Efficient approximation of Sparse Jacobians for time‐implicit reduced order models

This paper introduces a sparse matrix discrete interpolation method to effectively compute matrix approximations in the reduced order modeling framework. The sparse algorithm developed herein relies on the discrete empirical interpolation method and uses only samples of the nonzero entries of the matrix series. The proposed approach can approximate very large matrices, unlike the current matrix discrete empirical interpolation method, which is limited by its large computational memory requirements. The empirical interpolation indices obtained by the sparse algorithm slightly differ from the ones computed by the matrix discrete empirical interpolation method as a consequence of the singular vectors round‐off errors introduced by the economy or full singular value decomposition (SVD) algorithms when applied to the full matrix snapshots. When appropriately padded with zeros, the economy SVD factorization of the nonzero elements of the snapshots matrix is a valid economy SVD for the full snapshots matrix. Numerical experiments are performed with the 1D Burgers and 2D shallow water equations test problems where the quadratic reduced nonlinearities are computed via tensorial calculus. The sparse matrix approximation strategy is compared against five existing methods for computing reduced Jacobians: (i) matrix discrete empirical interpolation method, (ii) discrete empirical interpolation method, (iii) tensorial calculus, (iv) full Jacobian projection onto the reduced basis subspace, and (v) directional derivatives of the model along the reduced basis functions. The sparse matrix method outperforms all other algorithms. The use of traditional matrix discrete empirical interpolation method is not possible for very large dimensions because of its excessive memory requirements. Copyright © 2016 John Wiley & Sons, Ltd.     

C 1 natural element method for strain gradient linear elasticity and its application to microstructures

C 1 natural element method (C 1 NEM) is applied to strain gradient linear elasticity, and size effects on microstructures are analyzed. The shape functions in C 1 NEM are built upon the natural neighbor interpolation (NNI), with interpolation realized to nodal function and nodal gradient values, so that the essential boundary conditions (EBCs) can be imposed directly in a Galerkin scheme for partial differential equations (PDEs). In the present paper, C 1 NEM for strain gradient linear elasticity is constructed, and several typical examples which have analytical solutions are presented to illustrate the effectiveness of the constructed method. In its application to microstructures, the size effects of bending stiffness and stress concentration factor (SCF) are studied for microspeciem and microgripper, respectively. It is observed that the size effects become rather strong when the width of spring for microgripper, the radius of circular perforation and the long axis of elliptical perforation for microspeciem come close to the material characteristic length scales. For the U-shaped notch, the size effects decline obviously with increasing notch radius, and decline mildly with increasing length of notch.     

Static analysis of ultra-thin beams based on a semi-continuum model

A linear semi-continuum model with discrete atomic layers in the thickness direction was developed to investigate the bending behaviors of ultra-thin beams with nanoscale thickness.The theoretical results show that the deflection of an ultra-thin beam may be enhanced or reduced due to different relaxation coefficients.If the relaxation coefficient is greater/less than one,the deflection of micro/nano-scale structures is enhanced/reduced in comparison with macro-scale structures.So,two opposite types of size-dependent behaviors are observed and they are mainly caused by the relaxation coefficients.Comparisons with the classical continuum model,exact nonlocal stress model and finite element model (FEM) verify the validity of the present semi-continuum model.In particular,an explanation is proposed in the debate whether the bending stiffness of a micro/nano-scale beam should be greater or weaker as compared with the macro-scale structures.The characteristics of bending stiffness are proved to be associated with the relaxation coefficients.     

Vibration of quadrilateral embedded multilayered graphene sheets based on nonlocal continuum models using the Galerkin method

Free vibration analysis of quadrilateral multilayered graphene sheets(MLGS) embedded in polymer matrix is carried out employing nonlocal continuum mechanics.The principle of virtual work is employed to derive the equations of motion.The Galerkin method in conjunction with the natural coordinates of the nanoplate is used as a basis for the analysis.The dependence of small scale effect on thickness,elastic modulus,polymer matrix stiffness and interaction coefficient between two adjacent sheets is illustrated.The non-dimensional natural frequencies of skew,rhombic,trapezoidal and rectangular MLGS are obtained with various geometrical parameters and mode numbers taken into account,and for each case the effects of the small length scale are investigated.     

