基于模态价值分析的结构动力学模型降阶

针对小阻尼和无阻尼结构，提出了一种模态价值分析方法，用于在模态坐标下的动力学模型降阶问题。在得到系统动力学方程的基础上，首先进行模态分析，然后由模态参数和系统的输入输出矩阵计算各模态的价值，最后选择若干价值最大的模态向量形成变换矩阵对系统进行降阶。通过两个算例说明本文方法是对通常选择低频模态的降阶方法的改进，提高了低阶模型的精度。     

Multi-expansion modal reduction: A pragmatic semi–a priori model order reduction approach for nonlinear structural dynamics

A novel model order reduction scheme for nonlinear finite element (NL-FE) structural dynamics models is proposed in this work, which enables an overall accelerated model evaluation. The proposed multi-expansion modal (MEM) reduction employs a set of modal bases evaluated for a number of representative statically deformed configurations, combined with these static deformation patterns, to obtain a reduced order basis (ROB). This ROB is then exploited in an a priori element sampling to obtain a hyper-reduced model based on the energy-conserving sampling and weighting approach. The proposed element sampling scheme uses a nonnegative L₁ optimization of the element weights where an equality between the projected and hyper-reduced linearized stiffness matrix or static deformation forces is added. A procedure and several guidelines on the reduced order model setup and evaluation are presented, with specific attention to how the presented approach allows to exploit existing NL-FE frameworks. The MEM approach is validated numerically on an academic cantilever beam model and is shown to provide good accuracy and consistent convergence properties.     

Broadband model order reduction of polynomial matrix equations using single‐point well‐conditioned asymptotic waveform evaluation: derivations and theory

To effect a model order reduction (MORe) process on a system which has a polynomial matrix equation dependence on the MORe parameter, researchers generally take one of two approaches. The first is to linearize the system by introducing extra degrees of freedom and then to solve the resulting expanded, linear system with a method such as Lanczos or Arnoldi. The second approach is to work directly with the polynomial system and use a technique such as asymptotic waveform evaluation (AWE). Of course, each approach has advantages and disadvantages. In this paper, a new technique will be presented which has some desirable characteristics from both approaches and which is able to circumvent simultaneously some of their disadvantages. It can be shown that both the Arnoldi and the AWE methods are special cases of this new technique. Finally, numerical results will show the viability of the new method, which will be called the well‐conditioned asymptotic waveform evaluation (WCAWE) method. Copyright © 2003 John Wiley & Sons, Ltd.     

Linear vibrations of a rotating elastic beam with an attached point mass

The paper deals with linear vibrations of a thin elasticbeam with an attached point mass which is clamped radially to the inside of a rigid wheel rotating at a constant angular velocity. Material damping is also taken into consideration. A partial differential equation of motion with dynamical boundary conditions is derived. It is proved that in a sense the attached point mass plays a destabilizing role for any values of the problem parameters. The problem of the determination of critical parameter values is reduced to a second-order Sturm-Liouville problem with a singularity. Asymptotic representations of the stability threshold, as the ratio between the radius of the wheel and the length of the beam goes either to 0 or to +, are obtained. The dynamical stability of the rectilinear shape of the beam is studied by means of the direct Lyapunov method. A non-standard method of construction of Chetayev's function is proposed.     

An adaptive scheme for a class of interpolatory model reduction methods for frequency response problems

Frequency sweep problems arise in many structural dynamic, acoustic, and structural acoustic applications. In each case, they incur the evaluation of a frequency response function for a typically large number of frequencies. Because each function evaluation requires the solution of an often large‐scale system of equations, frequency sweep problems are computationally intensive. Interpolatory model order reduction is a powerful tool for reducing their cost. However, the performance of this tool depends on the location and number of the interpolation frequency points. It also depends on the number of consecutive frequency derivatives of the response function that are matched at each frequency point. So far, these two choices have been made in the literature in a heuristic manner. In contrast, this paper proposes an automatic adaptive strategy based on monitoring the Euclidean norm of the relative residual associated with the function to be evaluated over the frequency band of interest. More specifically, the number of interpolation points and the number of matched frequency derivatives are adaptively increased until the global Euclidean norm of the relative residual is reduced below a user‐specified tolerance. The robustness, accuracy, and computational efficiency of this adaptive strategy are highlighted with the solution of several frequency sweep problems associated with large‐scale structural dynamic, acoustic, and structural acoustic finite element models. Copyright © 2012 John Wiley & Sons, Ltd.     

