Implicit Integration Algorithm Research Articles

Approximation‐based implicit integration algorithm for the Simo‐Miehe model of finite‐strain inelasticity

AbstractWe propose a simple, efficient, and reliable procedure for implicit time stepping, regarding a special case of the viscoplasticity model proposed by Simo and Miehe (1992). The kinematics of this popular model is based on the multiplicative decomposition of the deformation gradient tensor, allowing for a combination of Newtonian viscosity and arbitrary isotropic hyperelasticity. The algorithm is based on approximation of precomputed solutions. Both Lagrangian and Eulerian versions of the algorithm with equivalent properties are available. The proposed numerical scheme is non‐iterative, unconditionally stable, and first order accurate. Moreover, the integration algorithm strictly preserves the inelastic incompressibility constraint, symmetry, positive definiteness, and w‐invariance. The accuracy of stress calculations is verified in a series of numerical tests, including non‐proportional loading and large strain increments. In terms of stress calculation accuracy, the proposed algorithm is equivalent to the implicit Euler method with strict inelastic incompressibility. The algorithm is implemented into MSC.MARC and a demonstration initial‐boundary value problem is solved.

High-order accurate multi-sub-step implicit integration algorithms with dissipation control for hyperbolic problems

High-order accurate multi-sub-step implicit integration algorithms with dissipation control for hyperbolic problems

An elastoplastic model with egg-shaped yield surface for coastal soft clay

A novel elastoplastic constitutive model with an egg-shaped yield surface (ESE model) is proposed herein to elucidate the strength-deformation characteristics of marine soft clay. One of the key improvements offered by this model is its ability to address the singularity associated with plastic strain increments at yield surface corners—a limitation commonly encountered in traditional elastoplastic models. Additionally, the proposed model demonstrates flexibility, as it can transform into various forms under different conditions. Subsequently, to evaluate the efficacy of this egg-shaped elastoplastic model, numerical simulations are performed within the finite element software ABAQUS using an implicit integration algorithm through the user material subroutine interface (UMAT). The simulations encompass standard triaxial consolidation tests involving saturated soft clay sourced from the Itsukaichi marine clay as well as two types of saturated kaolin clays. The outcomes derived from these simulations by the ESE model are compared with both empirical data and numerical simulations originating from the widely employed Modified Cam-Clay (MCC) model. The simulations of the ESE model for three conventional triaxial tests on clay demonstrate superior concordance with experimental measurements in comparison to the MCC model. This underscores the efficacy of the proposed ESE model in predicting the strength and deformation characteristics manifested by coastal soft clays under undrained conditions. Finally, the predictions for deformation in an adjacent tunnel according to the ESE model closely align with experimental measurements and prior to the simulations conducted using the MCC model and advanced models such as the bounding surface model and the unified model, thereby affirming its practical utility in engineering applications.

Combined anisotropic and cyclic constitutive model for laser powder bed fusion fabricated aluminum alloy

This study presents new methods to effectively model the anisotropic yielding and hardening behavior of laser powder bed fusion fabricated aluminum alloy under both monotonic and cyclic loading conditions. The proposed model combines the yield surface-interpolation method to accurately describe the anisotropic hardening rates in various directions, with the Chaboche kinematic hardening rule to precisely reflect the cyclic characteristics. For numerical implementation of the combined anisotropic and cyclic constitutive model, a fully implicit stress integration algorithm based on return mapping method is provided. Moreover, the multiple parameters associated with the model are categorized and identified in an uncoupled manner. The isotropic and cyclic hardening parameters are determined by an inverse method, and the stability of the optimization outcomes is validated by applying different starting points for the parameters. Particularly, the back-stress effect on the identification of anisotropic parameters associated with the stress invariant-based Hill48 yield function is considered for the first time. This consideration leads to an improved prediction accuracy compared to the identification of anisotropic parameters without considering back-stress effect. The combined anisotropic and cyclic constitutive model, along with the calibrated parameters, are proven capable of accurately reproducing the intricate deformation behavior of laser powder bed fusion fabricated AlSi10Mg.