弹塑性结构安定下限分析的无网格局部

将基于Voronoi结构的无网格局部Petrov-Galerkin法与减缩基技术相结合,建立了一种安定下限分析的新方法．为了克服移动最小二乘近似难以准确施加本质边界条件的缺点,采用了自然邻近插值构造试函数.通过引入基准载荷域上载荷角点的概念,消除了安定下限分析中由时间参数所引起的求解困难．利用减缩基技术,将安定分析问题化为一系列未知变量较少的非线性规划子问题．在每个非线性规划子问题中,自平衡应力场由一组带有待定系数的自平衡应力场基矢量的线性组合进行模拟,而这些自平衡应力场基矢量可应用弹塑性增量分析中的平衡迭代结果得到．算例结果证明了提出的分析方法的有效性．      

Research on the iterative method for model updating based on the frequency response function

Model reduction technique is usually employed in model updating process.In this paper,a new model updating method named as cross-model cross-frequency response function(CMCF) method is proposed and a new iterative method associating the model updating method with the model reduction technique is investigated.The new model updating method utilizes the frequency response function to avoid the modal analysis process and it does not need to pair or scale the measured and the analytical frequency response function,which could greatly increase the number of the equations and the updating parameters.Based on the traditional iterative method,a correction term related to the errors resulting from the replacement of the reduction matrix of the experimental model with that of the finite element model is added in the new iterative method.Comparisons between the traditional iterative method and the proposed iterative method are shown by model updating examples of solar panels,and both of these two iterative methods combine the CMCF method and the succession-level approximate reduction technique.Results show the effectiveness of the CMCF method and the proposed iterative method.     

A hybrid-stress element based on Hamilton principle

A novel hybrid-stress finite element method is proposed for constructing simple 4-node quadrilateral plane elements, and the new element is denoted as HH4-3fl here. Firstly, the theoretical basis of the traditional hybrid-stress elements, i.e., the Hellinger-Reissner variational principle, is replaced by the Hamilton variational principle, in which the number of the stress variables is reduced from 3 to 2. Secondly, three stress parameters and corresponding trial functions are introduced into the system equations. Thirdly, the displacement fields of the conventional bilinear isoparametric element are employed in the new models. Finally, from the stationary condition, the stress parameters can be expressed in terms of the displacement parameters, and thus the new element stiffness matrices can be obtained. Since the required number of stress variables in the Hamilton variational principle is less than that in the Hellinger-Reissner variational principle, and no additional incompatible displacement modes are considered, the new hybrid-stress element is simpler than the traditional ones. Furthermore, in order to improve the accuracy of the stress solutions, two enhanced post-processing schemes are also proposed for element HH4-3β. Numerical examples show that the proposed model exhibits great improvements in both displacement and stress solutions, implying that the proposed technique is an effective way for developing simple finite element models with high performance.     

Validation of a CIP-based tank for numerical simulation of free surface flows

A constrained interpolation profile CIP-based numerical tank is developed to simulate violent free surface flows.The numerical simulation is performed by the CIP-based Cartesian grid method,which is described in the present paper.The tangent of hyperbola for interface capturing(THINC) scheme is applied for capturing complex free surfaces.The new model is capable of simulating a flow with violently varied free surface.A series of computations are conducted to assess the developed algorithm and its versatility.These tests include the collapse of water column with and without an obstacle,sloshing in a fixed tank,the generation of regular waves in a tank,the generation of extreme waves in a tank.Excellent agreements are obtained when numerical results are compared with available analytical,experimental,and other numerical results.     

