旋转壳边界条件广义表达式

旋转壳的边界条件,传统的表达方式是在中面位移μ,μ,ω,ψ或相应的四个力共八个量当中给定四个.而以节圆广义位移作为基本未知数,一个节圆上未知数的数目超过四个[1][2][3][4].在这种情况下关于边界条件的处理问题尚无令人满意的解决办法.本文利用虚功原理,导出一组壳边广义量与非广义量关系公式.研究了七种类型常见边界,给出用广义力与广义位移表示的边界条件公式.每一种边界条件公式的数目可以和一个节圆上所采用的未知数数目相一致.有了这些公式,即可直接将边界条件代入广义位移法运动方程以求解广义位移.这样做,避免了文献[2]关于未知数的变换与逆变换过程,不仅道理上简明而且也简化了计算.有了边界条件广义表达式,使得旋转壳广义位移法在理论上也更为完善.     

Non-linear oscillations of a thin-walled composite beam with shear deformation

A geometrically non-linear theory is used to study the dynamic behavior of a thin-walled composite beam. The model is based on a small strain and large rotation and displacements theory, which is formulated through the adoption of a higher-order displacement field and takes into account shear flexibility (bending and warping shear). In the analysis of a weakly nonlinear continuous system, the Ritz’s method is employed to express the problem in terms of generalized coordinates. Then, perturbation method of multiple scales is applied to the reduced system in order to obtain the equations of amplitude and modulation. In this paper, the non-linear 3D oscillations of a simply-supported beam are examined, considering a cross-section having one symmetry axis. Composite is assumed to be made of symmetric balanced laminates and especially orthotropic laminates. The model, which contains both quadratic and cubic non-linearities, is assumed to be in internal resonance condition. Steady-state solution and their stability are investigated by means of the eigenvalues of the Jacobian matrix. The equilibrium solution is governed by the modal coupling and experience a complex behavior composed by saddle noddle, Hopf and double period bifurcations.     

三维非局部弹性场中裂纹问题的分析方法

通过求解得到了三维非局部弹性力学对称情形的单位集中不连续位移基本解·基于该基本解和三维局部（经典）弹性力学的不连续位移边界积分方程———边界元方法·提出了三维非局部弹性力学中的平片裂纹Ⅰ型问题的通用解法,并给出了算例     

Flexure Hinge Mechanisms Modeled by Nonlinear Euler-Bernoulli-Beams

A flexure hinge is an innovative engineering solution for providing relative motion between two adjacent stiff members by the elastic deformation of an arbitrary shaped flexible connector. In the literature, modeling of compliant mechanisms incorporating flexure hinges is mainly focused on linear methods. However, geometrically nonlinear effects cannot be ignored generally. This study presents a nonlinear modeling technique for flexure hinges based on the Euler-Bernoulli beam theory, in contrast to the predominant linear modeling approaches. Higher order beam elements of variable cross-section are employed to model the flexure hinge region. A Newton-Raphson scheme is applied to solve the resulting nonlinear system equations. The proposed approach reduces the overall degrees of freedom and is computationally efficient compared to commonly applied 3D finite element methods. A compliant displacement amplification mechanism is studied by means of the proposed method, where an excellent agreement with results of a reference solution is achieved. The modeling approach is suitable for the structural optimization of compliant mechanisms towards a less intuitive design process. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Elastic analytical solution of shallow tunnel owing to twin tunnelling based on a unified displacement function

A unified displacement function is used as the displacement boundary condition of the cross-section of each tunnel, which can be used to capture the asymmetrical deformation behaviors about the horizontal and vertical center line. Complex variable method is adopted to obtain the analytical solution of a single tunnel. Based on the Schwartz alternating method, the analytical solution of twin tunnels is obtained and the correctness of the analytical solution of twin tunnels is verified. A series of analyses are carried out to investigate the effects of the displacement boundary condition of the tunnel cross-section and the distance of twin tunnels on the ground surface settlement trough curve and the ground displacement field. Based on the proposed method, the parameters of displacement mode of the cross-section of twin tunnels are calibrated through field monitoring data of the Bangkok Subway Tunnel project, and the displacement field of the whole ground are analyzed. The proposed method can provides guidance in the conceptual stage of the design process of twin tunnels.     

