Implicit Numerical Integration Research Articles

Formula omitted]-CP: Open source dislocation density based crystal plasticity framework for simulating temperature- and strain rate-dependent deformation

This work presents an open source, dislocation density based crystal plasticity modeling framework, ρ-CP. A Kocks-type thermally activated flow is used for accounting for the temperature and strain rate effects on the crystallographic shearing rate. Slip system-level mobile and immobile dislocation densities, as well slip system-level backstress, are used as internal state variables for representing the substructure evolution during plastic deformation. A fully implicit numerical integration scheme is presented for the time integration of the finite deformation plasticity model. The framework is implemented and integrated with the open source finite element solver, Multiphysics Object-Oriented Simulation Environment (MOOSE). Example applications of the model are demonstrated for predicting the anisotropic mechanical response of single and polycrystalline hcp magnesium, strain rate effects and cyclic deformation of polycrystalline fcc OFHC copper, and temperature and strain rate effects on the deformation of polycrystalline bcc tantalum. Simulations of realistic Voronoi-tessellated microstructures as well as Electron Back Scatter Diffraction (EBSD) microstructures are demonstrated to highlight the model’s ability to predict large deformation and misorientation development during plastic deformation.

A SEIR Epidemic Model of Whooping Cough-Like Infections and Its Dynamically Consistent Approximation

Whooping cough is a highly transmitted disease around the world. According to the World Health Organization (WHO), 0.15 million cases had reported globally in 2018. Most of the Asian and African states are infected regions. Through the study, we investigated the whole population into the four classes susceptible (S), exposed (E), infected (I), and vaccinated or recovered (R). The transmission dynamics of whooping cough disease are studied analytically and numerically. Analytical analyses are positivity, boundedness, reproduction number, equilibria, and local and global stabilities. In numerical analysis, we developed an implicit numerical integration scheme consistent with the biological problem’s properties. The analysis of the implicit method for the said model is dynamically consistent, positive, and bounded. Furthermore, an implicit numerical integration scheme is suitable for studying a particular epidemic model such as the whooping cough disease.

Hug and hop: a discrete-time, nonreversible Markov chain Monte Carlo algorithm

Summary This article introduces the hug and hop Markov chain Monte Carlo algorithm for estimating expectations with respect to an intractable distribution. The algorithm alternates between two kernels, referred to as hug and hop. Hug is a nonreversible kernel that repeatedly applies the bounce mechanism from the recently proposed bouncy particle sampler to produce a proposal point that is far from the current position yet on almost the same contour of the target density, leading to a high acceptance probability. Hug is complemented by hop, which deliberately proposes jumps between contours and has an efficiency that degrades very slowly with increasing dimension. There are many parallels between hug and Hamiltonian Monte Carlo using a leapfrog integrator, including the order of the integration scheme, but hug is also able to make use of local Hessian information without requiring implicit numerical integration steps, and its performance is not terminally affected by unbounded gradients of the log-posterior. We test hug and hop empirically on a variety of toy targets and real statistical models, and find that it can, and often does, outperform Hamiltonian Monte Carlo.

Numerical Modeling and Simulations of Twinning-Induced Plasticity Using Crystal Plasticity Finite Element Method

In the current work, a fully implicit numerical integration scheme is developed for modeling twinning-induced plasticity using a crystal plasticity framework. Firstly, the constitutive formulation of a twin-based micromechanical model is presented to estimate the deformation behavior of steels with low stacking fault energy. Secondly, a numerical integration scheme is developed for discretizing constitutive equations through a fully implicit time integration scheme using the backward Euler method. A time sub-stepping algorithm and the two-norm convergence criterion are used to regulate time step size and stopping criterion. Afterward, a numerical scheme is implemented in finite element software ABAQUS as a user-defined material subroutine. Finally, finite element simulations are executed for observing the validity, performance, and limitations of the numerical scheme. It is observed that the simulation results are in good agreement with the experimental observations with a maximum error of 16% in the case of equivalent stress and strain. It is also found that the developed model is able to estimate well the deformation behavior, magnitude of slip and twin shear strains, and twin volume fraction of three different TWIP steels where the material point is subjected to tension and compression.

