A new method of fictitious viscous damping determination for the dynamic relaxation method

This paper develops a new method for calculating the viscous fictitious damping of the dynamic relaxation (DR) method to overcome one of the most crucial difficulties in its application – the low convergence rate. The DR formulation was derived by error minimizations between two successive iterations to deduce an optimum fictitious mass and viscous damping with the aid of the Stodola iterative process. The efficiency of the new method was verified by its application to a wide range of typical structures with strong nonlinearity. The results show that compared to the conventional DR algorithm such as kinetic approach, the new method improves the convergence rate considerably.     

Non-linear analysis of shells of revolution under arbitrary loads

A new finite element formulation is presented for the non-linear analysis of elastic doubly curved segmented and branched shells of revolution subject to arbitrary loads. The circumferential variations of all quantities are described by truncated Fourier series with an appropriate number of harmonic terms. A coupled harmonics approach is employed, in which coupling between different harmonics is dealt with directly rather than by the use of pseudo-loads. Key issues in the formulation, such as non-linear coupling and growth of harmonic modes, are carefully and systematically explained. This coupled harmonics approach allows an easy implementation of the arc-length method. As a result, post-buckling load–deflection paths can be traced efficiently and accurately. The formulation also employs a non-linear shell theory more complete than existing classical theories. The results from the present study are independently verified using ABAQUS, while those from other studies are found to be inaccurate in general.     

Collapse of long, noncircular, cylindrical shells under pure bending

This paper describes an extension of a method developed in a previous paper to determine the moment carrying capacity of elastoplastic noncircular cylindrical shells with infinite length by the finite element method. As a result of the shape change in the cross section of a shell during deformation, the bending moment reaches a global maximum value and then decreases as the bending curvature further increases. The shell would consequently collapse at the maximum moment. However, a bifurcation buckling may occur before the maximum moment can be developed. This bifurcation buckling could induce collapse of the shell under a moment less than the maximum. Determination of the likelihood that the bifurcation buckling would generate shell collapse may be made from the initial post-buckling behavior. An initial post-buckling analysis based on the J₂ deformation theory of plasticity has been developed in this paper. The finite element method with one spatial variable is used to locate the bifurcation point as well as to analyze the initial post-buckling behavior. Numerical examples of cylindrical shells with various cross-sectional shapes are shown. In particular, for a shell of square cross section, the moment at the bifurcation is much lower than the maximum value; however, the initial post-buckling analysis reveals that the state of equilibrium is still stable. Deep post-buckling analysis is required to determine the moment carrying capacity of a shell with such cross section.     

The efficiency of dynamic relaxation methods in static analysis of cable structures

Nine different schemes of dynamic relaxation method are compared in the paper. Schemes with viscous damping and schemes with kinetic damping are used. Kinetic damping with a peak in the middle of the time step and kinetic damping with the parabolic approximation of the peak are considered. They are also used in three different ways of cable approximation. The cable is approximated as a tension bar, a catenary and a parabolic cable element. The efficiency and stability of each method are compared to three selected 3D examples of cable structures and one existing structure. The effect of mass distribution along the structure is also of interest and is studied in the paper.     

基于GIMT和弧长参数化的QG-Ball曲线近似合并

曲线近似合并作为 CAGD 中复杂曲线设计的一种有效技术,一直备受学者们的关注,并在 CAD/CAM 领域得到了广泛的应用。针对现有带形状参数的广义 Ball 曲线难以合并的问题,提出了一种基于广 义逆矩阵理论(GIMT)和弧长参数化的 QG-Ball 曲线近似合并方法。首先,利用曲线近似弧长参数化算法计算出 QG-Ball 曲线弧长等分对应的配置点列(亦称等分点)和配置点参数值;其次,基于所得等弧长配置点列及其参 数值,再结合广义逆矩阵理论和曲线拟合方法,便可以直接得到计算合并后 QG-Ball 曲线控制顶点的一个显式 表达式;最后,利用连续函数的 L2 范数定义了一个度量曲线合并效果的误差计算公式,并给出了一些具有代 表性的数值算例及其合并误差。实例结果表明,所提出的方法可以高效地实现 QG-Ball 曲线的近似合并,不仅 易于操作、误差计算简单,而且能方便地推广到其他曲线的近似合并。     

