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.     

On a new integration scheme for von‐Mises plasticity with linear hardening

Limiting the discussion to an associative von‐Mises plasticity model with linear kinematic and isotropic hardening, we compare the performance of the classical radial return map algorithm with a new integration scheme based on the computation of an integration factor. The numerical examples clearly show the improved accuracy of the new method. Copyright © 2003 John Wiley & Sons, Ltd.     

一种改进的推广自适应相干累积算法

论文主要研究了LMS算法、ACI算法、GACI算法的原理和性能；并对GACI算法加以改进，以降低其参数对背景噪声变化的敏感性；将算法用来提取高斯白噪声背景下的线性调频脉冲信号，并用软件仿真器进行仿真。仿真结果表明了改进的GACI算法的优越性——参数对背景噪声功率变化的敏感性明显降低，并证明了其可靠性、可行性以及工程可实现性。     

A comprehensive implicit substepping integration scheme for multisurface plasticity

A complex elastoplastic model requires a robust integration procedure of the evolution equations. The performance of the finite element solution is directly affected by the convergence characteristics of the state-update procedure. Thereby, this study proposes a comprehensive numerical integration scheme to deal with generic multisurface plasticity models. This algorithm is based on the backward Euler method aiming at accuracy and stability, and on the Newton–Raphson method to solve the unconstrained optimization problem. In this scenario, a line search strategy is adopted to improve the convergence characteristics of the algorithm. The golden section method, an exact line search, is considered. Also, a substepping scheme is implemented to provide additional robustness to the state-update procedure. Therefore, this work contributes to computational plasticity presenting an adaptive substep size scheme and a consistent tangent modulus according to the substepping technique. Finally, some numerical problems are evaluated using the proposed algorithm. Single-surface and novel multisurface plasticity models are employed in these analyses. The results testify how the line search and substepping strategies can improve the robustness of the nonlinear analysis.     

A modified cutting-plane time integration scheme with adaptive substepping for elasto-viscoplastic models

Integration of stress-strain-time relationship is a key issue for the application of elasto-viscoplastic models to engineering practice. This article presents a novel adaptive substepping cutting-plane time integration scheme for elasto-viscoplastic models keeping the advantage of original cutting-plane (OCP) with only the first derivatives of loading surface required. The deficiency of OCP time integration algorithm is first discussed taking a simple overstress theory based elasto-viscoplastic modified Cam-Clay model (EVP-MCC) as example. To overcome this, a new algorithm is developed with three features: (1) an evolution function for the hardening variable of dynamic loading surface is innovatively deduced for the Taylor series approximation, (2) the elastic predictor is modified to account for the initial viscoplastic strain rate with more accuracy, and (3) a new adaptive substepping technique for restricting simultaneously both strain and time incremental sizes based on the overstress distance is proposed. For easy understanding, the proposed algorithm is first presented for one-dimensional condition, and then extended to three-dimensional condition. The new integrated EVP-MCC model using the proposed algorithm is examined by simulating laboratory tests at both levels of integration point and finite element with a good performance in terms of accuracy and convergence.     

Accuracy of a composite implicit time integration scheme for structural dynamics

A comprehensive study of the two sub‐steps composite implicit time integration scheme for the structural dynamics is presented in this paper. A framework is proposed for the convergence accuracy analysis of the generalized composite scheme. The local truncation errors of the acceleration, velocity, and displacement are evaluated in a rigorous procedure. The presented and proved accuracy condition enables the displacement, velocity, and acceleration achieving second‐order accuracy simultaneously, which avoids the drawback that the acceleration accuracy may not reach second order. The different influences of numerical frequencies and time step on the accuracy of displacement, velocity, and acceleration are clarified. The numerical dissipation and dispersion and the initial magnitude errors are investigated physically, which measure the errors from the algorithmic amplification matrix's eigenvalues and eigenvectors, respectively. The load and physically undamped/damped cases are naturally accounted. An optimal algorithm‐Bathe composite method (Bathe and Baig, 2005; Bathe, 2007; Bathe and Noh, 2012) is revealed with unconditional stability, no overshooting in displacement, velocity, and acceleration, and excellent performance compared with many other algorithms. The proposed framework also can be used for accuracy analysis and design of other multi‐sub‐steps composite schemes and single‐step methods under physical damping and/or loading. Copyright © 2016 John Wiley & Sons, Ltd.     

Extended overstress model and its implicit stress integration algorithm: Formulations,experiments, and simulations

The extended overstress model is formulated based on the subloading surface model with the smooth elastic-inelastic transition, which is called the subloading-overstress model. Therein, the rigorous translation rules of the elastic-core and the similarity-center, the limitation in the expansion of the subloading surface, and so forth are incorporated. The model possesses the basic structure capable of describing the monotonic/cyclic loading behaviors at the general rate of deformations from the quasi-static to the impact loading. The experiments were conducted using the spheroidal graphite cast iron under the various loading conditions. It was verified that the experimental results can be simulated accurately by the subloading-overstress model. Further, the complete implicit stress integration algorithm based on the return-mapping projection is formulated for the present subloading-overstress model and implemented into Abaqus through UMAT. Then, the deformation analyses of the R-notched cylinder were performed by the present algorithm. Consequently, the performability of the present algorithm is verified by the analyses of the boundary-value problem under the cyclic loadings.     

