并行算法。该算法的基本思想是首先对空间施行完全重叠型区域分解,然后各个处理器使用向后Euler格式独立并行求解关于时间t的常微分方程;在整个关于时间的迭代过程中,无需处理器间的通信,具有良好的并行性能。该算法中每个处理器所负责的子问题是一个全局问题,它定义在整个求解区域上,但绝大部分自由度来自其所负责的子区域,从而使得该算法稍加修改现有的串行程序即可实现相应的并行计算,实现简单,具有重要的使用价值。同时通过数值算例,在曙光集群并行机上编程实现了上述算法,验证了其有效性。     

基于单位分解法的实体壳广义单元模型

基于单位分解的广义有限元法的逼近空间由单位分解函数和局部覆盖函数构成,采用传统有限元形函数作为单位分解函数,其局部覆盖函数的定义不依赖于有限元网格.以十六结点六面体等参单元形函数作为单位分解函数,采用一阶多项式局部覆盖函数建立了十六结点六面体广义单元.在此基础上利用广义有限元法可以灵活构造各向异性逼近空间的特点,根据薄壳的变形特性,对壳体法向挠度和切向位移分别采用一阶和零阶多项式局部覆盖函数,构造了实体薄壳广义单元.计算结果表明：十六结点六面体广义单元和实体薄壳广义单元用于板壳结构分析时具有比相应的常规实体单元更高的收敛性和求解效率,且实体薄壳广义单元比十六结点六面体广义单元具有更高的求解效率.     

基于L^2-投影及H^1-投影进行不可压缩粘性流体数值模拟的比较：案例分析

本文的主要目的是讨论不可压缩粘性流体的Navier-Stokes方程的数值模拟。本文所用的方法是对时间用一阶精度算了分裂离散化,对空间度是用Uzawa方法对L2-投影及H1-投影求解Stokes问题,以及利用类波动方程方法求解平流问题。这两种投影格式都很容易实现。我们利用它们求解经典顶盖驱动方腔流问题直至雷诺数7500都取得了一致结果。当雷诺数处于区间[8575,8590](对应[8600,8625])时,运用L2-投影(对应H1-投影)得到的结果具有时间周期性,这表明Hopf分支的产生。当雷诺数为10000时,存在两个主导频率相互作用。     

基于L2-投影及日H1-投影进行不可压缩粘性流体数值模拟的比较:案例分析

本文的主要目的是讨论不可压缩粘性流体的Navier-Stokes方程的数值模拟.本文所用的方法足对时间用一阶精度算子分裂离散化,对空间度是用Uzawa方法对L2-投影及日H<,1>-投影求解Stokes问题,以及利用类波动方程方法求解平流问题.这两种投影格式都很容易实现.我们利用它们求解经典顶盖驱动方腔流问题直至雷诺数7500都取得了一致结果.当雷诺数处十区间[8575,8590](对应[8600,8625])时,运用L2-投影(对戍H1投影)得到的结果具有时间周期性,这表明Hopf分支的产生.当雷诺数为10000时,存在两个主导频率相互作用.     

The partition of unity finite element method for short wave acoustic propagation on non‐uniform potential flows

A novel numerical method is proposed for modelling time‐harmonic acoustic propagation of short wavelength disturbances on non‐uniform potential flows. The method is based on the partition of unity finite element method in which a local basis of discrete plane waves is used to enrich the conventional finite element approximation space. The basis functions are local solutions of the governing equations. They are able to represent accurately the highly oscillatory behaviour of the solution within each element while taking into account the convective effect of the flow and the spatial variation in local sound speed when the flow is non‐uniform. Many wavelengths can be included within a single element leading to ultra‐sparse meshes. Results presented in this article will demonstrate that accurate solutions can be obtained in this way for a greatly reduced number of degrees of freedom when compared to conventional element or grid‐based schemes. Numerical results for lined uniform two‐dimensional ducts and for non‐uniform axisymmetric ducts are presented to indicate the accuracy and performance which can be achieved. Numerical studies indicate that the ‘pollution’ effect associated with cumulative dispersion error in conventional finite element schemes is largely eliminated. Copyright © 2005 John Wiley & Sons, Ltd.     

Generalized finite element method using proper orthogonal decomposition

A methodology is presented for generating enrichment functions in generalized finite element methods (GFEM) using experimental and/or simulated data. The approach is based on the proper orthogonal decomposition (POD) technique, which is used to generate low‐order representations of data that contain general information about the solution of partial differential equations. One of the main challenges in such enriched finite element methods is knowing how to choose, a priori, enrichment functions that capture the nature of the solution of the governing equations. POD produces low‐order subspaces, that are optimal in some norm, for approximating a given data set. For most problems, since the solution error in Galerkin methods is bounded by the error in the best approximation, it is expected that the optimal approximation properties of POD can be exploited to construct efficient enrichment functions. We demonstrate the potential of this approach through three numerical examples. Best‐approximation studies are conducted that reveal the advantages of using POD modes as enrichment functions in GFEM over a conventional POD basis. Copyright © 2009 John Wiley & Sons, Ltd.     

