Implicit Integration Scheme Research Articles

Simulation of dynamic pulsing fracking in poroelastic media by a hydro-damage-mechanical coupled cohesive phase field model

This study develops a hydro-damage-mechanical fully coupled numerical method capable of modelling complex fluid-driven transient dynamic crack propagation in quasi-brittle poroelastic media. In this method, the fluid flow in both fractures and porous media is described by a fluid continuity equation with the modified Darcy-Poiseuille law based on the Biot's poroelastic theory. The fluid pressure and the inertial force of solids are coupled by governing equations of the mesh-insensitive phase-field regularized cohesive zone model that can simulate quasi-brittle multi-crack initiation, propagation, branching and merging without remeshing or crack tracking. The resultant displacement-pressure-damage coupled multiphysics system of equations is solved using an alternative minimization Newton-Raphson iterative algorithm with an implicit Newmark integration scheme within the finite element framework. The new method was first validated by a few 2D problems, including crack branching and deflecting in solids, fracking in a concrete cube, and consolidation and stress wave propagation in poroelastic media, subjected to various impulsive loadings. It was then applied to pressure pulsing fracking of a 40 m granite rock reservoir with extensive parametric studies of fluid viscosity and pulsing injection rate, mode and period. It was found that pulsing injection with higher fluid rates and lower fluid viscosities resulted in more developed crack patterns, and in particular, there existed an optimal pulsing injection period that could promote fracking under relatively low injection pressures. 3D horizontal well problems with multiple non-planar crack propagation and bifurcation were also successfully simulated to demonstrate the capacity and potential of the new method for engineering design and optimization of pressure pulsing fracking.

Experimental and theoretical study of stress relaxation in high-density polyethylene

Stress relaxation of high-density polyethylene is addressed both experimentally and theoretically. Two types of stress relaxation testing are carried out: uniaxial tensile testing at constant test specimen length and compression testing of a 3D structure producing inhomogeneous deformation fields and relaxation. A constitutive model for isotropic, semi-crystalline polymers is also proposed. The model has the ability to model stress relaxation at different time scales. The developed model was implemented as a user subroutine in Abaqus (UMAT). The implicit integration scheme including an algorithmic tangent modulus is described in detail. The material model is calibrated by use of the uniaxial tensile tests, and the model is then validated by simulating the compression tests of the 3D structure. The model is able to describe the uniaxial tension tests well, and the comparison between the simulations and experimental testing of the 3D structure shows very good agreement.

Implementation and verification of direct infinite elements in axisymmetric problems

In the field of dynamic soil-structure interaction, the simulation of geotechnical massifs under non-vanishing time-averaged loads is often challenging, depending on factors such as load variations, the 3D model representation and the chosen model boundaries. This study aims to evaluate the effectiveness of the method developed recently by the authors under non-vanishing time-averaged loads. The evaluation involves comparing the results of a homogeneous axisymmetric model at two control points and the simulation of an axisymmetric bilayer. The aim is to demonstrate the robustness of the method in contrast to widely used dissipative devices in scenarios involving sudden loading. Using an implicit integration scheme in the time domain, the method is shown to be effective and correlations with reliable solutions and comparisons with common dissipative devices for sudden-load models are presented. This analysis leads to the conclusion that the proposed procedure adeptly addresses the challenges associated with the simulation of geotechnical masses under non-vanishing time-averaged loads.

Modeling strain-induced martensitic transformation in austenitic stainless steels subjected to cryogenic temperatures using incremental mean-field homogenization schemes

