轴向均布载荷下压杆稳定问题的DQ解

叙述了微分求积法(differential quadrature method)的一般方法，研究用微分求积法求解在均布轴向载荷下细长杆的稳定问题．通过Newton-Raphson法求解非线性方程组，以及对问题进行线性假设后求解广义特征值方程，得到了精度很高的后屈曲挠度数值和临界载荷数值．与解析解和其他近似解相比，微分求积法具有较高的精度和简便性．     

A numerical analysis of partial slip problems under Hertzian contacts

In this study the tangential partial slip problems in Hertzian contact regions are treated by a numerical technique. The tangential loading may include tangential forces in the contact plane and a twisting moment normal to the contact plane. The Coulomb’s law of friction and the property that the direction of friction must oppose the relative motion lead to nonlinear equations. The Newton-Raphson method is utilized to solve these nonlinear equations. Numerical results for tangential tractions and sizes of stick and slip zones may be determined, and they agree with existing analytical results for circular contacts.     

力-热载荷下双金属复合管的屈曲失效研究

采用有限元方法研究了力-热载荷下双金属复合管的屈曲失效行为,通过三维有限元建模考虑了双金属复合管的准静态复合成型制造过程中产生的残余应力,分析了温度及内压两个主要参数对双金属复合管屈曲失效的影响。结果表明,高温导致材料发生软化,抑制了双金属复合管的屈曲;弯矩、内压及热载荷联合作用下,复合管内介质温度降低,复合管弯矩达到最大值对应的曲率减小,而弯曲承载能力增大,外基管的椭圆率也增大;内压变化对复合管的弯曲承载能力和外基管的椭圆率影响较小。     

On tensile instabilities and ellipticity loss in fiber-reinforced incompressible non-linearly elastic solids

Loss of ellipticity and associated failure in fiber-reinforced non-linearly elastic solids is examined for uniaxial plane deformations. We consider separately fiber reinforcement that either endows the material with additional stiffness only in the fiber direction or introduces additional stiffness under shear deformations. In the first case it is shown that loss of ellipticity under tensile loading in the fiber direction corresponds to a turning point of the nominal stress and requires concavity of the Cauchy stress–stretch curve. For the second example loss of ellipticity occurs after the nominal stress maximum and prior to a turning point of the Cauchy stress.     

Non-linear analysis of structures using two-point method

Non-linear algebraic equations must be solved by an iterative method, the non-linear equations being linearized by evaluating the non-linear terms with the known solution from the preceding iteration. The Newton-Raphson method, which is based on the Taylor series expansion and uses the tangent stiffness matrix, has been extensively used to solve non-linear problems. In this paper, a new Newton-Raphson algorithm is developed for analyses involving non-linear behavior. Our method, here named as a two-point method, is constructed as a predictor-corrector one, most frequently taking Newton's method in the first iteration. It should be noted that our concern in this research ignores the problem of passing limit points. The presented method incorporates the known information at each stage of the loading process to determine the subsequent unknown variables. Compared with the classic Newton-Raphson algorithm, it offers a strategy that can be deployed to reduce both the number of the iterations and the computing time involved in non-linear analysis of structures.     

Application of quasilinearization technique to the vertical channel flow and heat convection

The quasilinearization technique, also known as the generalized Newton-Raphson method, is employed to solve the problem of fluid injection through one side of a long vertical channel. With very approximate starting values, only a few iterations are needed to obtain a result with high accuracy. The results are compared with those obtained by the perturbation technique. Although the perturbation technique is perhaps the simplest and the most straight-forward method to achieve solutions to non-linear boundary value problems, the limitation on the magnitude of perturbed parameters hinders its applicability.     

优化迭代步长的两种改进增量谐波平衡法

增量谐波平衡法(IHB法)是一个半解析半数值的方法, 其最大优点是适合于强非线性系统振动的高精度求解. 然而, IHB法与其他数值方法一样, 也存在如何选择初值的问题, 如初值选择不当, 会存在不收敛的情况. 针对这一问题, 本文提出了两种基于优化算法的IHB法: 一是结合回溯线搜索优化算法(BLS)的改进IHB法(GIHB1), 用来调节IHB法的迭代步长, 使得步长逐渐减小满足收敛条件; 二是引入狗腿算法的思想并结合BLS算法的改进IHB法(GIHB2), 在牛顿-拉弗森(Newton-Raphson)迭代中引入负梯度方向, 并在狗腿算法中引入2个参数来调节BSL搜索方式用于调节迭代的方式, 使迭代方向沿着较快的下降方向, 从而减少迭代的步数, 提升收敛的速度. 最后, 给出的两个算例表明两种改进IHB法在解决初值问题上的有效性.      

Nonlinear Antiplane Deformation of an Elastic Body

The antiplane elastic deformation of a homogeneous isotropic prestretched cylindrical body is studied in a nonlinear formulation in actual–state variables under incompressibility conditions, the absence of volume forces, and under constant lateral loading along the generatrix. The boundary–value problem of axial displacement is obtained in Cartesian and complex variables and sufficient ellipticity conditions for this problem are indicated in terms of the elastic potential. The similarity to a plane vortex–free gas flow is established. The problem is solved for Mooney and Rivlin—Sonders materials simulating strong elastic deformations of rubber–like materials. Axisymmetric solutions are considered.     

