APPLICATION OF THE DUAL RECIPROCITY BOUNDARY ELEMENT METHOD TO SOLUTION OF NONLINEAR DIFFERENTIAL EQUATION

In this paper the dual reciprocity boundary element method is employed to solve nonlinear differential equation ∇² u+u+ɛu ³ =b. Results obtained in an example have a good agreement with those by FEM and show the applicability and simplicity of dual reciprocity method(DRM)in solving nonlinear differential equations.     

THE PROBLEMS OF THE NONLINEAR UNSYMMETRICAL BENDING FOR CYLINDRICALLY ORTHOTROPIC CIRCULA RPLATE（I）

ＴＨＥＰＲＯＢＬＥＭＳＯＦＴＨＥＮＯＮＬＩＮＥＡＲＵＮＳＹＭＭＥＴＲＩＣＡＬＢＥＮＤＩＮＧＦＯＲＣＹＬＩＮＤＲＩＣＡＬＬＹＯＲＴＨＯＴＲＯＰＩＣＣＩＲＣＵＬＡＲＰＬＡＴＥ（Ｉ）ＱｉｎＳｈｅｎｇ－Ｉｉ（秦圣立）ＨｕａｎｇＪｉａ－ｙｉｎ（黄家寅）（Ｑｕｆ...     

On the existence of the derivative of the volume average

A general theorem on the derivative of the volume average is formulated and proved. Conditions for the existence of the derivative are presented and discussed. This is done in order to give a better base to the theory of spatial averaging.Latin Letters E ₃ three-dimensional vector space over the field of real numbers - K, K(x) averaging domain - G, G _w, G_s open sets in E ₃; components of the two-phase system - C ¹(G) the set of functions 1-times continuously differentiable in G - W1/2(G) Sobolev space - V volume of the domain K - f function defined in G, G _w - K _infi ^sup* (x), K _infi ^sup– (x) special parts of K(x) Greek Letters boundary of G, G _w, G_s; w-s interface - _ij Kronecker delta - v unit outward normal of G, G _w - _j j-dimensional Lebesgue measure Other M closure of a set M in the metric space E ₃ - f phase average of f for the w-phase - (u, v) scalar product of u, v in E ₃ - one-sided derivatives     

The Laplace transform method for Burgers' equation

The Laplace transform method (LTM) is introduced to solve Burgers' equation. Because of the nonlinear term in Burgers' equation, one cannot directly apply the LTM. Increment linearization technique is introduced to deal with the situation. This is a key idea in this paper. The increment linearization technique is the following: In time level t, we divide the solution u(x, t) into two parts: u(x, t^k) and w(x, t), t^k⩽t⩽t^k+1, and obtain a time‐dependent linear partial differential equation (PDE) for w(x, t). For this PDE, the LTM is applied to eliminate time dependency. The subsequent boundary value problem is solved by rational collocation method on transformed Chebyshev points. To face the well‐known computational challenge represented by the numerical inversion of the Laplace transform, Talbot's method is applied, consisting of numerically integrating the Bromwich integral on a special contour by means of trapezoidal or midpoint rules. Numerical experiments illustrate that the present method is effective and competitive. Copyright © 2009 John Wiley & Sons, Ltd.     

High accuracy time and space transform method for advection–diffusion equation in an unbounded domain

A new numerical method called high accuracy time and space transform method (TSTM) is introduced to solve the advection–diffusion equation in an unbounded domain. By a spatial transform, the advection–diffusion equation in the unbounded domain Rⁿ is converted to one on the bounded domain [?1, 1]ⁿ, and the Laplace transform is applied to eliminate time dependency. The consequent boundary value problem is solved by collocation on Chebyshev points. To face the well‐known computational challenge represented by the numerical inversion of the Laplace transform, Talbot's method is applied, consisting of numerically integrating the Bromwich integral on a special contour by means of trapezoidal or midpoint rules. Numerical experiments illustrate that TSTM has exponential rate in time and space. Copyright © 2008 John Wiley & Sons, Ltd.     