大规模边界元模态分析的高效数值方法

随着大规模快速边界元计算技术的发展,在复杂结构的动态设计、振动与噪声分析中愈来愈多地采用边界元法,因此求解大规模边界元特征值问题、进行复杂结构和声场模态分析,成为工程应用中一个十分重要,但却极具挑战性的课题,目前国际上还没有十分有效的数值方法.本文针对边界元法中典型的非线性特征值问题,提出了一种通用、高效的数值解法,称为基于预解矩阵采样的Rayleigh-Ritz投影法,记为RSRR.首先,通过求解一系列频域边界元问题来构造特征向量搜索空间,进而可以采用Rayleigh-Ritz投影,将原问题转化为一个可以采用现有方法求解的小规模缩减特征值问题;其次,为了降低Rayleigh-Ritz投影过程的计算量,基于解析函数的Cauchy积分公式,构造了边界元系数矩阵的插值近似方法,以及缩减特征值问题系数矩阵的快速计算方法,给出了插值项数的估计策略;最后,将RSRR与声学快速边界元法结合,应用于大规模吸声结构的复模态分析.数值算例表明,RSRR方法能够可靠地求出给定频段内的全部特征值和特征向量,具有计算效率高、精度高、通用等优点.     

薄板分析的线性基梯度光滑伽辽金无网格法

薄板问题的控制方程为四阶微分方程,因而当采用伽辽金法进行分析时,形函数需要满足C$^{1}$连续性要求,且至少使用二次基函数才能保证方法的收敛性.无网格形函数虽然易于满足C$^{1}$连续性要求,但由于不是多项式,其二阶导数的计算较为复杂耗时,同时也对刚度矩阵的数值积分提出了更高的要求.本文提出了一种薄板分析的线性基梯度光滑伽辽金无网格法,该方法的基础是线性基无网格形函数的光滑梯度.在梯度光滑构造的理论框架内,无网格形函数的二阶光滑梯度可以表示为形函数一阶梯度的线性组合,因而可以提高形函数二阶梯度的计算效率.分析表明,线性基无网格形函数的光滑梯度不仅满足其固有的线性梯度一致性条件,还满足本属于二次基函数对应的额外高阶一致性条件,因此能够恰当地运用到薄板结构的伽辽金分析.此外,插值误差分析也很好地验证了线性基无网格光滑梯度的收敛特性.算例结果进一步表明,线性基梯度光滑伽辽金无网格法的收敛率与传统二次基伽辽金无网格法相当,但精度更高,同时刚度矩阵所需的高斯积分点数明显减少.      

基于分区加速和总体共轭梯度法的耦合界面数据传递问题研究

对于耦合动力学问题的分析过程,在界面上需频繁进行数据交换。为此,基于紧支径向基函数和多项式基函数推导了界面数据传递的插值算法,给出了传递矩阵的具体形式。通过分析时间复杂度,找出该算法在大节点量时效率不高的原因在于径向基矩阵的构造和传递矩阵的计算。为加快径向基矩阵的构造速度,提出分区加速处理以提高相关节点的搜索效率;为避免传递矩阵求解过程中的求逆运算,将其转化为多右端项的大型稀疏对称线性方程组问题,引入多右端项的总体共轭梯度迭代方法求解,并讨论了初始估计矩阵的选取方法。数值算例结果表明,结合使用分区加速原理和总体共轭梯度迭代方法,可在不损失插值精度的前提下显著提高求解效率。     

Assessment of composite failure and ultimate strength without experiment on composite

This paper attempts to estimate the ultimate strength of a laminated composite only based on its con- stituent properties measured independently. Three important issues involved have been systematically addressed, i.e., stress calculation for the constituent fiber and matrix materials, failure detection for the lamina and laminate upon the internal stresses in their constituents, and input data determination of the constituents from monolithic measurements. There are three important factors to influence the accuracy of the strength prediction. One is the stress concentration factor （SCF） in the matrix. Another is matrix plasticity. The third is thermal residual stresses in the constituents. It is these three factors, however, that have not been sufficiently well realized in the composite community. One can easily find out the elastic and strength parameters of a great many laminae and laminates in the current literature. Unfortunately, necessary information to determine the SCF, the matrix plasticity, and the thermal residual stresses of the composites is rare or incomplete. A useful design methodology is demonstrated in the paper.     

