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.     

Adaptive backward Euler time stepping with truncation error control for numerical modelling of unsaturated fluid flow

An automatic time stepping scheme with embedded error control is developed and applied to the moisture‐based Richards equation. The algorithm is based on the first‐order backward Euler scheme, and uses a numerical estimate of the local truncation error and an efficient time step selector to control the temporal accuracy of the integration. Local extrapolation, equivalent to the use of an unconditionally stable Thomas–Gladwell algorithm, achieves second‐order temporal accuracy at minimal additional costs. The time stepping algorithm also provides accurate initial estimates for the iterative non‐linear solver. Numerical tests confirm the ability of the scheme to automatically optimize the time step size to match a user prescribed temporal error tolerance. An important merit of the proposed method is its conceptual and computational simplicity. It can be directly incorporated into existing or new software based on the backward Euler scheme (currently prevalent in subsurface hydrologic modelling), and markedly improves their performance compared with simple fixed or heuristic time step selection. The generality of the approach also makes possible its use for solving PDEs in other engineering applications, where strong non‐linearity, stability or implementation considerations favour a simple and robust low‐order method, or where there is a legacy of backward Euler codes in current use. Copyright © 2001 John Wiley & Sons, Ltd.     

Degenerated shell element with composite implicit time integration scheme for geometric nonlinear analysis

A degenerated shell element with composite implicit time integration scheme is developed in the present paper to solve the geometric nonlinear large deformation and dynamics problems of shell structures. The degenerated shell element is established based on the eight‐node solid element, where the nodal forces, mass matrices, and stiffness matrices are firstly obtained upon virtual velocity principle and then translated to the shell element. The strain field is modified based on the mixed interpolation of tensorial components method to eliminate the shear locking, and the constitutive relation is modified to satisfy the shell assumptions. A simple and practical computational method for nonlinear dynamic response is developed by embedding the composite implicit time integration scheme into the degenerated shell element, where the composite scheme combines the trapezoidal rule with the three‐point backward Euler method. The developed approach can not only keep the momentum and energy conservation and decay the high frequency modes but also lead to a symmetrical stiffness matrix. Numerical results show that the developed degenerated shell element with the composite implicit time integration scheme is capable of solving the geometric nonlinear large deformation and dynamics problems of the shell structures with momentum and energy conservation and/or decay. Copyright © 2015 John Wiley & Sons, Ltd.     

A material point time integration procedure for anisotropic, thermo rheologically simple, viscoelastic solids

This study presents an effective and robust time integration procedure for general anisotropic, thermal rheologically simple viscoelasticity, that is suitable for implementation in a broad spectrum of general purpose nonlinear finite element programs. It features a judicious choice of state variables which record the extent of inelastic flow (creep), a stable backward Euler integration step, and a consistent tangent operator. Numerical examples involving homogeneous stress states such as uniaxial tension and simple shear, and non-uniform stress states such as a beam under tip load, were carried out by incorporating the present scheme into a general purpose FEM package. Excellent agreement with analytical results is observed.     

A robust and efficient substepping scheme for the explicit numerical integration of a rate‐dependent crystal plasticity model

This paper describes the development of efficient and robust numerical integration schemes for rate‐dependent crystal plasticity models. A forward Euler integration algorithm is first formulated. An integration algorithm based on the modified Euler method with an adaptive substepping scheme is then proposed, where the substepping is mainly controlled by the local error of the stress predictions within the time step. Both integration algorithms are implemented in a stand‐alone code with the Taylor aggregate assumption and in an explicit finite element code. The robustness, accuracy and efficiency of the substepping scheme are extensively evaluated for large time steps, extremely low strain‐rate sensitivity, high deformation rates and strain‐path changes using the stand‐alone code. The results show that the substepping scheme is robust and in some cases one order of magnitude faster than the forward Euler algorithm. The use of mass scaling to reduce computation time in crystal plasticity finite element simulations for quasi‐static problems is also discussed. Finally, simulation of Taylor bar impact test is carried out to show the applicability and robustness of the proposed integration algorithm for the modelling of dynamic problems with contact. Copyright © 2014 John Wiley & Sons, Ltd.     

Numerical integration and FEM-implementation of a viscoplastic Chaboche-model with static recovery

This paper is concerned with the implementation of a viscoplastic material model of the Chaboche type in the framework of the finite element method (FEM). The equations of the used constitutive law, that incorporates isotropic hardening, back stress evolution with static recovery terms and drag stress evolution, are introduced. A representation of their numerical integration using the implicit backward Euler method under the assumption of small deformations and an isothermal formulation follows. The use of the backward Euler method leads to a nonlinear algebraic system of three equations, which is solved by a combination of the Pegasus method and a fixed-point iteration. After considering the accuracy of the presented integration algorithm in form of iso-error maps, the derivation of the consistent viscoplastic tangent operator is shown. The integration scheme and the calculation of the consistent viscoplastic tangent operator are implemented in the commercial finite element code ABAQUS, using the possibility of the user-defined material subroutine (UMAT). Finally a numerical example in form of a notched bar under tension is presented.     