Shocks in local supersonic zones

The results of numerical integration of the Euler equations governing two-dimensional and axisymmetric flows of an ideal (inviscid and non-heat-conducting) gas with local supersonic zones are presented. The subject of the study is the formation of shocks closing local supersonic zones. The flow in the vicinity of the initial point of the closing shock is calculated on embedded, successively refined grids with an accuracy much greater than that previously achieved. The calculations performed, together with the analysis of certain controversial issues, leave no doubt that it is the intersection of C ^?-characteristics proceeding from the sonic line inside the supersonic zone that is responsible for the closing shock formation.     

The estimation of solution of the boundary value problem of the systems for quasi-linear ordinary differential equations

This paper deals.with the singular perturbation of the boundary value problem of the systems for quasi-linear ordinary differential equations where x,f, y, h, A, B and C all belong to Rm, and g is an n×n matrix function. Under suitable conditions we prove the existence of the solutions by diagonalization and the fixed point theorem and also estimate the remainder.     

Numerical method in dynamic response of nonlinear systems

A kind of modal synthesis techniques, which is applicable to vibration analysis for linear substructures with nonlinear coupling attachments, has been extended to nonlinear dynamic analysis of large complex structural systems^[9]. In this paper, a process is suggested to dynamic analysis of large complex structural systems with nonlinear characteristics of each substructure. At the end of this paper, an example shows the defendable accuracy of the results and high efficiency of this process.This paper was collected in Proceedings of Symposium on Nonlinear Mechanics, Vol. IV, 1982, Wuxi, China (in Chinese).     

Asymptotics of stresses and strain rates near the tip of a transverse shear crack in a material whose behavior is described by a fractional-linear law

An approximate solution of the problem of determining the fields of stresses and strain rates due to creep near the tip of a transverse shear crack in a material whose behavior is described by a fractional-linear law of the theory of steady-state creep is given. It is shown that the strain rates have a singularity of the type ∼ r^−α near the crack tip; the order of singularity α changes discretely, depending on the polar angle, and takes the values 1, 2/3, and 1/2. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 1, pp. 165–176, January–February, 2009.     

A Numerical Study Promoting Algebraic Models for the Lewis Number Effect in Atmospheric Turbulent Premixed Bunsen Flames

This numerical investigation carried out on turbulent lean premixed flames accounts for two algebraic – the Lindstedt–Vaos (LV) and the classic Bray–Moss–Libby (BML) – reaction rate models. Computed data from these two models is compared with the experimental data of Kobayashi et al. on 40 different methane, ethylene and propane Bunsen flames at 1 bar, where the mean flame cone angle is used for comparison. Both models gave reasonable qualitative trend for the whole set of data, in overall. In order to characterize quantitatively, firstly, corrections are made by tuning the model parameters fitting to the experimental methane–air (of Le = 1.0) flame data. In case of the LV model, results obtained by adjusting the pre-constant, i.e., reaction rate parameter, C_R, from its original value 2.6 to 4.0, has proven to be in good agreement with the experiments. Similarly, for the BML model, with the tuning of the exponent n, in the wrinkling length scale, L_y = C_l⋅ l_x(s_L/u′)ⁿ from value unity to 1.2, the outcome is in accordance with the measured data. The deviation between the measured and calculated data sharply rises from methane to propane, i.e., with increasing Lewis number. It is deduced from the trends that the effect of Lewis number (for ethylene–air mixtures of Le = 1.2 and propane–air mixtures of Le = 1.62) is missing in both the models. The Lewis number of the fuel–air mixture is related to the laminar flame instabilities. Second, in order to quantify for its influence, the Lewis number effect is induced into both the models. It is found that by setting global reaction rate inversely proportional to the Lewis number in both the cases leads to a much better numerical prediction to this set of experimental flame data. Thus, by imparting an important phenomenon (the Lewis number effect) into the reaction rates, the generality of the two models is enhanced. However, functionality of the two models differs in predicting flame brush thickness, giving scope for further analysis.     

