石英晶体板非线性高频振动的拓展伽辽金法分析

针对考虑几何和材料非线性的石英晶体板厚度剪切振动和弯曲振动的方程组,利用扩展伽辽金法对该方程组进行转化和求解,分别获得了强烈耦合的厚度剪切振动模态和弯曲振动模态的频率响应关系,绘制了不同振幅比和不同驱动电压影响下的频率响应曲线图。数值计算结果表明可以选取石英晶片的最佳长厚比尺寸来避免两种模态的强烈耦合。驱动电压的变化将引起石英晶体谐振器厚度剪切振动频率的明显改变,必须将振动频率的漂移值控制在常用压电声波器件的允许值之内。扩展伽辽金法对石英晶体板非线性振动方程组的求解为非线性有限元分析和偏场效应分析奠定了基础。     

Stabilized Nonconforming Mixed Finite Element Method for Linear Elasticity on Rectangular or Cubic Meshes

Based on the primal mixed variational formulation, a stabilized nonconforming mixed finite element method is proposed for the linear elasticity on rectangular and cubic meshes. Two kinds of penalty terms are introduced in the stabilized mixed formulation, which are the jump penalty term for the displacement and the divergence penalty term for the stress. We use the classical nonconforming rectangular and cubic elements for the displacement and the discontinuous piecewise polynomial space for the stress, where the discrete space for stress are carefully chosen to guarantee the well-posedness of discrete formulation. The stabilized mixed method is locking-free. The optimal convergence order is derived in the $L^2$-norm for stress and in the broken $H^1$-norm and $L^2$-norm for displacement. A numerical test is carried out to verify the optimal convergence of the stabilized method.     

On the Brauer p-dimension of Henselian discrete valued fields of residual characteristic p > 0

Let $(K,v)$ be a Henselian discrete valued field with residue field $\stackrel{?}{K}$ of characteristic $p>0$, and ${\text{Brd}}_p(K)$ be the Brauer p-dimension of K. This paper shows that ${\text{Brd}}_p(K)\ge n$ if $[\stackrel{?}{K}:{\stackrel{?}{K}}^p]=p^n$, for some $n\in \mathbb{N}$. It proves that ${\text{Brd}}_p(K)=\infty$ if and only if $[\stackrel{?}{K}:{\stackrel{?}{K}}^p]=\infty$.     

NUMERICAL SIMULATION OF FIBROBLAST DUROTAXIS MIGRATION INDUCED BY STIFFNESS GRADIENT OF SUBSTRATE1)

建立了一种细胞趋硬性迁移的理论模型和有限元分析框架,为连续变刚度人工基质的试验设计提供理论依据。考虑了细胞体的黏弹性属性,以及细胞与基质间的配受体动态反应过程,并以配受体合成时间为时间步长,将细胞运动方程化为静力学形式进行求解。对有限元过程提出一种动约束,便于消除其结构矩阵的奇异性。结果表明,模型能够模拟黏着斑内部力的快速波动现象,细胞的运动速度与观测数据一致,可有效模拟20,h以上的长时程问题。     

Short Syntheses of some ‘Decalin‐1,8‐diones’ and their Derivatives: Breaking the Pretended Symmetry

A cheap synthesis of the so‐called ‘decalin‐1,8‐diones’ started with the conjugate (1,4‐) addition of cyclohex‐2‐en‐1‐one derivatives to the γ‐position of the dilithium derivative (buta‐1,3‐diene‐1,1‐bis(olate)) of crotonic acid. Hydrogenation of these ‘1,4‐γ’ adducts and final cyclization afforded the enol tautomers of decalin‐1,8‐diones. Nucleophilic substitutions at these 3‐oxoenols by NH₃ or primary amines created only monoamino products (namely, 3‐oxoenamines) whose reactions with OPCl₃ yielded dihydro(1,3,2)oxazaphosphinin‐2‐one derivatives. The two regioisomers of a trimethyl‐3‐oxoenamine served as models for the constitutional assignments of the two rapidly interconverting (hence, individually NMR‐invisible), tautomeric trimethyl‐3‐oxoenols. Such methyl substitutions served to break the ‘pretended’ symmetry of ‘decalin‐1,8‐dione’. Hydrazine and 3‐oxoenols furnished oxygen‐free indazole derivatives whose N?H bonds exchanged with t_1/2=ca. 0.00035 s at ca. ?58(9) °C.     

