Reduced-basis Method在特征值问题中的应用

在特征值问题中引入了一种快速计算方法——缩减基法(Reduced-basis Method,RBM),介绍了RBM的计算原理和计算过程,给出了RBM误差估计的方法。以一塔式起重机塔身结构自由振动分析为例,修正设计参数时,由RBM得到结构的固有频率和振型及该计算结果的误差限。与有限元结果相比较,RBM不仅计算速度快,计算精度也很高,还可对结果进行误差估计。

A Real-Time Transient Analysis of a Functionally Graded Material Plate Using Reduced-Basis Methods

Based on the hybrid numerical method (HNM) combining with a reduced-basis method (RBM), the real-time transient response of a functionally graded material (FGM) plates is obtained. The large eigenvalue problem in wavenumber domain has been solved through real-time off-line/on-line calculation. At off-line stage, a reduced-basis space is constructed in sample wavenumbers according to the solved eigenvalue problems. The matrices independent of parameters are projected onto the reduced-basis spaces. At on-line stage, the reduced eigenvalue problems of the arbitrary wavenumbers are built. Subsequently, the responses in wavenumber domain are obtained by the approximated eigen-pairs. Because of the application of RBM, the computational cost of transient displacement analysis of FGM plate is decreased significantly, while the accuracy of the solution and the physics of the structure are still retained. The efficiency and validity of the proposed method are demonstrated through a numerical example.

Adaptive Reduced Basis Methods Applied to Structural Dynamic Analysis

The reduced basis methods (RBM) have been demonstrated as a promising numerical technique for statics problems and are extended to structural dynamic problems in this paper. Direct step-by-step integration and mode superposition are the most widely used methods in the field of the finite element analysis of structural dynamic response and solid mechanics. Herein these two methods are both transformed into reduced forms according to the proposed reduced basis methods. To generate a reduced surrogate model with small size, a greedy algorithm is suggested to construct sample set and reduced basis space adaptively in a prescribed training parameter space. For mode superposition method, the reduced basis space comprises the truncated eigenvectors from generalized eigenvalue problem associated with selected sample parameters. The reduced generalized eigenvalue problem is obtained by the projection of original generalized eigenvalue problem onto the reduced basis space. In the situation of direct integration, the solutions of the original increment formulation corresponding to the sample set are extracted to construct the reduced basis space. The reduced increment formulation is formed by the same method as mode superposition method. Numerical example is given in Section 5 to validate the efficiency of the presented reduced basis methods for structural dynamic problems.

面向数字孪生模型应用的油浸式变压器绕组温度POD-RBFLP降阶计算方法

了解油浸式电力变压器绕组的温度情况是保证其运行稳定性的关键,也是当前针对油浸式变压器数字孪生分析的必然需求。为了快速地获得变压器绕组的稳态温度,该文提出一种基于本征正交分解(proper orthogonal decomposition,POD)和包含线性多项式的径向基函数响应面法(radial basis function response surface method including linear polynomial,RBFLP)的降阶计算模型。首先,讨论POD方法的降阶特性,并设计一种基于留一法交叉验证的自适应获得快照矩阵方法,以提高计算精度及效率;其次,采用响应面方法建立POD模态系数与绕组工况的相关关系,旨在实现通过绕组工况快速获得POD模态系数,从而跳过对降阶模型的复杂非线性计算,进而高效重构绕组温度场。相关算例表明,该方法具有较好的计算精度和效率,在50组测试工况下,与全阶计算相比,误差不超过2.5 K,且总计算时间仅为1.45 s;最后,基于110 kV变压器绕组搭建温升试验平台,试验结果表明,降阶计算结果相较于试验结果,平均计算误差不超过2 K,且单步计算时间仅为0.03 s,相较于同等规模的全阶计算,计算效率有较大幅度地提升。

减基法在轨道-隧道-土体系统谐响应分析中的应用

为了提高轨道交通环境振动预测效率,提出基于减基法原理的轨道-隧道-土体系统谐响应分析方法,即将减基法应用到轨道-隧道-土体系统的谐响应分析中。结合有限元方法和理论解析方法,在既有计算频率样本的基础上,实现原求解域内系统频响函数的重采样计算。将本文方法与有限元方法和线性插值方法进行对比分析。研究结果表明:在不同规模模型条件下,所提出方法的计算误差要明显低于线性插值方法的计算误差。同时,相比于有限元求解方法,本文方法能够在不同线程条件下均保持较高的计算精度和效率;且随着模型规模增大,本文方法相比有限元法的加速比不断提高,最大加速比可以超过2000。

基于POD-RBF方法的管道内壁几何识别

针对天然气、石油等管道内部被腐蚀问题,基于本征正交分解-径向基函数(POD-RBF)提出了一种管道内壁几何识别方法.考虑静磁场并建立管道的简化有限元模型,构建变几何样本库,实现了POD-RBF对任意形状的响应预测.该方法在降阶分析的同时避免了迭代过程中因几何的改变需反复求解刚度矩阵,在很大程度上提高了计算效率.采用灰狼优化(GWO)算法对目标函数实施优化,避免了在变几何过程中灵敏度的求解.算例结果显示,该文方法可高效准确地反演管道内壁的几何形状,即使在引入噪声后GWO算法仍具有较好的稳定性.