K+ uptake by root systems grown in soil under salinity: I. A mathematical model

A novel theoretical model is proposed for K⁺ uptake by intact root systems from saline soil considering interactions with Na⁺, Ca²⁺ and Mg²⁺. The model assumes radial movement of ions towards the root governed by advection and diffusion flux mechanisms, and chemical exchange of the four cations according to Gapon isotherms, with Cl^? as the accompanying anion. Influx of K⁺ to the root surface is assumed as a function of its concentration in the soil solution at the root. This influx is governed by a saturable-cooperative term and a linear term for low and high K⁺ concentrations, respectively. Influx of Na⁺, above a critical value of its concentration, increases linearly with its concentration in the soil solution at the root surface. Uptake of Ca^{+ 2+} is controlled by the balance between influxes of anions and cations, which induces efflux of H⁺ or HCO ₃ ^? , and interacts with calcite in a calcareous soil. The model may provide information about the behavior of ions at the root-soil interface which cannot be measuredin situ.     

超强钢18NiC250在不同加载速率下的断裂韧性

为了揭示国产超强钢18NiC250的强度、断裂韧性随加载速率的变化规律,利用电子万能试验机和Hopkinson压杆,测试其在0.001～2 000 s-1的塑性流动应力应变曲线及在10-1~106 MPam1/2/s的断裂韧性,同时对断裂破坏机理进行了微观分析。结果表明:该材料的强度对加载速率不敏感,即流动应力基本保持在1.9 GPa;而断裂韧性很敏感,当加载速率由10-1 MPam1/2/s增大到106 MPam1/2/s时,断裂韧性降低了38.2%,断裂模式由韧窝断裂转变为解理断裂。     

Determination of dynamic yield strengths with embedded manganin gages in plate-impact and long-rod experiments

The dynamic yield strengths of three steels were determined at strain rates of about 10³ s⁻¹ and 10⁶ s⁻¹. The measurements at 10³ s⁻¹ were obtained by a new technique based on measurements of large amplitude elastic waves in long bars struck by rigid flyer plates. Embedded manganin gages were used to measure stress, and the gage records were long enough to observe subsequent reverberations between the bar free end and the plastically deformed impact end. The measurements at 10⁶ s⁻¹ were made with a slightly modified version of a conventional flyer-plate impact configuration. The data are combined with static results to show the behavior of these steels at strain rates of 10⁻³ s⁻¹ to 10⁶ s⁻¹.     

Precursor vacuum-ultraviolet radiation ahead of strong shock waves in a shock tube

The profile and excitation mechanism of vacuum-ultraviolet radiation emitted from shock wave is investigated in a shock tube. For shock wave in argon, the rdiation is due to resonant transition excited by argon-argon collision in the shock front with excitation cross section coefficientS ^*=1.0×10⁻¹⁷ cm²·ev⁻¹ and activation energyE ^*=11.4 ev. For shock wave in air the radition is emitted from a very thin shock layer in which the mechanism ofX ¹∑→b ¹∑ of N₂ is excited with excitation cross sectionQ=2×10⁻¹⁶cm² and activation energyE ^*=12.1 ev. Institute of Mechanics, Academia Sinica     