A boundary element method for the simulation of non-spherical bubbles and their interactions near a free surface

The basic principle and numerical technique for simulating two three-dimensional bubbles near a free surface are studied in detail by using boundary element method. The singularities of influence coefficient matrix are eliminated using coordinate transformation and so-called 4 π rule. The solid angle for the open surface is treated in direct method based on its definition. Several kinds of configurations for the bubbles and free surface have been investigated. The pressure contours during the evolution of bubbles are obtained in our model and can better illuminate the mechanism underlying the motions of bubbles and free surface. The bubble dynamics and their interactions have close relation with the standoff distances, buoyancy parameters and initial sizes of bubbles. Completely different bubble shapes, free surface motions, jetting patterns and pressure distributions under different parameters can be observed in our model, as demonstrated in our calculation results.     

自然单元法计算裂纹与材料边界问题

提出一种新的非凸边界上自然单元法形函数计算方法,通过边界结点限制点对间的邻点关系,对包括裂纹和材料边界在内的各种类型的非凸边界具有统一的处理原则,所得到的近似函数在边界结点间具有线性插值性.     

Rock deformation equations and application to the study on slantingly installed disc cutter

At present the mechanical model of the interac- tion between a disc cutter and rock mainly concerns indentation experiment, linear cutting experiment and tunnel boring machine （TBM） on-site data. This is not in line with the actual rock-breaking movement of the disc cutter and impedes to some extent the research on the rock-breaking mechanism, wear mechanism and design theory. Therefore, our study focuses on the interaction between the slantingly installed disc cutter and rock, developing a model in accordance with the actual rock-breaking movement. Displacement equations are established through an analysis of the velocity vector at the rock-breaking point of the disc cutter blade; the func- tional relationship between the displacement parameters at the rock-breaking point and its rectangular coordinates is established through an analysis of micro-displacement vectors at the rock-breaking point, thus leading to the geometric equations of rock deformation caused by the slantingly installed disc cutter. Considering the basically linear relationship between the cutting force of disc cutters and the rock deformation before and after the leap break of rock, we express the constitutive relations of rock deformation as generalized Hooke＇s law and analyze the effect of the slanting installa- tion angle of disc cutters on the rock-breaking force. This will, as we hope, make groundbreaking contributions to the development of the design theory and installation practice of TBM.     

NEW TREATMENT OF ESSENTIAL BOUNDARY CONDITIONS IN EFG METHOD BY COUPLING WITH RPIM

One of major difficulties in the implementation of meshfree methods using the moving least square (MLS) approximation, such as element-free Galerkin method (EFG), is the imposition of essential boundary conditions as the approximations do not pass through the nodal parameter values. Another class of meshfree methods based on the radial basis point interpolation can satisfy the essential boundary conditions exactly since its approximation function passes through each node in an influence domain and thus its shape functions possess the properties of delta function. In this paper, a coupled element-free Galerkin(EFG)-radial point interpolation method (RPIM) is proposed to enhance their advantages and avoid their disadvantages. Discretized equations of equilibrium are obtained in the RPIM region and the EFG region, respectively. Then a collocation approach is introduced to couple the RPIM and the EFG method. This method satisfies the linear consistency exactly and can maintain the stiffness matrix symmetric. Numerical tests show that this method gives reasonably accurate results consistent with the theory.     

