应用混合单元基光滑点插值法的声固耦合分析

基于有限元法的声固耦合计算模型由于刚度过硬,频率增高时计算结果容易出现较大的色散误差。为提高计算精度,提出将采用梯度光滑技术的混合单元基光滑点插值法(CSαPIM)应用于三维声学模拟,通过与边基光滑有限元(ESFEM)的二维板单元结合,对结构声振耦合系统进行模态和频率响应分析。结果表明梯度光滑可以有效软化系统刚度,提供更加准确的计算结果;同时,新方法采用三角形和四面体单元来分别离散结构域和声学域,对任意几何形状具有普遍适应性,可以减少复杂模型的前处理时间,具有很好的工程应用前景。     

板结构-声场耦合分析的有限元-径向插值/有限元法

为提高板结构-声场耦合分析的计算精度,将有限元-径向点插值法(Finite Element-Radial Point Interpolation,FE-RPIM)推广到板结构-声场耦合问题的结构域分析中,推导了FE-RPIM/FEM法分析板结构-声场耦合问题的计算公式。板结构-声场耦合分析的FE-RPIM/FEM法在流体域中采用标准的有限元插值函数;在结构域中采用有限元-径向点插值法,其形函数由等参单元形函数和径向点插值函数相结合构成,继承了有限元法的单元兼容性和径向点插值法的Kronecker性质,提高了插值精度。以六面体声场-结构耦合模型为研究对象进行分析,结果表明,与板结构-声场耦合问题分析的有限元/有限元法(Finite element method/Finite element method, FEM/FEM)和光滑有限元/有限元法(Smoothed Finite Element Method/Finite Element Method, SFEM/FEM)相比,FE-RPIM/FEM在分析板结构-声场耦合问题时具有更高的精度。     

基于点基局部光滑点插值法的结构动力分析

针对传统有限元法采用线性三角形单元刚度过硬、计算的固有频率值较大,四边形单元对复杂构件不能自动剖分,点基光滑点插值法刚度过软、会导致计算动力问题失败以及求解固有频率值过低的问题,提出了点基局部光滑点插值法(NPS-PIM)。该方法将有限元与点基光滑点插值法结合,对背景网格基础上形成的点基光滑域剖分、进行局部应变光滑,计算结构的动力问题。研究发现,该方法克服了点基光滑点插值法的时间不稳定性和求解固有频率值过低的缺陷;在采用同样线性三角形单元网格对问题域进行离散的情况下,计算得到的固有频率较传统有限元法有明显提高。该方法简便实用、易于实现,具有较好的工程应用前景。     

板结构-声场耦合分析的FE-LSPIM/FE法

为提高板结构-声场耦合分析的计算精度,将有限元-最小二乘点插值法(Finite Element-Least Square Point Interpolation Method,FE-LSPIM)推广到板结构-声场耦合问题的结构域分析中,提出了板结构-声场耦合问题分析的FE-LSPIM/FEM(Finite Element-Least Square Point Interpolation Method/Finite Element Method),推导了FELSPIM/FEM分析板结构-声场耦合问题的计算公式。此方法在结构域中应用四边形单元形函数和最小二乘点插值法进行局部逼近,继承了有限元法的单元兼容性和最小二乘插值法的二次多项式完备性,提高了结构域的计算精度;在流体域中应用标准有限元模型进行分析。以一六面体声场-结构耦合模型为研究对象进行分析,结果表明,与板结构-声场耦合问题分析的FEM/FEM和光滑有限元/有限元(Smoothed Finite Element Method/Finite Element Method,SFEM/FEM)相比,FE-LSPIM/FEM在分析板结构-声场耦合问题时具有更高的精度。     

改进型边光滑有限元在声固耦合中的应用研究

改进型边光滑有限元法(IES-FEM)是最近针对三维中频声学数值计算色散误差问题提出的一种改进数值计算方法。将该方法拓展应用于三维声固耦合问题的中高频预测。在建模过程中分别采用IES-FEM与边光滑有限元法(ES-FEM)用于构建声腔域与结构域的耦合离散方程。通过调整离散系统中刚度矩阵,使之适度软化来达到与质量矩阵之间的平衡以减少声固耦合系统中高频响应计算的色散误差。数值计算结果表明,相比于ES-FEM,所采用的耦合计算方法在声固耦合系统中高频响应计算方面具有较高的计算精度、计算效率和较好的收敛性。     