On the Choice of Methods for Numerically Integrating the Equations of Transients in Electric Power Systems

Algorithms for solving the system of differential-algebraic equations describing electromechanical transients in electric power systems are considered along with matters concerned with ensuring the reliability and required accuracy of the solution results. Classical explicit methods, methods for implicit numerical integration and simultaneous solution of differential-algebraic systems of equations are used to model dynamic processes in power systems. On the basis of the methods used, difference models of electric power system components are obtained. The equations of transients in electric power systems are written in a homogeneous coordinate basis with the use of nodal voltage equations. A comparative analysis of algorithms based on the explicit Runge-Kutta method and implicit methods of Euler, trapezoids, Euler with fitting coefficients is carried out. The analysis results have shown that in terms of stability and accuracy, the trapezoid method has the greatest advantages, which is not inferior to the 4th order Runge-Kutta method and allows calculations with a large integration step to be carried out. The effectiveness of the combined method is shown, in which the Euler method with fitting is used to solve the differential equations describing electromagnetic processes, and the trapezoid method is used to solve the equations of electromechanical motion.

On the Applicability of Approximate Rolling and Sliding Contact Algorithms in Anisothermal Problems with Thermal Softening

The residual stress approximation methods formulated by McDowell and Moyar, Jiang and Sehitoglu, and McDowell for rolling and sliding contact problems are reconsidered in the context of single anisothermal loading cycles and isotropic hardening. A consistent extention to incorporate thermal softening is developed and the generalized thermoelastoplastic algorithms are cast into a proper predictor–corrector formulation. Detailed explicit and implicit numerical integration strategies are presented and validated using specifically designed finite element models that conform to the underlying mechanical assumptions. Then, the applicability of the approximate algorithms to anisothermal problems with isotropic hardening and thermal softening is analyzed by assuming a rate-independent Johnson–Cook-type yield stress model and by comparing the obtained transient and residual stresses to results from full-scale finite element half-space models under varying loading and strain-hardening intensities. An in-depth, comparative discussion on the adequacy of the algorithms in conjunction with the justification of their respective mechanical simplifications follows. Sufficiently strong strain hardening is found to be a prerequisite for accurate predictions, and Jiang and Sehitoglu’s approach is deemed to be preferable for the considered type of problem. The conclusions drawn from the investigations are discussed in the context of common applications with particular emphasis on manufacturing process modeling and the corresponding guidelines are proposed for such use cases.

Deflection estimation of industrial robots with flexible joints

The application of industrial robots in manufacturing industries has received considerable concerns due to the high flexibility, multifunctionality, and cost-efficiency. It is well known that the robot positioning accuracy is susceptible to the load and motion of robots owing to the insufficient stiffness of robots. Therefore, the machining accuracy improvement has been a research focus in the robotic manufacturing industries in the last decade. To overcome the measurement difficulty of the joint torque and position as well as the complex dynamic coupling between rotors and links, two forward dynamics algorithms for the robot deflection estimation are proposed in this paper. The robot kinematics and dynamics algorithms considering the dynamic coupling between rotors and links are developed based on Lie theory. The forward dynamics equations of robots are solved via the proposed algorithms: the implicit numerical integration algorithm and numerical iterative estimation algorithm. When only the motor position is available, the implicit numerical integration algorithm is employed to solve the forward dynamics equations to estimate the joint torque and position. At the same time, when both the motor position and torque are available, the forward dynamics equations can be reorganized as algebraic equations and solved by the numerical iterative estimation algorithm. Simulations of a 6-DOF serial robot are performed to verify the accuracy of the proposed algorithms.

