重特征值及其特征向量摄动重分析方法探讨

本文基于已有研究结果,进一步探讨了线性广义特征值问题的重特征值及其特征向量摄动重分析理论和方法,阐述了重特征值情形下摄动重分析方法的一些重要特点,完善了重特征值的特征向量的二阶摄动算式,并论证了重特征值摄动法与相异特征值摄动法之间的统一性。还以算例说明了重特征值情形下摄动重分析算法的实施步骤及其有效性。     

强非线性自激振子同宿轨道的摄动分析方法

在双曲函数摄动法的基础上,推广双曲函数Lindstedt-Poincaré (L-P)法的适用范围,使之适用于定量分析一类含五次强非线性项的自激振子的同宿分岔和同宿解问题。以双曲函数系为基础推导出适用于高次非线性系统的摄动步骤,对极限环的同宿分岔参数进行摄动展开,给出同宿摄动解奇异项的定义,以消除同宿摄动解奇异项作为确定极限环同宿分岔点的条件,给出能够严格满足同宿条件的同宿轨道摄动解。算例表明,在相平面内该方法的结果与Runge-Kutta法数值周期轨道的逼近结果比较吻合。     

密集模态结构模态跃迁分析的简化摄动法

大跨空间结构的模态具有密集、低频的特性。结构在风作用下会产生非常复杂的动力学现象,模态跃迁就是其中的一种。以往对该类结构模态跃迁特性的研究几乎为一片空白。矩阵摄动分析法是研究变参数结构模态特性的重要方法,在已有矩阵摄动法基础上,该文提出分析密集低频结构模态跃迁现象的简化摄动法。阐述模态跃迁的机理,并加以证明。算例结果表明:简化摄动法用于密集低频结构模态跃迁的分析是有效、可行的,且简化了移位密集特征值摄动法的计算过程。     

摄动微分求积法解复合材料层合板非线性振动

以摄动法将层合板的非线性振动偏微分方程线性化,从摄动方程中分离出时域函数并求解。将场函数设定为含时域函数的待定函数,通过微分求积法对空间域的摄动方程进行离散,将其转化成一系列的线性方程并通过编程求解。算例结果分析说明算法精度良好且效率高。     

变密度环形薄膜的轴对称振动修正摄动解

采用修正摄动法研究了变密度环形薄膜的轴对称横向固有振动，并求得了确定其横向振动固有频率的特征方程，把该方法所得到的修正摄动解与有关文献所得结果进行比较，可知此修正摄动解不但计算简便，而且精度可与经典的打靶法与微分求积法的精度相当。     

耗能减震结构的摄动分析方法

针对用振型分解法对耗能减震结构进行反应分析中存在的计算精度差且耗时的问题,提出了求解振动方程特征值问题的改进一阶摄动的高精度模态展开法,并推导了强行解耦的振型分解的一阶摄动修正公式。最后通过算例证明,提出的摄动分析法是有效的。     

基于摄动法及等效线性化的耗能减震结构振型分解法

采用耗能减震结构是一种经济有效的提高结构抗震能力的方法。针对此类结构在基于等效线性化的振型分解反应谱法进行分析中存在的计算精度差且耗时的问题,提出了求解运动方程本征值问题的基于改进一阶摄动的模态高精度展开法,在此基础上,推导出强行解耦的振型分解法以及相应的一阶摄动修正。最后通过算例证明,提出的基于摄动法及等效线性化的耗能减震结构振型分解法是行之有效的。     

摄动DQ法分析板的大挠度热弯曲

本文给出了求解板的大挠度热弯曲问题的摄动DQ法。该方法由二阶摄动得到板的大挠度热弯曲问题的一组线性摄动方程后运用DQ法进行求解,具有良好的计算精度和计算效率。     

反对称角铺设层合矩形板在不同边界条件下的后屈曲问题

本文以板的无量纲最大挠度为摄动参数,应用摄动法研究了在几种支承情况下,反对称角铺设层合矩形板的后屈曲问题,并以梁的特征函数所构成的广义付里叶级数作为高阶摄动偏微分方程的近似解,本文方法简捷,计算收敛性好、数据可靠。     