A generalized preconditioned HSS method for non-Hermitian positive definite linear systems

Based on the HSS (Hermitian and skew-Hermitian splitting) and preconditioned HSS methods, we will present a generalized preconditioned HSS method for the large sparse non-Hermitian positive definite linear system. Our method is essentially a two-parameter iteration which can extend the possibility to optimize the iterative process. The iterative sequence produced by our generalized preconditioned HSS method can be proven to be convergent to the unique solution of the linear system. An exact parameter region of convergence for the method is strictly proved. A minimum value for the upper bound of the iterative spectrum is derived, which is relevant to the eigensystem of the products formed by inverse preconditioner and splitting. An efficient preconditioner based on incremental unknowns is presented for the actual implementation of the new method. The optimality and efficiency are effectively testified by some comparisons with numerical results.     

Experimental comparison of three-dimensional point and line modified incomplete factorizations

We examine how the variations of the coefficients of 3-dimensional (3D) partial differential equations (PDEs) influence the convergence of the conjugate gradient method, preconditioned by standard pointwise and linewise modified incomplete factorizations. General analytical spectral bounds obtained previously are applied, which displays the conditions under which good performances could be expected. The arguments also reveal that, if the total number of unknowns is very large or the number of unknowns in one direction is much larger than in both other ones, or if there are strong jumps in the variation of the PDE coefficients or fewer Dirichlet boundary conditions, then linewise preconditionings could be significantly more efficient than the corresponding pointwise ones. We also discuss reasons to explain why in the case of constant PDE coefficients, the advantage of preferring linewise methods to pointwise ones is not as pronounced as in 2D problems. Results of numerical experiments are reported. This revised version was published online in June 2006 with corrections to the Cover Date.     

Mixed finite element models for free vibrations of thin-walled beams

Simple mixed finite element models are developed for the free vibration analysis of curved thin-walled beams with arbitrary open cross section. The analytical formulation is based on a Vlasov's type thin-walled beam theory which includes the effects of flexural-torsional coupling, and the additional effects of transverse shear deformation and rotary inertia. The fundamental unknowns consist of seven internal forces and seven generalized displacements of the beam. The element characteristic arrays are obtained by using a perturbed Lagrangian-mixed variational principle. Only C⁰ continuity is required for the generalized displacements. The internal forces and the Lagrange multiplier are allowed to be discontinuous at interelement boundaries.

Numerical results are presented to demonstrate the high accuracy and effectiveness of the elements developed. The standard of comparison is taken to be the solutions obtained by using two-dimensional plate/shell models for the beams.     

SBFEM elements for thin-walled composite beams with arbitrary layup

The scaled boundary finite element method (SBFEM) is a semi-analytical method in which only the boundary is discretized. The results on the boundary are scaled into the domain with respect to a scaling center which must be “visible” from the whole boundary. For beam-like problems the scaling center can be selected at infinity and only the cross-section is discretized. Two new elements for thin-walled beams have been developed on the basis of the first order shear deformation theory. The beam sections are considered to be multilayered laminate plates with arbitrary layup. The arbitrary cross-section is discretized with beam elements of Timoshenko type. Using the virtual work principle gives the SBFEM equation, which is a system of differential equations of a gyroscopic type. The solution is calculated using the matrix exponential function. The elements have been tested and compared with a finite element model and they give good results. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An augmented mixed finite element method for 3D linear elasticity problems

In this paper we introduce and analyze a new augmented mixed finite element method for linear elasticity problems in 3D. Our approach is an extension of a technique developed recently for plane elasticity, which is based on the introduction of consistent terms of Galerkin least-squares type. We consider non-homogeneous and homogeneous Dirichlet boundary conditions and prove that the resulting augmented variational formulations lead to strongly coercive bilinear forms. In this way, the associated Galerkin schemes become well posed for arbitrary choices of the corresponding finite element subspaces. In particular, Raviart-Thomas spaces of order 0 for the stress tensor, continuous piecewise linear elements for the displacement, and piecewise constants for the rotation can be utilized. Moreover, we show that in this case the number of unknowns behaves approximately as 9.5 times the number of elements (tetrahedrons) of the triangulation, which is cheaper, by a factor of 3, than the classical PEERS in 3D. Several numerical results illustrating the good performance of the augmented schemes are provided.     