Comparison of implicit and explicit numerical integration schemes for a bounding surface soil model without elastic range

Standard integration schemes for rate constitutive equations were designed within the classical theory of plasticity. Consequently, they rely on the assumption that a yield criterion defines a range of purely elastic material behaviour. Many constitutive models for non-cohesive soils discard that assumption. They account for inelastic deformations without employing a yield criterion. This simplifies the formulation of the models but raises questions concerning their numerical implementation. In particular, it is not fully clear which algorithmic method is most appropriate for the numerical integration of constitutive models without elastic range. To investigate this, two different stress point algorithms for a critical state bounding surface model for sands were developed and implemented. The explicit method employs substepping to automatically control the local error. The implicit method updates stresses and state variables through a local Newton iteration, the Jacobian of which is computed by numerical differentiation. The two algorithms were compared by means of calculations at integration point level as well as with respect to a boundary value problem. The results show that for a given level of accuracy, the explicit update procedure is significantly more efficient than the implicit one. This holds regardless of initial state parameters and input strain increment magnitudes.

Fully implicit numerical integration of the Yoshida-Uemori two-surface plasticity model with isotropic hardening stagnation

The paper deals with the numerical investigation and implementation of the two-surface plasticity model (or bounding surface model). This plasticity theory allows to describe the deformation behavior under large strain cyclic plasticity and the material stress-strain responses at small-scale re-yielding after large pre-straining. A novel strategy to model the isotropic hardening stagnation is developed within a fully implicit integration scheme in order to speed up the computation and to improve the material description.

Implicit numerical integration of highly nonlinear plasticity models

When using an implicit numerical integration algorithm to implement a highly nonlinear elasto-plastic constitutive models in FEM simulations, some difficulties would encounter including Jacobian matrix singularity and non-convergence. Based on the composition of the elastoplastic constitutive model and the characteristics of the traditional implicit algorithms, this paper proposes three improved implicit algorithms, namely the Homotopy-Newton-CPPM algorithm, the CG-Newton-CPPM algorithm and the two-stage algorithm, which would contribute to improve the convergence and deal with singularity of the Jacobian matrix. Besides, the convergence and the computational efficiency as well as the accuracy of five algorithms, including a conventional explicit algorithm, a conventional implicit algorithm and three improved implicit algorithms, are compared with reference to numerical simulations of single element tests and multi-element analysis.

Coupled numerical simulation of low-cycle fatigue damage in metal components

A coupled cyclic plasticity-damage model is implemented for simulating low-cycle fatigue in metal components. Constitutive relations account for J2-flow theory with nonlinear kinematic/isotropic hardening, coupled with isotropic continuum damage mechanics. The damage potential is written in a general form, allowing for implementing different damage models. An implicit numerical integration scheme is developed and the incremental update of the internal variables is achieved through the solution of a single scalar equation. Consistent linearisation of the integration algorithm is provided explicitly to guarantee robustness of the proposed algorithm. The algorithm is implemented in a user subroutine and is inserted into a commercial finite element software. Its accuracy and computational efficiency are demonstrated through numerical simulation of large-scale experiments on metal piping components that failed under low-cycle fatigue loading. The numerical analyses are conducted using finite element models of different mesh density, implementing an appropriate simulation methodology. A simple and efficient damage evolution function is employed, regularised with respect to the element’s size, so that the numerical results present negligible mesh dependency. Excellent comparison is observed between experimental and numerical results in terms of global structural response, local strains and the number of cycles for developing through-thickness crack, indicating that the present formulation can be used as an efficient numerical tool for simulating inelastic damage and low-cycle fatigue in large-scale metal structural components.

An implicit updated lagrangian formulation for liquids with large surface energy

We present an updated Lagrangian discretization of surface tension forces for the simulation of liquids with moderate to extreme surface tension effects. The potential energy associated with surface tension is proportional to the surface area of the liquid. We design discrete forces as gradients of this energy with respect to the motion of the fluid over a time step. We show that this naturally allows for inversion of the Hessian of the potential energy required with the use of Newton's method to solve the systems of nonlinear equations associated with implicit time stepping. The rotational invariance of the surface tension energy makes it non-convex and we define a definiteness fix procedure as in [Teran et al. 2005]. We design a novel level-set-based boundary quadrature technique to discretize the surface area calculation in our energy based formulation. Our approach works most naturally with Particle-In-Cell [Harlow 1964] techniques and we demonstrate our approach with a weakly incompressible model for liquid discretized with the Material Point Method [Sulsky et al. 1994]. We show that our approach is essential for allowing efficient implicit numerical integration in the limit of high surface tension materials like liquid metals.