A partition of unity finite element method for obtaining elastic properties of continua with embedded thin fibres

The numerical analysis of large numbers of arbitrarily distributed discrete thin fibres embedded in a continuum is a computationally demanding process. In this contribution, we propose an approach based on the partition of unity property of finite element shape functions that can handle discrete thin fibres in a continuum matrix without meshing them. This is made possible by a special enrichment function that represents the action of each individual fibre on the matrix. Our approach allows to model fibre‐reinforced materials considering matrix, fibres and interfaces between matrix and fibres individually, each with its own elastic constitutive law. Copyright © 2010 John Wiley & Sons, Ltd.     

Singularity enrichment for complete sliding contact using the partition of unity finite element method

In this paper, the numerical modelling of complete sliding contact and its associated singularity is carried out using the partition of unity finite element method. Sliding interfaces in engineering components lead to crack nucleation and growth in the vicinity of the contact zone. To accurately capture the singular stress field at the contact corner, we use the partition of unity framework to enrich the standard displacement‐based finite element approximation by additional (enriched) functions. These enriched functions are derived from the analytical expression of the asymptotic displacement field in the vicinity of the contact corner. To characterize the intensity of the singularity, a domain integral formulation is adopted to compute the generalized stress intensity factor (GSIF). Numerical results on benchmark problems are presented to demonstrate the improved accuracy and benefits of this technique. We conduct an investigation on issues pertaining to the extent of enrichment, accurate numerical integration of weak‐form integrals and the rate of convergence in energy. The use of partition of unity enrichment leads to accurate estimations of the GSIFs on relatively coarse meshes, which is particularly beneficial for modelling non‐linear sliding contacts. Copyright © 2008 John Wiley & Sons, Ltd.     

Fast integration and weight function blending in the extended finite element method

Two issues in the extended finite element method (XFEM) are addressed: efficient numerical integration of the weak form when the enrichment function is self‐equilibrating and blending of the enrichment. The integration is based on transforming the domain integrals in the weak form into equivalent contour integrals. It is shown that the contour form is computationally more efficient than the domain form, especially when the enrichment function is singular and/or discontinuous. A method for alleviating the errors in the blending elements is also studied. In this method, the enrichment function is pre‐multiplied by a smooth weight function with compact support to allow for a completely smooth transition between enriched and unenriched subdomains. A method for blending step function enrichment with singular enrichments is described. It is also shown that if the enrichment is not shifted properly, the weighted enrichment is equivalent to the standard enrichment. An edge dislocation and a crack problem are used to benchmark the technique; the influence of the variables that parameterize the weight function is analyzed. The resulting method shows both improved accuracy and optimum convergence rates and is easily implemented into existing XFEM codes. Copyright © 2008 John Wiley & Sons, Ltd.     

基于有限元法的水声构件声性能预报

对水声构件的声性能进行有限元仿真研究,是因为水声构件的声性能不仅由材料本身的物理参数所决定,而且与其内部空腔形状与分布有关。通过反射波与入射波形成的驻波场声压可以直接计算出水声构件的吸声系数。对多种样品进行仿真与测量,两者结果较为一致。该法不受模型结构复杂性的限制,具有费时少、精度高的特点,为消声瓦等水声构件的声性能预报提供了保证。是预报水声构件声性能的一种便捷工具。     

Representation of singular fields without asymptotic enrichment in the extended finite element method

In this paper, we replace the asymptotic enrichments around the crack tip in the extended finite element method (XFEM) with the semi‐analytical solution obtained by the scaled boundary finite element method (SBFEM). The proposed method does not require special numerical integration technique to compute the stiffness matrix, and it improves the capability of the XFEM to model cracks in homogeneous and/or heterogeneous materials without a priori knowledge of the asymptotic solutions. A Heaviside enrichment is used to represent the jump across the discontinuity surface. We call the method as the extended SBFEM. Numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics show that the proposed method yields accurate results with improved condition number. A simple code is annexed to compute the terms in the stiffness matrix, which can easily be integrated in any existing FEM/XFEM code. Copyright © 2013 John Wiley & Sons, Ltd.     

A two-level finite element method and its application to the Helmholtz equation

A two-level finite element method is introduced and its application to the Helmholtz equation is considered. The method retains the desirable features of the Galerkin method enriched with residual-free bubbles, while it is not limited to discretizations using elements with simple geometry. The method can be applied to other equations and to irregular-shaped domains. © 1998 John Wiley & Sons, Ltd.     

Moving particle finite element method with global smoothness

We describe a new version of the moving particle finite element method (MPFEM) that provides solutions within a C⁰ finite element framework. The finite elements determine the weighting for the moving partition of unity. A concept of ‘General Shape Function’ is proposed which extends regular finite element shape functions to a larger domain. These are combined with Shepard functions to obtain a smooth approximation. The Moving Particle Finite Element Method combines desirable features of finite element and meshfree methods. The proposed approach, in fact, can be interpreted as a ‘moving partition of unity finite element method’ or ‘moving kernel finite element method’. This method possesses the robustness and efficiency of the C⁰ finite element method while providing at least C¹ continuity. Copyright © 2004 John Wiley & Sons, Ltd.