Improving convergence of incremental harmonic balance method using homotopy analysis method

We have deduced incremental harmonic balance an iteration scheme in the （IHB） method using the harmonic balance plus the Newton-Raphson method. Since the convergence of the iteration is dependent upon the initial values in the iteration, the convergent region is greatly restricted for some cases. In this contribution, in order to enlarge the convergent region of the IHB method, we constructed the zeroth-order deformation equation using the homotopy analysis method, in which the IHB method is employed to solve the deformation equation with an embedding parameter as the active increment. Taking the Duffing and the van der Pol equations as examples, we obtained the highly accurate solutions. Importantly, the presented approach renders a convenient way to control and adjust the convergence.     

A note on the pure torsion of a circular cylinder for a compressible nonlinearly elastic material with nonconvex strain-energy

The large deformation torsion problem for an elastic circular cylinder subject to prescribed twisting moments at its ends is examined for a particular homogeneous isotropic compressible material, namely the Blatz-Ko material. For this material, the displacement equations of equilibrium in three-dimensional elastostatics can lose ellipticity at sufficiently large deformations. For the torsion problem, it is shown that this occurs when the prescribed torque reaches a critical value. For values of the twisting moment greater than this critical value, there is an axial core of the cylinder on which ellipticity holds, surrounded by an annular region where loss of ellipticity has occurred. The physical implications in terms of localized shear bands are briefly discussed.     

Determining the Newton-Raphson basins of attraction in the electromagnetic Copenhagen problem

The Copenhagen problem where the primaries of equal masses are magnetic dipoles is used in order to determine the Newton-Raphson basins of attraction associated with the equilibrium points. The parametric variation of the position as well as of the stability of the Lagrange points are monitored when the value of the ratio λ of the magnetic moments varies in predefined intervals. The regions on the configuration (x,y) plane occupied by the basins of convergence are revealed using the multivariate version of the Newton-Raphson iterative scheme. The correlations between the basins of attraction of the libration points and the corresponding number of iterations needed for obtaining the desired accuracy are also illustrated. We perform a thorough and systematic numerical investigation by demonstrating how the dynamical quantity λ influences the shape, the geometry and also the degree of fractality of the attracting domains. Our numerical results strongly indicate that the ratio λ is indeed a very influential parameter in the electromagnetic binary system.     

Strong-form framework for solving boundary value problems with geometric nonlinearity

In this paper, we present a strong-form framework for solving the boundary value problems with geometric nonlinearity, in which an incremental theory is developed for the problem based on the Newton-Raphson scheme. Conventionally, the finite element methods (FEMs) or weak-form based meshfree methods have often been adopted to solve geometric nonlinear problems. However, issues, such as the mesh dependency, the numerical integration, and the boundary imposition, make these approaches computationally inefficient. Recently, strong-form collocation methods have been called on to solve the boundary value problems. The feasibility of the collocation method with the nodal discretization such as the radial basis collocation method (RBCM) motivates the present study. Due to the limited application to the nonlinear analysis in a strong form, we formulate the equation of equilibrium, along with the boundary conditions, in an incremental-iterative sense using the RBCM. The efficacy of the proposed framework is numerically demonstrated with the solution of two benchmark problems involving the geometric nonlinearity. Compared with the conventional weak-form formulation, the proposed framework is advantageous as no quadrature rule is needed in constructing the governing equation, and no mesh limitation exists with the deformed geometry in the incremental-iterative process.     

On the non-oscillation criterion for multiplicative anisotropic plasticity at large simple shear deformation

The criterion for non-oscillatory stresses under monotonic large simple shear deformation in the context of multiplicative anisotropic plasticity is discussed. In particular, evolving anisotropy combined with a Hill type of yield criterion is considered. It is shown that a sufficient, but not necessary, criterion for a non-oscillatory stress is ellipticity of the first Piola–Kirchhoff stress. Loss of ellipticity corresponds to a critical value h_cr of the generalized plastic modulus. Similarly, the absence of limit points on the stress–strain relation motivates an alternative criterion in terms of a critical value h_sh  h_cr. Finally, this criterion is demonstrated analytically as well as numerically for an important class of models with evolving anisotropy of the saturation type.     

索网天线找形分析的参变量变分有限元法

针对柔性空间索网天线的非线性特点,建立了基于参变量变分描述索网拉压非线性和共旋列式描述几何非线性的有限元控制方程,应用Lernke与改进牛顿法进行求解.通过对索网预张力平衡计算,证明改进牛顿法比Newton-Raphson法具有更强的收敛能力.进一步将力密度法迭代原理与有限元法结合应用于索网天线的非线性找形分析中,获得了理想的索网构型.本文的索网找形方法可广泛应用于空间索网天线结构的设计.     