Stabilized finite element methods for vertically averaged multiphase flow for carbon sequestration

A computationally efficient numerical model that describes carbon sequestration in deep saline aquifers is presented. The model is based on the multiphase flow and vertically averaged mass balance equations, requiring the solution of two partial differential equations – a pressure equation and a saturation equation. The saturation equation is a nonlinear advective equation for which the application of Galerkin finite element method (FEM) can lead to non‐physical oscillations in the solution. In this article, we extend three stabilized FEM formulations, which were developed for uncoupled systems, to the governing nonlinear coupled PDEs. The methods developed are based on the streamline upwind, the streamline upwind/Petrov–Galerkin and the least squares FEM. Two sequential solution schemes are developed: a single step and a predictor–corrector. The range of Courant numbers yielding smooth and oscillation‐free solutions is investigated for each method. The useful range of Courant numbers found depends upon both the sequential scheme (single step vs predictor–corrector) and also the time integration method used (forward Euler, backward Euler or Crank–Nicolson). For complex problems such as when two plumes meet, only the SU stabilization with an amplified stabilization parameter gives satisfactory results when large time steps are used. Copyright © 2016 John Wiley & Sons, Ltd.     

Computational modeling of growth

The present contribution is dedicated to the computational modeling of growth phenomena typically encountered in modern biomechanical applications. We set the basis by critically reviewing the relevant literature and classifying the existing models. Next, we introduce a geometrically exact continuum model of growth which is not a priori restricted to applications in hard tissue biomechanics. The initial boundary value problem of biomechanics is primarily governed by the density and the deformation problem which render a nonlinear coupled system of equations in terms of the balance of mass and momentum. To ensure unconditional stability of the required time integration procedure, we apply the classical implicit Euler backward method. For the spatial discretization, we suggest two alternative strategies, a node-based and an integration point–based approach. While for the former, the discrete balance of mass and momentum are solved simultaneously on the global level, the latter is typically related to a staggered solution with the density treated as internal variable. The resulting algorithms of the alternative solution techniques are compared in terms of stability, uniqueness, efficiency and robustness. To illustrate their basic features, we elaborate two academic model problems and a typical benchmark example from the field of biomechanics.     

Numerical modelling of thermomechanical solids with highly conductive energetic interfaces

Interfaces within a solid can play a significant role in the overall response of a body. The influence of an interface increases as the scale of the problem decreases. Furthermore, the thermomechanical properties of the interface can differ significantly from those of the surrounding bulk. Such effects are described here using interface elasticity theory. The objective of this contribution is to detail the computational aspects of modelling thermomechanical solids with highly conductive energetic interfaces. The interface is termed energetic in the sense that it possesses its own energy, entropy, constitutive relations and dissipation. The equations governing the fully nonlinear, transient problem are given. They are then solved using an efficient finite element scheme. Full details of the consistent linearisation of the nonlinear governing equations are provided. Key features of highly conductive energetic interfaces are then elucidated via a series of three‐dimensional numerical examples. In particular and in contrast to highly conductive thermal interfaces, it is clearly shown that a jump in the heat flux across the interface is possible even in the absence of a heat flux along the interface. Copyright © 2012 John Wiley & Sons, Ltd.     

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.     

Computational modeling of cardiac electrophysiology: A novel finite element approach

The key objective of this work is the design of an unconditionally stable, robust, efficient, modular, and easily expandable finite element‐based simulation tool for cardiac electrophysiology. In contrast to existing formulations, we propose a global–local split of the system of equations in which the global variable is the fast action potential that is introduced as a nodal degree of freedom, whereas the local variable is the slow recovery variable introduced as an internal variable on the integration point level. Cell‐specific excitation characteristics are thus strictly local and only affect the constitutive level. We illustrate the modular character of the model in terms of the FitzHugh–Nagumo model for oscillatory pacemaker cells and the Aliev–Panfilov model for non‐oscillatory ventricular muscle cells. We apply an implicit Euler backward finite difference scheme for the temporal discretization and a finite element scheme for the spatial discretization. The resulting non‐linear system of equations is solved with an incremental iterative Newton–Raphson solution procedure. Since this framework only introduces one single scalar‐valued variable on the node level, it is extremely efficient, remarkably stable, and highly robust. The features of the general framework will be demonstrated by selected benchmark problems for cardiac physiology and a two‐dimensional patient‐specific cardiac excitation problem. Copyright © 2009 John Wiley & Sons, Ltd.     

A Fictitious Time Integration Method for Multi-Dimensional Backward Heat Conduction Problems

In this article, we propose a new numerical approach for solving these multi-dimensional nonlinear and nonhomogeneous backward heat conduction problems (BHCPs). A fictitious time t is employed to transform the dependent variable u(x, y, z, t) into a new one by (1+t)u(x, y, z, t)=: v(x, y, z, t, t), such that the original nonlinear and nonhomogeneous heat conduction equation is written as a new parabolic type partial differential equation in the space of (x, y, z, t, t). In addition, a fictitious viscous damping coefficient can be used to strengthen the stability of numerical integration of the discretized equations by utilizing a group preserving scheme. Six numerical experiments illustrate that the present algorism can be employed to recover the initial data very well. Even under the very large noisy final data, the fictitious time integration method is also robust against noise.     

