Multimode Interaction in Axially Stiffened Cylindrical Shells

ABSTRACT

A number of studies show that nonlinear interaction between a shortwave local panel buckling mode and a longwave overall shell buckling mode usually causes strong imperfection sensitivity in stiffened shells. Interaction is particularly strong for simultaneous or nearly simultaneous modes. Mode interaction between two overall buckling modes, as well as interaction between two overall and one local mode, has received much less attention. The present study indicates that for certain imperfection combinations, such mode interactions are important. On the other hand, in most cases, interaction between one overall mode and one local mode governs.     

Orientation effects in extensional and squeezing flows of certain anisotropic fluids

The Ericksen-Leslie continuum theory of anisotropic fluids is here used to examine the behaviour of the orientation pattern within the bulk of a fluid that is undergoing an extensional-type of flow. The flow (which is irrotational and generally unsteady) is considered to be generated by application of prescribed normal stresses, and the orientation pattern (represented by a director field n) is taken to be spatially homogeneous but time-dependent. By means of a phase-plane analysis it is shown that, in stretching flows, the director eventually aligns parallel to the direction of imposed stretch, whereas in squeezing-type flows it eventually lies in the plane normal to the direction of squeezing. In both cases the lateral components of n may vary non-monotonically in time, before approaching their asymptotic values; also the lateral components of velocity may change sign during flow.A two-parameter classification is given of all possible modes of behaviour of these model fluids in these flows. Also analytical solutions are obtained for certain special cases, such as axisymmetric flow.     

Wave trapping in a two-dimensional sound-soft or sound-hard acoustic waveguide of slowly-varying width

In this paper we derive novel approximations to trapped waves in a two-dimensional acoustic waveguide whose walls vary slowly along the guide, and at which either Dirichlet (sound-soft) or Neumann (sound-hard) conditions are imposed. The guide contains a single smoothly bulging region of arbitrary amplitude, but is otherwise straight, and the modes are trapped within this localised increase in width.Using a similar approach to that in Rienstra (2003) [13], a WKBJ-type expansion yields an approximate expression for the modes which can be present, which display either propagating or evanescent behaviour; matched asymptotic expansions are then used to derive connection formulae which bridge the gap across the cut-off between propagating and evanescent solutions in a tapering waveguide. A uniform expansion is then determined, and it is shown that appropriate zeros of this expansion correspond to trapped mode wavenumbers; the trapped modes themselves are then approximated by the uniform expansion. Numerical results determined via a standard iterative method are then compared to results of the full linear problem calculated using a spectral method, and the two are shown to be in excellent agreement, even when ?, the parameter characterising the slow variations of the guide’s walls, is relatively large.     

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.     

A two scale method for modulated vibration modes of large,nearly repetitive,structuresUne méthode asymptotique à deux échelles pour les modes de vibrations modulés des longues structures presque répétitives

Nearly repetitive structures can present at least two kinds of vibration modes: localized modes and modulated ones. In this Note, the multiple scale method is applied to characterize a packet of modulated modes. In this respect, only small size problems are to be solved: periodic problems posed on a few basic cells and amplitude equations, which define a sort of homogenized model for this type of modes. It is established that the influence of the non-repetitive part of the structure is accounted by a boundary condition. To cite this article: E.M. Daya et al., C. R. Mecanique 331 (2003).     

Postbuckling and Imperfection Sensitivity Analysis of Axially Stiffened Cylindrical Shells with Mode Interaction

Abstract

Interaction of nearly simultaneous buckling modes in the presence of imperfections is studied. The investigation is concerned with axially stiffened cylindrical shells under axial compression. In these structures, two modes are of particular interest, namely an overall long-wave and a local shortwave buckling mode. Numerical results show that in some cases bending of the stringers in the local mode postbuckling solution plays an important role. Exclusion of this effect, as was done in a previous study by Byskov and Hutchinson, may lead to an overestimation of the carrying capacity of the shell. Furthermore, it is found that apparently reasonable approximations to the postbuckling fields associated with both the local and the overall mode, as well as with the overall mode alone, may lead to inexact values of the buckling load.     