Numerical integration of lattice rotation in polycrystal plasticity

A numerical algorithm to calculate the lattice rotation in polycrystal plasticity models, that is based on the analytical integration of the lattice rotation update equation, is presented. Implicit and explicit implementations perform equally well and are superior to the currently used numerical integration schemes. Copyright © 2001 John Wiley & Sons, Ltd.     

On the numerical integration of an advanced Gurson model

This article is focused on a new extended version of Gurson's model (J. Eng. Mater. Technol. 1977; 99 :2–15), its numerical integration scheme and its consistent tangent matrix being within an FE code. First, this new advanced Gurson model is proposed, which is an extension of the original to take into account plastic anisotropy and mixed (isotropic+kinematic) hardening. In this paper, only the growth phase of cavities is considered (the nucleation of new voids is ignored). Second, a new numerical algorithm for the integration of this new Gurson model is presented. The algorithm is implicit in all variables and is unconditionally stable. This algorithm is generic and could be used for other anisotropic yield functions and other hardening laws. Third, the consistent tangent matrix is computed in an explicit way by exact linearization of the constitutive equations. To check its efficiency and robustness, the proposed integration algorithm is compared, under some simplified assumptions and choices, with the algorithms of Aravas (Int. J. Numer. Meth. Engng 1987; 24 :1395–1416) and Kojic (Int. J. Numer. Meth. Engng 2002; 53 (12):2701–2720). The performance of the developed consistent modulus, compared to other techniques for the computation of the tangent matrix is assessed. The paper ends with numerical simulations of tensile tests on homogeneous and notched specimens. Copyright © 2010 John Wiley & Sons, Ltd.     

Generalized midpoint integration algorithms for J2 plasticity with linear hardening

We consider four schemes based on generalized midpoint rule and return map algorithm for the integration of the classical J₂ plasticity model with linear hardening. The comparison, aiming to establish which is the preferable scheme among the four considered, is both theoretical and numerical. On one side, extending and completing the existing results in the literature, we investigate the four schemes from the theoretical viewpoint, addressing in particular the existence of solution, long‐term behaviour, accuracy and stability. On the other hand, we develop an extensive set of numerical tests, based on pointwise stress–strain loading histories, iso‐error maps and initial boundary‐value problems. Copyright © 2007 John Wiley & Sons, Ltd.     

一种广义的自适应相干累积算法

根据最小均方误差算法（LMS）、自适应相干累积算法（ACI）及推广的自适应相干累积算法（Generalized Adaptive Coherent Integrator,GACI）在原理上的相关性和推广关系,提出了广义的自适应相干累积算法（IGACI）。以GACI为基础,在权系数迭代公式中引入更多的动量因子,即利用权系数的更多的历史信息,使被检测的单频或调频脉冲信号,在脉冲持续期间实现自适应相干累积,从而完成低输入信噪比条件下的信号检测,而无需或很少需要知道信号的先验信息。理论分析和仿真结果证明在低信噪比情况下,其性能优于GACI算法。     

Explicit momentum‐conserving integrator for dynamics of rigid bodies approximating the midpoint Lie algorithm

We reformulate the midpoint Lie algorithm, which is implicit in the torque calculation, to achieve explicitness in the torque evaluation. This is effected by approximating the impulse imparted over the time step with discrete impulses delivered at either the beginning of the time step or at the end of the time step. Thus, we obtain two related variants, both of which are explicit in the torque calculation, but only first order in the time step. Both variants are momentum conserving and both are symplectic. Consequently, drawing on the properties of the composition of maps, we introduce another algorithm that combines the two variants in a single time step. The resulting algorithm is explicit, momentum conserving, symplectic, and second order. Its accuracy is outstanding and consistently outperforms currently known implicit and explicit integrators. Copyright © 2005 John Wiley & Sons, Ltd.     

A dynamic algorithm for integration in the boundary element method

The discretization of the boundary in boundary element method generates integrals over elements that can be evaluated using numerical quadrature that approximate the integrands or semi-analytical schemes that approximate the integration path. In semi-analytical integration schemes, the integration path is usually created using straight-line segments. Corners formed by the straight-line segments do not affect the accuracy in the interior significantly, but as the field point approaches these corners large errors may be introduced in the integration. In this paper, the boundary is described by a cubic spline on which an integration path of straight-line segments is dynamically created when the field point approaches the boundary. The algorithm described improves the accuracy in semi-analytical integration schemes by orders of magnitude at insignificant increase in the total solution time by the boundary element method. Results from two indirect BEM and a direct BEM formulation in which the unknowns are approximated by linear and quadratic Lagrange polynomial and a cubic Hermite polynomial demonstrate the versatility of the described algorithm. © 1998 John Wiley & Sons, Ltd.     