Inorganic Molecular Electride Mg4O3: Structure,Bonding, and Nonlinear Optical Properties

Growing demands of material science and, in particular, in the field of nonlinear optics (NLO) encourage us to look for stable highly polarizable molecules with excess diffuse electrons. An unusual class of compounds called electrides comply with these requirements. Many attempts have been made, yet only few electrides have been synthesized as solids and none of them as molecular species. In this paper, a new theoretically designed molecular species with electride characteristics is reported. The idea of this molecular electride comes from the formation of electride-like features in the MgO crystal with defect F-centers. The geometry of the investigated molecule can be described as a Mg₄O₄ cube with one oxygen atom removed. In Mg₄O₃, two 3s electrons are pushed out from the inner area of the molecule forming a diffuse electride multicentered bond. Our calculations show that this electride-like cluster possesses a noticeably large first hyperpolarizability β=5733 au. At the same time, a complete cube Mg₄O₄ and Mg₄O₃²⁺ without electride electron pair have much smaller β: 0 au and 741 au, respectively. This fact indicates the decisive role of the electride electron pair in NLO properties. Additionally, vertical detachment energies of isomers (VDE), excitation energies ΔE, polarizabilities α, and IR spectra were calculated. These properties, including β, are supposed to be observable experimentally and can serve as indirect evidence of the stable molecular electride formation.     

The improvement of numerical modeling in the solution of incompressible viscous flow problems using finite element method based on spherical Hankel shape functions

In this paper, the finite element method with new spherical Hankel shape functions is developed for simulating 2‐dimensional incompressible viscous fluid problems. In order to approximate the hydrodynamic variables, the finite element method based on new shape functions is reformulated. The governing equations are the Navier‐Stokes equations solved by the finite element method with the classic Lagrange and spherical Hankel shape functions. The new shape functions are derived using the first and second kinds of Bessel functions. In addition, these functions have properties such as piecewise continuity. For the enrichment of Hankel radial basis functions, polynomial terms are added to the functional expansion that only employs spherical Hankel radial basis functions in the approximation. In addition, the participation of spherical Bessel function fields has enhanced the robustness and efficiency of the interpolation. To demonstrate the efficiency and accuracy of these shape functions, 4 benchmark tests in fluid mechanics are considered. Then, the present model results are compared with the classic finite element results and available analytical and numerical solutions. The results show that the proposed method, even with less number of elements, is more accurate than the classic finite element method.     

Syntheses and structures of two novel fluorescent metal–organic frameworks generated from a tridentate donor–acceptor motif ligand

The tridentate organic ligand 4,4′,4′′‐(4,4,8,8,12,12‐hexamethyl‐8,12‐dihydro‐4H‐benzo[9,1]quinolizino[3,4,5,6,7‐defg]acridine‐2,6,10‐triyl)tribenzoic acid ( H₃L ) has been synthesized (as the methanol 1.25‐solvate, C₄₈H₃₉NO₆·1.25CH₃OH). As a donor–acceptor motif molecule, H₃L possess strong intramolecular charge transfer (ICT) fluorescence. Through hydrogen bonds, H₃L molecules construct a two‐dimensional (2D) network, which pack together into three‐dimensional (3D) networks with an ABC stacking pattern in the crystalline state. Based on H₃L and M(NO₃)₂ salts (M = Cd and Zn) under solvothermal conditions, two metal–organic frameworks (MOFs), namely, catena‐poly[[triaquacadmium(II)]‐μ‐10‐(4‐carboxyphenyl)‐4,4′‐(4,4,8,8,12,12‐hexamethyl‐8,12‐dihydro‐4H‐benzo[9,1]quinolizino[3,4,5,6,7‐defg]acridine‐2,6‐diyl)dibenzoato], [Cd(C₄₈H₃₇NO₆)(H₂O)₃]_n, I , and poly[[μ₃‐4,4′,4′′‐(4,4,8,8,12,12‐hexamethyl‐8,12‐dihydro‐4H‐benzo[9,1]quinolizino[3,4,5,6,7‐defg]acridine‐2,6,10‐triyl)tribenzoato](μ₃‐hydroxido)zinc(II)], [Zn₂(C₄₈H₃₆NO₆)(OH)]_n, II , were synthesized. Single‐crystal analysis revealed that both MOFs adopt a 3D structure. In I , partly deprotonated HL ^2? behaves as a bidentate ligand to link a Cd^II ion to form a one‐dimensional chain. In the solid state of I , the existence of weak interactions, such as O—H…O hydrogen bonds and π–π interactions, plays an essential role in aligning 2D nets and 3D networks with AB packing patterns for I . The deprotonated ligand L ^3? in II is utilized as a tridentate building block to bind Zn^II ions to construct 3D networks, where unusual Zn₄O₁₄ clusters act as connection nodes. As a donor–acceptor molecule, H₃L exhibits fluorescence with a photoluminescence quantum yield (PLQY) of 70% in the solid state. In comparison, the PL of both MOFs is red‐shifted with even higher PLQYs of 79 and 85% for I and II , respectively.