板结构-声场耦合分析的FE-LSPIM/FE-LSPIM法

为提高板结构-声场耦合分析的计算精度,将有限元-最小二乘点插值法(Finite Element-Least Square Point Interpolation Method,FE-LSPIM)推广到板结构-声场耦合问题的分析中,提出了板结构-声场耦合问题分析的FELSPIM/FE-LSPIM方法,推导了FE-LSPIM/FE-LSPIM分析板结构-声场耦合问题的计算公式。FE-LSPIM/FE-LSPIM方法应用有限元单元形函数和最小二乘点插值法进行局部逼近,继承了有限元法的单元兼容性和最小二乘插值法的二次多项式完备性,提高了计算精度。以一六面体声场-结构耦合模型为研究对象进行分析,结果表明,与板结构-声场耦合问题分析的FEM/FEM和光滑有限元/有限元(Smoothed Finite Element Method/Finite Element Method,SFEM/FEM)相比,FELSPIM/FE-LSPIM在分析板结构-声场耦合问题时具有更高的精度。     

光滑有限元的声学研究：时域和频域分析

摘　要：在使用有限元进行声场的数值模拟中,存在着两个主要误差,一个是数值方法中常规的插值误差,另外一个是计算声学中所特有的耗散误差(dispersion error),后者则是影响声学模拟仿真置信度的最重要因素。产生耗散误差的本质原因是由于有限元的数值模型刚度“偏硬”造成的。为了控制耗散误差,最重要的是使数值模型更好的反映真实模型。本文采用了一种基于边光滑的有限元方法(ES-FEM)来对声场的时域和频域进行数值模拟研究。该方法只采用对复杂问题域适应性很强的三角形网格,通过引进基于边的广义梯度光滑技术,能够使得有限元系统得到适当的“软化”。关于时域和频域的算例表明了在使用同样网格的情况下,本方法在声学模拟中的精度都要比有限元模型的高。     

陀螺转子动力学系统的时间有限元内点法

将保辛的时间有限元法(FEM)应用于陀螺转子动力学系统,给出了陀螺转子动力学系统时间有限元法的时间单元刚度阵列式和非齐次外力的表达式,以及辛时间传递矩阵。在此基础上,提出了精度更高的时间有限元内点法(IDTFEA),该方法既继承了时间有限元保辛的优良特性,又大大提高了数值计算精度,具有非常明显的优越性。算例给出了该方法和Newmark方法的比较结果,表明该方法的优越性。     

运动介质中奇异边界元积分式的精确求解

采用边界元方法求解与运动介质相关声学问题时,难点之一是如何精确计算场点与源点重合所导致的奇异积分式。论文提出一种将具有奇性的单元面积分式拆分为奇性和非奇性积分部分分别进行计算的新方法。对奇性积分部分,经过严格的数学推导给出解析解;而对非奇性积分部分则通过高斯积分法处理。新方法可有效地提高边界元计算精度和效率,对运动介质中的有关声学问题的边界元数值计算具有重要意义。     

钢铝结构连接耦合损耗因子计算方法研究

统计能量法已广泛用于解决高频振动与噪声问题,耦合损耗因子是其关键参数之一。研究基于统计能量法、有限元法和功率输入法,提出一种钢铝连接耦合损耗因子数值计算方法,并运用MATLAB编写了计算软件。通过对含过渡接头的T型连接结构进行数值和实验研究,验证了该法的可靠性。这对提高钢铝混合结构的噪声预报精度及指导其结构声学设计具有重要意义。     

A cell-based smoothed radial point interpolation method (CS-RPIM) for static and free vibration of solids

A cell-based smoothed radial point interpolation method (CS-RPIM) based on the generalized gradient smoothing operation is proposed for static and free vibration analysis of solids. In present method, the problem domain is first discretized using triangular background cells, and each cell is further divided into several smoothing cells. The displacement field function is approximated using RPIM shape functions which have Kronecker delta function property. Supporting node selection for shape function construction uses the efficient T2L-scheme associated with edges of the background cells. The system equations are derived using the generalized smoothed Galerkin (GS-Galerkin) weak form, and the essential boundary conditions are imposed directly as in the finite element method (FEM). The effects of the number of divisions smoothing cells on the solution properties of the CS-RPIM are investigated in detail, and preferable numbers of smoothing cells is recommended. To verify the accuracy and stability of the present formulation, a number of numerical examples are studied to demonstrate numerically the efficiency of the present CS-RPIM.     

