A generalized variational principle of composite shallow shells and its application to the folded shell

In this paper, a generalized variational principle of elastodynamics in composite shallow shells with edge beams is presented, and its equivalence to corresponding basic equations, ridge conditions and boundary conditions is proved. Then this variational principle is applied to the folded shell structure. By means of double series, the approximate analytical solutions for statics and dynamics under common boundary conditions are obtained. The comparison of our results with FEM computations and experiments shows the analytical solutions have good convergence and their accuracy is quite satisfactory.     

广义有限差分法在含阻抗边界空腔声学分析中的应用

声学分析在噪声控制、室内隔音等工程计算中有着重要的作用.由于现实生活中的声学模型往往伴随着吸声材料,因此分析含阻抗边界条件的声学问题显得十分必要.广义有限差分法是一种新型区域型无网格数值离散方法,该方法基于多元函数泰勒级数展开式和加权最小二乘拟合,将控制方程中未知参量的各阶偏导数表示为相邻节点函数值的线性组合.本文首次...     

A domain decomposition method for stochastic analysis of acoustic fields with hybrid and localized uncertainties

An efficient domain decomposition method (DDM) is proposed for the dynamic analysis of stochastic acoustic fields with hybrid and localized uncertainties. The hybrid and localized uncertainties refer to the parameters that are associated with local properties of the acoustic fields and meanwhile are subjected to different kinds of randomness. To take advantage of the locally distributed feature of uncertain parameters, the full acoustic domain is divided into several sub-domains, along with each localized uncertain parameter being assigned to one specific sub-domain. In each sub-domain, the deterministic Helmholtz equation is transformed to a weak integral form and the discretized governing equation is obtained by employing Chebyshev orthogonal polynomials as admissible functions. The random or interval perturbation technique is applied to the individual governing equation according to the respective uncertainty type, whereby the stochastic governing equation is established. The original acoustic field is eventually recovered by the introduction of penalty functions to impose sound pressure continuity on the interfaces of sub-domains, and the (intervals of) sound pressure, together with its expectation and variance, can be subsequently obtained. The accuracy and efficiency of the proposed method are verified in several numerical examples by comparisons with the results given by brute force Monte Carlo simulations, and the DDM-based independent way of modelling and analysis proves to be quite effective and flexible for uncertainty quantification in acoustic fields.     

阶梯圆柱形耦合声场的特征正交-里兹法建模及声学特性分析

针对阶梯圆柱形耦合声场建模问题,提出基于特征正交-里兹能量原理的声学建模方法.该方法利用二维特征正交多项式和周向傅里叶级数表征阶梯圆柱形耦合声场子分段的声压函数,从能量角度考虑邻近子声场间声学连续性条件,并结合里兹法获得耦合声场的声学特性.基于本建模方法对不同分段的耦合声场开展声学特性分析,结果表明,本建模方法在保证计算准确性的基础上有效提高了计算效率,且对任意阶梯分段的圆柱形耦合声场普遍适用;圆柱形耦合声场固有频率会随着腔体外径增大而普遍增大,而腔深的影响规律相反;降低声学边界阻抗可抑制声学响应幅值,为此类声场的噪声控制提供了设计依据.     

On variational formulations with rigid-body constraints for finite elasticity: Applications to 2D and 3D finite element simulations

We investigate a recently proposed variational principle with rigid-body constraints and present an extension of its implementation in three dimensional finite elasticity problems. Through numerical examples, we illustrate that the proposed variational principle with rigid-body constraints is applicable to both single field and mixed finite elements of arbitrary order and geometry, e.g. triangular/tetrahedral and quadrilateral/hexagonal elements, in two and three dimensions. Moreover, we demonstrate that, as compared to the commonly adopted approach of discretizing the rigid domains with standard finite elements, the proposed formulation requires neither discretization nor numerical integration in the interior of each rigid domain. As a comparative result, the variational formulation may reduce the total number of degrees of freedom of the resulting finite element system and provide improved accuracy.     

