几何精确NURBS有限元中边界条件施加方式对精度影响的三维计算分析

非均匀有理B样条(NURBS)有限元法把计算机辅助几何设计(CAGD)中的NURBS几何构形方法与有限元方法有机结合起来,有效消除了有限元离散模型的几何误差,提高了计算精度。但是由于NURBS基函数不是插值函数,直接在控制节点上施加位移边界条件会引起较大误差。本文详细讨论了NURBS基函数的插值特性,在NURBS有限元分析中采用罚函数法施加位移边界条件,提高了收敛率和计算精度。结合典型三维弹性力学问题,对两种施加位移边界条件的方法进行了对比和分析。计算结果表明,直接施加位移边界条件会导致收敛率和精度的明显降低,而基于罚函数法的NURBS有限元分析则能达到最优收敛率,并具有更高的精度。     

等几何分析中采用Nitsche法施加位移边界条件

等几何分析使用NURBS基函数统一表示几何和分析模型,消除了传统有限元的网格离散误差,容易构造高阶连续的协调单元.对于结构分析,选择合适的几何参数可以得到光滑的应力解,避免了后置处理的应力磨平.但是由于NURBS基函数不具备插值性,难以直接施加位移边界条件.针对这一问题,提出一种基于Nitsche变分原理的边界位移条件"弱"处理方法,它具有一致稳定的弱形式,不增加自由度,方程组对称正定和不会产生病态矩阵等优点.同时给出方法的稳定性条件,并通过求解广义特征值问题计算稳定性系数.最后,数值算例表明Nitsche方法在h细化策略下能获得最优收敛率,其结果要明显优于在控制顶点处直接施加位移约束.     

IMPOSING DISPLACEMENT BOUNDARY CONDITIONS WITH NITSCHE＇S METHOD IN ISOGEOMETRIC ANALYSIS

等几何分析使用 NURBS 基函数统一表示几何和分析模型, 消除了传统有限元的网格离散误差, 容易构造高阶连续的协调单元. 对于结构分析, 选择合适的几何参数可以得到光滑的应力解, 避免了后置处理的应力磨平. 但是由于 NURBS 基函数不具备插值性, 难以直接施加位移边界条件. 针对这一问题, 提出一种基于 Nitsche 变分原理的边界位移条件“弱”处理方法, 它具有一致稳定的弱形式, 不增加自由度, 方程组对称正定和不会产生病态矩阵等优点. 同时给出方法的稳定性条件, 并通过求解广义特征值问题计算稳定性系数. 最后, 数值算例表明 Nitsche 方法在h细化策略下能获得最优收敛率, 其结果要明显优于在控制顶点处直接施加位移约束.}     

椭圆微孔端面机械密封泄漏率与几何收敛点的关系

采用解析法和数值计算相结合的方法,解释椭圆微孔端面机械密封的方向角所引起的几何特性改变如何影响到泄漏率.首先,得到了不同方向角下,椭圆沿速度方向上的几何收敛点的解析解.然后,基于质量守恒的JFO空化边界条件建立数值计算模型,分析了液膜的压力分布,并使用最高压力点的坐标来近似表征高压区的位置,得到了方向角与最高压力点的关系.接着,通过对比,发现几何收敛点和最大压力点位置一致,并分析了原因.最后,分析了在不同结构参数和操作参数下方向角对泄漏率的影响规律,通过分析得到,方向角的改变,使几何收敛点的位置发生改变,从而改变高低压区的分布,当高压区靠近泄漏出口时,泄漏率大;当高压区远离泄漏出口时,泄漏率小.     

An Augmented Formulation for Mechanical Systems with Non-Generalized Coordinates: Application to Rigid Body Contact Problems

In computational multibody algorithms, the kinematic constraintequations that describe mechanical joints and specified motiontrajectories must be satisfied at the position, velocity andacceleration levels. For most commonly used constraint equations, onlyfirst and second partial derivatives of position vectors with respect tothe generalized coordinates are required in order to define theconstraint Jacobian matrix and the first and second derivatives of theconstraints with respect to time. When the kinematic and dynamicequations of the multibody systems are formulated in terms of a mixedset of generalized and non-generalized coordinates, higher partialderivatives with respect to these non-generalized coordinates arerequired, and the neglect of these derivatives can lead to significanterrors. In this paper, the implementation of a contact model in generalmultibody algorithms is presented as an example of mechanical systemswith non-generalized coordinates. The kinematic equations that describethe contact between two surfaces of two bodies in the multibody systemare formulated in terms of the system generalized coordinates and thesurface parameters. Each contact surface is defined using twoindependent parameters that completely define the tangent and normalvectors at an arbitrary point on the body surface. In the contact modeldeveloped in this study, the points of contact are searched for on lineduring the dynamic simulation by solving the nonlinear differential andalgebraic equations of the constrained multibody system. It isdemonstrated in this paper that in the case of a point contact andregular surfaces, there is only one independent generalized contactconstraint force despite the fact that five constraint equations areused to enforce the contact conditions.     