从衰减型课程范式转向生加型课程范式——课程多样化的一种探讨

现代中国的主流课程范式基本上是一种衰减型课程范式,进入新时代,应该致力于人类的升级,因而转向生加型课程范式。这种新的课程范式的观念基础主要包括:人性观念;自由发展观念;自化思想和自创生理论;正义观念。其主要价值包括:充分发挥各类有关人员的积极作用;充分实现人类生命的增值;大力促进人类生态的优化。向生加型课程范式转变的关键方略包括:形成前瞻性的课程理念;形成立体化的课程系统运行机制;构建生加型的课程框架;构建生加型的学习评价体系;提供相关的指导、支持和保障。

Reduced Basis Approaches in Time-Dependent Non-Coercive Settings for Modelling the Movement of Nuclear Reactor Control Rods

In this work,two approaches,based on the certified Reduced Basis method,have been developed for simulating the movement of nuclear reactor control rods,in time-dependent non-coercive settings featuring a 3D geometrical framework.In particular,in a first approach,a piece-wise affine transformation based on subdomains division has been implemented for modelling the movement of one control rod.In the second approach,a“staircase”strategy has been adopted for simulating themovement of all the three rods featured by the nuclear reactor chosen as case study.The neutron kinetics has been modelled according to the so-called multi-group neutron diffusion,which,in the present case,is a set of ten coupled parametrized parabolic equations(two energy groups for the neutron flux,and eight for the precursors).Both the reduced order models,developed according to the two approaches,provided a very good accuracy comparedwith high-fidelity results,assumed as“truth”solutions.At the same time,the computational speed-up in the Online phase,with respect to the fine“truth”finite element discretization,achievable by both the proposed approaches is at least of three orders of magnitude,allowing a real-time simulation of the rod movement and control.

REDUCED BASIS METHOD FOR PARAMETRIZED ELLIPTIC ADVECTION-REACTION PROBLEMS

In this work we consider the Reduced Basis method for the solution of parametrized advection-reaction partial differential equations. For the generation of the basis we adopt a stabilized finite element method and we define the Reduced Basis method in the ＂primal- dual＂ formulation for this stabilized problem. We provide a priori Reduced Basis error estimates and we discuss the effects of the finite element approximation on the Reduced Basis error. We propose an adaptive algorithm, based on the a posteriori Reduced Basis error estimate, for the selection of the sample sets upon which the basis are built; the idea leading this algorithm is the minimization of the computational costs associated with the solution of the Reduced Basis problem. Numerical tests demonstrate the efficiency, in terms of computational costs, of the ＂primal-dual＂ Reduced Basis approach with respect to an ＂only primal＂ one. Parametrized advection-reaction partial differential equations, Reduced Basis method, ＂primal-dual＂ reduced basis approach, Stabilized finite element method, a posteriori error estimation.

A Non-Intrusive Reduced Basis EKI for Time Fractional Diffusion Inverse Problems

In this study,we consider an ensemble Kalman inversion(EKI)for the numerical solution of time fractional diffusion inverse problems(TFDIPs).Computational challenges in the EKI arise from the need for repeated evaluations of the forward model.We address this challenge by introducing a non-intrusive reduced basis(RB)method for constructing surrogate models to reduce computational cost.In this method,a reduced basis is extracted from a set of full-order snapshots by the proper orthogonal decomposition(POD),and a doubly stochastic radial basis function(DSRBF)is used to learn the projection coefficients.The DSRBF is carried out in the offline stage with a stochastic leave-one-out cross-validation algorithm to select the shape parameter,and the outputs for new parameter values can be obtained rapidly during the online stage.Due to the complete decoupling of the offline and online stages,the proposed non-intrusive RB method–referred to as POD-DSRBF–provides a powerful tool to accelerate the EKI approach for TFDIPs.We demonstrate the practical performance of the proposed strategies through two nonlinear time-fractional diffusion inverse problems.The numerical results indicate that the new algorithm can achieve significant computational gains without sacrificing accuracy.