Variational asymptotic micromechanics modeling of heterogeneous porous materials

This paper presents a numerical technique to predict the effective elastic properties of heterogeneous fluid-filled porous media where the heterogeneity may result from dissimilar solid and fluid phase properties or due to mismatch in porous microstructure. The technique is based on the variational asymptotic method of homogenization where finite element method is employed for discretization. Biot’s theory of poroelasticity is used to describe porous media where both solid and fluid phase motions (u ? U formulation) are considered with associated strain measures. The method estimates the poroelastic constitutive law in single analysis which makes it very efficient compared to other finite element based homogenization techniques. The method is also general enough to compute all 28 elements of an anisotropic constitutive matrix. Other than estimating the effective properties the micro-stress/strain distribution is also obtained at no additional cost.The method is successfully applied for homogenization of porous media, fluid-filled cavity and finally for effective property estimation of bone lamella. In absence of any other direct method of porous media homogenization, the present technique is compared with classical homogenization methods with fluid approximated as solid of very high Poisson’s ratio. The suitability of this approximation and various other alternatives are also discussed. It is shown that the present homogenization method can be an efficient tool for bone property estimation where fluid-filled porous hierarchical micro-/nanostructure must be respected at all steps.     

Design of a tuned vibration absorber for a slender hollow cylindrical structure

Abstract

In this paper, details of the design work for a tuned vibration absorber to be used on a hollow cylindrical structure is presented. The vibration problem is of resonant type and the tuned vibration absorber is designed to suppress the displacement vibration response of the free end of the slender hollow structure dominated by the contribution of its lowest transverse vibration modes. The structure is modeled using a commercial finite element software. Finite element model of the structure is verified using experimentally obtained frequency response functions and modal parameters. Effective parameters of the tuned vibration absorber design are then determined based on finite element analysis simulations of the vibration suppression performance of the tuned vibration absorber as it is used on the structure. Details of the tuned vibration absorber design are determined and a prototype is fabricated. Prototype tuned vibration absorber is then characterized experimentally both as a standalone system and also as it is used on the main structure. Vibration reduction performance of the physical prototype of the tuned vibration absorber is also compared with its vibration reduction performance estimated from finite element analysis simulations so that the analysis based design process can be validated.

Communicated by Dumitru Caruntu.     

含裂纹体的单位分解扩展无网格方法

不连续体的数值模拟尤其是动态裂纹的追踪问题一直是工程界研究的热点和难点问题。无网格方法仅仅需要结点信息,非常适合于求解这类问题。基于单位分解思想,在移动最小二乘近似函数(MLS)中根据裂纹面的不连续位移增加一个Heaviside函数,在裂尖则增加四个扩展函数描述渐进裂纹位移场;应用Galerkin方法推导了平衡方程的离散线性方程,并给出了求解裂纹问题应力强度因子的计算公式。与其他类型的扩展无网格相比,在裂尖处近似函数不需要使用可视准则,很容易生成r1/2奇异;另一个优势是影响域并没有因为裂纹的存在而改变,不会降低方程的稀疏性,求解效率较高。数值算例表明,该方法能方便有效地模拟不连续问题,具有十分广阔的应用空间。     

L2Roe: a low dissipation version of Roe's approximate Riemann solver for low Mach numbers

A modification of the Roe scheme called L²Roe for low dissipation low Mach Roe is presented. It reduces the dissipation of kinetic energy at the highest resolved wave numbers in a low Mach number test case of decaying isotropic turbulence. This is achieved by scaling the jumps in all discrete velocity components within the numerical flux function. An asymptotic analysis is used to show the correct pressure scaling at low Mach numbers and to identify the reduced numerical dissipation in that regime. Furthermore, the analysis allows a comparison with two other schemes that employ different scaling of discrete velocity jumps, namely, LMRoe and a method of Thornber et al. To this end, we present for the first time an asymptotic analysis of the last method. Numerical tests on cases ranging from low Mach number (M_∞=0.001) to hypersonic (M_∞=5) viscous flows are used to illustrate the differences between the methods and to show the correct behavior of L²Roe. No conflict is observed between the reduced numerical dissipation and the accuracy or stability of the scheme in any of the investigated test cases. Copyright © 2015 John Wiley & Sons, Ltd.     

