A sub-stepping approach for elasto-plasticity with rotational hardening

Within the framework of the finite element method, this paper presents new algorithms implementing implicit stress integration and consistent tangent matrix calculations for an elasto-plastic model with rotational hardening. The sub-stepping technique is used for both the numerical integration of the constitutive relations and determination of the consistent tangent matrix in order to overcome the convergence difficulty arising from the complexity of the elasto-plastic model with rotational hardening. The integration of the constitutive relations and the computation of the consistent tangent matrix are incorporated into a unique procedure. Numerical tests are carried out and discussed to demonstrate the global accuracy and stability of the presented algorithms.     

Formulation of Lucas–Kanade Digital Image Correlation Algorithms for Non‐contact Deformation Measurements: A Review

Digital image correlation (DIC) metrology has been increasingly used in a wide range of experimental mechanics research and applications. The DIC algorithm used so far is however limited mostly to the classic forward additive Lucas–Kanade type. In this paper, a survey is given about the formulation of other types of Lucas–Kanade DIC algorithms that have been appeared in computer vision, robotics, medical image analysis literature and so on. Concise notations consistent with the finite deformation kinematics analysis in continuum mechanics are used to describe all Lucas–Kanade DIC algorithms. An intermediate image is introduced as a frame of reference to clarify the so‐called compositional algorithms in a two‐frame DIC analysis. Explicit examples about the additive and compositional updating of deformation parameters are given for affine deformation mapping. Extensions of these algorithms to the so‐called consistent or symmetric types are also presented. The equivalency of final numerical solutions using additive, compositional and inverse compositional algorithms is shown analytically for the case of affine deformation mapping. In particular, the inverse compositional algorithm for affine image subset deformation is highlighted for its superior computational efficiency. While computationally less efficient, consistent and symmetric algorithms may be more robust and less biased and their potentials in experimental mechanics applications remain to be explored. The unified formulation of these Lucas–Kanade DIC algorithms collected all together in this paper can serve as a useful guide for researchers in experimental mechanics to further evaluate the merits as well as limitations of these non‐classic algorithms for image‐based precision displacement measurement applications.     

Analysis of high‐order finite elements for convected wave propagation

In this paper, we examine the performance of high‐order finite element methods (FEM) for aeroacoustic propagation, based on the convected Helmholtz equation. A methodology is presented to measure the dispersion and amplitude errors of the p‐FEM, including non‐interpolating shape functions, such as ‘bubble’ shape functions. A series of simple test cases are also presented to support the results of the dispersion analysis. The main conclusion is that the properties of p‐FEM that make its strength for standard acoustics (e.g., exponential p‐convergence, low dispersion error) remain present for flow acoustics as well. However, the flow has a noticeable effect on the accuracy of the numerical solution, even when the change in wavelength due to the mean flow is accounted for, and an approximation of the dispersion error is proposed to describe the influence of the mean flow. Also discussed is the so‐called aliasing effect, which can reduce the accuracy of the solution in the case of downstream propagation. This can be avoided by an appropriate choice of mesh resolution. Copyright © 2013 John Wiley & Sons, Ltd.     

Very high-order accurate finite volume scheme for the convection-diffusion equation with general boundary conditions on arbitrary curved boundaries

Obtaining very high-order accurate solutions in curved domains is a challenging task as the accuracy of discretization methods may dramatically reduce without an appropriate treatment of boundary conditions. The classical techniques to preserve the nominal convergence order of accuracy, proposed in the context of finite element and finite volume methods, rely on curved mesh elements, which fit curved boundaries. Such techniques often demand sophisticated meshing algorithms, cumbersome quadrature rules for integration, and complex nonlinear transformations to map the curved mesh elements onto the reference polygonal ones. In this regard, the reconstruction for off-site data method, proposed in the work of Costa et al, provides very high-order accurate polynomial reconstructions on arbitrary smooth curved boundaries, enabling integration of the governing equations on polygonal mesh elements, and therefore, avoiding the use of complex integration quadrature rules or nonlinear transformations. The method was introduced for Dirichlet boundary conditions and the present article proposes an extension for general boundary conditions, which represents an important advance for real context applications. A generic framework to compute polynomial reconstructions is also developed based on the least-squares method, which handles general constraints and further improves the algorithm. The proposed methods are applied to solve the convection-diffusion equation with a finite volume discretization in unstructured meshes. A comprehensive numerical benchmark test suite is provided to verify and assess the accuracy, convergence orders, robustness, and efficiency, which proves that boundary conditions on arbitrary smooth curved boundaries are properly fulfilled and the nominal very high-order convergence orders are effectively achieved.     

