Conservative integration of rigid body motion by quaternion parameters with implicit constraints

An angular momentum and energy‐conserving time integration algorithm for rigid body rotation is formulated in terms of the quaternion parameters and the corresponding four‐component conjugate momentum vector via Hamilton's equations. The introduction of an extended mass matrix leads to a symmetric set of eight state‐space equations of motion. The extra inertial parameter serves as a multiplier on the kinematic constraint, and it is demonstrated that convergence characteristics are improved by selecting this parameter somewhat larger than the inertial moments. External loads enter these equations via the set of momentum equations. Initially, the normalization of the quaternion array is introduced via a Lagrange multiplier. However, this Lagrange multiplier can be expressed explicitly in terms of the gradient of the external load potential, and elimination of the Lagrange multiplier from the final format leaves only an explicit projection applied to the external load potential gradient. An algorithm is developed by forming a finite increment of the Hamiltonian. This procedure identifies the proper selection of increments and mean values, and leads to an algorithm with conservation of momentum and energy. Implementation, conservation properties, and accuracy of the algorithm are illustrated by examples with a flying box and a spinning top. Copyright © 2012 John Wiley & Sons, Ltd.     

一种结构动力方程的并行直接积分方法

本文提出了一种求解大型结构动力方程的新的并行直接积分方法。该方法在L.Brusa和L.Nigro提出的一步(one-step)直接积分方法的基础上,引进并行运算步骤。并行运算步骤是通过将动力积分方程子结构化,同时进行组集和凝聚实现的。该方法在西安交通大学ELXSI-6400并行机上程序实现,计算结果表明能有效地求解大型结构有限元动力方程的并行直接积分方法。     

Inelastic post buckling behavior of very short column

Abstract

This paper presents an unconditionally stable implicit algorithm for the direct integration of a linear structural dynamic equation of motion. The algorithm is based on two simultaneous difference equations and a weighting factor G for solving displacement at the next time step. Unconditional stability is proven for the weighting factor G in the range from –0.466 to 0.140 in all undamped and damped cases. The unconditionally stable range of G can be extended, from –1.0 to 0.333, for certain types of structure. The amplitude decay and period variation are used as the basic parameters to compare the accuracy of the present algorithm with various other integration methods. A spring‐mass‐dashpot model is applied to illustrate the algorithm for transient and quasi‐static analyses.     

基于三次样条插值的精细积分法

结合指数矩阵的精细算法,提出了一类基于三次样条插值的精细积分方法。针对结构动力学方程一般解中的积分项,考虑在一个时间步长内激励为线性和正余弦两种变化形式,通过对积分项中的指数矩阵进行三次样条插值函数模拟,得到一组新的被积函数,最后通过多次分部积分,构造了一类新的高精度计算格式。在三次样条插值函数构造过程中引入了指数矩阵的精细算法,有效避免了中间过程中有效数字的丢失,同时还有效解决了HPD-F算法中涉及的矩阵求逆问题,大大增加了算法的数值稳定性。数值算例显示了该方法的有效性。     

On time integration of viscoplastic constitutive models suitable for creep

Creep of critical components such as electrical solder connections may occur over long periods of time. Efficient numerical simulations of such problems generally require the use of quasi‐static formulations with conjugate‐gradient techniques for solving the large number of algebraic equations. Implicit in the approach is the need to solve the constitutive equation several times for large time steps and for loading directions that may have no resemblance to the actual solution. Therefore, an unconditionally stable and efficient algorithm for solving the constitutive equation is essential for the overall efficiency of the solution procedure. Unfortunately, constitutive equations suitable for simulating the materials of interest are notoriously difficult to solve numerically and most existing algorithms have a stability limit on the time step which may be several orders of magnitude smaller than the desired time step. Here an algorithm is proposed which is a combination of the use of a trapezoidal rule and an iterative Newton–Raphson method for solving implicitly the non‐linear equations. The key to the success of the proposed approach is to always use an initial guess based on the steady‐state solution to the constitutive equation. A representative viscoplastic constitutive equation is used as a model for illustrating the approach. The algorithm is developed and typical numerical results are provided to substantiate the claim that stability has been achieved. Copyright © 2001 John Wiley & Sons, Ltd.     

