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Table of Content
25 September 2008, Volume 40 Issue 5
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Research paper
Numerical investigation into the mechanism of underexpanded supersonic jet instability
2008, 40(5): 577584. DOI:
10.6052/04591879200852006651
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The numerical investigation into the instability of the underexpanded supersonic jets is presented in this paper by solving the axisymmetric NS equations with the NND finite difference scheme. Numerical results demonstrated that there are three kinds of the instability for the underexpanded 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 underexpanded supersonic jets, the repetitivelygenerated quasisteady 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): 585592. DOI:
10.6052/04591879200852007445
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In this paper, the introduction of porosity into the onedimensional 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 Roetype 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 highresolution characteristics.
Gust response analysis using cfdbased reduced order models
2008, 40(5): 593598. DOI:
10.6052/04591879200852007357
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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 CFDbased ROM (reduced order model) is firstly constructed in state space. Gust response analysis is then investigated by coupling structural equations with CFDbased ROM. The case of a squareedged 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 CFDbased 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 ZhanXi
2008, 40(5): 599604. DOI:
10.6052/04591879200852007141
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Porous media is firstly simplified as a bunch of constricted capillary tubes related to grain diameter, particle arrangement, porethroat 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 Ushape sandpack are investigated. The results demonstrate the linear decreasing pressure distribution of steady foams in porous media and the influencing factors, i.e., porethroat structure, flowing rate and quality of foams, interfacial tension of gas and liquid, bubble size in pore media. Furthermore, the results show further that porethroat 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): 605610. DOI:
10.6052/04591879200852007057
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A numerical method for an ionized hypersonic air flow with radiation under thermochemical nonequilibrium condition has been developed with multispecies NS 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 MUSESC were shown and compared well with the reference data. The influence of radiation was also discussed.
Inverse estimation of heat source term in threedimensional transient heat conduction problem
Weiqi Qian
2008, 40(5): 611618. DOI:
10.6052/04591879200852007256
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Inverse estimation of heat source term in threedimensional heat conduction problem from surface temperature measurements is a typical Inverse Heat Conduction Problem (IHCP). In this paper, based on the threedimensional (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 illposedness 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): 619628. DOI:
10.6052/04591879200852007240
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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): 629635. DOI:
10.6052/04591879200852007155
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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): 636645. DOI:
10.6052/04591879200852007145
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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, ReissnerMindlin 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 FourierBessel 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 edgefixed/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): 646653. DOI:
10.6052/04591879200852007455
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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): 654662. DOI:
10.6052/04591879200852007274
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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): 663671. DOI:
10.6052/04591879200852007311
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The acoutic field excited by a monopole pointsource was numerically investigated with a model of a transversely isotropic fluidsaturated poroelastic formation. The wavefields inside and outside the borehole are formulated in the frequencywavenumber 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 pseudoRayleigh 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 pseudoRayleigh wave at the lowfrequency limit is not always equal to the shear wave velocity in transversely isotropic formations. Meanwhile, the excitation spectra and waveforms in the timespace domain are acquired for Stoneley wave, pseudoRayleigh wave and twoleakymodes wave, contributed by the poles and two branchcuts (the quasishear and the quasifast compressional branchcuts). The results showed that the contribution of leakymode poles can not be omitted in the full waveforms.
A coupled elastoplastic microplane damage model for jointed rock masses at great depth
2008, 40(5): 672683. DOI:
10.6052/04591879200852007089
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Rockwall excavated at great depth often experiences large ductile deformation and may be failure due to high compressive insitu 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 elastoplastic 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 elastoplastic 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 nonideal constraints
Qi Zhaohui
2008, 40(5): 684694. DOI:
10.6052/04591879200852007577
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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 treetype multibody system can be measured by the transmission of the inertia mass and forces. The nonideal constraint forces and the constraint forces at the cutjoints in a closedloop multibody system can be viewed as external applied forces, therefore, any mutibody system corresponds to an equivalent treetype 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 closedloop flexible multibody system with nonideal 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 nonideal 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): 695700. DOI:
10.6052/04591879200852007104
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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 RiemannHilbert problem by means of selfsimilar 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 closedform solutions can also be simply obtained with Muskhelishvili's theory.
Fracture analysis of pzt4 compact tension specimen
Haijun Li Feng Liu T.C. Wang
2008, 40(5): 701706. DOI:
10.6052/04591879200852007625
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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 PZT4 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 singleparameter fracture criterion for the piezoelectric material. Then, in terms of the PZT4 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 PZT4 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 PZT4 compact tension specimen can be regarded to be composed of a notch and an ideal crack (socalled microcrack) in front of the notch. The model of this assumed crack profile has been analyzed by the finite element method.
Damage failure analysis for composite bolted joints subjected to static loading
Danyong Wang
2008, 40(5): 707715. DOI:
10.6052/04591879200852007207
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A progressive damage method for predicting the static strength and damage failure of mechanically fastened joints in fibrereinforced plastics is established in the paper based on the classical laminate theory, the threedimensional Hashintype failure criteria , the failure criteria of the structure and material property degradation rules related to the correlation among four basic damage mechanisms. Damage accumulation in composite bolted joints was investigated and compared with the Xradiographs of fractured specimens under multistep loading. Numerical results have been demonstrated that the method established in the paper is capable of predicting the ultimate tensile strength and the failure mode of joints and can be applied to simulate the whole process of failure initiation, propagation and catastrophic failure of the structure. We find an excellent agreement between the results from the present scheme and the experimental data.
Frequency sensitivity analysis for dynamic structural systems reliability
2008, 40(5): 716720. DOI:
10.6052/04591879200852008060
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A method of frequency reliability sensitivity for dynamic structural systems is presented in the paper based on the frequency reliability theory. With the criterion that the difference in value between the natural frequency and driving frequency is controlled, the reliability mode and the safety probability of the random systems are then defined as series mode to perform the frequency reliability sensitivity analysis. A case of resonant problem in the dynamic structural systems is carried out by using the random perturbation technique, the reliability theory and sensitivity approach. Numerical results have demonstrated that the proposed method is efficient and accurate. This scheme is helpful for the design and optimization of dynamic structural systems.