Approaches to improve the convergence and efficiency of the implicit stress integration algorithm for the unified hardening model

To realize the numerical calculation of the unified hardening (UH) model for soils, the closest point projection method (CPPM) is used to do the stress integration in the finite element analysis. A package of approaches is proposed to improve the convergence and efficiency of the algorithm. An implicit differentiation method, which can greatly save the computational cost and storage space, is employed to calculate the second derivatives of the 3D yield function generalized by the transformed stress (TS) method. In addition, the denominator of the UH law is equal to 0 at the characteristic state and critical state, while the TS equation contains a square root and may be invalid for tensile stress. These singular problems are solved by introducing subtle treatments without changing the calculation results of the UH model. The consistent tangent operator corresponding to the stress integration can be expressed in a concise form. Finally, a series of simple loading tests and finite element simulations of the bearing capacity of a rigid circular foundation are conducted to evaluate the accuracy, convergence and efficiency of the proposed algorithm.

Numerical predictions of low cyclic fatigue and creep-fatigue behavior of P92 steel under non-isothermal-mechanical loading

Low-Cycle Fatigue (LCF) and creep-fatigue-interaction (CFI) behavior of a ferritic-martensitic stainless steel P92 is studied under strain-controlled non-isothermal -mechanical fatigue loading in the temperature range of 300°C∼620°C and 500°C∼620°C, respectively. The strain-controlled tests with and without the tensile and compression holding times are simulated for different mechanical strains amplitudes. An improved Unified Viscoplastic constitutive Model (UVM) is proposed based on the framework of modified Chaboche non-linear kinematic hardening rule. The improved UVM incorporates strain rate sensitivity, temperature rate dependence, static recovery associated with plastic strain, mean stress evolution and cyclic softening of strain range dependence, and so on. Furthermore, the expressions of the implicit stress integration algorithm and the consistent tangential modulus are deduced on the basis of radial return mapping and backward Euler integration methods, respectively, and the improved UVM is implemented into the ABAQUS software using the user material subroutine UMAT. The simulation results of the UVM correlate well with the experimental data.

On the modeling of cavitation and yielding in rubber-toughened amorphous polymers: Fully implicit implementation and optimization-based calibration

In the present contribution, a finite strain visco-elastic visco-plastic (three-dimensional) constitutive model is proposed to predict the nonlinear response of neat, porous and rubber-toughened amorphous polymers. In this context, a well-known nucleation law is modified and coupled with a nucleation criterion to suppress nucleation under compression and shear loadings, as well as to identify its material parameters directly. The growth of both the nucleated and initial voids is ruled by a modified version of the Gurson’s potential, ensuring the coherence in the energy dissipation between the macro and microscales. The post-yield strain hardening of the material is modeled with the well-established eight-chain model. A fully implicit integration algorithm is formulated under the infinitesimal strains functional format, and both constitutive state update and consistent tangent modulus are properly established. A three-stage optimization-based calibration procedure is designed to identify the model’s material parameters efficiently. This decoupled calibration strategy is only possible due to the well-defined influence of the model’s material parameters in the numerical response. The predictive capability of the proposed constitutive model is validated against literature results for polycarbonate and three rubber-filled polycarbonate blends. Moreover, the numerical response of the model is assessed under different triaxial states and is further compared with the homogenized response of a Representative Volume Element of the voided-amorphous polymer microstructure. The results highlight that the model can capture the typical behavior of amorphous polymers, namely yield stress, strain-softening and strain hardening under different loading conditions, as well as the decrease of stress associated with the increase of the void volume fraction. The additional numerical analyses conducted, validated some constitutive assumptions. The efficiency of the calibration procedure and the overall numerical strategy is also demonstrated.