A novel accurate and computationally efficient integration approach to viscoplastic constitutive model

A novel, accurate, and computationally efficient integration approach is developed to integrate small strain viscoplastic constitutive equations involving nonlinear coupled first-order ordinary differential equations. The developed integration scheme is achieved by a combination of the implicit backward Euler difference approximation and the implicit asymptotic integration. For the uniaxial loading case, the developed integration scheme produces accurate results irrespective of time steps. For the multiaxial loading case, the accuracy and computational efficiency of the developed integration scheme are better than those of either the implicit backward Euler difference approximation or the implicit asymptotic integration. The simplicity of the developed integration scheme is equivalent to that of the implicit backward Euler difference approximation since it also reduces the solution of integrated constitutive equations to the solution of a single nonlinear equation. The algorithm tangent constitutive matrix derived for the developed integration scheme is consistent with the integration algorithm and preserves the quadratic convergence of the Newton–Raphson method for global iterations.     

Hybrid state‐space time integration in a rotating frame of reference

A time integration algorithm is developed for the equations of motion of a flexible body in a rotating frame of reference. The equations are formulated in a hybrid state‐space, formed by the local displacement components and the global velocity components. In the spatial discretization the local displacements and the global velocities are represented by the same shape functions. This leads to a simple generalization of the corresponding equations of motion in a stationary frame in which all inertial effects are represented via the classic global mass matrix. The formulation introduces two gyroscopic terms, while the centrifugal forces are represented implicitly via the hybrid state‐space format. An angular momentum and energy conserving algorithm is developed, in which the angular velocity of the frame is represented by its mean value. A consistent algorithmic damping scheme is identified by applying the conservative algorithm to a decaying response, which is rendered stationary by an increasing exponential factor that compensates the decay. The algorithmic damping is implemented by introducing forward weighting of the mean values appearing in the algorithm. Numerical examples illustrate the simplicity and accuracy of the algorithm. Copyright © 2011 John Wiley & Sons, Ltd.     

基于Hamilton体系多层各向异性材料表面波传播的数值解法

针对层状各向异性材料,着重研究波传播问题中的表面波。将位移分量和应力分量组合成混合状态向量,使研究体系从Lagrange体系转换成Hamilton体系。用精细算法和扩展的Wittrick-Williams算法求解波传播问题的特征值,并着重对表面波做了分析。给出了该方法在各向同性、多层各向异性材料的数值算例,计算结果具有非常高的精度。      

动力学方程的解析逐步积分法

提出了求解动力学方程的一个新型的逐步积分法。基于动力学方程的解析齐次解,构造出动力学方程解的一般积分表达式,借助于显式、自起动、预测－校正的单步四阶精度的积分算法,离散方程右端的等价荷载项,给出了一个新的解析逐步积分方法格式。如果用分块求解,其刚度阵、质量阵等将有较小的规模,将使计算效率更高。算例表明本文方法比中心差分法、Ｎｅｗｍａｒｋ、Ｗｉｌｓｏｎ－θ、Ｈｏｕｂｏｌｔ法等有较高的精度,本文结果更接近解析解。本文方法也适用于非线性,因为本计算格式是显示,因此不需要迭代求解。     

基于参数二次规划与精细积分方法的动力弹塑性问题分析

给出了将参数二次规划方法与精细积分方法相结合进行结构弹塑性动力响应分析的一条新途径。基于参变量变分原理与有限元参数二次规划方法建立了动力弹塑性问题的求解方程，方法对于关联与非关联问题的求解在算法上是完全一致的。对于动力非线性方程求解则进一步采用了被线性问题分析所广泛采用的精细积分方法，推导了方法在动力弹塑性问题求解上的算法列式。所给出的数值算例在验证本文理论与算法的同时，进一步证实了精细积分方法在动力学分析中所具有的各种良好性态。     

GEOMETRICALLY NON-LINEAR AND ELASTOPLASTIC THREE-DIMENSIONAL SHEAR FLEXIBLE BEAM ELEMENT OF VON-MISES-TYPE HARDENING MATERIAL