反对称角铺设层合矩形板在不同边界条件下的后屈曲问题

本文以板的无量纲最大挠度为摄动参数,应用摄动法研究了在几种支承情况下,反对称角铺设层合矩形板的后屈曲问题,并以梁的特征函数所构成的广义付里叶级数作为高阶摄动偏微分方程的近似解,本文方法简捷,计算收敛性好、数据可靠。     

Bifurcation indicators

Summary In this paper we propose bifurcation indicators for linear or nonlinear eigenvalue problems. These indicators are the determinants of a reduced stiffness matrix. They measure the intensity of the response of the system to perturbation forces. The numerical computation of the indicators is done by a direct method and by an Asymptotic Numerical Method.     

Solving hyperelastic material problems by asymptotic numerical method

This paper presents a numerical algorithm based on a perturbation technique named asymptotic numerical method (ANM) to solve nonlinear problems with hyperelastic constitutive behaviors. The main advantages of this technique compared to Newton–Raphson are: (a) a large reduction of the number of tangent matrix decompositions; (b) in presence of instabilities or limit points no special treatment such as arc-length algorithms is necessary. The ANM uses high order series approximation with auto-adaptive step length and without need of any iteration. Introduction of this expansion into the set of nonlinear equations results into a sequence of linear problems with the same linear operator. The present work aims at providing algorithms for applying the ANM to the special case of compressible and incompressible hyperelastic materials. The efficiency and accuracy of the method are examined by comparing this algorithm with Newton–Raphson method for problems involving hyperelastic structures with large strains and instabilities.     

Studies of Newmark method for solving nonlinear systems: (II) Verification and guideline

Abstract

Numerical properties of the Newmark method in the solution of nonlinear systems derived in the accompanying paper are thoroughly confirmed with numerical examples herein. It seems that analytical results can reveal the insight of the Newmark method in the step‐by‐step solution of linear and nonlinear systems. Although the constant average acceleration method is unconditionally stable for linear elastic systems these explorations confirm that it might lead to instability for nonlinear systems. In addition, numerical accuracy for period distortion and amplitude change is also shown to be consistent with the analytical predictions. Therefore, the performance of the Newmark method in the step‐by‐step solution of nonlinear systems is well investigated. As a result, a rough guideline to yield accurate solutions for the use of step‐by‐step integration methods to solve nonlinear systems is proposed.     

Model reduction for nonlinear multiscale parabolic problems using dynamic mode decomposition

In this article, a model reduction technique is presented to solve nonlinear multiscale parabolic problems using dynamic mode decomposition. The multiple scales and nonlinearity bring great challenges for simulating the problems. To overcome this difficulty, we develop a model reduction method for the nonlinear multiscale dynamic problems by integrating constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM) with dynamic mode decomposition (DMD). CEM-GMsFEM has shown great efficiency to solve linear multiscale problems in a coarse space. However, using CEM-GMsFEM to directly solve multiscale nonlinear parabolic models involves dynamically computing the residual and the Jacobian on a fine grid. This may be very computationally expensive because the evaluation of the nonlinear term is implemented in a high-dimensional fine scale space. As a data-driven method, DMD can use observation data and give an explicit expression to accurately describe the underlying nonlinear dynamic system. To efficiently compute the multiscale nonlinear parabolic problems, we propose a CEM-DMD model reduction by combing CEM-GMsFEM and DMD. The CEM-DMD reduced model is a coarsen linear model, which avoids the nonlinear solver in the fine space. It is crucial to judiciously choose observation in DMD. Only proper observation can render an accurate DMD model. In the context of CEM-DMD, we introduce two different observations: fine scale observation and coarse scale observation. In the construction of DMD model, the coarse scale observation requires much less computation than the fine scale observation. The CEM-DMD model using the coarse scale observation gives a complete coarse model for the nonlinear multiscale dynamic systems and significantly improves the computation efficiency. To show the performance of the CEM-DMD using the different observations, we present a few numerical results for the nonlinear multiscale parabolic problems in heterogeneous porous media.     