A constitutive model for amorphous thermoplastics from low to high strain rates: Formulation and computational aspects

In this paper, a recently proposed finite strain visco-elastic visco-plastic (three-dimensional) constitutive model is extended to predict the nonlinear response of amorphous polymers from low to high strain rates. The model accounts for the influence of distinct molecular mechanisms, which become active at different deformation rates. Therefore, the constitutive equations include two relaxation phenomena to describe the strain rate sensitivity of amorphous polymers. Well-established rheological elements are adopted to define visco-elasticity (generalized Maxwell elements) and visco-plasticity (Eyring dashpots). In addition, strain hardening is modeled with a plasticity-induced (nonlinear) hardening element which is extended to distinguish between the contribution of the two transitions. From a computational viewpoint, a fully implicit integration algorithm is derived, and a highly efficient implementation is obtained. It is shown that it is possible to reduce the return mapping system of equations to only two independent (scalar) nonlinear equations. A four-stage optimization-based calibration procedure is proposed to identify the model’s material parameters in a completely unsupervised way. The predictive capability of the constitutive model is validated against literature results for polycarbonate and poly(methyl methacrylate), accounting for temperature and strain rate dependencies under different loading conditions. The results show that the model can capture the transition in the yield behavior and predict the post-yield large strain behavior over a wide range of strain rates. The efficiency of the calibration procedure and the overall numerical strategy is also demonstrated. Despite the adiabatic conditions observed under high strain rates, the model replicates the associated effect of temperature through strain rate dependency.

Three optimal families of three‐sub‐step dissipative implicit integration algorithms with either second, third, or fourth‐order accuracy for second‐order nonlinear dynamics

SummaryThis paper reviews the published composite three‐sub‐step implicit algorithms all of which adopt the trapezoidal rule in the first sub‐step. Three optimal families of three‐sub‐step implicit algorithms are developed to achieve second, third, and fourth‐order accuracy. The present second‐ and third‐order methods achieve identical effective stiffness matrices, thus embedding optimal spectral characteristics. Besides, both of them impose two dissipative parameters ( and ) to control numerical dissipation imposed in the second and third sub‐steps. The parameter adjusts overall numerical dissipation in the whole integration schemes, while the firstly used parameter can change numerical dissipation in the second sub‐step. The numerical example has shown the superiority of controlling middle sub‐step dissipation via . The present fourth‐order method is moderately dissipative due to achieving higher‐order accuracy, but it presents a more reasonable sub‐step division than the published fourth‐order three‐sub‐step trapezoidal rule. Linear and nonlinear examples are simulated to show the numerical performance and superiority of the three novel methods. This paper recommends using the proposed second‐ and third‐order three‐sub‐step methods to solve various dynamic problems.

Implicit integration of an anisotropic egg-shaped elastoplastic model for saturated soft clay

PurposeThe purpose of this paper is to suggest an implicit integration method for updating the constitutive relationships in the newly proposed anisotropic egg-shaped elastoplastic (AESE) model and to apply it in ABAQUS.Design/methodology/approachThe implicit integration algorithm based on the Newton–Raphson method and the closest point projection scheme containing an elastic predictor and plastic corrector are implemented in the AESE model. Then, the integration code for this model is incorporated into the commercial finite element software ABAQUS through the user material subroutine (UMAT) interface to simulate undrained monotonic triaxial tests for various saturated soft clays under different consolidation conditions.FindingsThe comparison between the simulated results from ABAQUS and the experimental results demonstrates the satisfactory performance of this implicit integration algorithm in terms of effectiveness and robustness and the ability of the proposed model to predict the characteristics of soft clay.Research limitations/implicationsThe rotational hardening rule in the AESE model together with the implicit integration algorithm cannot be considered.Originality/valueThe singularity problem existing in most elastoplastic models is eliminated by the closed, smooth and flexible anisotropic egg-shaped yield surface form in the AESE model. In addition, this notion leads to an efficient implicit integration algorithm for updating the highly nonlinear constitutive equations for unsaturated soft clay.

An efficient finite strain constitutive model for amorphous thermoplastics: Fully implicit computational implementation and optimization-based parameter calibration

In this paper, a new visco-elastic visco-plastic (three-dimensional) constitutive model is consistently formulated under isothermal conditions to describe the nonlinear behavior of thermoplastic polymers at finite strains. The constitutive equations include three well-established rheological elements to have well-defined parametric behavior and capture the experimental response. In particular, a visco-elastic generalized Maxwell element, a visco-plastic Eyring dashpot, and a plasticity-induced (nonlinear) hardening element are selected and consistently combined. A fully implicit integration algorithm is derived, and highly efficient implementation is obtained by simplifying the system of equations to a single (scalar) nonlinear residual equation. A new two-stage optimization-based calibration procedure is developed to identify the material parameters in a completely unsupervised way. The constitutive model is validated against results available in the literature for polycarbonate, accounting for temperature and strain rate dependencies. The results show the efficiency of the overall numerical strategy and demonstrate that quadratic convergence rates are attained. Despite the strongly nonlinear finite strain response, it is possible to observe an excellent agreement for all stages of deformation.