场地地震反应一维非线性计算模型

本文运用Ramberg-Osgood或Hardin-Drnevich等动力本构关系与广义Masing准则考虑土料在地震等产生的不规则加载条件下的非线性滞变特征,将逐步积分法与不平衡荷载转移法相结合提出了动力非线性分析的一种有效增量迭代计算格式,从而建立了求解场地地震反应的真非线性分析方法;编制了同时可以进行所建议的真非线性分析与传统的等价线性化运算的通用计算程序,进而通过算例分析进行了讨论。     

抛物问题的Mortar元方法

本文提出并分析了抛物问题的mortar元方法。对时间变量采用向后Euler格式,对空间变量采用mortar元近似。得到L2模及能量模误差估计。     

The creep of an elastoviscoplastic beam under a bending loading

Thermoplastics start to manifest a nonlinear mechanical behavior from relatively low loading levels. Under a bending solicitation, which generates a nonuniform stress field, the material behavior becomes more challenging. Indeed, a flexed specimen may have different behaviors from one point to another according to the local stress state. In the present work, a six-parameter rheological model is used to simulate the nonlinear behavior of an elastoviscoplastic beam, subjected to a three-point bending load. In the framework of Euler–Bernoulli theory, the mathematical formulation of a bent beam behavior involves the bending curvature function. This function allows the determination of the strain and stress fields along and through the beam. However, when the beam reaches the viscoplastic stage, the differential equation providing the bending curvature of the beam requires a numerical integration, which has been accomplished in this work. This theoretical modeling approach is supported by experimental creep tests carried out on polyamide specimens (PA6). The testing results are qualitatively consistent with the predictions of the proposed rheological model.     

Interaction of time stepping and convection schemes for unsteady flow simulation

For the time dependent flow simulation, despite the existence of extensive literature dealing with either the convection or the unsteady terms, their interaction has not been adequately investigated. In order to shed more light on this important issue, three time stepping methods, including the first-order backward Euler scheme, the second-order Crank-Nicolson scheme, and the third-order Adams-Bashforth/Adams-Moulton predictor-corrector scheme are studied along with four convection schemes, including the first-order upwind, the second-order upwind, the second-order central differencing, and QUICK schemes. The Burgers equation of both linear and nonlinear forms is used as the test problem, aided by the von Neumann stability analysis and the FFT spectral analysis. The results indicate that a second-or higher-order accuracy for both time and space discretizations can produce satisfactory results for smooth solution profiles. Overall, among the schemes tested, either a combination of first-order upwind for convection and Crank-Nicolson for time, or a combination of second-order upwind for convection and backward Euler for time performs better. It appears that by selectively utilizing the dispersive and diffusive characteristics of the time stepping and convection schemes in complementary manners, overall accuracy can be improved.     

A finite element constitutive update scheme for anisotropic,viscoelastic solids exhibiting non‐linearity of the Schapery type

This study presents a new scheme for performing integration point constitutive updates for anisotropic, small strain, non‐linear viscoelasticity, within the context of implicit, non‐linear finite element structural analysis. While the basic scheme has been presented earlier by the authors for linear viscoelasticity, the present work illustrates the generality of the underlying fundamentals by extending to Schapery's non‐linear model. The method features a judicious choice of state variables, a stable backward Euler integration step, and a consistent tangent operator. Its greatest strength lies in ready incorporation into existing FEM codes. Numerical examples involving homogeneous stress states such as uniaxial extension and simple shear, and non‐uniform stress states such as a beam under tip load, were carried out by incorporating the present scheme into a general purpose FEM package. Excellent agreement with analytical results is observed. Copyright © 1999 John Wiley & Sons, Ltd.     

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 semi‐implicit integration scheme for a combined viscoelastic‐damage model of plastic bonded explosives

This paper presents a new implementation of a constitutive model commonly used to represent plastic bonded explosives in finite element simulations of thermomechanical response. The constitutive model, viscoSCRAM, combines linear viscoelasticity with isotropic damage evolution. The original implementation was focused on short duration transient events; thus, an explicit update scheme was used. For longer duration simulations that employ significantly larger time step sizes, the explicit update scheme is inadequate. This work presents a new semi‐implicit update scheme suitable for simulations using relatively large time steps. The algorithm solves a nonlinear system of equations to ensure that the stress, damaged state, and internal stresses are in agreement with implicit update equations at the end of each increment. The crack growth is advanced in time using a sub‐incremental explicit scheme; thus, the entire implementation is semi‐implicit. The theory is briefly discussed along with previous explicit integration schemes. The new integration algorithm and its implementation into the finite element code, Abaqus, are detailed. Finally, the new and old algorithms are compared via simulations of uniaxial compression and beam bending. The semi‐implicit scheme has been demonstrated to provide higher accuracy for a given allocated computational time for the quasistatic cases considered here. Published 2014. This article is a US Government work and is in the public domain in the USA.