Investigation of integration algorithms for rate-dependent crystal plasticity using explicit finite element codes

The work presented here concerns the use of rate-dependent crystal plasticity into explicit dynamic finite element codes for structural analysis. Different integration or stress update algorithms for the numerical implementation of crystal plasticity, two explicit algorithms and a fully-implicit one, are described in detail and compared in terms of convergence, accuracy and computation time. The results show that the implicit time integration is very robust and stable, provided low enough convergence tolerance is used for low strain-rate sensitivity coefficients, while being the slowest in terms of CPU time. Explicit methods prove to be fast, stable and accurate. The algorithms are then applied to two structural analyses, one concerning flat rolling of a polycrystalline slab and another on the response of a multicrystalline sample under uniaxial tensile condition. The results show that the explicit algorithms perform well with simulation times much smaller compared to their implicit counterpart. Finally, mesh sensitivity for the second structural analysis is investigated and shows to slightly affect the global response of the structure.     

AN ALGORITHM FOR INTEGRATING THE SPIN ON CONVECTED BASES

A new algorithm is proposed for integrating the spin in the frame of large deformation analysis. The method is based on the integration of a matrix relation, obtained from an adapted decomposition of the deformation gradient, and directly written in convected co-ordinates. The input of this algorithm is either the Gram matrix at any time (if a kinematical method is used, for instance) or more generally, the incremental deformation gradient, and the output is the required rotation matrix on convected bases. The result takes a very simple form in the important case of classical shells.     

High-order cubature rules for tetrahedra

In this paper, we construct new high-order numerical integration schemes for tetrahedra, with positive weights and integration points that are in the interior of the domain. The construction of cubature rules is a challenging problem, which requires the solution of strongly nonlinear algebraic (moment) equations with side conditions given by affine inequality constraints. We present a robust algorithm based on a sequence of three modified Newton procedures to solve the constrained minimization problem. In the literature, numerical integration rules for the tetrahedron are available up to order p=15. We obtain integration rules for the tetrahedron from p=2 to p=20, which are computed using multiprecision arithmetic. For p≤15, our approach provides integration rules that have the same or fewer number of integration points than existing rules; for p=16 to p=20, our rules are new. Numerical tests are presented that verify the polynomial-precision of the cubature rules. Convergence studies are performed for the integration of exponential, rational, weakly singular and trigonometric test functions over tetrahedra with flat and curved faces. In all tests, improvements in accuracy is realized as p is increased, though in some cases nonmonotonic convergence is observed.     

Controlling the onset of numerical fracture in parallelized implementations of the material point method (MPM) with convective particle domain interpolation (CPDI) domain scaling

The material point method is well suited for large‐deformation problems in solid mechanics but requires modification to avoid cell‐crossing errors as well as extension instabilities that lead to numerical (nonphysical) fracture. A promising solution is convected particle domain interpolation (CPDI), in which the integration domain used to map data between particles and the background grid deforms with the particle, based on the material deformation gradient. While eliminating the extension instability can be a benefit, it is often desirable to allow material separation to avoid nonphysical stretching. Additionally, large stretches in material points can complicate parallel implementation of CPDI if a single particle domain spans multiple computational patches. A straightforward modification to the CPDI algorithm allows a user‐specified scaling of the particle integration domain to control the numerical fracture response, which facilitates parallelization. Combined with particle splitting, the method can accommodate materials with arbitrarily large failure strains. Used with a smeared damage/softening model, this approach will prevent nonphysical numerical fracture in situations where the material should remain intact, but the effect of a single velocity field on localization may still produce errors in the post‐failure response. Details are given for both 2‐D and 3‐D implementations of the scaling algorithm. Copyright © 2015 John Wiley & Sons, Ltd.     

基于有限质点法的结构屈曲行为分析

有限质点法是一种新颖的结构分析方法,它以向量力学和数值计算为基础,将结构离散为质点群,采用牛顿第二定律描述这些质点的运动。该方法中引入移动基础架构求解结构单元内力。采用显示时间积分求解运动方程,避免了迭代求解非线性方程组。该文采用有限质点法分析结构的屈曲行为。以空间杆单元为例推导了有限质点法计算公式。利用自编程序,对空间杆系结构的屈曲行为进行了模拟。在不经过任何特殊处理的情况下,该方法不仅可以越过屈曲极值点,而且能够跟踪结构屈曲后的行为。此外,该方法无需分级加载即可模拟结构的屈曲行为,结构荷载可以在计算分析的初始步全部加在结构上,更符合实际情况。通过算例验证表明有限质点法在结构屈曲行为模拟中的适用性和真实性。     

