#### Table of Content

25 September 2008, Volume 40 Issue 5
Research paper
Numerical investigation into the mechanism of under-expanded supersonic jet instability
2008, 40(5):  577-584.  DOI: 10.6052/0459-1879-2008-5-2006-651
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The numerical investigation into the instability of the under-expanded supersonic jets is presented in this paper by solving the axisymmetric N-S equations with the NND finite difference scheme. Numerical results demonstrated that there are three kinds of the instability for the under-expanded supersonic jets according to their shock cell structure patterns: (1) the regular reflection shock cell with a single shear layer, (2) the Mach reflection shock cell with two shear layers and (3) the curved Mach stem with a relatively high expansion ratio. Moreover, for the under-expanded supersonic jets, the repetitively-generated quasi-steady shock cells in the jets indicate their stable development and the disappearance of these shock cells implies that the jets are about to fall into instable states.
One dimensional shallow water equations with porosity and their numerical discretization scheme
Zhili Wang Yongjun Lu Yanfen Geng
2008, 40(5):  585-592.  DOI: 10.6052/0459-1879-2008-5-2007-445
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In this paper, the introduction of porosity into the one-dimensional shallow water equations is investigated to lead to modified expressions for the fluxes and source terms subjected to the effects of buildings and other structures. An extra porosity source term appears in the momentum equation. The numerical discretization scheme for shallow water equations with porosity is presented using finite volume method and Roe-type approximate Riemann solver. The source terms of the bed slope and the porosity are both decomposed in the characteristic direction to make the scheme exactly satisfy the conservative property. Several cases have been brought, such as the dam break with variable porosity, the steady flow through continue porosity and the damming heights in different pier width, to show that the scheme can be used to simulate the influence of buildings and other structures in the channel with harmonious, robust and high-resolution characteristics.
Gust response analysis using cfd-based reduced order models
2008, 40(5):  593-598.  DOI: 10.6052/0459-1879-2008-5-2007-357
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Gust response analysis plays an important role in large aircraft design. Most of existing aerodynamic models for gust response analysis are linear, not suitable for transonic nonlinear aeroelastic analysis. In this paper, based on CFD technique and system identification method, a CFD-based ROM (reduced order model) is firstly constructed in state space. Gust response analysis is then investigated by coupling structural equations with CFD-based ROM. The case of a square-edged gust response for a typical wing is presented to obtain the responses of generalized coordinates and root bending moment (RBM) and to validate the correctness and precision of the CFD-based ROM method, compared well with the direct approach of a coupled CFD/CSD methodology
Numerical simulation and experimental investigation on steady flow foam in porous media
PANG Zhan-Xi
2008, 40(5):  599-604.  DOI: 10.6052/0459-1879-2008-5-2007-141
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Porous media is firstly simplified as a bunch of constricted capillary tubes related to grain diameter, particle arrangement, pore-throat scaling ratio and saturation of immobile water. Based on the deformation characteristics of a single bubble in throat and pore of porous media, pressure distribution of fluid phase is derived from the conservation of momentum in a single pore unit, to obtain the equation for pressure distribution in porous media. Moreover, the experiments on flow foam in U-shape sandpack are investigated. The results demonstrate the linear decreasing pressure distribution of steady foams in porous media and the influencing factors, i.e., pore-throat structure, flowing rate and quality of foams, interfacial tension of gas and liquid, bubble size in pore media. Furthermore, the results show further that pore-throat structure and foam quality have the most influence on the blocking ability of foams in porous media.