ON THE MAXIMAL LYAPUNOV EXPONENT FOR A REAL NOISE PARAMETRICALLY EXCITED CO-DIMENSION TWO BIFURCATION SYSTEM (Ⅰ)

For a real noise parametrically excited co-dimension two bifurcation system on three-dimensional central manifold, a model of enhanced generality is developed in the present paper by assuming the real noise to be an output of a linear filter system, namely, a zero-mean stationary Gaussian diffusion process that satisfies the detailed balance condition. On such basis, asymptotic expansions of invariant measure and maximal Lyapunov exponent for the relevant system are established by use of Arnold asymptotic analysis approach in parallel with the eigenvalue spectrum of Fokker-Planck operator. Foundation item: the National Natural Science Foundation of China (19602016)     

ANALYTICAL SOLUTION FOR MODE Ⅲ DYNAMIC RUPTURE OF STANDARD LINEAR VISCOELASTIC SOLID WITH NONLINEAR DAMPING

Introducing the nonlinear Rayleigh damping into the governing equation of the Mode Ⅲ dynamic rupture for standard viscoelastic solid, this equation is a partial differential and integral equation. First, eliminating the integral term, a PDE of third-order is obtained. Then, applying the small parameter expansion method, linearized asymptotic governing equation for each order of the small parameter is obtained. Dividing the third-order PDE into an elastic part with known solution, the rest part pertains to viscous effect which is neither a Mathieu equation nor a Hill one. The WKBJ method is still adopted to solve it analytically.     

Numerical investigation of supersonic plate flutter using the exact aerodynamic theory

In classical investigations of panel flutter it is usually assumed that the gas pressure acting on the plate can be calculated within the framework of the piston theory, an approximation exact for high Mach numbers. The loss of stability revealed in these investigations is of the “coupled” type, involving the interaction of two oscillation modes. Recently, the use of asymptotic methods revealed another single-mode type of stability loss, which cannot be obtained within the framework of the piston theory. In the present study this type of stability loss is investigated numerically using the Bubnov-Galerkin method.     

The two-dimensional transmission and reflection of a solitary wave

In the present paper, limited to the discussion of weak non-linear shallow water waves, the transmission and reflection of a planar soliton on a two-dimensional structure are considered. The whole flow field is divided mainly into two subfields. One is in the vicinity of the structure, called the inner field; the other is far from the structure, called the outer field. In the outer field, according to its definition, the influence of the structure on the flow is negligible; to the order O(α, β) the governing equation for the flow is replaced by the Boussinesq equation. In the inner field the effect of the structure on the flow is significant, so the full Laplace equation is adopted as the governing equation for the flow field. Then the matched asymptotic expansion method is employed to connect smoothly the inner and outer solutions. Owing to the irregularity of the bottom of the structure, the boundary element method is incorporated. As an example, the case in which the incoming wave is a solitary wave is calculated and the time histories of transmitted and reflected waves are plotted.     

A fast mesh deformation method for marine propeller flow

ABSTRACT

A fast mesh deformation method for propeller flow is developed based on the elastic solid method. The flow field of a propeller is assumed to be fulfilled with a kind of pseudo elastic solid which does not influence the flow. The vibration equation for the propeller blade-pseudo elastic solid system is derived. During fluid-structure coupling, the nodal displacement for the blade and the flow mesh is computed by modal superposition of the first several modes. Fluid-structure coupling is performed for a highly skewed propeller. The computing time for the dynamic mesh by the present method is about 0.017% of the computing time by the existing elastic solid method. The computing time for the fluid-structure coupling using the present method is 52% less than the computing time by the existing elastic solid method.     