A Hybrid, Coupled Approach to the Continuous-Discrete Kalman Filter

In this letter we present a novel approach to continuous-discrete (CD) Kalman filtering. Unlike the EKF or other hybrid UKF-EKF filters, the novel approach does not require direct calculation of system Jacobians and instead uses unscented transforms (UTs) to extract a pair of matrices, each made up of a linear combination of derivatives (with respect to the state), that are used in its place. More specifically, they are used (1) in parts of the filtering process, and (2) in the linearly implicit numerical integration scheme of the filter's state propagation stage. Extracting these matrices from the filter's UTs and using them in both the filtering process and model simulation, or what we refer to as coupling the filter to model simulation, avoids having to calculate further function evaluations for standard implicit methods, making the process of state estimation for stiff systems more efficient. Another benefit of the proposed approach is that it offers UKF accuracy levels but improved numerical stability over the UKF. This is because it uses symmetric and positive-definite (PD) representations of the state covariance propagation and measurement update equations.

Experimental program and numerical assessment for determination of stress triaxiality and J3 effects on AA6101-T4

This article proposes the study of the mechanical behavior of AA6101-T4 at fracture for different stress states. Firstly, an experimental program has conducted including the design and manufacture of six different types of specimens to perform mechanical tests considering some levels of stress triaxiality and normalized third invariant. The specimens are the smooth and notched cylindrical bars under tensile loading and rectangular cross-section specimens under pure shear and combined tensile/shear loading conditions. Furthermore, the Gao’ model assuming non-linear isotropic hardening has used to describe the mechanical behavior of the material in axisymmetric and tridimensional problems until the facture, regarding an implicit numerical integration scheme in a finite elements framework. Posteriorly, reaction curves obtained numerically have compared with curves observed in the experimental tests. Then, it has proposed a discussion about the influence of the stress triaxiality and normalized third invariant on the level of the accumulated plastic strain at fracture. Finally, the equivalent plastic strain at fracture has been analyzed as a fracture indicator in order to observe the correct fracture onset.

Development of implicit method for numerical modeling of turbomachine blade thermoelastic vibrations

There was a tendency in recent decades to increase the combustion temperature in gas-turbine engines (GTE). This allows increasing both the efficiency of the engine and the output power. In modern engines, the temperature of the exhaust gas is already significantly higher than the melting temperature of blade material. In this regard, in the design of GTE turbines there is a need to use numerical methods that allow the most reliable modeling of unsteady aerothermoelastic effects. One of the components of the aerothermoelastic problem is to integrate the unsteady equations of thermoelasticity together with the equations of aerodynamics. As these equations must be solved together with a single step in time, implicit numerical integration methods should be preferred. The object of research is the unsteady interaction of thermoelastic vibrations of the turbine blades and gas flow. This paper presents an implicit numerical method for modeling thermoelastic vibrations of the GTE turbine flow parts, including turbine blades with cooling channels. The method is based on equations of linear thermoelasticity, which are integrated by the finite element method. The investigated volume is divided into cells, forming a calculation grid with hexahedrons with additional nodes. The compute nodes are selected so that one element has 20 nodes. The approximation of the parameters in the element is performed using third-degree polynomials. Time integration is also performed with third-order accuracy. The results of testing the method on test problems, as well as comparing the results of the vibrations simulation of the standard configuration blades with the results of other authors are shown. The discrepancy of the results does not exceed 0.4 % for the test problem and 0.7 % for the blade vibrations. The obtained results indicate that the presented method can be used for numerical simulation of the unsteady thermoelastic vibrations of the gas-turbine engine flow parts.