Stress singularity analysis for orthotropic V‐notches in the generalised plane strain state

The singularity for the V‐notch under the generalised plane deformation is investigated by the combination of the asymptotic analysis with the interpolating matrix method developed by part of the authors before. The displacement asymptotic expansions at the vicinity of the V‐notch vertex are introduced into the equilibrium equations, which are transformed into a set of characteristic ordinary differential equations with respect to the notch singularity orders. The boundary conditions and interfacial compatibility conditions are also represented by the combination of the singularity orders and characteristic angular functions. The determination of the singularity orders and characteristic angular functions are transformed into solving the ordinary differential equations with variable coefficients, which are solved by the interpolating matrix method. The present method is suitable for the singularity analysis for isotropic and orthotropic V‐notches. It is versatile for analysing the stress singularity of single material V‐notches, bi‐material V‐notches, interface edges and cracks. The correctness of the results by the proposed method is ensured by the comparison with the published ones.     

A C1 finite element including transverse shear and torsion warping for rectangular sandwich beams

A new three‐noded C¹ beam finite element is derived for the analysis of sandwich beams. The formulation includes transverse shear and warping due to torsion. It also accounts for the interlaminar continuity conditions at the interfaces between the layers, and the boundary conditions at the upper and lower surfaces of the beam. The transverse shear deformation is represented by a cosine function of a higher order. This allows us to avoid using shear correction factors. A warping function obtained from a three‐dimensional elasticity solution is used in the present model. Since the field consistency approach is accounted for interpolating the transverse strain and torsional strain, an exact integration scheme is employed in evaluating the strain energy terms. Performance of the element is tested by comparing the present results with exact three‐dimensional solu‐tions available for laminates under bending, and the elasticity three‐dimensional solution deduced from the de Saint‐Venant solution including both torsion with warping and bending. In addition, three‐dimensional solid finite elements using 27 noded‐brick elements have been used to bring out a reference solution not available for sandwich structures having high shear modular ratio between skins and core. A detailed parametric study is carried out to show the effects of various parameters such as length‐to‐thickness ratio, shear modular ratio, boundary conditions, free (de Saint‐Venant) and constrained torsion. Copyright © 1999 John Wiley & Sons, Ltd.     

压电材料切口奇性指数计算

利用一种数值方法分析压电材料切口尖端包括奇异应力场和奇异电位移场在内的双重奇异性。基于切口尖端的位移场按幂级数渐近展开假设, 从应力平衡方程和Maxwell方程出发, 推导出关于压电材料切口奇性指数的特征方程组, 同时将切口的力学和电学边界条件转化为奇性指数和特征函数的组合表达, 从而将压电材料双重奇性分析问题转化为在相应边界条件下微分方程组的特征值求解问题, 采用插值矩阵法, 可以一次性地计算出压电材料切口的各阶奇性指数。裂纹作为切口的特例, 其尖端的电弹性奇性指数亦可以根据本法求出。     

A finite element approach combining a reduced‐order system,Padé approximants,and an adaptive frequency windowing for fast multi‐frequency solution of poro‐acoustic problems

In this work, a solution strategy is investigated for the resolution of multi‐frequency structural‐acoustic problems including 3D modeling of poroelastic materials. The finite element method is used, together with a combination of a modal‐based reduction of the poroelastic domain and a Padé‐based reconstruction approach. It thus takes advantage of the reduced‐size of the problem while further improving the computational efficiency by limiting the number of frequency resolutions of the full‐sized problem. An adaptive procedure is proposed for the discretization of the frequency range into frequency intervals of reconstructed solution. The validation is presented on a 3D poro‐acoustic example. Copyright © 2013 John Wiley & Sons, Ltd.     

Complex Gaussian functions expansion method applied to truncated Gaussian beams

An analytical expression for the beam propagation factor (M ² factor) of truncated Gaussian beams was derived by using the complex Gaussian functions expansion method. The reasonability of the approximation of complex Gaussian functions expansion method is studied, and a comparison of this method with the generalised truncated second-order moments method and the asymptotic analysis method is also made. In general, an easy analytical expression for the M ² factor of truncated laser beams can be derived by using the complex Gaussian functions expansion method. The M ² factor obtained by using the complex Gaussian functions expansion method is more consistent with that in practice than that obtained by using two other methods. The analytical results obtained by using the complex Gaussian functions expansion method can be reduced to that for the non-truncated case when the truncation parameter is sufficiently large. Therefore, the complex Gaussian functions expansion method is a suitable approximation method for studying the M ² factor of truncated laser beams.     