Nearest lattice point algorithms on semi k-reduced basis

In this paper, we firstly generalize the relations among the basis vectors of LLL reduced basis to semi k-reduced basis. Then we analyze the complexities of the nearest plane algorithm and round-off algorithm on semi k-reduced basis, which, compared with L. Babai's results on LLL reduced basis, have better approximate ratios and contain almost the same time complexities.

基于POD降维外推差分算法的热传导模型研究

针对一维热传导问题,基于奇异值分解和特征投影本征分解(proper orthogonal decomposition,POD)法通过提取特征模态,建立了一种极少自由度、较高精度的降维外推仿真模型。给出模型降维近似解分量的数学推导、算法过程以及误差分析,实现了温度场的快速计算。最后,通过数值例子,将POD计算结果与有限差分法(FDM)计算结果进行了对比。结果表明:POD方法在不同的时间步长、空间步长及定解条件下,均能捕捉到传热过程的准确信息,平均计算速度比传统有限差分法计算速度提高了200倍,有效缩短了计算机模拟时间。考核了低阶模型的准确性,并说明了低阶方程可以定性的反映原高维系统的传热特性。同时,POD所得结果与FDM结果间的最大相对误差为0.15%,满足工程计算精度要求。所提出的POD降维外推算法方案,不但扩展了POD特征空间,而且可以逐步改进数值求解步骤,弥补了POD方法的不足,验证了利用POD降维算法研究传热问题的可行性与有效性。对于实现复杂传热模型的高效准确的分析与仿真数值求解过程有一定的理论参考价值。

水下推进电机的自适应降阶热模型

为了提升水下推进电机的综合性能,提出自适应降阶模型来分析电机的动态多模态热场域性能特性,解决现有算法在处理高维复杂问题时资源耗费大和易陷入局部最优的问题。在时域中模型采用“无方程”的非侵入式动态模态分解方法,从时间序列数据中提取主要动态模态,实现系统数据的降维和模态分解,精确描述系统行为和预测未来状态;在空间域中,结合基于改善期望的自适应策略和径向基函数进行参数化近似和降阶。通过评估设计空间中各候选点的预测不可信度,迭代提升模型质量,平衡空间全局探索与局部探究间的关系。多步骤协作建模框架能够从有限的大规模模拟数据集中进行精确的场域解预测。通过实验系统验证了模型在常规和异常状态下温度变化规律的有效性和可靠性,结果表现出高准确性和稳定性。

基于混合双层自组织径向基函数神经网络的优化学习算法

针对传统方法采用先训练后测试两阶段学习机制极易导致的过拟合或欠拟合问题,提出一种基于混合双层自组织径向基函数神经网络的优化学习(hybrid bilevel self-organizing radial basis function neural network optimization learning,Hb-SRBFNN-OL)算法。首先,将训练过程和测试过程集成到一个统一的框架中,规避过拟合或欠拟合问题。其次,基于进化学习机制,提出上下2层的交互式优化学习算法,上层基于网络复杂度和测试误差自组织调整网络结构,下层采用列文伯格-马夸尔特(Levenberg Marquardt,LM)算法作为优化器对自组织径向基函数神经网络(self-organizing radial basis function neural network,SO-RBFNN)的连接权值进行优化。最后,利用来自多个子网络的综合信息生成模型的最终输出,加速网络全局收敛。为验证所提方法的可行性,分别在多个分类和预测任务中进行了测试实验。结果表明,在与传统神经网络结构相似甚至更好的测试和分类精度下,该方法不仅能实现更快的训练收敛,而且能进化成更精简紧凑的径向基函数神经网络(radial basis function neural network,RBFNN)模型。尤其在污水处理过程中总磷的质量浓度预测实验中,测试集中均方根误差(root mean squared error,RMSE)最高可降低48.90%,实际场景实验结果验证了所提算法的精确性更佳且泛化能力更强。

一种基于Galerkin映射-减基法的结构参数反求方法

针对结构特性参数识别问题,提出了一种基于Galerkin映射-减基法的参数反求方法。采用该方法对结构进行有限元建模,在观测点位置通过对向量子空间的夹角进行评判,构造出非奇异的观测点基矩阵及相应的整体基矩阵。通过使向量离散L2范数最小化途径构建结构参数的减基识别模型,进而利用信赖域优化技术求解得出问题的结构参数。算例表明,该方法在反求精度和总体计算效率方面优于传统方法。