A three-dimensional elastoplastic beam element being capable of incorporating large displacement and large rotation is developed and examined. Elastoplastic constitutive equations are applied to the beam element based upon the assumption of small deformational strain leading to a material formulation which is completely objective for the application of stress update procedures. The continuum-type equations of plastic model of J₂ mixed hardening are transformed into the beam equations by satisfying beam hypotheses. An effective stress update algorithm is proposed to integrate elastoplastic rate equations by means of the so-called multistep method which is a method of successive control of residuals on yield surfaces. It avoids severe divergence when the displacement increments become large which is usual for the continuation methods. Material tangent stiffness matrix is derived by using consistent elastoplastic modulus resulting from the integration algorithm and is combined with geometric tangent stiffness matrix. Different from other elements, the present element is shear flexible and can satisfy the plasticity condition in a pointwise fashion. A great number of numerical examples are analysed and compared with the literature. The proposed beam element is verified to be not only quite accurate but also very effective for the analyses of pre-buckling and large deflection collapse of spatial framed structures.     

Consistent linearization and finite element implementation of an incrementally objective canonical form return mapping algorithm for large deformation inelasticity

The present paper focuses on the consistent linearization and finite element implementation of an incrementally objective canonical form return mapping algorithm. A general and modular algorithmic setting, suited for almost any rate constitutive equations, is presented where the finite deformation consistent tangent modulus is obtained as a by‐product of the integration algorithm. Numerical examples illustrate the good performance of the proposed formulation, especially for large deformation increments with noteworthy superimposed rotation, where the consistent formulation converges quadratically in a reasonable number of iterations. Copyright © 2008 John Wiley & Sons, Ltd.     

Automatic consistency integrations for the stress update of J2 elastoplastic rate constitutive equations with combined hardening

A novel method is introduced to study numerical integrations of J₂ elastoplastic rate constitutive equations with general combined hardening. The basic idea is to transform the usual time rate constitutive equations into those with reference to the equivalent plastic strain. By virtue of tensorial matrix operations, we show that these transformed equations may be converted to a linear differential system governing the shifted stress and the plastic multiplier. From this system, we derive explicit integrations for the shifted stress and then for the back stress and the Cauchy stress. We demonstrate that these results are accurate up to within a third order term of the equivalent plastic strain increment. In particular, for pure kinematic hardening, we show that the integrations obtained can achieve automatic enforcement of both the plastic consistency condition and the loading condition, thus bypassing the numerical treatment of the latter two. Furthermore, we explain that, with the new algorithm for the stress update, the continuum tangent moduli may be used to ensure a quadratic rate of convergency in Newton's iteration scheme for the balance equation. Numerical examples suggest that the new algorithm may be more accurate and efficient than the widely used return algorithm. Copyright © 2014 John Wiley & Sons, Ltd.     

Formulation and performance of variational integrators for rotating bodies

Variational integrators are obtained for two mechanical systems whose configuration spaces are, respectively, the rotation group and the unit sphere. In the first case, an integration algorithm is presented for Euler’s equations of the free rigid body, following the ideas of Marsden et al. (Nonlinearity 12:1647–1662, 1999). In the second example, a variational time integrator is formulated for the rigid dumbbell. Both methods are formulated directly on their nonlinear configuration spaces, without using Lagrange multipliers. They are one-step, second order methods which show exact conservation of a discrete angular momentum which is identified in each case. Numerical examples illustrate their properties and compare them with existing integrators of the literature. Financial support for this work has been provided by grant DPI2006-14104 from the Spanish Ministry of Education and Science.     

Explicit multistep time integration for discontinuous elastic stress wave propagation in heterogeneous solids

A multistep explicit time integration algorithm is presented for tracking the propagation of discontinuous stress waves in heterogeneous solids whose subdomain-to-subdomain critical time step ratios range from tens to thousands. The present multistep algorithm offers efficient and accurate computations for tracking discontinuous waves propagating through such heterogeneous solids. The present algorithm, first, employs the partitioned formulation for representing each subdomain, whose interface compatibility is enforced via the method of the localized Lagrange multipliers. Second, for each subdomain, the governing equations of motion are decomposed into the extensional and shear components so that tracking of waves of different propagation speeds is treated with different critical step sizes to significantly reduce the computational dispersion errors. Stability and accuracy analysis of the present multistep time integration is performed with one-dimensional heterogeneous bar. Analyses of the present algorithm are also demonstrated as applied to the stress wave propagation in one-dimensional heterogeneous bar and in heterogeneous plain strain problems.     