Numerical study of ionized hypersonic air flow with radiation
2008, 40(5):  605-610.  DOI: 10.6052/0459-1879-2008-5-2007-057
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A numerical method for an ionized hypersonic air flow with radiation under thermo-chemical nonequilibrium condition has been developed with multi-species N-S equations based on unstructured mesh. In the algorithm, there are 11 chemical species (O$_{2}$, N$_{2}$, O, N, NO, NO$^{+}$, N$^+$, O$^+$, N$_2^+$, O$_2^+$, e$^{ - }$), 20 chemical reactions, and two temperature models subjected to thermal nonequilibrium effect. Radiation transfer equation (RTE) was solved by finite volume method. In particular, the Delta, Epsilon, Beta prime and Gamma prime bands of NO were introduced in this paper. The numerical results of MUSES-C were shown and compared well with the reference data. The influence of radiation was also discussed.
Inverse estimation of heat source term in three-dimensional transient heat conduction problem
Weiqi Qian
2008, 40(5):  611-618.  DOI: 10.6052/0459-1879-2008-5-2007-256
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Inverse estimation of heat source term in three-dimensional heat conduction problem from surface temperature measurements is a typical Inverse Heat Conduction Problem (IHCP). In this paper, based on the three-dimensional (3D) numerical simulation of transient heat conduction problem with the Finite Volume Method (FVM), the 3D transient IHCP is generally converted into an optimization problem to inversely estimate the heat source with Adjoint Equation Method (AEM) and Conjugate Gradient Method (CGM). A typical case has been shown that the two estimation methods are feasible and not sensitive to the measurement noise. Moreover, the selection of stopping criterion for the inverse estimation algorithm is further investigated. The results demonstrate that the closer agreement between the estimated and exact heat sources hasn't made with the decreasing value of the objective function due to the ill-posedness of the IHCP, which can be overcome with the proper stopping criterion to simulate the regularization effect in the IHCP.
A holistic thermomechanical model of skin thermal pain
2008, 40(5):  619-628.  DOI: 10.6052/0459-1879-2008-5-2007-240
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Skin thermal pain is one of the most common problems for human in everyday life and in thermal therapies. However, its underlying physical mechanisms are not clearly understood, and there are few attempts to model it. In this paper, a holistic mathematical model for quantifying skin thermal pain sensation is developed. The model is composed of three interconnected parts: peripheral modulation of noxious stimuli, which converts the energy from a noxious thermal stimulus into electrical energy via nerve impulses; transmission, which transports these neural signals from the site of transduction in the skin to the spinal cord and brain; and modulation and perception in the spinal cord and brain. With the holistic model, a direct relationship is built between the level of thermal pain sensation and the character of noxious stimuli.
Invariant tensor function representations of constitutive equations for the isotropic and rate independent materials
2008, 40(5):  629-635.  DOI: 10.6052/0459-1879-2008-5-2007-155
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This paper combines the internal variable theory and the tensor function representation theory to establish the constitutive equations of the deformation theory and the increment theory for the isotropic and rate independent materials. In the equations, there are three complete and irreducible base tensors, that is, the stress tensor of the zero order, the first order and the second order power, to show that the principal axes of plastic strain and its increment are coincident with those of the stresses. With the orthogonalization of the base tensors, the geometrical explanation of the constitutive equations is obtained in the principal stress space. Furthermore, the coefficients in the constitutive equations of deformation theory (or increment theory) can be derived with three invariants of stresses and plastic strain ( or plastic strain increments). Meanwhile, the present constitutive equations may reduce to classical deformation theory (or plastic potential theory), and be consistent to the singular yield surface theory.
Simplified theory and analytical solutions for functionally graded piezoelectric circular plate
Yaochen Li Feng Qi Zheng Zhong
2008, 40(5):  636-645.  DOI: 10.6052/0459-1879-2008-5-2007-145
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Simplified theory and analytical solutions for a functionally graded piezoelectric circular plate subjected to axisymmetric loading was presented with some assumptions, such as Kirchhoff assumption, Reissner-Mindlin assumption, and the assumption that material properties vary exponentially across the thickness of the plate. We derived the solution of the rotation angle of the normal line of the neutral layer and the solution of the electric potential in the plate expressed by Fourier-Bessel series, in the cases that the edges of the plate was fixed or simply supported and grounded. The presented solutions were obviously simple compared with some exact analytical solutions and easy to perform the numerical analysis. The numerical results for the edge-fixed/grounded plate were applied to validate the simplified theory presented in the paper.