Postbuckling of multilayer cylindrical and spherical shell panels reinforced with graphene platelet by isogeometric analysis

The present work fills a gap on the postbuckling behavior of multilayer functionally graded graphene platelet reinforced composite (FG-GPLRC) cylindrical and spherical shell panels resting on elastic foundations subjected to central pinching forces and pressure loadings. Based on a higher-order shear deformation theory and the von Kármán’s nonlinear strain–displacement relations, the governing equations of the FG-GPLRC cylindrical and spherical shell panels are established by the principle of virtual work. The non-uniform rational B-spline (NURBS) based isogeometric analysis (IGA), the modified arc-length method and the Newton’s iteration method are employed synthetically to obtain nonlinear load–deflection curves for the panels numerically. Several comparative examples are performed to test reliability and accuracy of IGA and arc-length method in present formulation and programming implementation. Parametric investigations are carried out to illustrate the effects of dispersion type of the graphene platelet (GPL), weight fraction of the GPL, thickness of the panel, radius of the panel and parameters of elastic foundation on the load–deflection curves of the FG-GPLRC shell panels. Some complex load–deflection curves of the FG-GPLRC cylindrical and spherical shell panels resting on elastic foundations may be useful for future references.

     

Accelerated discrete velocity method for axial-symmetric flows of gaseous mixtures as defined by the McCormack kinetic model

An accelerated discrete velocity method is presented to calculate the steady axial-symmetric flows of gaseous mixtures defined by the McCormack kinetic model. The scheme is formulated in cylindrical coordinates. Diffusion equations for the macroscopic velocity and the heat-flow are derived on the basis of the projected McCormack equations. The solutions of the kinetic equations are carried out iteratively by using the discrete velocity method. The diffusion equations are also solved in each stage of the iteration in order to accelerate the scheme. Pressure driven flows of He/Xe and Ne/Ar mixtures through a cylindrical tube are simulated in order to study the computational performance of the approach. It is shown that the required number of iterations and the computational times are significantly reduced at intermediate and large values of the rarefaction parameter by using the accelerated method. In the hydrodynamic limit, the flow rates of the components converge to the corresponding slip flow results. Flows driven by mole fraction gradient are also successfully simulated. Typical velocity and heat-flow profiles for pressure driven flow of He/Xe mixture are shown and commented on.     

On post-buckling analysis and experimental correlation of cylindrical composite shells with Reissner-Mindlin-Von Kármán type facet model

In this paper post-buckling analysis of carbon fibre reinforced plastic cylindrical shells under axial compression is considered. Reissner-Mindlin-Von Kármán type shell facet model is used in the computations. The effect of geometric imperfection shape and amplitude on nonlinear analysis results is discussed. Numerical-experimental correlation is performed using the results of experimental buckling tests found in the literature. Results show that bringing the diamond shape geometric imperfection in the model significantly improves the correlation and gives good accuracy in simulating cylindrical shell post-buckling behaviour.     

Buckling and post-buckling analysis of extensible beam-columns by using the differential quadrature method

Buckling and post-buckling analysis of extensible beam-columns is performed numerically in this paper. It was experienced earlier that in some cases the numerical integration would not produce the convergent post-buckling solutions, especially under high loads. Therefore, a new differential quadrature (DQ) based iterative numerical integration method is proposed to solve post-buckling differential equations of extensible beam-columns. Six cases, including five classical Euler buckling cases, are analyzed. Critical loads and convergent post-buckling solutions under different applied loads are obtained. The results are compared with the existing multiple scales solutions. It is found that under high applied loads, the small rotation assumption in obtaining multiple scales solutions is no longer valid. The proposed iterative DQ based numerical integration method can yield reliable and accurate post-buckling solutions even at high applied loads for the cases investigated.     

A priori error estimates for explicit finite element for linear elasto-dynamics by Galerkin method and central difference method

The explicit finite element method for transient dynamics of linear elasticity is formulated by using Galerkin method for space and the central difference method for time. An a priori error estimate is derived and the optimal rate of convergence for displacement similar to the linear elliptic problem is found. The error estimation is extended to velocity, internal (strain) energy and kinetic energy for engineering applications. The approximation error of initial data is analyzed. The error estimate is refined for a class of engineering applications with zero initial deformation and initial force. Examples of a 1-D rod axial vibration and a 2-D plate in-plane vibration are solved using linear elements as verification.     

