Implicit Numerical Integration Research Articles

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.

Two approaches for automating analysis of electromagnetic processes in non-linear circuits with valves

This article analyses and compares two approaches related to automating modeling of valve electric circuits with piecewise linear approximation of valve characteristics. The first approach is based on operator equivalent circuits’ analytical formulas and on analytical expressions programming which describe equivalent circuits on each conduction interval of valve elements. The second approach provides implementation of a system for modeling valve converters based on implicit numerical integration formulas.

Application of the two-loop procedure in multibody dynamics with contact and constraint

This paper focuses on the performance of the two two-loop implicit sparse matrix numerical integration (TLISMNI) methods as the numerical solution of index-3 differential algebraic equations (DAEs) of motion arising in stiff multibody dynamics with contact and constraint. The original TLISMNI method proposed for the first time by Shabana and Hussein uses the Newmark method or Hilber/Hughes/Taylor (HHT) method as the implicit integration algorithm [TLISMNI (Newmark/HHT) method]. These two integration algorithms consider the accelerations and Lagrange multipliers as basic unknowns and their numerical accuracy is no more than order 2. In order to have a higher integral precision, we propose a new two-loop implicit procedure based on an extended backward differentiation formula scheme [TLISMNI (EBDF) method] which considers coordinates and velocities as basic unknowns. The whole structure of this proposed method is different from the traditional one due to the different unknowns. Furthermore, we apply these two kinds of methods, TLISMNI (Newmark/HHT) method and TLISMNI (BDF/EBDF) method, to the contact and constraint problems of flexible multibody dynamics while few studies of TLISMNI methods have been done on these non-smooth problems. The Baumgarte method and HHT-α method are also used to solve the same contact and constraint problem, and the advantages and disadvantages of all these methods above are compared in this study. Results show that the proposed method only needs to perform less iterations to satisfy the same tolerance of error than the TLISMNI (HHT) method does. In some cases the Baumgarte method and HHT-α method cannot give reasonable results while the TLISMNI methods perform well when the contact and constraint problems are involved. Moreover, the numerical experiment also indicates that the numerical damping property of the HHT method employed in both the TLISMNI method and the GGL formulation (used in the HHT-α method) can behave in a different way to damp out high frequency oscillations induced by impact.

Simulation of spatial multibody systems with friction

ABSTRACTA formulation for modeling and simulation of friction effects in spatial multibody systems is presented. Constraint reaction forces on rigid bodies that are connected by joints that support friction are derived as functions of Lagrange multipliers, using D’Alembert’s principle. Friction forces acting on bodies are calculated as a function of joint geometry, constraint reaction forces that are functions of Lagrange multipliers, and relative velocities at constraint contact points that are determined by system kinematics. Friction forces are implemented in index 0 differential-algebraic equations of motion that are solved numerically using explicit and implicit numerical integration methods. Spatial examples are presented, yielding accurate results and demonstrating that the systems are not stiff, even in the presence of friction and stiction.

Modeling of mechanical behavior of amorphous solids undergoing fatigue loadings, with application to polymers

An approach suitable for modeling viscoelastic-viscoplastic response with isothermal fatigue damage in amorphous solids is proposed. The theory explicitly accounts for frame-indifference and dependence of the free energy on both the viscoelastic-viscoplastic deformation and fatigue damage in a thermodynamically consistent manner. The damage evolution per se is formulated by utilizing an endurance surface that shifts in an effective stress space independent on damage. The idea is suitable for solids in which the fatigue behavior is ductile, i.e. localized damage during the creation of micro-cracks governs majority (up to 95%) of the total fatigue life. Based on implicit numerical integration, the solution procedure is presented, and its capability for technologically important polycarbonate (PC) polymer is addressed. To simulate the fatigue in real specimens, the approach is implemented in a finite-element program. A microscopic, rectangular region representing a RVE of a test specimen is investigated. Simulations, in accordance with experimental observations, indicate that damage develops in small zones around involved inhomogeneities while majority of the material remains undamaged for most of the fatigue life. The results also show that fatigue life can be predicted using a single point at which fatigue most intensively initiates.