A hybrid vertex-centered finite volume/element method for viscous incompressible flows on non-staggered unstructured meshes

This paper proposes a hybrid vertex-centered finite volume/finite element method for solution of the two dimensional (2D) incompressible Navier-Stokes equations on unstructured grids.An incremental pressure fractional step method is adopted to handle the velocity-pressure coupling.The velocity and the pressure are collocated at the node of the vertex-centered control volume which is formed by joining the centroid of cells sharing the common vertex.For the temporal integration of the momentum equations,an implicit second-order scheme is utilized to enhance the computational stability and eliminate the time step limit due to the diffusion term.The momentum equations are discretized by the vertex-centered finite volume method (FVM) and the pressure Poisson equation is solved by the Galerkin finite element method (FEM).The momentum interpolation is used to damp out the spurious pressure wiggles.The test case with analytical solutions demonstrates second-order accuracy of the current hybrid scheme in time and space for both velocity and pressure.The classic test cases,the lid-driven cavity flow,the skew cavity flow and the backward-facing step flow,show that numerical results are in good agreement with the published benchmark solutions.     

A new beam element for analyzing geometrical and physical nonlinearity

Based on Timoshenko＇s beam theory and Vlasov＇s thin-walled member theory, a new model of spatial thin-walled beam element is developed for analyzing geometrical and physical nonlinearity, which incorporates an interior node and independent interpolations of bending angles and warp and takes diversified factors into consideration, such as traverse shear deformation, torsional shear deformation and their coupling, coupling of flexure and torsion, and the second shear stress. The geometrical nonlinear strain is formulated in updated Lagarange （UL） and the corresponding stiffness matrix is derived. The perfectly plastic model is used to account for physical nonlinearity, and the yield rule of von Mises and incremental relationship of Prandtle-Reuss are adopted. Elastoplastic stiffness matrix is obtained by numerical integration based on the finite segment method, and a finite element program is compiled. Numerical examples manifest that the proposed model is accurate and feasible in the analysis of thin-walled structures.     

非线性MEMS微梁的重心有理插值迭代配点法分析

文章利用重心有理插值迭代配点法分析计算非线性MEMS微梁问题。通过处理MEMS微梁的几何通过假设初始函数,将微梁非线性控制方程转换为线性化微分方程,建立逼近非线性微分方程的线性化迭代格式。采用重心有理插值配点法求解线性化微分方程,提出了数值分析MEMS微梁非线性弯曲问题的重心插值迭代配点法。给出了非线性微分方程的直接线性化和Newton线性化计算公式,详细讨论了非线性积分项的计算方法和公式。利用重心有理插值微分矩阵,建立了矩阵-向量化的重心插值迭代配点法的计算公式。数值算例结果表明,重心插值迭代配点法求解微梁非线性弯曲问题,具有计算公式简单、程序实施方便和计算精度高的特点。     

Development of optimized interpolation schemes with spurious modes minimization

This work constitutes a fraction of a more extensive effort, which ultimate objective is the development of advanced aeroacoustics hybrid methods. Within this framework, we here focus on the interpolation step, on which generally rely all coupling processes that link altogether the various stages constituting any given hybrid method. In that regard, previous works by the present authors had revealed the intrinsic limitations and subsequent side effects (e.g., signal degradation) that weight on usual high‐order interpolation schemes, whether the latter are of centered or noncentered nature, as well as optimized in an acoustic sense or not. Based on the outcomes of such study, here, a novel optimization technique for interpolation schemes is proposed. Such a technique, which is designed hereafter as the interpolation by parts (IBP), allows interpolating accurately a given signal, while minimizing its possible degradation. As a result, compared with its standard counterpart, any IBP‐optimized interpolation scheme exhibits improved characteristics, such as a spurious modes generation that is greatly reduced (up to a 99% factor). Such improved characteristics are here validated on the basis of three test cases (of 1D, 2D and 3D nature), which illustrates the potentialities offered by the IBP optimization technique. Copyright © 2015 John Wiley & Sons, Ltd.