Weakly enforced essential boundary conditions for NURBS‐embedded and trimmed NURBS geometries on the basis of the finite cell method

Enforcing essential boundary conditions plays a central role in immersed boundary methods. Nitsche's idea has proven to be a reliable concept to satisfy weakly boundary and interface constraints. We formulate an extension of Nitsche's method for elasticity problems in the framework of higher order and higher continuity approximation schemes such as the B‐spline and non‐uniform rational basis spline version of the finite cell method or isogeometric analysis on trimmed geometries. Furthermore, we illustrate a significant improvement of the flexibility and applicability of this extension in the modeling process of complex 3D geometries. With several benchmark problems, we demonstrate the overall good convergence behavior of the proposed method and its good accuracy. We provide extensive studies on the stability of the method, its influence parameters and numerical properties, and a rearrangement of the numerical integration concept that in many cases reduces the numerical effort by a factor two. A newly composed boundary integration concept further enhances the modeling process and allows a flexible, discretization‐independent introduction of boundary conditions. Finally, we present our strategy in the framework of the modeling and isogeometric analysis process of trimmed non‐uniform rational basis spline geometries. Copyright © 2013 John Wiley & Sons, Ltd.     

Analysis of orthotropic behavior in convected coordinate frames

The aim of this contribution is to present a theoretical and numerical model applicable to large strain analysis of orthotropic bodies. This three-dimensional constitutive law which uses the concept of convected coordinate frame is devoted to materials presenting elastic orthotropic behaviors in the large deformation field such as the elastomer-fabric composite materials. The proposed model is implemented in a finite element code and numerical examples are given to demonstrate the effectiveness of the model and the numerical algorithms. Finally, the model is compared to experimental results obtained from a bulge test.     

Finite element implementation of non‐linear elastoplastic constitutive laws using local and global explicit algorithms with automatic error control

Implicit stress integration algorithms have been demonstrated to provide a robust formulation for finite element analyses in computational mechanics, but are difficult and impractical to apply to increasingly complex non‐linear constitutive laws. This paper discusses the performance of fully explicit local and global algorithms with automatic error control used to integrate general non‐linear constitutive laws into a non‐linear finite element computer code. The local explicit stress integration procedure falls under the category of return mapping algorithm with standard operator split and does not require the determination of initial yield or the use of any form of stress adjustment to prevent drift from the yield surface. The global equations are solved using an explicit load stepping with automatic error control algorithm in which the convergence criterion is used to compute automatically the coarse load increment size. The proposed numerical procedure is illustrated here through the implementation of a set of elastoplastic constitutive relations including isotropic and kinematic hardening as well as small strain hysteretic non‐linearity. A series of numerical simulations confirm the robustness, accuracy and efficiency of the algorithms at the local and global level. Published in 2001 by John Wiley & Sons, Ltd.     

Enhancing particle swarm optimization algorithm using two new strategies for optimizing design of truss structures

This work develops an augmented particle swarm optimization (AugPSO) algorithm using two new strategies,: boundary-shifting and particle-position-resetting. The purpose of the algorithm is to optimize the design of truss structures. Inspired by a heuristic, the boundary-shifting approach forces particles to move to the boundary between feasible and infeasible regions in order to increase the convergence rate in searching. The purpose of the particle-position-resetting approach, motivated by mutation scheme in genetic algorithms (GAs), is to increase the diversity of particles and to prevent the solution of particles from falling into local minima. The performance of the AugPSO algorithm was tested on four benchmark truss design problems involving 10, 25, 72 and 120 bars. The convergence rates and final solutions achieved were compared among the simple PSO, the PSO with passive congregation (PSOPC) and the AugPSO algorithms. The numerical results indicate that the new AugPSO algorithm outperforms the simple PSO and PSOPC algorithms. The AugPSO achieved a new and superior optimal solution to the 120-bar truss design problem. Numerical analyses showed that the AugPSO algorithm is more robust than the PSO and PSOPC algorithms.     