Analyses of wrinkled and slack membranes through an error reproducing mesh-free method

A mesh-free approximation of large deformations of flexible membrane structures within the tension field theory is considered in this paper. A modification of the wrinkling theory, originally proposed by Roddeman et al. (1987) [Roddeman, D.G., Drukker, J., Oomens, C.W.J., Janssen, J.D., 1987, The wrinkling of thin membranes: Part I—theory; Part II—numerical analysis. ASME J. Appl. Mech. 54, 884–892.], is proposed to study the behaviour of an isotropic membrane under the mixed state of stress (taut, wrinkled and slack). Using the facts that the state of stress is not uniform across an element and that the deformation gradient is a spatially continuous (and possibly non-differentiable) tensor, the proposed model uses a continuously modified deformation gradient to capture the location and orientation of wrinkles more precisely. While the deformation gradient need not be everywhere-differentiable in a wrinkled membrane, it is argued that the fictive non-wrinkled (non-slack) surface may be looked upon as an everywhere-taut surface in the limit as the minor (and major) principal tensile stresses over the wrinkled (slack) portions go to zero. Accordingly, the modified deformation gradient is thought of as the limit of a sequence of everywhere-differentiable tensors. The weighted residual from the governing equations are presently solved via a mesh-free method, where the entire domain is discretized only by a set of grid points. A non-uniform-rational-B-spline (NURBS) based error reproducing kernel method (ERKM) has been used to approximate the field variable over the domain. The first step in the method is to approximate a function and its derivatives through NURBS basis functions. However, since NURBS functions neither reproduce any polynomial nor interpolate the grid points (also referred to as control or nodal points), the approximated functions result in uncontrolled errors over the domain including the grid points. Accordingly the error functions in the NURBS approximation and its derivatives are reproduced via a family of non-NURBS basis functions. The non-NURBS basis functions are constructed using a polynomial reproduction condition and added to the NURBS approximation of the function obtained in the first step. Several numerical examples on wrinkled and/or slack membranes are also provided.     

Hamiltonian structures of dynamics of a gyrostat in a gravitational field

The dynamics of a gyrostat in a gravitational field is a fundamental problem in celestial mechanics and space engineering. This paper investigates this problem in the framework of geometric mechanics. Based on the natural symplectic structure, non-canonical Hamiltonian structures of this problem are derived in different sets of coordinates of the phase space. These different coordinates are suitable for different applications. Corresponding Poisson tensors and Casimir functions, which govern the phase flow and phase space structures of the system, are obtained in a differential geometric method. Equations of motion, as well as expressions of the force and torque, are derived in terms of potential derivatives. We uncover the underlying Lie group framework of the problem, and we also provide a systemic approach for equations of motion. By assuming that the gravitational field is axis-symmetrical and central, SO(2) and SO(3) symmetries are introduced into the general problem respectively. Using these symmetries, we carry out two reduction processes and work out the Poisson tensors of the reduced systems. Our results in the central gravitational filed are in consistent with previous results. By these reductions, we show how the symmetry of the problem affects the phase space structures. The tools of geometric mechanics used here provide an access to several powerful techniques, such as the determination of relative equilibria on the reduced system, the energy-Casimir method for determining the stability of equilibria, the variational integrators for greater accuracy in the numerical simulation and the geometric control theory for control problems.     

On the Use of NURBS Functions for Displacement Derivatives Measurement by Digital Image Correlation

In this paper, we propose to investigate the potential improvement of using Non-Uniform Rational B-Spline (NURBS) functions for displacement measurements by digital image correlation (DIC). The aim is at improving the performance of DIC to capture with low uncertainty and low noise levels not only the displacement field but also its derivatives. Indeed, when the displacement field is used to feed constitutive law identification procedures, displacement derivatives are required and thus may be measured with robustness. Two examples illustrate the potential of NURBS for DIC: a compressive test on a wood sample and a bending test on a steel beam. For the latter, beam kinematics are adopted and NURBS are used in order to capture the variation of the curvature (second derivative of the displacement) along the beam axis. For these two examples, an error study based on a decomposition of the error into the correlation error and the interpolation error, is carried out and shows the great potential of NURBS functions for DIC.     