Topology optimization of continuum structures with materials exhibiting different tensile and compressive properties
Hongwu Zhang
2008, 40(5):  646-653.  DOI: 10.6052/0459-1879-2008-5-2007-455
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The effects of the difference between the tensile and compressive properties (TCPs) of materials on the final topologies of structures are investigated with floating reference intervals in the optimization case of a bone growing process based on Wolff's law. There are two aspects in the difference on TCPs of basal materials to have equivalent effects on the structural topology optimization: one is the difference between their tensile and compressive Young's moduli, , the other is the variation of the intervals of their references strains in bone growing process under the tensile or compressive state . The numerical cases verify this equivalence and further reveal a more reasonable structure topology subjected to the differences on TCPs of the materials.
Reliability sensitivity analysis based on subset simulation and importance sampling
Shufang Song Zhenzhou Lu
2008, 40(5):  654-662.  DOI: 10.6052/0459-1879-2008-5-2007-274
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Reliability sensitivity algorithm is presented on the basis of subset simulation and importance sampling due to the small failure probability highly experienced in engineering. Firstly, a small failure probability is expressed as a product of larger conditional failure probabilities of some intermediate failure events. Secondly, the larger conditional failure probabilities can be estimated efficiently by constructing the importance sampling density functions of the intermediate failure events. Thirdly, the reliability sensitivity is transformed into the partial derivatives of conditional failure probabilities with respect to the distribution parameters of the basic variables in the paper. The estimation of the reliability sensitivity and its variance are then derived for the presented algorithm. The results from several cases show that the present method is efficient, precise and applicable to the structural system with single and multiple failure modes.
Wave components of acoustic logs in transversely isotropic porous formations
He Xiao Hengshan HU
2008, 40(5):  663-671.  DOI: 10.6052/0459-1879-2008-5-2007-311
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The acoutic field excited by a monopole point-source was numerically investigated with a model of a transversely isotropic fluid-saturated poroelastic formation. The wavefields inside and outside the borehole are formulated in the frequency-wavenumber domain. All the poles of the acoustic function are obtained to correspond to several modes of guide waves: the normal modes (including Stoneley wave and pseudo-Rayleigh wave) and two leaky modes. The numerical results showed that the contribution of the lowest order leaky mode can not be ignored since its imaginary part has the same magnitude order as that of the normal mode. Moreover, the velocity of pseudo-Rayleigh wave at the low-frequency limit is not always equal to the shear wave velocity in transversely isotropic formations. Meanwhile, the excitation spectra and waveforms in the time-space domain are acquired for Stoneley wave, pseudo-Rayleigh wave and two-leaky-modes wave, contributed by the poles and two branchcuts (the quasi-shear and the quasi-fast- compressional branchcuts). The results showed that the contribution of leaky-mode poles can not be omitted in the full waveforms.
A coupled elasto-plastic microplane damage model for jointed rock masses at great depth
2008, 40(5):  672-683.  DOI: 10.6052/0459-1879-2008-5-2007-089
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Rockwall excavated at great depth often experiences large ductile deformation and may be failure due to high compressive in-situ stresses. Such phenomena are closely related to anisotropic distribution of joints in rock masses and mechanical response of the rock block and the joint. To take into account the shear resistance of joint surface and anisotropic inelastic deformation related to secondary joints formation, a coupled elasto-plastic microplane damage model for jointed rock masses is developed in the framework of microplane model and damage mechanics. Each microplane of a jointed rock mass is regarded as a binary medium, which is composed of rock and joint. Joint connectivity on the microplane is introduced as the damage variable to characterize area damage of the rock mass on that orientation due to presence of the joints. By adopting different yield function and damage evolution law for the microplane under tensile and compressive normal stress respectively, the coupled mechanisms of inelastic deformation and damage evolution are modeled. Rate form constitutive relation of stresses and strains on microplanes is developed based on classical plastic theory. According to the kinematic constraint condition of microplane model, macroscopic constitutive relation is obtained through directional integration along all microplanes. The present elasto-plastic microplane damage model is implemented to the commercial FEM software MARC through a user subroutine. Simulations with respect to the uniaxial tension and compression of jointed rock masses and problem of wellbore stability under inner mud pressure are presented respectively. It has demonstrated that the coupled effect between anisotropic inelastic deformation and damage evolution of jointed rock masses can be characterized efficiently with the model.