A robust stress update algorithm for elastoplastic models without analytical derivation of the consistent tangent operator and loading/unloading estimation

AbstractA robust and concise implicit stress integration algorithm of elastoplastic models is presented. It does not require the loading/unloading estimation and analytical derivation operation for the stress update. First, the elastoplastic stress update problem is recast into an unconstrained minimization problem by utilizing the smooth function to bypass the loading/unloading estimation. Then, the object problem is solved by the line search method instead of the Newton method for better convergence. The consistent tangent operator is evaluated by the complex step derivative approximation without the subtraction cancellation error, which provides the quadratic convergence rate of global iteration. The rationality of the numerical consistent tangent operator is validated by the one obtained by the analytical derivation. A recently developed non‐orthogonal elastoplastic (NEP) clay model is implemented using the new algorithm. The algorithm is confirmed through comparing the numerical solution and the analytical one for a cavity expansion problem. The algorithm performance is assessed based on a series of geotechnical boundary value problems. It is found that the new algorithm is more robust than the one employed by ABAQUS. The source code of the model implementation can be downloaded from https://github.com/zhouxin615.

The viscoplastic constitutive model of TC4 alloy under high temperature

TC4 alloy is widely used in the high temperature environment. This paper proposed an ununified viscoplastic constitutive model according to the uniaxial creep experiment and high temperature tensile experiment of TC4 alloy. The inelastic strain rate was decomposed into two parts: creep strain rate and plastic strain rate. The creep strain was calculated by multiaxial ductility exhaustion creep damage law, and the plastic strain was get from the yield function containing the damage and the kinematic hardening effect. The implicit stress integration algorithm and the consistent tangent modulus of this model were derived. The material constants of the proposed model were determined by using genetic algorithm. The proposed constitutive model was compiled in a UMAT subroutine of finite element software Abaqus and 3D eight nodes isoparametric brick element was employed to simulate the creep and the tensile behavior of TC4 alloy. Form the result, the proposed constitutive model can predict the three stages of creep curves and the softening stage of tensile curves accurately.

Implicit Integration Algorithm for Solving Evolution of Microstructural Vectors Based on Eulerian Formulation in Plane Stress Condition

Abstract This paper presents a mathematical formulation and implicit numerical algorithm for solving the integral of a three-dimensional momentum balance based on the inelastic evolution of microstructural vectors for thin plates in Eulerian formulation. A recent theoretical discussion (Lee and Rubin, 2020, “Modeling Anisotropic Inelastic Effects in Sheet Metal Forming Using Microstructural Vectors—Part I: Theory,” Int. J. Plast., 134, p. 102783. 10.1016/j.ijplas.2020.102783) showed that Eulerian constitutive equation based on microstructural vectors for thin plates has the advantage of capturing the anisotropic behavior of the material axis with insensitivity to the randomness of the reference configuration. However, all the discussions were theoretically conducted only at a local material point in homogeneous deformation conditions, which do not require consideration of the momentum balance with flexible velocity gradients in a three-dimensional volume. For usability, numerical algorithms are needed to solve evolution of the microstructural vectors in the three-dimensional space. This paper presents the first numerical algorithm to solve the inelastic evolution of microstructural vectors in the Eulerian formulation. A generalized material coordinated system is matched to the microstructural vectors in a three-dimensional space by considering the Eulerian constitutive equations insensitive to the superposed rigid body motions (SRBM). Numerical algorithms were then introduced to implicitly solve the nonlinear momentum balance, evolution of the microstructural vectors, and tangent modulus. The formula and numerical algorithms were validated by predicting the tension tests when the principal loading angle varied from the reference axis. The results show that the proposed numerical algorithm can describe the evolution of the microstructure based on the Eulerian formulation.