参数化流动传热问题的模型降阶方法研究

高精度的数值模型是构建反应堆堆芯数字孪生的基础,但传统的高分辨率数值模型的计算效率无法满足数字孪生的需求。为显著提高计算效率,综合利用伽辽金投影法和径向基函数插值法构造了参数化流动传热问题的降阶模型。该模型以典型的数值解作为学习样本,通过本征正交分解获得各物理场分布的主模态作为缩减基,然后将控制方程在缩减基张成的空间上进行投影以获得自由度个数显著降低的降阶模型。螺旋十字型燃料冷却剂的测试算例表明,相较于全阶模型,该降阶模型可实现高达3~4个数量级的加速效果,降阶模型的速度场与压力场的相对误差均小于10%。

F[x]-lattice basis reduction algorithm and multisequence synthesis

By means of F[x]-lattice basis reduction algorithm, a new algorithm is presented for synthesizing minimum length linear feedback shift registers (or minimal polynomials) for the given mul-tiple sequences over a field F. Its computational complexity is O(N2) operations in F where N is the length of each sequence. A necessary and sufficient condition for the uniqueness of minimal polynomi-als is given. The set and exact number of all minimal polynomials are also described when F is a finite field.

Isolation of Cr(Ⅵ) reducing bacteria from industrial effuents and their potential use in bioremediation of chromium containing wastewater

The present study was aimed to assess the ability of Bacillus sp.JDM-2-1 and Staphylococcus capitis to reduce hexavalent chromium into its trivalent form.Bacillus sp.JDM-2-1 could tolerate Cr（Ⅵ）（4800 μg/mL） and S.capitis could tolerate Cr（Ⅵ）（2800 μg/mL）.Both organisms were able to resist Cd^2＋（50 μg/mL）,Cu^2＋（200 μg/mL）,Pb^2＋（800 μg/mL）,Hg^2＋（50 μg/mL） and Ni2＋（4000 μg/mL）.S.capitis resisted Zn^2＋ at 700 μg/mL while Bacillus sp.JDM-2-1 only showed resistance up to 50 μg/mL.Bacillus sp.JDM-2-1 and S.capitis showed optimum growth at pH 6 and 7,respectively,while both bacteria showed optimum growth at 37°C.Bacillus sp.JDM-2-1 and S.capitis could reduce 85% and 81% of hexavalent chromium from the medium after 96 h and were also capable of reducing hexavalent chromium 86% and 89%,respectively,from the industrial effuents after 144 h.Cell free extracts of Bacillus sp.JDM-2-1 and S.capitis showed reduction of 83% and 70% at concentration of 10 μg Cr（Ⅵ）/mL,respectively.The presence of an induced protein having molecular weight around 25 kDa in the presence of chromium points out a possible role of this protein in chromium reduction.The bacterial isolates can be exploited for the bioremediation of hexavalent chromium containing wastes,since they seem to have a potential to reduce the toxic hexavalent form to its nontoxic trivalent form.

基于POD-RBFN降阶模型的串列叶栅流场预测

通过传统实验或CFD手段获取流场信息的方法往往需要耗费大量资源或时间,这在需要快速获取大量流场信息时产生的成本是无法接受的,发展比传统CFD更快速的流场预测方法具有重要意义。采用本征正交分解(POD)方法对样本流场进行模态分解,提取流场的主导模态;而后采用径向基函数神经网络(RBFN)响应POD基函数的系数,实现流场降阶预测模型的构建,并在模型中采用基于函数响应偏差的自适应抽样方法;通过某串列叶栅非定常流场数据对预测模型进行验证。结果表明:本文构建的POD-RBFN混合模型可以快速准确地预测出串列叶栅的流场参数分布;与静态采样相比,本文采用的自适应采样方法在采样效率上表现出明显优势,同样重构精度所需的样本数降低了25%左右。

Algebraic Solution for the Forward Displacement Analysis of the General 6-6 Stewart Mechanism

The solution for the forward displacement analysis（FDA） of the general 6-6 Stewart mechanism（i.e., the connection points of the moving and fixed platforms are not restricted to lying in a plane） has been extensively studied, but the efficiency of the solution remains to be effectively addressed. To this end, an algebraic elimination method is proposed for the FDA of the general 6-6 Stewart mechanism. The kinematic constraint equations are built using conformal geometric algebra（CGA）. The kinematic constraint equations are transformed by a substitution of variables into seven equations with seven unknown variables. According to the characteristic of anti-symmetric matrices, the aforementioned seven equations can be further transformed into seven equations with four unknown variables by a substitution of variables using the Grobner basis. Its elimination weight is increased through changing the degree of one variable, and sixteen equations with four unknown variables can be obtained using the Grobner basis. A 40th-degree univariate polynomial equation is derived by constructing a relatively small-sized 9 × 9 Sylvester resultant matrix. Finally, two numerical examples are employed to verify the proposed method. The results indicate that the proposed method can effectively improve the efficiency of solution and reduce the computational burden because of the small-sized resultant matrix.