Biorthogonal Rational Functions and the Generalized Eigenvalue Problem

We present some general results concerning so-called biorthogonal polynomials of R_II type introduced by M. Ismail and D. Masson. These polynomials give rise to a pair of rational functions which are biorthogonal with respect to a linear functional. It is shown that these rational functions naturally appear as eigenvectors of the generalized eigenvalue problem for two arbitrary tri-diagonal matrices. We study spectral transformations of these functions leading to a rational modification of the linear functional. An analogue of the Christoffel–Darboux formula is obtained.     

On the spectral representation of symmetric stable processes

The so-called spectral representation theorem for stable processes linearly imbeds each symmetric stable process of index p into L^p (0 < p ≤ 2). We use the theory of L^p isometries for 0 < p < 2 to study the uniqueness of this representation for the non-Gaussian stable processes. We also determine the form of this representation for stationary processes and for substable processes. Complex stable processes are defined, and a complex version of the spectral representation theorem is proved. As a corollary to the complex theory we exhibit an imbedding of complex L^q into real or complex L^p for 0 < p < q ≤ 2.     

竖向地震对建筑结构抗震性能的影响研究

应用弹塑性时程分析方法研究了组合墙结构在竖向和水平地震共同作用下的抗震性能.研究结果表明结构的最大层间位移由于受到竖向地震的影响有向结构中部转移的趋势.而且,层间位移增大.     

三变量通用阈值逻辑门的设计

讨论了逻辑函数的谱系数分类,给出了特征阈值逻辑函数的定义.在此基础上提出了三变量通用阈值逻辑门的设计方法,分别得到了单输出和带反相双输出的三变量通用阈值逻辑门,并给出了用三变量通用阈值逻辑门实现三变量阈值逻辑函数的实例.     

离子液体作为Si3N4-Ti3SiC2摩擦副润滑剂的摩擦学性能研究

采用SRV摩擦磨损试验机在室温及100 ℃下考察了两种离子液体（L-B106 和L-P106）、丙三醇、水作为Si₃N₄-Ti₃SiC₂摩擦副润滑剂的摩擦学行为,利用扫描电子显微镜（SEM）及X光电子能谱（XPS）对磨损表面进行了分析.结果表明：室温、20 N条件下,两种离子液体和丙三醇抗磨和减摩性能相当,室温、100 ℃条件下,L-P106相较于L-B106具有更好的润滑性能,且其抗磨和减摩性能均优于丙三醇,作为Si₃N₄-Ti₃SiC₂摩擦副润滑剂具有在苛刻环境条件下使用的应用前景. XPS分析结果表明：Ti₃SiC₂材料在摩擦过程中在摩擦热作用下生成了SiO_x、TiO₂,进而有效提高了Ti₃SiC₂摩擦副材料的抗磨损性能;此外,离子液体中的活性元素在Si₃N₄-Ti₃SiC₂摩擦副表面发生了复杂的摩擦化学反应,生成了由氟化钛、磷酸钛及硼酸钛等组成的具有减摩和抗磨性能的边界润滑膜.     

Analysis of the anisotropy of group velocity error due to spatial finite difference schemes from the solution of the 2D linear Euler equations

Numerical differencing schemes are subject to dispersive and dissipative errors, which in one dimension, are functions of a wavenumber. When these schemes are applied in two or three dimensions, the errors become functions of both wavenumber and the direction of the wave. For the Euler equations, the direction of flow and flow velocity are also important. Spectral analysis was used to predict the error in magnitude and direction of the group velocity of vorticity–entropy and acoustic waves in the solution of the linearised Euler equations in a two‐dimensional Cartesian space. The anisotropy in these errors, for three schemes, were studied as a function of the wavenumber, wave direction, mean flow direction and mean flow Mach number. Numerical experiments were run to provide confirmation of the developed theory. Copyright © 2012 John Wiley & Sons, Ltd.     

Discrete filter operators for large‐eddy simulation using high‐order spectral difference methods