An adaptive radial basis algorithm (ARBF) for expensive black-box mixed-integer constrained global optimization

Response surface methods based on kriging and radial basis function (RBF) interpolation have been successfully applied to solve expensive, i.e. computationally costly, global black-box nonconvex optimization problems. In this paper we describe extensions of these methods to handle linear, nonlinear, and integer constraints. In particular, algorithms for standard RBF and the new adaptive RBF (ARBF) are described. Note, however, while the objective function may be expensive, we assume that any nonlinear constraints are either inexpensive or are incorporated into the objective function via penalty terms. Test results are presented on standard test problems, both nonconvex problems with linear and nonlinear constraints, and mixed-integer nonlinear problems (MINLP). Solvers in the TOMLAB Optimization Environment () have been compared, specifically the three deterministic derivative-free solvers rbfSolve, ARBFMIP and EGO with three derivative-based mixed-integer nonlinear solvers, OQNLP, MINLPBB and MISQP, as well as the GENO solver implementing a stochastic genetic algorithm. Results show that the deterministic derivative-free methods compare well with the derivative-based ones, but the stochastic genetic algorithm solver is several orders of magnitude too slow for practical use. When the objective function for the test problems is costly to evaluate, the performance of the ARBF algorithm proves to be superior.     

Substructured formulations of nonlinear structure problems – influence of the interface condition

We investigate the use of non‐overlapping domain decomposition (DD) methods for nonlinear structure problems. The classic techniques would combine a global Newton solver with a linear DD solver for the tangent systems. We propose a framework where we can swap Newton and DD so that we solve independent nonlinear problems for each substructure and linear condensed interface problems. The objective is to decrease the number of communications between subdomains and to improve parallelism. Depending on the interface condition, we derive several formulations that are not equivalent, contrarily to the linear case. Primal, dual and mixed variants are described and assessed on a simple plasticity problem. Copyright © 2015 John Wiley & Sons, Ltd.     

求解非定常N-S方程的稳定化分数步长法研究

本文研究了求解非定常Navier-Stokes方程的稳定化分数步长法.首先,通过一阶精度的算子分裂,将非线性项和不可压缩条件分裂到两个不同的子问题中,并对非线性项采用Oseen迭代.格式分为两步:第一步求解一个线性椭圆问题;第二步求解一个广义的Stokes问题.这两个子问题关于速度都满足齐次Dilichlet边界条件.同时,在格式的第二步添加了局部稳定化项,使用等阶序对来加强数值解的稳定性.通过能量估计方法,对速度与压力做了收敛性分析和误差估计.最后,数值实验验证了方法的有效性.     

圆柱壳几何大变形非线性频率求解的渐近摄动法

为解决圆柱壳在工作状态中由几何大变形而引起的弱非线性振动问题，将渐近摄动法引入求解考虑几何非线性的薄壁圆柱壳振动频率。首先，应用Donnell＇s简化壳理论获得了考虑几何大变形情况下具有位移三次项的非线性频率方程，把位移及频率以非线性参数的幂级数形式展开，并令同次幂的非线性项系数相等，由此得到非线性频率一次近似值与初始振幅的一系列耦合代数方程，引入Galerkin＇s方法对非线性频率方程进行解耦正交并忽略其中的永年项，考虑了对应实数根，各阶频率对应的振幅间不存在相互耦合的内共振现象，最终在引入小参数后用摄动法求出了非线性频率的一次近似解。计算结果表明，几何非线性使薄壁圆柱壳产生硬化，其非线性频率升高，并同时讨论了线性、非线性频率与节径数及初始位移之间的关系。     

非线性声波方程的几种解法对比

在医学诊疗领域及微、介观损伤的无损检测行业中,经常需要对介质的材料非线性系数进行表征,以得到局部区域更加精细的力学性能变化.文章在简述各向同性固体和理想流体介质中的非线性声波方程的基础上,证实了它们具有相同的形式,这表明它们的解也应具有相同的形式和性质.介绍了求解非线性声波方程的五种方法,包括有限差分、有限元、摄动法、...     

A finite difference perturbation procedure for non-linear elastic stability problems

A finite difference perturbation scheme is developed which allows a simple solution to a wide class of non-linear difurcation problems. The analysis shows that in order to determine the inital post bucking behaviour accurately, it is not necessary to solve more than the linear eigenvalue difference equation with similar accuracy.