Chemical characteristics and Pb isotopic compositions of PM2.5 in Nanchang,China

In mid-September 2013, PM_2.5 samples were collected at six sites in Nanchang, Jiangxi Province, China, to quantify nine water-soluble ions (Ca²⁺, Mg²⁺, K⁺, Na⁺, NH₄⁺, SO₄²⁻, Cl⁻, F⁻, NO₃⁻), 29 trace elements (Ba, Zn, Pb, Ni, Mo, Cr, Cu, Sr, Sb, Rb, Cd, Bi, Zr, V, Ga, Li, Y, Nb, W, Cs, Tl, Sc, Co, U, Hf, In, Re, Be, and Ta), and to characterize Pb isotopic ratios (²⁰⁷Pb/²⁰⁶Pb, ²⁰⁸Pb/²⁰⁶Pb, and ²⁰⁷Pb/²⁰⁴Pb) for identifying the main source(s) of Pb. The results showed that the average daily PM_2.5 concentration (53.16 ± 24.17) μg/m³ was within the secondary level of the Chinese ambient air quality standard. The combined concentrations of SO₄²⁻, NH₄⁺, and NO₃⁻ to total measured water-soluble ion concentrations in PM_2.5 ranged from 79.40% to 95.18%, indicating that anthropogenic sources were significant. Coal combustion and vehicle emissions were both contributors to PM_2.5 based on the NO₃⁻/SO₄²⁻ ratios. Wushu School experienced the lowest concentrations of PM_2.5 and most trace elements among the six sampling sites. Enrichment factor results showed that Tl, Cr, In, Cu, Zn, Pb, Bi, Ni, Sb, and Cd in PM_2.5 were affected by anthropogenic activities. Cluster analysis suggested that Cd, Sb, Pb, Re, Zn, Bi, Cs, Tl, Ga, and In were possibly related to coal combustion and vehicle exhaust, while Ni, Nb, Cr, and Mo may have originated from metal smelting. Pb isotopic tracing showed that coal dust, cement dust, road dust and construction dust were the major Pb sources in PM_2.5 in Nanchang. Combined, these sources contributed an average of 72.51% of the Pb measured, while vehicle exhaust accounted for 27.49% of Pb based on results from a binary Pb isotope mixed model.     

The mechanisms governing the activation of dislocation sources in aluminum at different strain rates

This article examines the time to activate Frank–Read sources in response to macroscopic strain rates ranging from 10¹ s⁻¹ to 10¹⁰ s⁻¹ in aluminium under athermal conditions. We develop analytical models of the bowing of a pinned dislocation segment as well as numerical simulations of three dimensional dislocation dynamics. We find that the strain rate has a direct influence on both the activation time and the source strength of Frank–Read sources at strain rates up to 10⁶ s⁻¹, and the source strength increases in almost direct proportion to the strain rate. This contributes to the increase in the yield stress of materials at these strain rates. Above 10⁶ s⁻¹, the speed of the bowing segments reaches values that exceed the domain of validity of the linear viscous drag law, and the drag law is modified to account for inertial effects on the motion of the dislocation. As a result the activation times of Frank–Read sources reach a finite limit at strain rates greater than 10⁸ s⁻¹, suggesting that Frank–Read sources are unable to operate before homogeneous nucleation relaxes elastic stresses at the higher strain rates of shock loading. Elastodynamic calculations are carried out to compare the contributions of Frank–Read sources and homogeneous nucleation of dislocations to plastic relaxation. We find that at strain rates of 5×10⁷ s⁻¹ homogeneous nucleation becomes the dominant generation mechanism.     

Precursor ionization ahead of strong shock waves in argon

The mechanism of precursor ionization ahead of strong shock waves has been studied in a low density shock tube. The experimental results are illustrated with Arrhenius plots with kink points dividing them into two parts with apparent activation energy ratio 1:2, namely with the values 7.7 eV and 15.3 eV, and varying with first and third power of the density respectively. A model is proposed to interpret the facts where the process taking place in the precursor region, is a two step photo ionization accompanied with the drift flow effect of the gas relative to the shock wave or the ionization recombination effect according to whether the shock speed and initial density are low enough. The product of the A-A collision excitation cross section coefficientS ^* multiplied by the radiation cross sectionQ ^* of ArgonS ^*×Q ^*=1×10⁻³⁶ (cm⁴eV⁻¹) and the three body recombination coefficient of Argon at room temperaturek _ra =1×10⁻²⁴ (cm⁻⁶s⁻¹). The project supported by the National Natural Science Foundation of China     

Phase evolution of （K,Na）NbO3 powder prepared by high temperature mixing under hydrothermal conditions