Model order reduction for hyperelastic materials

In this paper, we develop a novel algorithm for the dimensional reduction of the models of hyperelastic solids undergoing large strains. Unlike standard proper orthogonal decomposition methods, the proposed algorithm minimizes the use of the Newton algorithms in the search of non‐linear equilibrium paths of elastic bodies. The proposed technique is based upon two main ingredients. On one side, the use of classic proper orthogonal decomposition techniques, that extract the most valuable information from pre‐computed, complete models. This information is used to build global shape functions in a Ritz‐like framework. On the other hand, to reduce the use of Newton procedures, an asymptotic expansion is made for some variables of interest. This expansion shows the interesting feature of possessing one unique tangent operator for all the terms of the expansion, thus minimizing the updating of the tangent stiffness matrix of the problem. The paper is completed with some numerical examples in order to show the performance of the technique in the framework of hyperelastic (Kirchhoff–Saint Venant and neo‐Hookean) solids. Copyright © 2009 John Wiley & Sons, Ltd.     

A nonparametric probabilistic approach for quantifying uncertainties in low‐dimensional and high‐dimensional nonlinear models

A nonparametric probabilistic approach for modeling uncertainties in projection‐based, nonlinear, reduced‐order models is presented. When experimental data are available, this approach can also quantify uncertainties in the associated high‐dimensional models. The main underlying idea is twofold. First, to substitute the deterministic reduced‐order basis (ROB) with a stochastic counterpart. Second, to construct the probability measure of the stochastic reduced‐order basis (SROB) on a subset of a compact Stiefel manifold in order to preserve some important properties of a ROB. The stochastic modeling is performed so that the probability distribution of the constructed SROB depends on a small number of hyperparameters. These are determined by solving a reduced‐order statistical inverse problem. The mathematical properties of this novel approach for quantifying model uncertainties are analyzed through theoretical developments and numerical simulations. Its potential is demonstrated through several example problems from computational structural dynamics. Copyright © 2016 John Wiley & Sons, Ltd.     

从 Levinson 高阶梁理论的一致变分到高次翘曲梁理论

将矩形截面梁的截面翘曲位移设定为3次Legendre多项式的形式,利用弹性力学平面应力问题分项的不完全的广义变分原理,导出高次翘曲梁理论,得到形式简单易求解的方程。由于引入轴向拉伸的情况,使梁的平面内变形问题得以统一;计及了梁表面剪切荷载的作用,并严格满足表面剪应力边界条件;通过引入轴向位移约束参考点间距离的概念对梁端翘曲约束作更精致地描述,且使得该理论包含了变分一致或者不一致的高阶剪切梁理论。该理论的推导还表明,Levinson梁理论的变分不一致仅仅局限于有转角约束的梁端。通过算例,将高次翘曲梁理论与弹性力学平面应力问题以及Timoshenko梁理论、Levinson梁理论进行比较,初步显示出该理论的优越性。     

An efficient numerical solution to the multinormal integral

The multinormal integral which has important applications in structural reliability is evaluated by application of a recent, asymptotic formula for the probability content of general intersections in the standard normal space together with a recursive scheme proposed earlier by Hohenbichler. It is suitable for arbitrary correlations, computationally efficient even in higher dimensions and most accurate for small or large probabilities.     

Adaptive training of local reduced bases for unsteady incompressible Navier–Stokes flows

This report presents a numerical study of reduced‐order representations for simulating incompressible Navier–Stokes flows over a range of physical parameters. The reduced‐order representations combine ideas of approximation for nonlinear terms, of local bases, and of least‐squares residual minimization. To construct the local bases, temporal snapshots for different physical configurations are collected automatically until an error indicator is reduced below a user‐specified tolerance. An adaptive time‐integration scheme is also employed to accelerate the generation of snapshots as well as the simulations with the reduced‐order representations. The accuracy and efficiency of the different representations is compared with examples with parameter sweeps. Copyright © 2015 John Wiley & Sons, Ltd.