无条件稳定的显式降维非线性有限元

针对传统降维非线性有限元计算速度与精确度难以兼顾的问题,提出了一种无条件稳定的显式迭代算法。基于泰勒展开式得到速度、加速度的三阶精度差分表达式从而获得新的有限元显式迭代方程,并分析其单自由度系统下的传递矩阵谱半径。改进迭代方程使谱半径始终小于1从而满足无条件稳定的要求。实验表明,改进后的显式迭代算法在等效阻尼比的精度上优于中心差分法和隐式迭代法;在降维非线性有限元模型计算中的计算耗时优于隐式迭代方法,提高了降维非线性有限元的迭代计算速度。模型在降维后维度数值较高时,仍能维持良好的计算耗时和帧率,保证了模型的精确度。     

A coupled boundary element/finite difference method for fluid–structure interaction with application to dynamic analysis of outer hair cells

Outer hair cells (OHC) in the inner ear, which resemble fluid-filled and fluid-surrounded cylinders, are known to exhibit motility and play a critical role in our keen sense of hearing. In this study, we investigate the OHC frequency response using a mathematical model of the OHC, which consists of a two-layered anisotropic cylindrical lateral wall, and both the intracellular and extracellular fluids. We use the boundary integral equations to model the intracellular and extracellular fluids, and these are coupled to the anisotropic cylindrical shell equations (discretized using the finite difference method). Since the geometry is axisymmetric, the dynamic analysis is performed by decomposing the motion into Fourier modes in the circumferential direction. For the boundary element method, this leads to two sequences of line integrals along the generator of the domain, and the singular kernels need to be evaluated with elliptic integrals. The coupled fluid–structure equations are solved for several modes of deformation (axisymmetric, cylindrical beam-bending, and pinched modes), and the frequency responses are obtained. The frequency response of the model with viscous fluid is found to be significantly different from that using inviscid fluid. For the small length scale of the OHC (which is of micron size), the viscosity of the fluid is found to have significant damping effects on the OHC frequency response.     

Maximization of the crushing energy absorption of tubes

This paper concerns the development of crashworthiness maximization techniques for tubular structures, and the application to the axial crushing problem of cylindrical tubes as well as square tubes. In the program system presented in this study, an explicit finite element code, DYNA3D is adopted for simulating complicated crushing behaviour of tubular structures. The response surface approximation technique is applied to construct an approximated design subproblem in the preassigned design space by using the technique of design-of-experiment. The approximated subproblem is solved by the usual mathematical programming technique. These optimization processes are repeated until the given convergence conditions are satisfied. Moreover, a comparison of the crushing energy absorption between cylindrical tubes and square tubes is discussed.     

ReALE: A Reconnection Arbitrary-Lagrangian–Eulerian method in cylindrical geometry

This paper deals with the extension to the cylindrical geometry of the recently introduced Reconnection algorithm for Arbitrary-Lagrangian–Eulerian (ReALE) framework. The main elements in standard ALE methods are an explicit Lagrangian phase, a rezoning phase, and a remapping phase. Usually the new mesh provided by the rezone phase is obtained by moving grid nodes without changing connectivity of the underlying mesh. Such rezone strategy has its limitation due to the fixed topology of the mesh. In ReALE we allow connectivity of the mesh to change in rezone phase, which leads to general polygonal mesh and permits to follow Lagrangian features much better than for standard ALE methods. Rezone strategy with reconnection is based on using Voronoi tesselation machinery. In this work we focus on the extension of each phase of ReALE to cylindrical geometry. The Lagrangian, rezone with reconnection and remap phases are revamped to take into account the cylindrical geometry. We demonstrate the efficiency of our ReALE in cylindrical geometry on series of numerical examples.     

复杂圆柱壳结构参数化建模方法及模型库设计

针对船舶的主要结构形式——复杂圆柱壳结构,为研究其快速建模方法以实现船舶结构的优化设计,基于隐式参数化模型概念提出参数化模型表征方法,建立其分类和命名体系,给出其参数化建模方式和流程,利用数值仿真工具完成其参数化模型库设计和开发,并应用于某船舶结构的快速建模.算例表明:复杂圆柱壳参数化模型库设计合理,满足复杂圆柱壳结构快速建模和修改的要求.     