Modeling of masonry structures using a new 3D cohesive interface material model considering dilatancy softening

This study presents a new 3D multi-yield surfaces material model for masonry mortar joints, which can be used with interface elements for finite element (FE) modeling of masonry structures to capture various failure mechanisms. This model is featured with (1) two hyperbolic yield surfaces, capable of capturing various failure modes of mortar joints, including tensile cracking, shear sliding, and compressive crushing, (2) an unassociated flow rule to capture the ‘dilatancy’ phenomenon in the mortar joints, and (3) the dilatancy softening and variation of mode II fracture energy under different normal stress levels. An implicit Euler backward integration algorithm, combined with local-global Newton-Raphson (NR) solver, is adopted to achieve the predictor-corrector return mapping in the numerical formulation. The error-based auto-adaptive sub-stepping algorithm is employed to achieve the robustness and efficiency in the integration procedure. The developed model is implemented in the general-purpose FE software ABAQUS with the specialized capability of modeling masonry structures. The developed model is validated through three brick-mortar-brick assemblages and three un-reinforced masonry (URM) walls under in-plane (IP) loading or out-of-plane (OOP) loading. The importance of appropriate modeling of dilatancy on simulating the IP and OOP behavior of URM walls is highlighted. The validation results show that the developed constitutive model is capable of modeling masonry structures at various scales with improved accuracy by considering dilatancy softening.

A closest point projection method for stress integration of 3D sand models generalised by transformed stress method

ABSTRACT It is significant to generalise constitutive models from the triaxial space to the multi-axial space for soils. Different from the popular approaches by a g(θ) function to interpolate failure curve, the transformed stress (TS) method can bring the effect of the third stress invariant and stress-induced anisotropy into constitutive models without adding any extra parameters. The closest point projection method (CPPM) is the most popular implicit integration algorithm for the implementation of constitutive model in the finite element analysis. The disadvantage of the CPPM is that it requires the deduction of the quadratic derivative of the plastic potential function, which is extremely difficult for a complex plastic potential. This study employed the TS method with the spatially mobilised plane criterion to generalise a state-dependent sand model, in which the analytical expression of the quadratic derivative of the plastic potential can be obtained. A novel CPPM integration scheme was proposed for the generalised model, which employed an accurate integration of the non-linear elastic constitutive law and a sub-stepping technique to improve the elastic trial state before the Newton’s iteration. The numerical examples demonstrated that the proposed CPPM algorithm has the higher efficiency and better robustness than the conventional cutting plane algorithm.

Unified viscoplasticity model for type 316 stainless steel subjected to isothermal and anisothermal-mechanical loading

316FR-type low carbon stainless steel is usually used for many advanced gas cooled reactor (AGR) power plants which are subjected to operating temperatures in the range of 500–650 °C under thermo-mechanical loading, which may lead to isothermal low cycle fatigue (LCF) and thermal mechanical fatigue (TMF) damage. The strain controlled thermo-mechanical fatigue test with and without holding time was carried out for 316FR-type stainless steel under the temperature cycling between 500 °C and 650 °C, and the strain ranges are Δε = ± 0.4%, ±0.8%, ±1.0% and ±1.2%, respectively. Within the framework of Abdel Karim and Ohno model, an improved unified viscoplastic constitutive model (UVM) is proposed to simulate the cyclic plastic behavior of 316FR stainless steel. The improved UVM includes strain rate sensitivity, temperature rate dependence, static recovery, average stress evolution and strain range dependence of cyclic hardening. And then based on the radial return mapping and backward Euler integration methods, the implicit stress integration algorithm and the new expressions of the consistent tangent modulus are derived respectively, the improved UVM is implemented by a User Material Subroutine into ABAQUS software again. The good correlation between simulation results and experimental data is obtained.