Numerical simulation of flow‐induced noise using LES/SAS and Lighthill's acoustic analogy

We present a finite element (FE) formulation of Lighthill's acoustic analogy for the hybrid computation of noise generated by turbulent flows. In the present approach, the flow field is computed using large eddy simulation and scale adaptive simulation turbulence models. The acoustic propagation is obtained by solving the variational formulation of Lighthill's acoustic analogy with the FE method. In order to preserve the acoustic energy, we compute the inhomogeneous part of Lighthill's wave equation by applying the FE formulation on the fine flow grid. The resulting acoustic nodal loads are then conservatively interpolated to the coarser acoustic grid. Subsequently, the radiated acoustic field can be solved in both time and frequency domains. In the latter case, an enhanced perfectly matched layer technique is employed, allowing one to truncate the computational domain in the acoustic near field, without compromising the numerical solution. Our hybrid approach is validated by comparing the numerical results of the acoustic field induced by a corotating vortex pair with the corresponding analytical solution. To demonstrate the applicability of our scheme, we present full 3D numerical results for the computed acoustic field generated by the turbulent flow around square cylinder geometries. The sound pressure levels obtained compare well with measured values. Copyright © 2009 John Wiley & Sons, Ltd.     

Trapped modes,surface waves and Wood's anomalies for gently sloped periodic boundaries

The two-dimensional problem on the diffraction of acoustic waves by a sound hard periodic grating is considered. We construct examples of surface waves (solutions of the homogeneous problem exponentially decaying at infinity) for various values of the frequency. These results are obtained while applying the variational approach for low frequencies, and a method based on introducing the augmented scattering matrix for high frequencies.     

FUNDAMENTAL SOLUTION BASED GRADED ELEMENT MODEL FOR STEADY-STATE HEAT TRANSFER IN FGM

A novel hybrid graded element model is developed in this paper for investigating thermal behavior of functionally graded materials (FGMs). The model can handle a spatially varying material property field of FGMs. In the proposed approach, a new variational functional is first constructed for generating corresponding finite element model. Then, a graded element is formulated based on two sets of independent temperature fields. One is known as intra-element temperature field defined within the element domain; the other is the so-called frame field defined on the element boundary only. The intra-element temperature field is constructed using the linear combination of fundamental solutions, while the independent frame field is separately used as the boundary interpolation functions of the element to ensure the field continuity over the interelement boundary. Due to the properties of fundamental solutions, the domain integrals appearing in the variational functional can be converted into boundary integrals which can significantly simplify the calculation of generalized element stiffness matrix. The proposed model can simulate the graded material properties naturally due to the use of the graded element in the finite element (FE) model. Moreover, it inherits all the advantages of the hybrid Trefftz finite element method (HT-FEM) over the conventional FEM and boundary element method (BEM). Finally, several examples are presented to assess the performance of the proposed method, and the obtained numerical results show a good numerical accuracy.     

不连续Reissner矩形板的自由振动

用分区加权残值法研究Ressner矩形板在几何形状,边界条件等有突变时的自由振动问题,将研究对象按照结构几何形状和边界条件的具体情况划分为若干子域,在每个子域内用不同的试函数代入该域的内的控制方程到内部残值,并代入板的边界条件和各子域的协调条件得到边界残值和连续性残值,然后用最小二乘法消除残值,得到特征方程,文中讨论了该方法的收敛性和计算精度,求解了开孔矩形板的固有频率,并与已有结果进行了比较,结果表明：？该方法收敛性好,精度较高,适用范围广。     

Study on calculation methods for acoustic radiation of axisymmetric structures in finite water depth

In this paper, a kind of FEM–WSM (Finite Element Method–Wave​ Superposition Method) is used to calculate the acoustic radiation of axisymmetric structures in finite water depth. FEM is used to solve the dry modes of axisymmetric structures, and WSM is applied along with the dry mode method to consider fluid–structure interaction effects and calculate the acoustic radiated field. This method combines the advantages of FEM and WSM. On one hand, it is suitable for complex or large axisymmetric structures on the one hand. On the other hand, it has higher computational efficiency than the FEM, and the computational domain size for the water is not limited. As long as the Green’s function is tailored for the boundary condition, the acoustic radiated field of axisymmetric structures in more complex ocean acoustic environments can be calculated by using this method. Besides, a least-square method is used to reduce the distortion resulting from computational errors of the modal estimates. The influence of the number of source and field points and the finite element mesh density on the calculation accuracy are discussed, eliciting some disciplinary conclusions. Using a spherical shell and a capsule shell as models, the results from the present approach, a semi-analytical method, and the crude FEM are compared to verify the correctness and efficiency. Based on numerical examples, the influence of the sea surface and the seafloor on the acoustic radiated field of structures in finite water depth is also analyzed.     