A modified Gurson-type plasticity model at finite strains: formulation, numerical analysis and phase-field coupling

The modeling of failure in ductile materials must account for complex phenomena at the micro-scale, such as nucleation, growth and coalescence of micro-voids, as well as the final rupture at the macro-scale, as rooted in the work of Gurson (J Eng Mater Technol 99:2–15, 1977). Within a top–down viewpoint, this can be achieved by the combination of a micro-structure-informed elastic–plastic model for a porous medium with a concept for the modeling of macroscopic crack discontinuities. The modeling of macroscopic cracks can be achieved in a convenient way by recently developed continuum phase field approaches to fracture, which are based on the regularization of sharp crack discontinuities, see Miehe et al. (Comput Methods Appl Mech Eng 294:486–522, 2015). This avoids the use of complex discretization methods for crack discontinuities, and can account for complex crack patterns. In this work, we develop a new theoretical and computational framework for the phase field modeling of ductile fracture in conventional elastic–plastic solids under finite strain deformation. It combines modified structures of Gurson–Tvergaard–Needelman GTN-type plasticity model outlined in Tvergaard and Needleman (Acta Metall 32:157–169, 1984) and Nahshon and Hutchinson (Eur J Mech A Solids 27:1–17, 2008) with a new evolution equation for the crack phase field. An important aspect of this work is the development of a robust Explicit–Implicit numerical integration scheme for the highly nonlinear rate equations of the enhanced GTN model, resulting with a low computational cost strategy. The performance of the formulation is underlined by means of some representative examples, including the development of the experimentally observed cup–cone failure mechanism.

An improved and fully implicit multi-group non-local electron transport model and its validations

The combined effect of thermal flux inhibition and non-local electron heat flux in the radiation hydrodynamics (RHD) simulation of laser-driven systems can be accurately predicted by using non-local electron transport (NLET) models. These models can avoid commonly used space and time-independent ad-hoc flux-limiting procedures. However, the use of classical electron collision frequency in these models is rigorously valid for high temperature non-degenerate plasmas. In laser-driven systems, the electron thermal energy transport is important in regions between the critical density and ablation surface where the plasma is partially degenerate. Therefore, an improved model for electron collision frequency in this regime is required to accurately predict the thermal energy transport. Previously, we have reported an improved single group non-local electron transport model by using a wide-range electron collision frequency model valid from warm-dense matter (WDM) to fully ionized plasmas. In this work, we have extended this idea into a two-dimensional multi-group non-local electron transport (MG-NLET) model. Moreover, we have used a fully implicit numerical integration scheme in which the models for multi-group thermal radiation transport, laser absorption, electron-ion thermal energy relaxation and ion heat conduction are included in a single step. The performance of this improved MG-NLET model has been assessed by comparing the simulated foil trajectories with the reported experimental data for laser-driven plastic foils. The results indicate that the improved model yields results that are in better agreement with the experimental data.

TLISMNI/Adams algorithm for the solution of the differential/algebraic equations of constrained dynamical systems

This paper examines the performance of the 3rd and 4th order implicit Adams methods in the framework of the two-loop implicit sparse matrix numerical integration method in solving the differential/algebraic equations of heavily constrained dynamical systems. The variable-step size two-loop implicit sparse matrix numerical integration/Adams method proposed in this investigation avoids numerical force differentiation, ensures satisfying the nonlinear algebraic constraint equations at the position, velocity, and acceleration levels, and allows using sparse matrix techniques for efficiently solving the dynamical equations. The iterative outer loop of the two-loop implicit sparse matrix numerical integration/Adams method is aimed at achieving the convergence of the implicit integration formulae used to solve the independent differential equations of motion, while the inner loop is used to ensure the convergence of the iterative procedure used to satisfy the algebraic constraint equations. To solve the independent differential equations, two different implicit Adams integration formulae are examined in this investigation; a 3rd order implicit Adams-Moulton formula with a 2nd order explicit predictor Adams Bashforth formula, and a 4th order implicit Adams-Moulton formula with a 3rd order explicit predictor Adams Bashforth formula. A standard Newton–Raphson algorithm is used to satisfy the nonlinear algebraic constraint equations at the position level. The constraint equations at the velocity and acceleration levels are linear, and therefore, there is no need for an iterative procedure to solve for the dependent velocities and accelerations. The algorithm used for the error check and step-size change is described. The performance of the two-loop implicit sparse matrix numerical integration/Adams algorithm developed in this investigation is evaluated by comparison with the explicit predictor-corrector Adams method which has a variable-order and variable-step size. Simple and heavily constrained dynamical systems are used to evaluate the accuracy, robustness, damping characteristics, and effect of the outer-loop iterations of the proposed implicit schemes. The results obtained in this investigation show that the two-loop implicit sparse matrix numerical integration methods proposed in this study can be more efficient for stiff systems because of their ability to damp out high-frequency oscillations. Explicit integration methods, on the other hand, can be more efficient in the case of non-stiff systems.