The combination of a high‐order unstructured spectral difference (SD) spatial discretization scheme with sub‐grid scale (SGS) modeling for large‐eddy simulation is investigated with particular focus on the consistent implementation of a structural mixed model based on the scale similarity hypothesis. The difficult task of deriving a consistent formulation for the discrete filter within the SD element of arbitrary order led to the development of a new class of three‐dimensional constrained discrete filters. The discrete filters satisfy a set of selected criteria and are completely local within the SD element. Their weights can be automatically computed at run time from the number of solution points within each element and the expected filter cutoff length scale. The novel discrete filters can be applied to any SGS model involving explicit filtering and to a broad class of high‐order discontinuous finite element numerical schemes. The code is applied to the computation of turbulent channel flows at three Reynolds numbers, namely Re_τ = 180, 395, and 590 (based on the friction velocity u_τ and channel half‐width δ). Results from computations with and without the SGS model are compared against results from direct numerical simulation. The numerical experiments suggest that the results are sensitive to the use of the SGS model, even when a high‐order numerical scheme is used, especially when the grid resolution is kept relatively low and mostly in terms of resolved Reynolds stresses. Results obtained using existing filters based on the projection of the solution over lower‐order polynomial bases are also shown and demonstrate that these filters are inadequate for SGS modeling purposes, mostly because of their inability to enforce the selected cutoff length scale with sufficient accuracy. The use of the similarity mixed formulation proved to be particularly accurate in reproducing SGS interactions, confirming that its well‐known potential can be realized in conjunction with state‐of‐the‐art high‐order numerical schemes.Copyright © 2012 John Wiley & Sons, Ltd.     

DYNAMICAL APPROACH STUDY OF SPURIOUS STEADY-STATE NUMERICAL SOLUTIONS OF NONLINEAR DIFFERENTIAL EQUATIONS II. GLOBAL ASYMPTOTIC BEHAVIOR OF TIME DISCRETIZATIONS ∗

SUMMARY

The global asymptotic nonlinear behavior of 11 explicit and implicit time discretizations for four 2 × 2 systems of first-order autonomous nonlinear ordinary differential equations (ODEs) is analyzed. The objectives are to gain a basic understanding of the difference in the dynamics of numerics between the scalars and systems of nonlinear autonomous ODEs and to set a baseline global asymptotic solution behavior of these schemes for practical computations in computational fluid dynamics. We show how “numerical” basins of attraction can complement the bifurcation diagrams in gaining more detailed global asymptotic behavior of time discretizations for nonlinear differential equations (DCs). We show how in the presence of spurious asymptotes the basins of the true stable steady states can be segmented by the basins of the spurious stable and unstable asymptotes. One major consequence of this phenomenon which is not commonly known is that this spurious behavior can result in a dramatic distortion and, in most cases, a dramatic shrinkage and segmentation of the basin of attraction of the true solution for finite time steps. Such distortion, shrinkage and segmentation of the numerical basins of attraction will occur regardless ofthe stability ofthe spurious asymptotes, and will occur for unconditionally stable implicit linear multistep methods. In other words, for the same (common) steady-state solution the associated basin of attraction of the DE might be very different from the discretized counterparts and the numerical basin of attraction can be very different from numerical method to numerical method. The results can be used as an explanation for possible causes of error, and slow convergence and nonconvergence of steady-state numerical solutions when using the time-dependent approach for nonlinear hyperbolic or parabolic PDEs.     

Large bending behavior of creased paperboard. II. Structural analysis

The mechanics of a paradigmatic typical carton corner with five creases is analyzed theoretically, in closed form. A general kinematical analysis of the mechanism (in finite rotation) is presented, assuming the versor of the intermediate crease, s, as a 2-degree-of-freedom Lagrangian parameter. The rotation θ_c of the cth crease is derived, together with the existence domain and a discussion of the singular configurations.The actions, driving the carton during a prescribed quasi-static erection program, are derived in a very efficient manner using the Virtual Works Equation, taking into account a non-linear anholonomic bending constitutive law of the creased paperboard. In particular, the active and reactive components of the moment ?, driving s along its path, are identified. No resort to the tangent stiffness computation is required. Some numerical examples illustrate the rotation and the driving forces obtained for both monotone-loading and complex loading–unloading erection paths.The presented results, “exact” within the scope of the restrictive hypotheses assumed, may be used in a preliminary design approach as well as a benchmark for more realistic FEM or CAE simulators.     

基于迟滞行为的2D-SiC/SiC复合材料组份力学性能分析

基于剪滞理论, 建立了单向纤维增强陶瓷基复合材料的加卸载理论模型, 分析了基体长碎块和短碎块对材料迟滞力学行为的不同影响. 通过拉伸循环加卸载试验, 获得了2D-SiC/SiC 复合材料的迟滞应力—应变行为.依据材料基体损伤特点, 将试验结果代入长碎块对应理论推导结果, 计算得到了4 个表征材料组份性能的参数:基体开裂应力为90 MPa, 热残余应力为19 MPa, 界面脱粘能为3.1 Jm2, 界面滑移力为74 MPa. 最后结合少量短碎块的存在对试验结果的影响, 定性分析了计算结果的偏差. 结果表明, 获得的材料组分性能参数具有较小的分散性, 并能够准确表征材料整体的力学行为.