Design and Implementation of a Pointer System Controller

The development of a pointer system as used for space structures, is discussed. The pointer has to keep its signal on a stationary target. The pointer itself is mounted on a supporting system which is excited in a plane either parallel or normal to the pointer-target beam. A fuzzy logic control system has been adopted due to the geometric nonlinearities of the problem.     

Geometry and differentiability requirements in multibody railroad vehicle dynamic formulations

The dynamic equations of multibody railroad vehicle systems can be formulated using different sets of generalized coordinates; examples of these sets of coordinates are the absolute Cartesian and trajectory coordinates. The absolute coordinate based formulations do not require introducing an intermediate track coordinate system since all the absolute coordinates are defined in the global system. On the other hand, when the trajectory coordinates are used, a track coordinate system that follows the motion of a body in the railroad vehicle system is introduced. This track coordinate system is defined by the track geometry and the distance traveled by the body along the track centerline. The configuration of the body with respect to the track coordinate system is defined using five coordinates; two translations and three Euler angles. In this paper, the formulations based on the absolute and trajectory coordinates are compared. It is shown that these two sets of coordinates require different degrees of differentiability and smoothness. When an elastic contact formulation is used to study the wheel/rail dynamic interaction, there are significant differences in the order of the derivatives required in both formulations. In fact, as demonstrated in this study, in the absence of a contact constraint formulation, higher order derivatives with respect to geometric parameters are still required when the equations are formulated using the trajectory coordinates. The formulation of the constraints used in the analysis of the wheel/rail contact is discussed and it is shown that when the absolute coordinates are used, only third order derivatives need to be evaluated. The relationship between the track frame used in railroad vehicle dynamics and the Frenet frame used in the theory of curves to describe the curve geometry is also discussed in this paper. Based on the analysis presented in this paper, the advantages and drawbacks of a hybrid method which employs both the absolute and trajectory coordinates and planar contact conditions in order to reduce the number of contact constraints and relax the differentiability requirements are discussed. In this method, the absolute coordinates are used to formulate the equations of motion of the railroad vehicle system. The absolute coordinate solution can be used to determine the trajectory coordinates and their time derivatives. Using the trajectory coordinates, the motion of the body in the vehicle with respect to the track coordinate system can be predicted and used in the formulation of the planar contact model.     

Dynamic analysis of geometrically nonlinear robot manipulators

In this paper, a method for the dynamic analysis of geometrically nonlinear elastic robot manipulators is presented. Robot arm elasticity is introduced using a finite element method which allows for the gross arm rotations. A shape function which accounts for the combined effects of rotary inertia and shear deformation is employed to describe the arm deformation relative to a selected component reference. Geometric elastic nonlinearities are introduced into the formulation by retaining the quadratic terms in the strain-displacement relationships. This has lead to a new stiffness matrix that depends on the rotary inertia and shear deformation and which has to be iteratively updated during the dynamic simulation. Mechanical joints are introduced into the formulation using a set of nonlinear algebraic constraint equations. A set of independent coordinates is identified over each subinterval and is employed to define the system state equations. In order to exemplify the analysis, a two-armed robot manipulator is solved. In this example, the effect of introducing geometric elastic nonlinearities and inertia nonlinearities on the robot arm kinematics, deformations, joint reaction forces and end-effector trajectory are investigated.     

Experimental analysis of nonlinear resonances in piezoelectric plates with geometric nonlinearities

Piezoelectric devices with integrated actuation and sensing capabilities are often used for the development of electromechanical systems. The present paper addresses experimentally the nonlinear dynamics of a fully integrated circular piezoelectric thin structure, with piezoelectric patches used for actuation and other for sensing. A phase-locked loop control system is used to measure the resonant periodic response of the system under harmonic forcing, in both its stable and unstable parts. The single-mode response around a symmetric resonance as well as the coupled response around an asymmetric resonance, involving two companion modes in 1:1 internal resonance, is accurately measured. For the latter, a particular location of the patches and additional signal processing is proposed to spatially discriminate the response of each companion mode. In addition to a hardening behavior associated with geometric nonlinearities of the plate, a softening behavior predominant at low actuation amplitudes is observed, resulting from the material piezoelectric nonlinearities.