Recursive method for modeling flexible multibody systems with non-ideal constraints
Qi Zhaohui
2008, 40(5):  684-694.  DOI: 10.6052/0459-1879-2008-5-2007-577
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With the recursive kinematics relationship between the two bodies interconnected with a joint, it is showed that the interaction between the outermost body and the inboard body in a tree-type multibody system can be measured by the transmission of the inertia mass and forces. The non-ideal constraint forces and the constraint forces at the cut-joints in a closed-loop multibody system can be viewed as external applied forces, therefore, any mutibody system corresponds to an equivalent tree-type system. Moreover, the general accelerations of the system appearing in the constraint equations can be systematically replaced by the Lagrange multipliers which are related to the constraint forces. Based on the above ideas, a recursive method is presented, by which, any closed-loop flexible multibody system with non-ideal constraints can be dynamically reduced to an equivalent system with only one body in a recursive manner. Consequently, the inverse of the mass matrix of multibody systems is not involved in the calculations of the general accelerations, and the number of equations to solve the non-ideal constraint forces can be reduced to minimum. An example is analyzed to illustrate and validate the presented method in details.
Brief Report
Dynamic propagation of asymmetrical mode III interface crack
2008, 40(5):  695-700.  DOI: 10.6052/0459-1879-2008-5-2007-104
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Dynamic propagation of asymmetrical mode III interface crack were investigated with the theory of complex functions. It can be translated into a Riemann-Hilbert problem by means of self-similar functions, and the universal representations of analytical solution were obtained when there is $Pt / x$ or $Px^3 / t^2$ load at the origin of the crack coordinates. The closed-form solutions can also be simply obtained with Muskhelishvili's theory.
Fracture analysis of pzt-4 compact tension specimen
Haijun Li Feng Liu T.C. Wang
2008, 40(5):  701-706.  DOI: 10.6052/0459-1879-2008-5-2007-625
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Based on a complex potential theory of linear piezoelectric materials and series expansion of the complex potential, a general method is proposed for solving the crack problem of a finite piezoelectric plate. By this method, fracture parameters such as the stress intensity factor, the electric displacement intensity factor and energy release rate et al. can be easily evaluated under the impermeable boundary condition. Numerical results of fracture parameters demonstrate that the proposed method gives satisfactory results compared with the existing solutions given by Woo and Wang for a central crack in a finite piezoelectric plate. This method is identified to be an accurate and efficient method. An experiment with a PZT-4 compact tension specimen has been modeled under the impermeable boundary condition by the present method, and the critical values of fracture parameters are greatly varied and not the material constants. Therefore, these fracture parameters are not used as the single-parameter fracture criterion for the piezoelectric material. Then, in terms of the PZT-4 compact tension specimen, the influence of permeability of the crack has been studied based on the results of finite element calculations. The finite element model of the PZT-4 compact tension specimen with an actual crack profile is analyzed for two different electric permeability of the medium in the crack cavity corresponding to an impermeable medium and silicone oil. The largest applied negative electric field and the corresponding mechanical load for fracture are considered. Finally, the crack profile in the PZT-4 compact tension specimen can be regarded to be composed of a notch and an ideal crack (so-called microcrack) in front of the notch. The model of this assumed crack profile has been analyzed by the finite element method.