Structural–acoustic sensitivity analysis of radiated sound power using a finite element/ discontinuous fast multipole boundary element scheme

The complete interaction between the structural domain and the acoustic domain needs to be considered in many engineering problems, especially for the acoustic analysis concerning thin structures immersed in water. This study employs the finite element method to model the structural parts and the fast multipole boundary element method to model the exterior acoustic domain. Discontinuous higher‐order boundary elements are developed for the acoustic domain to achieve higher accuracy in the coupling analysis. Structural–acoustic design sensitivity analysis can provide insights into the effects of design variables on radiated acoustic performance and thus is important to the structural–acoustic design and optimization processes. This study is the first to formulate equations for sound power sensitivity on structural surfaces based on an adjoint operator approach and equations for sound power sensitivity on arbitrary closed surfaces around the radiator based on the direct differentiation approach. The design variables include fluid density, structural density, Poisson's ratio, Young's modulus, and structural shape/size. A numerical example is presented to demonstrate the accuracy and validity of the proposed algorithm. Different types of coupled continuous and discontinuous boundary elements with finite elements are used for the numerical solution, and the performances of the different types of finite element/continuous and discontinuous boundary element coupling are presented and compared in detail. Copyright © 2016 John Wiley & Sons, Ltd.     

二维水下声辐射问题的改进径向基点插值法研究

径向基点插值法是一种典型的无网格数值计算方法,在分析声学问题时,相比于传统有限元法能更好地抑制频散误差,且在相同的节点分布下通常可以得到更精确的数值解。本文提出一种改进的节点选取方案用于构造插值形函数,即改进径向基点插值法。该方案采取一个简单而直接的格式,可确保在进行数值积分时同一背景积分单元中的被积函数是连续可微的,从而减小数值积分误差,得到比原始径向基点插值法更精确的数值解。同时,为了处理外声场问题,本文采用DtN映射技术将无限域截断为有界计算域,满足索默菲尔德辐射条件。数值试验表明,相比于传统有限元法和原始径向基点插值法,本文改进方法具有更高的计算精度和计算效率,在研究水下声辐射问题时具有良好的应用前景。     

弹性力学广义混合变分原理及有限元广义混合法

近些年弹性力学中出现一种新型的变分原理,称为广义混合变分原理.特点是其泛函中包含某些可以任意选择的附加函数,称为分裂因子.新原理将弹性理论中现有的各主要变分原理都统一在一个框架中,并揭示出它们之间更深一层的相互关系.在应用方面,它提供了一个新的数学手段以建立有限元分析中的新模式.这些新模式已经显示出它们的优点:适当调节分裂因子,它们给出更好的数值解答,特别是,可用它们来处理有限元方法中棘手的病态问题.本文综述了线性及非线性弹性理论中的这种新型变分原理并就其在有限元中的应用作了讨论.      

Acoustic simulation using a novel approach for reducing dispersion error

It is well‐known that the traditional finite element method (FEM) fails to provide accurate results to the Helmholtz equation with the increase of wave number because of the ‘pollution error’ caused by numerical dispersion. In order to overcome this deficiency, a gradient‐weighted finite element method (GW‐FEM) that combines Shepard interpolation and linear shape functions is proposed in this work. Three‐node triangular and four‐node tetrahedral elements that can be generated automatically are first used to discretize the problem domain in 2D and 3D spaces, respectively. For each independent element, a compacted support domain is then formed based on the element itself and its adjacent elements sharing common edges (or faces). With the aid of Shepard interpolation, a weighted acoustic gradient field is then formulated, which will be further used to construct the discretized system equations through the generalized Galerkin weak form. Numerical examples demonstrate that the present algorithm can significantly reduces the dispersion error in computational acoustics. Copyright © 2016 John Wiley & Sons, Ltd.     

A point assembly method for stress analysis for two-dimensional solids

A novel point assembly method (PAM) is presented for stress analysis for two-dimensional solids. In the present method, the boundaries of the problem domain are represented by a set of discrete points, and the domain itself is represented by properly scattered points. The displacement in the influence triangular areas of a point is interpolated by the displacements at the point and pairs of surrounding points using shape functions. The shape functions used in this work are obtained in the same way as those of a triangular element in the conventional finite element method (FEM). A variational (weak) form of the equilibrium equation is used to produce a set of system equations. These equations are assembled for all the points in the domain, and solved for the displacement field. Stresses and strains at a point are then computed using the displacements obtained for the point and pairs of the surrounding points. A PAM program with an automatic point-searching algorithm has been developed in fortran. Patch tests and convergence studies have been carried out to verify the convergence of the present method and program. Examples are also presented to demonstrate the efficiency and accuracy of the present method compared with analytical solutions as well as the conventional FEM solutions.     