A generalized midpoint algorithm for the integration of a generalized plasticity model for sands

This paper describes a method of performing the integration of generalized plasticity models, in which, unlike classical elastoplasticity, the yield surface is not explicitly defined. The algorithm is based on a generalized midpoint scheme and is applied to a specific generalized plasticity model for sands, in which a hyperelastic formulation is introduced to describe the reversible component of the soil response instead of the hypoelastic approach originally proposed. In this way, an efficient integration scheme is developed in the elastic strain space. The consistent, algorithmic tangent operator is derived. Isoerror maps are generated to study the local accuracy of the numerical integration algorithm. Results from a series of numerical examples based on the simulation of drained triaxial tests are given to illustrate the accuracy and convergence properties of the algorithm, both at the local and at the global level. Finally an example is given of the simulation of a cyclic triaxial test to illustrate the improvement on accuracy caused by the use of a hyperelastic law into the constitutive equations, as opposed to the hypoelastic formulation initially adopted in the model. Copyright © 2008 John Wiley & Sons, Ltd.     

无域积分的弹塑性边界元法的非线性互补方法

该文引入非线性互补方法来求解边界元法的弹塑性问题,其中方程组由内部点应力方程和反映塑性本构定律的互补函数形成。涉及的域积分采用径向积分法转化为边界积分。通过受内压的厚壁圆筒的应力、位移和荷载-位移情况表明了该算法的精度。     

State‐space time integration with energy control and fourth‐order accuracy for linear dynamic systems

A fourth‐order accurate time integration algorithm with exact energy conservation for linear structural dynamics is presented. It is derived by integrating the phase‐space representation and evaluating the resulting displacement and velocity integrals via integration by parts, substituting the time derivatives from the original differential equations. The resulting algorithm has an exact energy equation, in which the change of energy is equal to the work of the external forces minus a quadratic form of the damping matrix. This implies unconditional stability of the algorithm, and the relative phase error is of fourth‐order. An optional high‐frequency algorithmic damping is constructed by optimal combination of three different damping matrices, each proportional to either the mass or the stiffness matrix. This leads to a modified form of the undamped algorithm with scalar weights on some of the matrices introducing damping of fourth‐order in the frequency. Thus, the low‐frequency response is virtually undamped, and the algorithm remains third‐order accurate even when algorithmic damping is included. The accuracy of the algorithm is illustrated by an application to pulse propagation in an elastic medium, where the algorithmic damping is used to reduce dispersion due to the spatial discretization, leading to a smooth solution with a clearly defined wave front. Copyright © 2005 John Wiley & Sons, Ltd.     

Assumed strain nodally integrated hexahedral finite element formulation for elastoplastic applications

In this work, a linear hexahedral element based on an assumed strain finite element technique is presented for the solution of plasticity problems. The element stems from the Nodally Integrated Continuum Element (NICE) formulation and its extensions. Assumed gradient operators are derived via nodal integration from the kinematic‐weighted residual; the degrees of freedom are only the displacements at the nodes. The adopted constitutive model is the classical associative von Mises plasticity model with isotropic and kinematic hardening; in particular, a double‐step midpoint integration algorithm is adopted for the integration and solution of the relevant nonlinear evolution equations. Efficiency of the proposed method is assessed through simple benchmark problems and comparison with reference solutions. Copyright © 2014 John Wiley & Sons, Ltd.     

A simple integration algorithm for a non‐associated anisotropic plasticity model for sheet metal forming

In this paper, an anisotropic material model based on a non‐associated flow rule and nonlinear mixed isotropic‐kinematic hardening is developed. The quadratic Hill48 yield criterion is considered in the non‐associated model for both yield function and plastic potential to account for anisotropic behavior. The developed model is integrated based on fully implicit backward Euler's method. The resulting problem is reduced to only two simple scalar equations. The consistent local tangent modulus is obtained by exact linearization of the algorithm. All numerical development was implemented into user‐defined material subroutine for the commercial finite element code ABAQUS/Standard. The performance of the present algorithm is demonstrated by numerical examples. Copyright © 2015 John Wiley & Sons, Ltd.