Numerical-Asymptotic Expansion Matching for Computing a Viscous Flow Around a Sharp Expansion Corner

A computational technique which is based on a numerical-asymptotic expansion matching for computing the local singular behavior of a viscous flow around a sharp right-angle expansion corner is presented. Moffatt's (1964) asymptotic solution is extended and a matching with a time-marching finite-difference scheme of the Navier--Stokes equations is formulated. Local mesh refinement around the corner is required to meet the validity of the asymptotic solution. Flows in an expanding channel with expansion ratio D/d=3 at various Reynolds numbers 1≤Re≤700 are simulated. The results are compared with those from a standard finite-difference scheme that uses second-order forward/backward differences near the corner. It is found that the results of the standard scheme converge toward those of the present technique as the level of local refinement near the corner is increased. The time-dependent parameters of the first two terms of the asymptotic solution at the steady-state solution are also described for various cases of Re and D/d. It is demonstrated that the present method enhances the accuracy of the simulations and requires less refinements near the corners to achieve converged numerical results. Received 14 August 2000 and accepted 25 October 2001     

Sparse matrix implicit numerical integration of the Stiff differential/algebraic equations: Implementation

The finite element absolute nodal coordinate formulation (ANCF) is often used in modeling very flexible bodies in multibody system (MBS) applications. This formulation leads to a constant mass matrix, allowing for an efficient sparse matrix implementation. Nonetheless, the use of the ANCF finite elements to model stiff structures can lead to high frequencies associated with ANCF coupled deformation modes, as discussed in the literature. Implicit numerical integration methods can be effectively used to develop efficient procedures for the solution of MBS differential/algebraic equations. Most existing implicit integration algorithms, however, require numerical differentiation of the equations of motion, and some of these integration methods do not ensure that the kinematic algebraic constraint equations are satisfied at all levels (position, velocity, and acceleration). Because of these limitations, existing implicit integration methods can be less accurate and less efficient when used to solve large scale MBS applications. In order to circumvent this problem, the two-loop implicit sparse matrix numerical integration (TLISMNI) method was proposed for the solution of MBS differential/algebraic equations. The TLISMNI method does not require numerical differentiation of the forces and allows for an efficient sparse matrix implementation. This paper discusses TLISMNI implementation issues including the step size selection, the error control, and the effect of the numerical damping. The relation between the step size selection and the structure stiffness is also discussed. The use of the computer implementation described in this paper is demonstrated by solving very stiff structure problems using the Hilber?CHughes?CTaylor (HHT) method, which includes numerical damping. An eigenvalue analysis and Fast Fourier Transform (FFT) are performed in order to identify the fundamental modes of deformation and demonstrate that the contributions of these fundamental modes can be erroneously damped out when some other implicit integration methods are used. The TLISMNI method, on the other hand, captures the contributions of these fundamental modes. The results, obtained using the TLISMNI method, are compared with the results obtained using other methods including the implicit HHT-I3 and the explicit Adams integration methods. The results obtained show that the TLISMNI method can be five times faster than the other two methods when no numerical damping is considered.     

Effet de l'orientation d'un champ magnétique horizontal sur la stabilité de l'écoulement de HadleyEffet of the orientation of a horizontal magnetic field on the stability of Hadley flow

A numerical study based on the linear stability analysis is undertaken, in order to determine the influence of a horizontal magnetic field on the marginal modes occuring in a fluid layer subjected to a horizontal temperature gradient. A particular interest is devoted to the influence of the magnetic field orientation on both nature and critical values of the unstable modes. Calculations show, that when it is subjected to such a magnetic field, this type of flow, known as Hadley flow, can present oblique waves, hitherto non-existent when no magnetic field is applied and even when a vertical, a transverse or a longitudinal magnetic field is imposed. A new asymptotic behavior is also observed for the stabilizing effects. To cite this article: S. Kaddeche et al., C. R. Mecanique 331 (2003).     

Steady thermocapillary-buoyant convection in a shallow annular pool. Part 2: Two immiscible fluids

This work is devoted to the study of steady thermocapillary-buoyant convection in a system of two horizontal superimposed immiscible liquid layers filling a lateral heated thin annular pool.The governing equations are solved using an asymptotic theory for the aspect ratios ε→ 0.Asymptotic solutions of the velocity and temperature fields are obtained in the core region away from the cylinder walls.In order to validate the asymptotic solutions,numerical simulations are also carried out and the results are compared to each other.It is found that the present asymptotic solutions are valid in most of the core region.And the applicability of the obtained asymptotic solutions decreases with the increase of the aspect ratio and the thickness ratio of the two layers.For a system of gallium arsenide (lower layer) and boron oxide (upper layer),the buoyancy slightly weakens the thermocapillary convection in the upper layer and strengthens it in the lower layer.