一种求解三维声腔声压分布问题的无网格法

利用传统有限元法求解声压分布问题常常受到污染误差和色散误差的困扰。加权最小二乘无网格法（MWLS）是一种基于移动最小二乘（MLS）近似的无网格方法,求解声腔声压分布问题具有低色散、高精度的特点。然而传统的MLS近似有时容易产生病态矩阵,利用加权正交基函数构建改进的移动最小二乘（IMLS）近似,得到的系统方程为非病态的。本文基于改进的加权最小二乘无网格法（IMWLS）求解三维声腔内部声压分布。计算得到的声压分布和声压频响曲线都与参考值十分吻合,峰值误差和污染误差都比FEM的小,计算成本相比无单元伽辽金法显著降低。计算结果表明IMWLS相比传统的FEM,能在更高的频段内达到高精度,并且相比EFGM能大幅提高计算效率。     

分段线性强化问题的全量无迭代解

本文应用[1]提出的塑性全量理论中的控制变量(也称参变量)变分原理,对各级分段线性强化规律建立了相应的本构状态方程以及有限元求解公式,可使分段线性强化全量问题的数值解不需迭代.对于一般的(?)—(?)曲线,适当选择分段数目可达到足够的精度。本文给出了有一定代表性的手工及数值算例。     

基于区域分解的环肋圆柱壳-圆锥壳组合结构振动分析

提出了一种区域分解法来分析不同边界条件下环肋骨圆柱壳-圆锥壳组合结构的振动特性.首先把组合壳体分解为自由的圆柱壳、圆锥壳段;视环肋骨为离散元件,根据肋骨与圆柱壳段之间的变形协调条件,将肋骨的动能和应变能附加于圆柱壳段能量泛函中.然后基于分区广义变分和最小二乘加权残值法将所有分区界面的位移协调方程引入到组合壳体的能量泛函中.圆柱壳段、圆锥壳段位移变量的周向和轴向分量分别采用Fourier级数和Chebyshev多项式展开.以自由-自由、自由-固支和固支-固支边界条件的环肋骨组合壳体为例,采用区域分解法分析了其自由振动及在不同激励下的振动响应.通过与有限元软件ANSYS结果进行对比,发现两种方法计算结果非常吻合,验证了区域分解方法的计算精度和高效性.     

Topology optimization of a plate coupled with acoustic cavity

Optimization of the topology of a plate coupled with an acoustic cavity is presented in an attempt to minimize the fluid–structure interactions at different structural frequencies. A mathematical model is developed to simulate such fluid–structure interactions based on the theory of finite elements. The model is integrated with a topology optimization approach which utilizes the moving asymptotes method. The obtained results demonstrate the effectiveness of the proposed approach in simultaneously attenuating the structural vibration and the sound pressure inside the acoustic domain at several structural frequencies by proper redistribution of the plate material.Experimental verification is carried out by manufacturing topology optimized plates and monitoring their vibration and sound radiation into a rigid acoustic cavity. The measured sound pressure and plate vibration are found to be in good agreement with the predictions of the mathematical model.The presented theoretical and experimental techniques present valuable tools in the design of a wide variety of critical structures which must operate quietly when subjected to fluid loading.     

裂缝分析的比例边界有限元与有限元耦合的虚拟结构面模型

比例边界有限元法SBFEM(Scaled Boundary Finite Element Method)是一种半解析数值方法,在裂缝分析特别是强度因子计算上具有相当高的精度。本文提出了一种用于裂缝分析的基于虚拟结构面的SBFEM与常规FEM的耦合分析方法。首先选取裂缝周边一定范围的计算域,并将结构分成不含裂缝区域和含裂缝区域两部分。然后,对不含裂缝区域,采用FEM进行网格离散;对含裂缝区域,采用SBFEM进行网格离散;两者相互独立,在这两个域内,分别采用各自相应的位移模式。最后通过在SBFEM网格的外边界设置虚拟耦合结构面的模式,实现有限元网格和比例边界有限元网格的耦合。通过两个经典的含裂缝平板的算例研究,探讨了本文方法在I型开裂和混合型开裂分析中,影响应力强度因子精度的因素。算例表明,SBFEM具有的降维和半解析性质,使本文方法在裂缝分析中的前处理简单易行,且计算结果具有相当高的计算精度。