Dispersion error reduction for acoustic problems using the edge‐based smoothed finite element method (ES‐FEM)

The paper reports a detailed analysis on the numerical dispersion error in solving 2D acoustic problems governed by the Helmholtz equation using the edge‐based smoothed finite element method (ES‐FEM), in comparison with the standard FEM. It is found that the dispersion error of the standard FEM for solving acoustic problems is essentially caused by the ‘overly stiff’ feature of the discrete model. In such an ‘overly stiff’ FEM model, the wave propagates with an artificially higher ‘numerical’ speed, and hence the numerical wave‐number becomes significantly smaller than the actual exact one. Owing to the proper softening effects provided naturally by the edge‐based gradient smoothing operations, the ES‐FEM model, however, behaves much softer than the standard FEM model, leading to the so‐called very ‘close‐to‐exact’ stiffness. Therefore the ES‐FEM can naturally and effectively reduce the dispersion error in the numerical solution in solving acoustic problems. Results of both theoretical and numerical studies will support these important findings. It is shown clearly that the ES‐FEM suits ideally well for solving acoustic problems governed by the Helmholtz equations, because of the crucial effectiveness in reducing the dispersion error in the discrete numerical model. Copyright © 2010 John Wiley & Sons, Ltd.     

A Smoothed Finite Element Method for Mechanics Problems

In the finite element method (FEM), a necessary condition for a four-node isoparametric element is that no interior angle is greater than 180° and the positivity of Jacobian determinant should be ensured in numerical implementation. In this paper, we incorporate cell-wise strain smoothing operations into conventional finite elements and propose the smoothed finite element method (SFEM) for 2D elastic problems. It is found that a quadrilateral element divided into four smoothing cells can avoid spurious modes and gives stable results for integration over the element. Compared with original FEM, the SFEM achieves more accurate results and generally higher convergence rate in energy without increasing computational cost. More importantly, as no mapping or coordinate transformation is involved in the SFEM, its element is allowed to be of arbitrary shape. Hence the restriction on the shape bilinear isoparametric elements can be removed and problem domain can be discretized in more flexible ways, as demonstrated in the example problems.     

3D mass-redistributed finite element method in structural–acoustic interaction problems

A 2D mass-redistributed finite element method (MR-FEM) for pure acoustic problems was recently proposed to reduce the dispersion error. In this paper, the 3D MR-FEM is further developed to solve more complicated structural–acoustic interaction problems. The smoothed Galerkin weak form is adopted to formulate the discretized equations for the structure, and MR-FEM is applied in acoustic domain. The global equations of structural–acoustic interaction problems are then established by coupling the MR-FEM for the acoustic domain and the edge-based smoothed finite element method for the structure. The perfect balance between the mass matrix and stiffness matrix is able to improve the accuracy of the acoustic domain significantly. The gradient smoothing technique used in the structural domain can provide a proper softening effect to the “overly-stiff” FEM model. A number of numerical examples have demonstrated the effectiveness of the mass-redistributed method with smoothed strain.     

An enhanced cell‐based smoothed finite element method for the analysis of Reissner–Mindlin plate bending problems involving distorted mesh

In this work, an enhanced cell‐based smoothed finite element method (FEM) is presented for the Reissner–Mindlin plate bending analysis. The smoothed curvature computed by a boundary integral along the boundaries of smoothing cells in original smoothed FEM is reformulated, and the relationship between the original approach and the present method in curvature smoothing is established. To improve the accuracy of shear strain in a distorted mesh, we span the shear strain space over the adjacent element. This is performed by employing an edge‐based smoothing technique through a simple area‐weighted smoothing procedure on MITC4 assumed shear strain field. A three‐field variational principle is utilized to develop the mixed formulation. The resultant element formulation is further reduced to a displacement‐based formulation via an assumed strain method defined by the edge‐smoothing technique. As the result, a new formulation consisting of smoothed curvature and smoothed shear strain interpolated by the standard transverse displacement/rotation fields and smoothing operators can be shown to improve the solution accuracy in cell‐based smoothed FEM for Reissner–Mindlin plate bending analysis. Several numerical examples are presented to demonstrate the accuracy of the proposed formulation.Copyright © 2013 John Wiley & Sons, Ltd.     