Quaternion-based knowledge graph neural network for social recommendation

In recent years, the surge in the number of users in recommender systems has brought unprecedented opportunities and challenges to research on recommender systems. The development of the graph neural network makes social recommendation based on graph data more advantageous, but it also brings problems with parameterization. Although the recommender system contains a large amount of data, such as users, items, and user-item interactions, the social relationship data of users in social recommendation are still sparse. To solve the problem of parameterization, we propose a novel quaternion-based knowledge graph neural network for social recommendation (QSoR), which applies hypercomplex space to the social recommendation. Without affecting the recommendation quality, QSoR utilizes the expressibility of the quaternion and weight sharing mechanism of the Hamilton product to reduce the parameters during training. To solve the problem of the sparsity of users’ social relationship data, we propose a quaternion-based explicit and implicit social relationships integration algorithm, which obtains users’ implicit social relationships from the user-item graph as a supplement for users’ social information. We design a double-end attention mechanism when obtaining the user’s preferences and social relationships, which makes full use of the item information and the interaction information between the user and the neighbor nodes. In this way, we can obtain users’ personalized preferences, distinguish the importance of user relationships, and obtain neighbors with a high degree of influence on users, thus improving the accuracy of recommendation results. The social relationship graph and the user-item graph are effectively combined in the quaternion space to make personalized recommendations for users. Experiments on three public datasets, Yelp, Last-FM, and Epinions, demonstrate the validity of the proposed model QSoR.

An implicit integration algorithm based on invariants for isotropic elasto‐plastic models of the Cosserat continuum

AbstractA Finite Element (FE) procedure based on a fully implicit backward Euler predictor/corrector scheme for the Cosserat continuum is here presented. The integration algorithm is suitable for yield and plastic potential surfaces with general shape in the deviatoric plane. The key element of the integration scheme is the spectral decomposition of the stress tensor, which is achieved, despite the lack of symmetry, because of the mathematical structure of the yield function and the set of invariants chosen as independent variables. It is also shown that the choice of invariants enables considerable mathematical simplifications, which result in the reduction of the system of equations and unknowns of the elasto‐plastic problem from 19 to 1, and to rigorously handle the discontinuity at the apex of the surfaces. The algorithm has been implemented in a proprietary FE programme, and used for the constitutive model recently proposed by the same authors in this journal for the Cosserat continuum, which allows to set various classical failure criteria as yield and plastic potential surfaces. Numerical analyses have been conducted to simulate a biaxial compression test and a shallow strip footing resting on a Tresca, Mohr–Coulomb, Matsuoka–Nakai and Lade–Duncan soil. The benefits of the Cosserat continuum over the Cauchy/Maxwell medium are discussed considering mesh refinement, non‐associated flow and softening behaviour.

A re‐evaluation of overshooting in time integration schemes: The neglected effect of physical damping in the starting procedure

AbstractAn important property of implicit time integration algorithms for structural dynamics is their tendency to “overshoot” the exact solution in the first few steps of the computed response due to high‐frequency components in the initial excitations. The typical analysis technique for overshooting involves the study of asymptotic response of the algorithm's first step in the limiting high frequency case. This article finds that the prior analysis of overshooting in much of the engineering literature is incomplete in that it neglects the effect of physical damping. With physical damping included, first‐order overshooting components enter into several well‐known time integration algorithms which were previously thought to exhibit zero‐order overshooting in displacement. The Newmark method, Wilson‐ method, Bazzi‐ method, HHT‐ method, WBZ‐ method, and three parameter optimal/generalized‐ method are analyzed, as well as the generalized single‐step single‐solve (GSSSS) framework which encompasses all of the prior schemes and other new and optimal algorithms and designs based upon the issues under consideration. The additional overshooting component is eliminated in the novel amended GSSSS V0 family (which is noteworthy and cannot be derived by conventional means), while the numerically dissipative schemes in the GSSSS U0 family (encompassing traditional methods such as the HHT‐ method, WBZ‐ method, and three parameter optimal/generalized‐ method among other new algorithm designs) are shown to be irremediable as the additional overshooting component from physical damping enters into the second step of the response, which is a wholly new finding. Numerical verifications of the overshooting analysis are performed for SDOF and MDOF structures with and without physical damping, and practical recommendations are given for the solution of MDOF problems on the basis of algorithm design and selection for various initial conditions.