Effects of time integration schemes on discontinuous deformation simulations using the numerical manifold method

Numerical stability by using certain time integration scheme is a critical issue for accurate simulation of discontinuous deformations of solids. To investigate the effects of the time integration schemes on the numerical stability of the numerical manifold method, the implicit time integration schemes, ie, the Newmark, the HHT‐α, and the WBZ‐α methods, and the explicit time integration algorithms, ie, the central difference, the Zhai's, and Chung‐Lee methods, are implemented. Their performance is examined by conducting transient response analysis of an elastic strip subjected to constant loading, impact analysis of an elastic rod with an initial velocity, and excavation analysis of jointed rock masses, respectively. Parametric studies using different time steps are conducted for different time integration algorithms, and the convergence efficiency of the open‐close iterations for the contact problems is also investigated. It is proved that the Hilber‐Hughes‐Taylor‐α (HHT‐α), Wood‐Bossak‐Zienkiewicz‐α (WBZ‐α), Zhai's, and Chung‐Lee methods are more attractive in solving discontinuous deformation problems involving nonlinear contacts. It is also found that the examined explicit algorithms showed higher computational efficiency compared to those implicit algorithms within acceptable computational accuracy.     

Numerically consistent regularization of force‐based frame elements

Recent advances in the literature regularize the strain‐softening response of force‐based frame elements by either modifying the constitutive parameters or scaling selected integration weights. Although the former case maintains numerical accuracy for strain‐hardening behavior, the regularization requires a tight coupling of the element constitutive properties and the numerical integration method. In the latter case, objectivity is maintained for strain‐softening problems; however, there is a lack of convergence for strain‐hardening response. To resolve the dichotomy between strain‐hardening and strain‐softening solutions, a numerically consistent regularization technique is developed for force‐based frame elements using interpolatory quadrature with two integration points of prescribed characteristic lengths at the element ends. Owing to manipulation of the integration weights at the element ends, the solution of a Vandermonde system of equations ensures numerical accuracy in the linear‐elastic range of response. Comparison of closed‐form solutions and published experimental results of reinforced concrete columns demonstrates the effect of the regularization approach on simulating the response of structural members. Copyright © 2008 John Wiley & Sons, Ltd.     

Convergence analysis of a parallel interfield method for heterogeneous simulations with dynamic substructuring

In order to predict the dynamic response of a complex system decomposed by computational or physical considerations, partitioned procedures of coupled dynamical systems are needed. This paper presents the convergence analysis of a novel parallel interfield procedure for time‐integrating heterogeneous (numerical/physical) subsystems typical of hardware‐in‐the‐loop and pseudo‐dynamic tests. The partitioned method is an extension of the method originally proposed by Gravouil and Combescure which utilizes a domain decomposition enforcing the continuity of the velocity at interfaces. In particular, the merits of the new method which can couple arbitrary Newmark schemes with different time steps in different subdomains and advance all the substructure states simultaneously are analysed in terms of accuracy and stability. All theoretical results are derived for single‐ and two‐degrees‐of‐freedom systems, as a multi‐degree‐of‐freedom system is too difficult to analyse mathematically. However, the insight gained from the analysis of these coupled problems and the conclusions drawn are confirmed by means of the numerical simulation on a four‐degrees‐of‐freedom system. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Probabilistic analysis of linear elastic cracked structures with uncertain damage

This paper presents a simple and reliable method for the probabilistic characterization of the linear elastic response of cracked structures with uncertain damage. In particular, truss and frame structures with edge cracks of uncertain depth and location are considered. The method of analysis originates from an approach recently appeared in the literature, which is generalized to treat structures with cracks affected by uncertainty. According to this approach, the uncertainties are transformed into superimposed deformations depending on the distribution of internal forces and an iterative procedure is established to solve the resultant equations. The procedure is optimally tuned based on the convergence analysis. Several numerical tests evidence excellent accuracy and convergence qualities also in the case of multicracked structures with large fluctuation of damage.     

Hamilton's law of variable mass system and time finite element formulations for time‐varying structures based on the law

Traditional principles of mechanics are primarily conceived for constant mass systems, which are only valid if mass is gained or lost at null velocity with respect to an inertial reference frame for variable mass systems, thus the numerical algorithms for time‐varying structures based on these principles are only suitable for this special case. In this paper, Hamilton's law of variable mass system is derived based on Meshchersky's fundamental equation, and two classes of novel time finite element formulations for linear systems with arbitrary continuous time‐varying parameters are developed based on the previous law. The formulations are verified extensively through numerical examples in which the convergence and effectiveness of algorithms are evaluated. Numerical examples demonstrate that compared with the algorithms for time‐varying structures that developed based on traditional principles of mechanics, the proposed algorithms provide extended capabilities in both time‐varying mass problems that mass is gained or lost at any velocity (such as rocket problem) and moving‐mass problems (such as vehicle‐bridge interaction problem) besides the time‐varying stiffness and damping problems, the proposed algorithms have a wider range of application. In particular, Hamilton's law of variable mass system provides a solid theoretical foundation for further research on the algorithm design for time‐varying structures. Copyright © 2014 John Wiley & Sons, Ltd.