接触分析的光滑模型及迭代算法

利用变分不等式和基于信息熵的凝聚函数把有摩擦接触问题模型化为一个标准的凸二次规划问题,极大地简化了这一复杂的问题,同时引入摩擦方向约束并构造了以无摩擦解为初值的迭代算法,在较摩擦系数时计算也能保证收敛,算例表明算法高效可靠。     

Finite element method for viscoelastic flows based on the discrete adaptive viscoelastic stress splitting and the discontinuous Galerkin method: DAVSS-G/DG

Accurate and robust finite element methods for computing flows with differential constitutive equations require approximation methods that numerically preserve the ellipticity of the saddle point problem formed by the momentum and continuity equations and give numerically stable and accurate solutions to the hyperbolic constitutive equation. We present a new finite element formulation based on the synthesis of three ideas: the discrete adaptive splitting method for preserving the ellipticity of the momentum/continuity pair (the DAVSS formulation), independent interpolation of the components of the velocity gradient tensor (DAVSS-G), and application of the discontinuous Galerkin (DG) method for solving the constitutive equation. We call the method DAVSS-G/DG. The DAVSS-G/DG method is compared with several other methods for flow past a cylinder in a channel with the Oldroyd-B and Giesekus constitutive models. Results using the Streamline Upwind Petrov–Galerkin method (SUPG) show that introducing the adaptive splitting increases considerably the range of Deborah number (De) for convergence of the calculations over the well established EVSS-G formulation. When both formulations converge, the DAVSS-G and DEVSS-G methods give comparable results. Introducing the DG method for solution of the constitutive equation extends further the region of convergence without sacrificing accuracy. Calculations with the Oldroyd-B model are only limited by approximation of the almost singular gradients of the axial normal stress that develop near the rear stagnation point on the cylinder. These gradients are reduced in calculations with the Giesekus model. Calculations using the Giesekus model with the DAVSS-G/DG method can be continued to extremely large De and converge with mesh refinement.     

混凝土本构模型及其动态有限元算法研究

混凝土的力学性质及其本构研究是当前力学领域的一个重要课题。混凝土的实验数据表明其力学性质表现出很强的各向异性及并且在高加载速率的情况下，混凝土的性质物美价廉 时时明显不同。这就要求我们采用应变率相关的弹粘弹性本构模型来描述混凝土的力学行为。本语文从Ｏｔｔｏｓｅｎ的四参数混凝土模型出发，引进损伤和应变率的影响，参考关联塑性的理论，改进了混凝土的本构模型。为了把混凝土的数值模拟推向实用，本文概括混凝土     

AN APPLICATION OF THE CBS SCHEME IN THE FLUID-MEMBRANE INTERACTION

提出了一种不可压缩流体与弹性薄膜耦合问题的特征线分裂有限元解法. 首先, 给出了流场和结构的控制方程. 然后, 对流场、结构以及流固耦合的具体求解过程进行了描述. 其中, 流场求解采用改进特征线分裂方法和双时间步方法相结合的隐式求解方式, 并利用艾特肯加速法对每个时间步的迭代收敛过程进行了加速处理;结构部分的空间离散和时间积分分别采用伽辽金有限元方法和广义方法, 并通过牛顿迭代法对所得非线性代数方程组进行了求解;流场网格的更新采用弹簧近似法;流场、结构两求解模块之间采用松耦合方式.最后, 采用该方法对具有弹性底面的方腔顶盖驱动流问题进行了求解, 验证了算法的准确性和稳定性.此外, 计算结果表明艾特肯加速法可以显著地提高双时间步方法迭代求解过程的收敛速度.     

基于牛顿-拉夫逊的拉压不同模量问题的数值求解

针对拉压模量不同引起的材料本构非线性,首先,通过引入改进的Heaviside函数将本构方程连续光滑化;然后,基于特征值与特征向量的求导策略,推导有限元求解模型中切线刚度矩阵的列式;最终,提出基于牛顿-拉夫逊迭代格式的拉压不同模量问题的数值求解算法。数值算例验证了本文算法比传统算法具有更稳定的收敛性和更高的求解精度,特别适合于工程分析中大规模计算问题的求解。     

On the Eigenvalues of the Fourth-Order Constitutive Tensor and Loss of Strong Ellipticity in Elastoplasticity

The eigenvalues of the fourth-order constitutive tangent modulus and the corresponding acoustic tensors are analyzed. Explicit expressions of the eigenvalues are made for the nonsymmetric tangent modulus tensor, and in the case of the deviatoric associative rule for the symmetric part of the tangent modulus and its acoustic tensor. In this context, a rate independent infinitesimal elastoplastic model is considered. The expressions of the plastic hardening modulus are summarized for the different local stability criteria (loss of second order work positiveness, loss of ellipticity, and loss of strong ellipticity). The critical hardening modulus and orientation are discussed in detail in the case of loss of ellipticity and loss of strong ellipticity. This analysis is based on the geometric method and linear, isotropic elasticity and deviatoric associative flow rule. In particular, the critical orientation for the loss of strong ellipticity and the classical shear band localization are compared.