Decomposed optimization time integrator for large-step elastodynamics

Simulation methods are rapidly advancing the accuracy, consistency and controllability of elastodynamic modeling and animation. Critical to these advances, we require efficient time step solvers that reliably solve all implicit time integration problems for elastica. While available time step solvers succeed admirably in some regimes, they become impractically slow, inaccurate, unstable, or even divergent in others --- as we show here. Towards addressing these needs we present the Decomposed Optimization Time Integrator (DOT), a new domain-decomposed optimization method for solving the per time step, nonlinear problems of implicit numerical time integration. DOT is especially suitable for large time step simulations of deformable bodies with nonlinear materials and high-speed dynamics. It is efficient, automated, and robust at large, fixed-size time steps, thus ensuring stable, continued progress of high-quality simulation output. Across a broad range of extreme and mild deformation dynamics, using frame-rate size time steps with widely varying object shapes and mesh resolutions, we show that DOT always converges to user-set tolerances, generally well-exceeding and always close to the best wall-clock times across all previous nonlinear time step solvers, irrespective of the deformation applied.

Explicit, fully implicit and forward gradient numerical integration of a hyperelasto-viscoplastic constitutive model for amorphous polymers undergoing finite deformation

Following the growing use of amorphous polymers in an expanding range of applications, interest for numerical modeling of polymer behavior has greatly increased. Together with reliable constitutive models, stable, accurate and rapid integration algorithms valid for large deformations need to be developed. Here, in the framework of hyperelasto-viscoplasticity and multiplicative split formulation, three integration algorithms (explicit, fully implicit and forward gradient) are generated for a constitutive polymer model and respective stability is investigated. The algorithms are furthermore implemented in a commercial Finite Element code and simulation of a full field tensile test is shown to capture the actual deformation behavior of polymers.

Analysis of high-frequency ANCF modes: Navier–Stokes physical damping and implicit numerical integration

This paper is concerned with the study of the high-frequency modes resulting from the use of the finite element absolute nodal coordinate formulation (ANCF) in multibody system (MBS) applications. The coupling between the cross-sectional deformations and bending and extension of ANCF beam and plate elements produces high-frequency modes which negatively impact the computational efficiency. In this paper, two new and fundamentally different approaches are proposed to efficiently solve stiff systems of differential/algebraic equations by filtering and/or damping out ANCF high-frequency modes. A new objective large rotation and large deformation viscoelastic constitutive model defined by the Navier–Stokes equations, widely used for fluids, is proposed for ANCF solids. The proposed Navier–Stokes viscoelastic constitutive model is formulated in terms of a diagonal damping matrix, allows damping out insignificant high-frequency modes, and leads to zero energy dissipation in the case of rigid body motion. The second approach, however, is numerical and is based on enhancing the two-loop implicit sparse matrix numerical integration (TLISMNI) method by introducing a new stiffness detection error control criterion. The new criterion avoids unnecessary reductions in the time step and minimizes the number of TLISMNI outer loop iterations required to achieve convergence. The TLISMNI method ensures that the MBS algebraic constraint equations are satisfied at the position, velocity, and acceleration levels, efficiently exploits sparse matrix techniques, and avoids numerical force differentiation. The performance of the TLISMNI/Adams algorithm using the proposed error criterion is evaluated by comparison with the TLISMNI/HHT method and the explicit predictor–corrector, variable-order, and variable step-size Adams methods. Several numerical examples are used to evaluate the accuracy, efficiency, and damping characteristics of the new nonlinear viscoelastic constitutive model and the TLISMNI procedure.

Implicit Integration of the Unified Yield Criterion in the Principal Stress Space

AbstractAn implicit numerical integration algorithm is presented for the unified yield criterion, which could encompass most other yield criteria. The modification matrix, which is used to convert ...

Analysis of Numerical Methods to Include Dynamic Constraints in an Optimal Power Flow Model

The optimization of the operation of power systems including steady state and dynamic constraints is efficiently solved by Transient Stability Constrained Optimal Power Flow (TSCOPF) models. TSCOPF studies extend well-known optimal power flow models by introducing the electromechanical oscillations of synchronous machines. One of the main approaches in TSCOPF studies includes the discretized differential equations that represent the dynamics of the system in the optimization model. This paper analyzes the impact of different implicit and explicit numerical integration methods on the solution of a TSCOPF model and the effect of the integration time step. In particular, it studies the effect on the power dispatch, the total cost of generation, the accuracy of the calculation of electromechanical oscillations between machines, the size of the optimization problem and the computational time.