Closed-Loop Inverse Kinematics Algorithm with Implicit Numerical Integration

Closed-Loop Inverse Kinematics Algorithm with Implicit Numerical Integration

A fully implicit numerical integration of the relativistic particle equation of motion

Relativistic strongly magnetized plasmas are produced in laboratories thanks to state-of-the-art laser technology but can naturally be found around compact objects such as neutron stars and black holes. Detailed studies of the behaviour of relativistic plasmas require accurate computations able to catch the full spatial and temporal dynamics of the system. Numerical simulations of ultra-relativistic plasmas face severe restrictions due to limitations in the maximum possible Lorentz factors that current algorithms can reproduce to good accuracy. In order to circumvent this flaw and repel the limit to$\unicode[STIX]{x1D6FE}\approx 10^{9}$, we design a new fully implicit scheme to solve the relativistic particle equation of motion in an external electromagnetic field using a three-dimensional Cartesian geometry. We show some examples of numerical integrations in constant electromagnetic fields to prove the efficiency of our algorithm. The code is also able to follow the electric drift motion for high Lorentz factors. In the most general case of spatially and temporally varying electromagnetic fields, the code performs extremely well, as shown by comparison with exact analytical solutions for the relativistic electrostatic Kepler problem as well as for linearly and circularly polarized plane waves.

An index 0 differential-algebraic equation formulation for multibody dynamics: Nonholonomic constraints

ABSTRACTA method is presented for formulating and numerically integrating index 0 differential-algebraic equations of motion for multibody systems with holonomic and nonholonomic constraints. Tangent space coordinates are defined in configuration and velocity spaces as independent generalized coordinates that serve as state variables in the formulation. Orthogonal dependent coordinates and velocities are used to enforce position, velocity, and acceleration constraints to within specified error tolerances. Explicit and implicit numerical integration algorithms are presented and used in solution of three examples: one planar and two spatial. Numerical results verify that accurate results are obtained, satisfying all three forms of kinematic constraint to within error tolerances embedded in the formulation.

Sensitivity of polycrystal plasticity to slip system kinematic hardening laws for Al 7075-T6

The prediction of formation and early growth of microstructurally small fatigue cracks requires use of constitutive models that accurately estimate local states of stress, strain, and cyclic plastic strain. However, few research efforts have attempted to systematically consider the sensitivity of overall cyclic stress-strain hysteresis and higher order mean stress relaxation and plastic strain ratcheting responses introduced by the slip system back-stress formulation in crystal plasticity, even for face centered cubic (FCC) crystal systems. This paper explores the performance of two slip system level kinematic hardening models using a finite element crystal plasticity implementation as a User Material Subroutine (UMAT) within ABAQUS (Abaqus unified FEA, 2016) [1], with fully implicit numerical integration. The two kinematic hardening formulations aim to reproduce the cyclic deformation of polycrystalline Al 7075-T6 in terms of both macroscopic cyclic stress-strain hysteresis loop shape, as well as ratcheting and mean stress relaxation under strain- or stress-controlled loading with mean strain or stress, respectively. The first formulation is an Armstrong-Frederick type hardening-dynamic recovery law for evolution of the back stress [2]. This approach is capable of reproducing observed deformation under completely reversed uniaxial loading conditions, but overpredicts the rate of cyclic ratcheting and associated mean stress relaxation. The second formulation corresponds to a multiple back stress Ohno-Wang type hardening law [3] with nonlinear dynamic recovery. The adoption of this back stress evolution law greatly improves the capability to model experimental results for polycrystalline specimens subjected to cycling with mean stress or strain. The relation of such nonlinear dynamic recovery effects are related to slip system interactions with dislocation substructures.

A viscoelastic-viscoplastic-damage model for creep and recovery of a semicrystalline thermoplastic

Thermoplastics subjected to static mechanical loads tend to creep, i.e., deform with respect to time. Motivated by creep experiments, a finite strain material model is proposed. The approach models recoverable and irrecoverable deformations through viscoelastic and viscoplastic contributions, respectively. The viscoelasticity is based on a generalized Maxwell model and the viscoplastic component has a quadratic pressure-dependent flow potential and a typical creep viscosity function. An isotropic damage formulation models the degradation of material properties with respect to time. However, a newly proposed recovery variable hinders the damage evolution especially in long-term processes. A reduced predictor-corrector scheme is presented for an efficient solution of the implicit numerical integration of the model in the context of Finite Element calculations. Creep experiments with constant load and cyclic loading-unloading are shown to characterize the mechanical long-term behavior of polyoxymethylene, a semicrystalline thermoplastic. Finite Element calculations of the experiments with the proposed material model demonstrate its capability to model the long-term deformation behavior of polyoxymethylene.