A hardening orthotropic plasticity model for non-frictional composites: Rate formulation and integration algorithm

The rate formulation and an unconditionally stable integration algorithm for a single-surface hardening anisotropic elastoplasticity model are developed to obtain an effective medium constitutive model for a class of periodic elastoplastic composites. Because the plasticity model features an intrinsically anisotropic failure criterion and hardening law, it requires more material parameters (15) than does its isotropic analogue (2). To address this issue and to assess how well the new model serves as an effective medium constitutive model for a class of composites, results from numerical elastoplastic homogenization computations are utilized for free-parameter estimation. It is shown that the new model provides a good fit to the homogenization data, and the model's excellent performance in a two-dimensional finite element setting is demonstrated by performing computations involving in-plane loading of an elastoplastic masonry wall. The wall is modelled first as a heterogeneous medium with all of its microstructure, and subsequently as a homogeneous effective medium with the new plasticity model.     

A novel ‘optimal’ exponential‐based integration algorithm for von‐Mises plasticity with linear hardening: Theoretical analysis on yield consistency,accuracy, convergence and numerical investigations

In this communication we propose a new exponential‐based integration algorithm for associative von‐Mises plasticity with linear isotropic and kinematic hardening, which follows the ones presented by the authors in previous papers. In the first part of the work we develop a theoretical analysis on the numerical properties of the developed exponential‐based schemes and, in particular, we address the yield consistency, exactness under proportional loading, accuracy and stability of the methods. In the second part of the contribution, we show a detailed numerical comparison between the new exponential‐based method and two classical radial return map methods, based on backward Euler and midpoint integration rules, respectively. The developed tests include pointwise stress–strain loading histories, iso‐error maps and global boundary value problems. The theoretical and numerical results reveal the optimal properties of the proposed scheme. Copyright © 2006 John Wiley & Sons, Ltd.     

A simple explicit single step time integration algorithm for structural dynamics

In this article, a new single-step explicit time integration method is developed based on the Newmark approximations for the analysis of various dynamic problems. The newly proposed method is second-order accurate and able to control numerical dissipation through the parameters of the Newmark approximations. Explicitness and order of accuracy of the proposed method are not affected in velocity-dependent problems. Illustrative linear and nonlinear examples are used to verify performances of the proposed method.     

大规模动力系统高精度増维精细积分方法快速算法

考虑高精度増维精细积分法求解大规模动力系统快速算法。为提高増维精细积分方法求解大规模动力系统精度,将非齐次项近似为高阶多项式,形成高精度増维精细积分方法;为减少计算时间、提高计算效率,提出高精度増维精细积分方法快速算法。算例表明,通过提高非齐次项近似阶数可显著提高计算精度,快速算法可使计算效率呈量级提高,高精度快速算法适合大规模动力系统长时间推进计算。     

A class of generalized mid-point algorithms for the Gurson–Tvergaard material model

We investigate the generalized mid-point algorithms for the integration of elastoplastic constitutive equations for the pressure-dependent Gurson–Tvergaard yield model. By exact linearization of the algorithms and decomposition of the stresses into hydrostatic and deviatoric parts, a formula for explicitly calculating the consistent tangent moduli with the generalized mid-point algorithms is derived for the Gurson–Tvergaard model. The generalized mid-point algorithms, together with the consistent tangent moduli, have been implemented into ABAQUS via the user material subroutine. An analytical solution of the Gurson–Tvergaard model for the plane strain tension case is given and the performances of the generalized mid-point algorithms have been assessed for plane strain tension and hydrostatic tension problems and compared with the exact solutions. We find that, in the two problems considered, the generalized mid-point algorithms give reasonably good accuracy even for the case using very large time increment steps, with the true mid-point algorithm (α = 0·5) the most accurate one. Considering the extra non-symmetrical property of the consistent tangent moduli of the algorithms with α < 1, the Euler backward algorithm (α = 1) is, perhaps, the best choice.     

基于隐函数模型的光学微腔传输光谱拟合算法

光学微腔品质因子高、灵敏度高,在精密生物传感方面有广阔的应用前景。针对洛伦兹拟合算法不能很好地拟合光学微腔输出端非对称波形和劈裂模式波形的问题,提出了隐函数模型算法。该算法首先建立模板波形,然后经平移、放缩理论实现模板波形操作,利用Levenberg-Marquardt (LM)算法优化参数值,能够实现对称波形、非对称波形和劈裂模式波形数据拟合。通过搭建光学微腔数据采集系统,采用高斯、洛伦兹和隐函数模型算法对不同折射率溶液的实验数据进行拟合。结果表明：隐函数模型算法比前两种算法的MSE低1个数量级,且拟合优度(R2)达到了0.99,拟合效果较好;隐函数模型算法谐振频率误差最小,谐振频率偏移量最大,对应的灵敏度最高,有利于提高光学微腔灵敏度。