Austenitic stainless steels are commonly used as structural materials in high-field superconducting magnet systems because they retain high strength, ductility, and toughness at very low temperatures, and they are paramagnetic or antiferromagnetic under the Néel temperature in their fully austenitic state. However, they are susceptible to strain-induced martensitic transformation, especially at cryogenic temperatures, which modifies the material properties, induces volume changes and additional strain hardening, and leads to ferromagnetic behavior. Thus, accurate predictions of the structural performance of these materials at very low temperatures are of great interest for the conception and design of these cryo-magnetic systems. In this paper, we propose an adequate constitutive model for the evolving bi-phase material—austenite and martensite—based on a Hill-type incremental formulation. Two different versions of the model are proposed based on the linear mean-field homogenization scheme: Mori-Tanaka and Self-Consistent. Moreover, a rate-independent nonlinear mixed kinematic-isotropic hardening law is used for each phase, and the martensitic transformation is described by the nonlinear kinetic law proposed by Olson and Cohen (1975). The constitutive model is implemented in ABAQUS/Standard through a UMAT user subroutine, for which a return mapping algorithm based on the implicit backward Euler integration scheme is used and a closed-form expression of the consistent Jacobian tensor is provided. The Mori-Tanaka and Self-Consistent approaches are evaluated in terms of their ability to describe the mechanical behavior of the bi-phase aggregate by comparing the predictions of the homogenization schemes with unit-cell finite element calculations with an explicit description of the martensite inclusions and the austenite matrix. The comparison is carried out for different stress states with controlled triaxiality and Lode parameter under monotonic and cycling loading, paying special attention to the evolution of the mechanical fields in each phase. The unit-cell calculations are performed for both constant and evolving martensite volume fractions. In addition, numerical simulations of tensile tests on samples subjected to different initial temperatures are carried out for the transforming bi-phase material and the results are compared to experimental data for AISI 304L and AISI 316LN steels.

Multiblock parallel high-order implicit residual smoothing time scheme for compressible Navier–Stokes equations

In direct and large eddy simulations, very small space steps are used close to the solid walls in order to resolve the boundary-layer structures. Due to the restrictive CFL stability criteria of explicit time-stepping schemes, the maximum allowable time step is also very small, leading to high computational costs, notably for converging flow statistics. The use of an implicit integration scheme may overcome this limitation at the price of an increased computational cost per step. Moreover, the most commonly used fully implicit schemes induce higher errors due to the necessary approximations and poor dispersion and dissipation properties. As a compromise, a fourth-order implicit residual smoothing scheme (IRS4), successfully validated for a finite volume solver in Cinnella nad Content (2016); Hoarau and Cinnella (2020), has been introduced in a multiblock high-order finite-difference solver. Several improvements are proposed to ensure better dissipation properties, a more efficient treatment of physical boundaries and an accurate and stable parallel multiblock implementation. For moderate CFL numbers, a similar accuracy as the explicit method is obtained with substantial savings in terms of computational time.

Thermomechanical behaviour of soils and soil-structure interfaces in thermo-active geo-structures