Hysteretic damping of structures vibrating at resonance: An iterative complex eigensolution method based on damping-stress relation

In this paper the damping is examined as an engineering property used in analysis and design of structures and machines. The design engineer needs to know not only the stresses of his structure or machine, under steady state conditions but also the stresses under resonance conditions. Then the material damping, as a function of the stress of the structure, has an important role to play and ignoring the damping the calculated stresses are far from reality. The nonlinearity here is due to the dependence of the hysteretic damping on the stress of the structure. Specifically here two problems are investigated in the following way:Firstly the direct problem is solved. The direct problem is to find the maximum bending stress at the resonance when the relation of the dissipating energy (or of the hysteretic damping) vs. the bending stress is known in advance. To perform this calculation, a useful tool for the design engineer, the structure is modelled using the continuum mechanics analytical approach or the finite elements (FE) method. Then the eigenvalues are calculated and using an iterative procedure the real stress. The procedure presented here is called iterative complex eigensolution method (ICEM). Secondly the inverse problem is solved. The inverse problem is to find the relation between the hysteretic damping and the bending stress. For this purpose the logarithmic decrement is experimentally measured, the eigenvalues and the maximum bending stress of the structure, excited at the eigenvalue, when the damping is the same as the measured one, are computed using the finite elements method. Once the bending stresses are found in each discrete element of the structure, then the mathematical expression of the relation of the dissipating energy and the stresses can be specified by minimizing a suitably formed objective function.     

A method for synthesis of optimal weight structures

A method to synthesize structures of optimal or least weight is presented and demonstrated for structures employing cylindrical tube members. The method employs a unique reduction in complexity from the general n-dimensional problem to a two dimensional “design plane”. Comparison with previous authors' results are included.     

Finite elements for post-buckling analysis. II—Application to composite plate assemblies

In a companion paper the authors presented a convenient formulation for the stability analysis of structures using the finite element method. The main assumptions are linear elasticity, a linear fundamental path and the existence of distinct critical loads. The formulation developed is known as the W-formulation, where the energy is written in terms of a sliding set of incremental coordinates measured with respect to the fundamental path. In the present paper a number of applications of finite elements for post-buckling analysis on composite plate assemblies are presented. Thin-walled composite plates, I-beams, angle sections, and a specially designed box-beam with flanges (unicolumn) are studied in post-buckling when axially loaded. The results are in good agreement with previous studies. Moreover, a parametric study involving critical buckling load and geometry is presented for the case of the unicolumn.     

Homoclinic and heteroclinic orbits underlying the post-buckling of axially-compressed cylindrical shells

A structural system with an unstable post-buckling response that subsequently restabilizes has the potential to exhibit homoclinic connections from the fundamental equilibrium state to itself over a range of loads, and heteroclinic connections between fundamental and periodic equilibrium states over a different (smaller) range of loads. It is argued that such equilibrium configurations are important in the interpretation of observed behaviour, and govern the minimum possible post-buckling loads.

To illustrate this, the classical problem of a long thin axially-compressed cylindrical shell is revisited from three different perspectives: asymptotic conjecture, analogy with nonlinear dynamics, and numerical continuation analysis of a partial spectral decomposition of the underlying equilibrium equations. The nonlinear dynamics analogy demonstrates that the structure of the heteroclinic connections is more complicated than that indicated by the asymptotics: this is confirmed by the numerics. However, when the asymptotic portrayal is compared to the numerics, it turns out to be surprisingly accurate in its Maxwell-load prediction of the practically-significant first minimum to appear in the post-buckling regime.     

基于等效模态阻尼的复合隔振系统振动计算方法

基于线性黏弹性假设,将应变能阻尼理论推广到复合隔振系统的等效模态阻尼计算中,运用Python和Abaqus编制相应的计算程序,该程序可考虑材料阻尼的频变特性。以多种材料组成的船舶双层复合隔振系统为算例,计算其等效模态阻尼和隔振器等效阻尼系数。分别采用直接积分法和模态叠加法计算系统振动响应,对比设备、筏架、船底壳的振动加速度响应,验证基于等效模态阻尼的模态叠加法的准确性。结果表明,该方法可以准确计算复杂组合模型的模态阻尼,算例的振动响应计算结果一致性较好,用模态叠加法可以大幅提高复合隔振系统稳态振动响应的计算效率。