An edge‐based smoothed finite element method for 3D analysis of solid mechanics problems

The edge‐based smoothed finite element method (ES‐FEM) was proposed recently in Liu, Nguyen‐Thoi, and Lam to improve the accuracy of the FEM for 2D problems. This method belongs to the wider family of the smoothed FEM for which smoothing cells are defined to perform the numerical integration over the domain. Later, the face‐based smoothed FEM (FS‐FEM) was proposed to generalize the ES‐FEM to 3D problems. According to this method, the smoothing cells are centered along the faces of the tetrahedrons of the mesh. In the present paper, an alternative method for the extension of the ES‐FEM to 3D is investigated. This method is based on an underlying mesh composed of tetrahedrons, and the approximation of the field variables is associated with the tetrahedral elements; however, in contrast to the FS‐FEM, the smoothing cells of the proposed ES‐FEM are centered along the edges of the tetrahedrons of the mesh. From selected numerical benchmark problems, it is observed that the ES‐FEM is characterized by a higher accuracy and improved computational efficiency as compared with linear tetrahedral elements and to the FS‐FEM for a given number of degrees of freedom. Copyright © 2013 John Wiley & Sons, Ltd.     

Theoretical aspects of the smoothed finite element method (SFEM)

This paper examines the theoretical bases for the smoothed finite element method (SFEM), which was formulated by incorporating cell‐wise strain smoothing operation into standard compatible finite element method (FEM). The weak form of SFEM can be derived from the Hu–Washizu three‐field variational principle. For elastic problems, it is proved that 1D linear element and 2D linear triangle element in SFEM are identical to their counterparts in FEM, while 2D bilinear quadrilateral elements in SFEM are different from that of FEM: when the number of smoothing cells (SCs) of the elements equals 1, the SFEM solution is proved to be ‘variationally consistent’ and has the same properties with those of FEM using reduced integration; when SC approaches infinity, the SFEM solution will approach the solution of the standard displacement compatible FEM model; when SC is a finite number larger than 1, the SFEM solutions are not ‘variationally consistent’ but ‘energy consistent’, and will change monotonously from the solution of SFEM (SC = 1) to that of SFEM (SC → ∞). It is suggested that there exists an optimal number of SC such that the SFEM solution is closest to the exact solution. The properties of SFEM are confirmed by numerical examples. Copyright © 2006 John Wiley & Sons, Ltd.     

Dispersion free analysis of acoustic problems using the alpha finite element method

The classical finite element method (FEM) fails to provide accurate results to the Helmholtz equation with large wave numbers due to the well-known “pollution error” caused by the numerical dispersion, i.e. the numerical wave number is always smaller than the exact one. This dispersion error is essentially rooted at the “overly-stiff” feature of the FEM model. In this paper, an alpha finite element method (α-FEM) is then formulated for the acoustic problems by combining the “smaller wave number” model of FEM and the “larger wave number” model of NS-FEM through a scaling factor ${a\in [0,1]}The classical finite element method (FEM) fails to provide accurate results to the Helmholtz equation with large wave numbers due to the well-known “pollution error” caused by the numerical dispersion, i.e. the numerical wave number is always smaller than the exact one. This dispersion error is essentially rooted at the “overly-stiff” feature of the FEM model. In this paper, an alpha finite element method (α-FEM) is then formulated for the acoustic problems by combining the “smaller wave number” model of FEM and the “larger wave number” model of NS-FEM through a scaling factor a ? [0,1]{a\in [0,1]} . The motivation for this combined approach is essentially from the features of “overly-stiff” FEM model and “overly-soft” NS-FEM model, and accurate solutions can be obtained by tuning the α-FEM model. A technique is proposed to determine a particular alpha with which the α-FEM model can possess a very “close-to-exact” stiffness, which can effectively reduce the dispersion error leading to dispersion free solutions for acoustic problems. Theoretical and numerical studies shall demonstrate the excellent properties of the present α-FEM.     

A cell-based smoothed radial point interpolation method with virtual nodes for three-dimensional mid-frequency acoustic problems

It is well known that the finite element method (FEM) encounters dispersion errors in coping with mid-frequency acoustic problems due to its “overly stiff” nature. By introducing the generalized gradient smoothing technique and the idea of condensed shape functions with virtual nodes, a cell-based smoothed radial point interpolation method is proposed to solve the Helmholtz equation for the purpose of reducing dispersion errors. With the properly selected virtual nodes, the proposed method can provide a close-to-exact stiffness of continuum, leading to a conspicuous decrease in dispersion errors and a significant improvement in accuracy. Numerical examples are examined using the present method by comparing with both the traditional FEM using four-node tetrahedral elements (FEM-T4) and the FEM model using eight-node hexahedral elements with modified integration rules (MIR-H8). The present cell-based smoothed radial point interpolation method has been demonstrated to possess a number of superiorities, including the automatically generated tetrahedral background mesh, high computational efficiency, and insensitivity to mesh distortion, which make the method a good potential for practical analysis of acoustic problems.