Transparent boundary conditions for the harmonic diffraction problem in an elastic waveguide

This work concerns the numerical finite element computation, in the frequency domain, of the diffracted wave produced by a defect (crack, inclusion, perturbation of the boundaries, etc.) located in a 3D infinite elastic waveguide. The objective is to use modal representations to build transparent conditions on some artificial boundaries of the computational domain. This cannot be achieved in a classical way, due to non-standard properties of elastic modes. However, a biorthogonality relation allows us to build an operator, relating hybrid displacement/stress vectors. An original mixed formulation is then derived and implemented, whose unknowns are the displacement field in the bounded domain and the normal component of the normal stresses on the artificial boundaries. Numerical validations are presented in the 2D case.     

An Augmented Beam Element for Structural Dynamics Using Unit Deflection Shapes Gained on a 2D Finite Element Mesh of the Cross Section

Wave behaviour of line shaped structures can be analysed if they are simply shaped (beams, rods, etc.) and if the frequency or wavelength is within a range where the underlying theories are still applicable. If the frequencies and wavelengths exceed these limits, the modelling of these line or beam shaped structures is only possible using volume elements which results in considerably high computation time and modelling effort. The idea of this work is to keep the beam like modelling of structures but enhance the solution space such that arbitrary but reasonable deflection shapes of the cross section can be covered. The unit deflection shapes are defined and computed on the basis of a 2D finite element mesh of the beam cross section: warpshapes for primary and secondary torsion and shear forces, Eigenmodes for the problem of a plate in membrane and a plate in bending action, derived warpshapes for in plane shapes and shapes which cover the lateral strains. By this transformation of unknowns the number of system degrees of freedom can be reduced considerably. Expensive pre- and postprocessing steps can be done in parallel which opens the gate for massive performance gains. This method was validated in previous publications of the authors and shall now be extended using unit deflection shapes gained from eigenmodes of an infinite Fourier transformed structure obtained by solving a quadratic eigenvalue problem for the wavenumber k_x. The results of both methods shall be compared. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Boundary element analysis of uncoupled transient thermo-elastic problems with time- and space-dependent heat sources

A boundary element method (BEM) for the analysis of two- and three-dimensional uncoupled transient thermo-elastic problems involving time- and space-dependent heat sources is presented. The domain integrals are efficiently treated using the Cartesian transformation and the radial integration methods without considering any internal cells. Similar to the dual reciprocity method (DRM), some internal points without any connectivity are considered; however, in contrast to the DRM, any arbitrary mesh-free interpolation method can be used in the present formulation. There is no need to find any particular solutions and the shape functions in the mesh-free interpolation method can be arbitrary and sufficiently complicated. Unlike the DRM, the generated system of equations contains the unknowns only on the boundary. After finding the primary unknowns on the boundary, the temperature, displacement, and stress components at all internal points can directly be found without solving any system of equations. Three examples with different forms of heat sources are presented to demonstrate the efficiency and accuracy of the proposed method. Although the proposed BEM is mathematically more complicated than domain methods, such as the finite element method (FEM), it is more efficient from a modelling viewpoint since only the surface mesh has to be generated in the presented method.     

Analytic solutions of a reducible strain gradient elasticity model for solid cylinder with a cavity and its application in zonal failure

In this work, the reducibility condition of the fourth-order equilibrium equation in the strain gradient elasticity (SGE) model for solid cylinder with a cavity is obtained. When the reducibility condition is satisfied, the analytic displacement, generalized radial stress, and generalized angular stress can be solved out, and according to the higher-order coefficients, internal length scale, and Lamé constants, the displacement, generalized radial stress, and generalized angular stress are classified into four types: (1) conventional elasticity solution, (2) quasiperiodic SGE solution, (3) monotonous SGE solution, and (4) non-real-number solution. Quasiperiodic generalized radial stress and generalized angular stress are used to explain the occurrence of zonal failure of surrounding rock of a circular roadway. Numerical analysis with MATLAB is applied to study the influence of loading on zonal failure of surrounding rock of a circular raodway.     