An extended GTN model for indentation-induced damage

The damage of metallic alloys induced by tensile deformation can be well described by Gurson-Tvergaard-Needleman (GTN) model. However, damage growth under intense shear loading, which has been found in micro-/macro-indentation tests, cannot be captured by this model. In this paper, an extension to GTN model is proposed to explain the damage induced by spherical indentation deformation. The nucleation of the secondary voids and the shear softening and localization due to the existing voids are taken into account. An independent law is proposed for the evolution of damage as a function of porosity. An implicit numerical integration using the generalized mid-point algorithm is formulated in the presented paper. Based on the extended GTN model, indentation simulations are performed. It is found that the coexistence of the shear softening and the nucleation of the secondary void mechanism is adequate to explain the damage induced by indentation as a result of the good agreement between the experimental measurement and the simulation prediction.

Assessment of third invariant elasto-plastic models: Mathematical aspects, numerical strategies and comparative results

This contribution presents a study of Hosford’s and Gao et al.’s elasto-plastic models each influenced by the third invariant of the deviatoric stress tensor (J3) in their formulations. The first stage of this work is a review of the concepts of Mechanics of Materials and Constitutive Modeling Theory. Subsequently, Hosford’s and Gao et al.’s models with isotropic hardening are assessed taking into account their mathematical and numerical aspects. Their yield surface’s shape and consequent convexity issues are investigated through a preliminary analysis of the effects that the third invariant (J3) has on their formulations. A return mapping algorithm is then proposed and tested for Hosford’s and Gao et al.’s elasto-plastic models based on the operator splitting methodology. The proposed algorithm was implemented through an implicit numerical integration method in an academic finite element environment, along with its consistent tangent matrix. Finally, the numerical results obtained from the proposed models are compared to experimental data available in the literature. This research assesses the performance and precision of the proposed constitutive formulations to correctly describe the mechanical behavior of an aircraft aluminum alloy and 1045 alloy steel under realistic stress and strain fields. In order to do so, different specimens capable of generating distinct stress states within high and low stress triaxiality regions are taken into account. As a result, this work demonstrates that Gao et al.’s yield criterion may generate non-convex yield surfaces when the influence of the third invariant is especially strong. In addition, it is observed that Hosford’s constitutive model presents better agreements between the obtained numerical results and the experimental data collected from the literature. The most important finding however is that the third invariant should be essential to any formulation.

An index 0 Differential-Algebraic equation formulation for multibody dynamics: Holonomic constraints

ABSTRACTThe Lagrange multiplier form of index 3 differential-algebraic equations of motion for holonomically constrained multibody systems is transformed using tangent space generalized coordinates to an index 0 form that is equivalent to an ordinary differential equation. The index 0 formulation includes embedded tolerances that assure satisfaction of position, velocity, and acceleration constraints and is solved using established explicit and implicit numerical integration methods. Numerical experiments with two spatial applications show that the formulation accurately satisfies constraints, preserves invariants due to conservation laws, and behaves as if applied to an ordinary differential equation.

A consistency elasto-viscoplastic damage model for progressive failure analysis of composite laminates subjected to various strain rate loadings

This paper is concerned with the development and validation of a consistency elasto-viscoplastic damage model that accounts for the strain rate-dependent plastic response and the progressive post-failure behaviour of composite materials. The model proposed is suitable for progressive failure analysis of composite materials and structures subjected to loadings at various strain rates. A strain rate-dependent yield criterion is adopted; whereas the standard Kuhn–Tucker conditions for plastic loading and unloading remain valid. Also, the plastic consistency condition for strain rate-dependent material is satisfied. The relevant strain-driven implicit numerical integration procedure for the proposed material model is developed. A tangent stiffness operator including the strain rate-dependent effects and consistent with the integration procedure developed is derived to ensure the quadratic convergence rate pertaining to the Newton–Raphson method in a finite element analysis. The implicit integration algorithm is implemented in the finite element code Abaqus through the user-defined material subroutine UMAT. The validation of the proposed model is performed on S2/8552 glass/epoxy off-axis block composite specimens subjected to compression at various strain rates. It is shown that the strain rate-dependent response of composite materials is well captured and the predicted results agree well with the experimental data reported in the literature.

Optimal design for concrete pavement situated above box culvert: experimental and numerical study

This study examined the pavement distress data obtained from the field as well as the effects of the soil cover depth, joint position and reinforced slab length on the behaviour of concrete pavement using the finite element method. An ultimate strength model with a similarity of 1/2 was designed and constructed. Finite element analysis incorporating an implicit direct numerical integration scheme was conducted to obtain an optimal design for the pavement structure above a box culvert. An accelerated pavement test was conducted for validation, in which a realistic traffic load was imposed on the pavement model constructed in the laboratory. The strain histories were measured at each section and compared with those of numerical analysis. The optimal joint position and reinforced slab length that can minimise the damage to the pavement structure above a box culvert were also determined.