（Na, K）NbO3 （KNN） powders were successfully prepared by high temperature mixing method （HTMM） under hydrothermal conditions to study the effect of reaction time on the formation of KNN for three K＋/（K＋ ＋Na＋） ratios of 0.6, 0.7 and 0.8. The products were characterized by X-ray diffraction （XRD）, scanning electron microscope （SEM）, transmission electron microscopy （TEM） and selected area electron diffraction （SAED）, to show the change of phase and morphology of the as-prepared particles with the K＋/（K＋ ＋ Na＋） molar ratio in the solution. Pure Na-rich KNN monoclinic phase and pure K-rich KNN orthorhombic phase could be obtained quickly after mixing the solutions at high temperature when the K＋/（K＋ ＋Na＋） molar ratio was either 0.6 or 0.8. When the K＋/（K＋ ＋Na＋） molar ratio was 0.7, however, the K-rich KNN orthorhombic phase grain formed first, followed by the Na-rich KNN monoclinic phase grain, with the two phases coexisting in the final product.     

Certain basic issues in finite-deformation continuum plasticity

Summary I consider the kinematical implications embedded in the decomposition of the deformation gradient F into two constituents, F^* (say, elastic) and F^p (say, plastic), in the form F=F^* F^p. Whatever the origin, nature, and physical meaning of these three deformation gradients, the decomposition can always be recast in the form F==V^* Q U^p, where V^* and U^p are stretch tensors, respectively associated with F^* and F^p, and Q is a unique rotation tensor. It then follows that the (elastic) spin associated with V^* and the (plastic) spin associated with U^p are fully defined by their corresponding deformation rates. I give explicit expressions defining elastic and plastic spins in terms of the elastic and plastic deformation rates, respectively. I show that, if the total velocity gradient and either the elastic or the plastic (but not both) deformation rate tensor, are given, then all the remaining kinematical rate quantities can be calculated explicitly, with no constitutive or other assumptions. I give explicit expressions for the rate of change of V^*, Q, and U^p, from which these quantities can be incrementally but directly obtained. In particular, I give explicit expressions for the elastic spin, plastic spin, and the rigid spin. Constitutive implications of these purely kinematical results are briefly discussed in connection with rate-independent and rate-dependent phenomenological elastoplasticity, and, particularly, kinematical hardening.

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Sommario Si considerano le implicazioni cinematiche insite nella decomposizione del gradiente di deformazione F in due componenti, F^* (per esempio, la parte elastica) ed P^p (per esempio, la parte plastica), nella forma F=F^*F^p. Qualunque sia l'origine, la natura e il significato fisico di questi tre gradienti, la decomposizione si può sempre riscrivere nella forma F=U^* Q U^p, in cui V^* ed U^p sono tensori di deformazione associati rispettivamente con F^* ed F^p, e Q è un tensore di rotazione univocamente definito. Ne segue che lo spin (elastico) associato con U^* e lo spin (plastico) associato con U^p sono completamente definiti dalle corrispondenti derivate delle deformazioni: di questa dipendenza si forniscono le espressioni esplicite. Si mostra quindi che, se sono dati il gradiente della velocità totale e il tensore delle deformazioni elastiche o quello delle deformazioni plastiche (ma non entrambi), tutte le altre quantità cinematiche possono essere calcolate esplicitamente, senza alcuna ulteriore ipotesi, neanche sulla legge costitutiva. Si forniscono ancora espressioni per i gradienti di V^*, Q ed U^p, dalle quali queste quantità si possono ottenere direttamente, per incrementi successivi: in particolare, si danno espressioni esplicite per lo spin elastico, plastico e rigido. Si discutono brevemente le implicazioni costitutive di questi risultati puramente cinematici, in connessione con vari aspetti fenomenologici dell'elasto-plasticità, e in particolare l'incrudimento cinematico.

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Invited Lecture presented at New Developments in Structural Mechanichs, International Meeting in memory of Professor Manfredi Romano; Catania, Italy, 4–6 July 1990.     

ON THE CRITICAL POINTS OF THE MAP Fp:X→‖AXB-C‖pp

Suppose A,B and C are the bounded linear operators on a Hilbert space H, when A has a generalized inverse A^- such that (AA^-)^*=AA^- and B has a generalized inverse B^- such that (B^-B)^*=B^-B,the general characteristic forms for the critical points of the map F_p:X^→‖AXB-C‖_p^p(1p=2. Similarly, the same question has been discussed for several operators.