A spatial shear deformable beam finite element based on the absolute nodal coordinate formulation

In the present paper a three-dimensional beam finite element undergoing large deformations is proposed. Since the definition of the proposed finite element is based on the absolute nodal coordinate formulation (ANCF), no rotational coordinates occur in the formulation. In the current approach, the orientation of the cross section is parameterized by means of slope vectors. Since those are no unit vectors, the cross-section can deform, similar to existing thick beam and shell elements. The nodal displacements and the directional derivatives of the displacements are chosen as nodal coordinates, but in contrast to standard ANCF elements, the proposed formulation is based on the two transversal slope vectors per node only. Different approaches for the virtual work of elastic forces are presented: a continuum mechanics based formulation, as well as a structural mechanics based formulation, which is in accordance with classical nonlinear beam finite elements. Since different interpolation functions as in standard ANCF elements are used, a much better convergence rate (up to order four) can be obtained. Therefore, the present element has high potential for application in geometrically nonlinear problems. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Multi-scale modeling of beam-like structures: A new boundary condition concept for the RVE

In this work a coupled two-scale beam model using Timoshenko beam elements [1] with finite displacements on the macro scale and fully non-linear 3D brick elements on the micro scale is proposed. The calculation is carried out with the so-called FE² concept. To achieve the coupling between the beam and the brick elements, the algorithm from [2] is adapted. Within the degenerated concept of the Timoshenko beam, the introduction of a pure shear deformation leads to significant problems concerning the equilibrium condition on the micro scale. Applying this deformation mode on the RVE with periodic boundary conditions results in a rigid body rotation. Using linear displacement boundary conditions instead, the wrapping deformation is suppressed on the boundary, leading to a length dependency in the torsional deformation mode. In addition, the shear forces introduce a bending moment, which depends on the length of the RVE and adds spurious normal stresses and a length dependency of the shear stiffness. To overcome these problems, periodic boundary conditions are applied and the displacement assumptions are modified such that the shear deformation is achieved with force pairs on both ends of the RVE. The resulting model leads to length independent results in tension, bending and torsion and a domain which is able to produce a pure shear stress state. Consequently, only this domain of the model should be homogenized which can be accomplished by modifying the variations in the algorithm [2]. The concept is validated by simple linear and non-linear test problems. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

多层快速多极子方法的快速插值

多层快速多极子方法(MLFMM)可用来加速迭代求解由Maxwell方程组或Helmholtz方程导出的积分方程,其复杂度理论上是O(NlogN),N为未知量个数.MLFMM依赖于快速计算每层的转移项,以及上聚和下推过程中的层间插值.本文引入计算类似N体问题的一维快速多极子方法(FMM1D).基于FMM1D的快速Lagr...     

Transformated Hypergraphs in Synthesis of Vibrating Beam-Systems

In this paper the characteristics of subsystems obtained by the approximate and exact method were compared, because purpose is to answer to the question ­ if the method can be used to nominate the characteristics of mechatronic systems. In this paper frequency-modal analysis have been presented for the mechanical system, that means flexibly vibrating free beam. The model of the beam was presented in the five-vertex hypergraph, which in case of approximate frequency-modal analysis we can imitate in three-vertex hypergraph. Such formulation maybe the introduction to synthesis of flexibly vibration complex beam systems with constant cross-section. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A ROBUST DISCRETIZATION OF THE REISSNER-MINDLIN PLATE WITH ARBITRARY POLYNOMIAL DEGREE

A numerical scheme for the Reissner-Mindlin plate model is proposed.The method is based on a discrete Helmholtz decomposition and can be viewed as a generalization of the nonconforming finite element scheme of Arnold and Falk[SIAM J.Numer.Anal.,26(6):1276-1290,1989].The two unknowns in the discrete formulation are the in-plane rotations and the gradient of the vertical displacement.The decomposition of the discrete shear variable leads to equivalence with the usual Stokes system with penalty term plus two Poisson equations and the proposed method is equivalent to a stabilized discretization of the Stokes system that generalizes the Mini element.The method is proved to satisfy a best-approximation result which is robust with respect to the thickness parameter t.