     

A unified nonlinear formulation for plate and shell theories

A general geometrically exact nonlinear theory for the dynamics of laminated plates and shells under-going large-rotation and small-strain vibrations in three-dimensional space is presented. The theory fully accounts for geometric nonlinearities by using the new concepts of local displacements and local engineering stress and strain measures, a new interpretation and manipulation of the virtual local rotations, an exact coordinate transformation, and the extended Hamilton principle. Moreover, the model accounts for shear coupling effects, continuity of interlaminar shear stresses, free shear-stress conditions on the bonding surfaces, and extensionality. Because the only differences among different plates and shells are the initial curvatures of the coordinates used in the modeling and all possible initial curvatures are included in the formulation, the theory is valid for any plate or shell geometry and contains most of the existing nonlinear and shear-deformable plate and shell theories as special cases. Five fully nonlinear partial-differential equations and corresponding boundary and corner conditions are obtained, which describe the extension-extension-bending-shear-shear vibrations of general laminated two-dimensional structures and display linear elastic and nonlinear geometric coupling among all motions. Moreover, the energy and Newtonian formulations are completely correlated in the theory.     

Multipatch discontinuous Galerkin isogeometric analysis of composite laminates

This paper is concerned with the isogeometric analysis (IGA) of composite laminates under cylindrical bending. Non-uniform rational B-splines (NURBS) are employed as basis functions for both geometric and computational implementations. In order to account for multiple domains, each lamina is modeled as a single NURBS patch. This multipatch representation corresponds to decomposition of the computational domain (composite laminate) into non-overlapping subdomains. As NURBS patches are discontinuous across their boundaries, a standard FEA-like procedure does not work for multipatch IGA; an additional numerical technique is required for coupling NURBS patches. Therefore, in this paper, one of the discontinuous Galerkin (DG) methods, namely symmetric interior penalty Galerkin formulation, is employed to allow for discontinuities. For numerical calculations, a composite laminate with stacking sequences $$0^{\circ }{/}90^{\circ }$$ and $$0^{\circ }{/}90^{\circ }{/}0^{\circ }$$, respectively, is adopted. The stresses are calculated along the thickness of the composite laminate, subjected to a sinusoidal load, and they are compared with the analytical solutions. It is shown that DG–IGA gives a better approximation in comparison with the standard IGA.     

边界面法分析三维实体线弹性问题

本文利用以边界积分方程为理论基础的边界面法分析三维实体的线弹性问题。在该方法中,边界积分和场变量插值都是在实体边界曲面的参数空间里进行。积分点的几何数据,如坐标、雅可比、外法向量都是直接由曲面算得,而不是通过单元插值近似,从而避免了几何误差。另外,该方法的实现是直接基于CAD模型中的边界表征数据结构,可以做到与CAD系统无缝集成。在分析中,避免对结构作几何上的简化,结构的所有局部细节都按照实际形状尺寸作为三维实体处理。应用实例表明,本文方法可以简单有效地模拟具有细小特征的复杂结构,可以直接基于三维弹性理论求解薄型壳体结构,可以获得比有限元法更精确的计算结果。     

Chebyshev spectral variational integrator and applications

The Chebyshev spectral variational integrator(CSVI) is presented in this paper. Spectral methods have aroused great interest in approximating numerically a smooth problem for their attractive geometric convergence rates. The geometric numerical methods are praised for their excellent long-time geometric structure-preserving properties.According to the generalized Galerkin framework, we combine two methods together to construct a variational integrator, which captures the merits of both methods. Since the interpolating points of the variational integrator are chosen as the Chebyshev points,the integration of Lagrangian can be approximated by the Clenshaw-Curtis quadrature rule, and the barycentric Lagrange interpolation is presented to substitute for the classic Lagrange interpolation in the approximation of configuration variables and the corresponding derivatives. The numerical float errors of the first-order spectral differentiation matrix can be alleviated by using a trigonometric identity especially when the number of Chebyshev points is large. Furthermore, the spectral variational integrator(SVI) constructed by the Gauss-Legendre quadrature rule and the multi-interval spectral method are carried out to compare with the CSVI, and the interesting kink phenomena for the Clenshaw-Curtis quadrature rule are discovered. The numerical results reveal that the CSVI has an advantage on the computing time over the whole progress and a higher accuracy than the SVI before the kink position. The effectiveness of the proposed method is demonstrated and verified perfectly through the numerical simulations for several classical mechanics examples and the orbital propagation for the planet systems and the Solar system.