In recent years, the demand for sustainable, low carbon-emission energy solutions has risen, leading to an increased interest in shallow geothermal energy systems. These systems, such as ground source heat pump (GSHP) systems, harness the Earth's stable temperature at shallow depths as a heat sink and source of energy. Energy piles are a type of GSHP system that performs both structural and thermal energy-exchange functions [e.g. 1]. Despite the technological advantages of these systems, understanding the thermomechanical behavior of soils and soil-structure interfaces remains a challenge, limiting safe, efficient, and cost-effective designs. This research aimed to address these challenges by developing thermomechanical constitutive models for soils, implementing these models in boundary-value solvers, and analyzing the experimental results from direct shear tests performed on soils to understand their thermomechanical behavior at the soil-structure interface. By comprehensively examining the complex interactions between energy piles and surrounding soils, the research results offer a foundation for improved design standards and implementation.
 The methodology followed the following three main directions:
 Development of thermomechanical constitutive models for soils: A thermomechanical constitutive model based on the thermodynamic framework of Hyperelasticity-Hyperplasticity was developed to capture the major thermomechanical behaviors of fine-grained soils when subjected to temperature variations [2]. By utilizing a flexible yield surface within the framework, a more precise representation of soil behavior can be simulated. This framework was extended to develop a dual-surface thermomechanical model with an additional yield surface and a temperature-dependent kinematic rule to capture the shakedown behavior of soils subjected to heating-cooling cycles more accurately [3], as illustrated in Figure 1 (left) where two triaxial tests are simulated. The enhanced model addresses the need for improved representation of diverse soil behaviors and their implications for the performance of energy pile systems.
 Numerical aspects to improve simulation capability: A new flexible yield function was introduced that addresses the common issue of undesired elastic domains and erratic or divergent gradients in the numerical implementation of constitutive models with flexible yield surfaces and plastic potentials for use in implicit stress integration schemes [4]. By providing a robust and efficient framework for return mapping algorithms, this new yield function supports more precise simulations of soil behavior. An advanced numerical algorithm was devised to implement the thermomechanical constitutive model within a boundary-value solver such as the finite-element method [5]. The algorithm incorporates Gibbs energy potential, Lode angle dependency, and additional features to ensure accuracy, robustness, effectiveness, and convergence during simulations.
 Experimental analysis of the soil-structure interface including the impact of initial shear stress and thermal cycles: Laboratory-scale direct shear tests were conducted on fine- and coarse-grained soils to investigate their thermomechanical behavior at the interface with a concrete structure [6, 7]. A focus here was on the impact of thermal cycles; in particular, under realistic working stress conditions. Figure 1 (right) presents example results, where thermal cycles are applied in a temperature controlled shear box, with a clay-concrete interface which had a constant shear stress. Thermal creep is observed, which also has a hardening effect on future shear loading. By exploring the variations in soil behavior under differing temperature and loading conditions, the research deepens the understanding of the complexity inherent in soil-pile interactions for energy pile applications.
 This comprehensive investigation of the thermomechanical behavior of soils and soil-structure interfaces in the context of thermo-active geo-structures represents an essential step forward in optimizing energy pile systems for diverse applications. The developed constitutive models, numerical algorithms, and experimental analyses contribute valuable insights for improving energy pile design, ultimately leading to more efficient, safe, and cost-effective solutions.
 Future research should continue refining our understanding of the complex processes governing soil behavior in energy pile systems, with emphasis on consolidating the developed models and their implementation for a range of soil types and conditions. Additionally, further experimental studies should focus on elucidating the interactions between soils, piles, and the broader environment to identify potential opportunities for maximizing the performance of energy pile systems through innovative design and implementation.

Two Simple Numerical Implementation Methods for Damage-Coupled Viscoplastic Constitutive Model

Abstract This paper is concerned with the two simple numerical implementation methods for a damage-coupled Chaboche-type viscoplastic constitutive model. By considering the damage variable as a constant in each incremental step, the return-mapping procedure is reduced to the solution of only one nonlinear scalar equation. Depending on the use of damage value in the current or prior incremental state, the two methods are named the backward difference implicit integration scheme and the two-step explicit integration scheme respectively. These two numerical algorithms are implemented into the ansys software by developing the usermat subroutine and verified by comparing them with available experimental data. Several numerical examples on the Gauss point level are studied in terms of stability, accuracy, computational efficiency, and applicability for further numerical observation. In addition to higher computational efficiency and lower memory requirements, the two methods can be easily extended to other damage models due to their simplicity.

An Optimization‐based SPH Solver for Simulation of Hyperelastic Solids

AbstractThis paper proposes a novel method for simulating hyperelastic solids with Smoothed Particle Hydrodynamics (SPH). The proposed method extends the coverage of the state‐of‐the‐art elastic SPH solid method to include different types of hyperelastic materials, such as the Neo‐Hookean and the St. Venant‐Kirchoff models. To this end, we reformulate an implicit integration scheme for SPH elastic solids into an optimization problem and solve the problem using a general‐purpose quasi‐Newton method. Our experiments show that the Limited‐memory BFGS (L‐BFGS) algorithm can be employed to efficiently solve our optimization problem in the SPH framework and demonstrate its stable and efficient simulations for complex materials in the SPH framework. Thanks to the nature of our unified representation for both solids and fluids, the SPH formulation simplifies coupling between different materials and handling collisions.

An anisotropic egg-shaped bounding surface model for saturated soft clay

An anisotropic egg-shaped bounding surface plasticity model (AESBS model) is proposed to describe cyclic and monotonic behaviors of saturated clays. The model is based on an innovative egg-shaped bounding surface, which not only eliminates the singularity present in most plasticity models and facilitates numerical calculation but also can be degenerated into various configurations. Moreover, to improve its representation when considering the initial consolidated state, an anisotropic tensor is incorporated into the bounding surface. Furthermore, a novel generalized isotropic hardening rule that combines isotropic and kinematic hardening is applied. It regards reversal stress points as homological mapping centers, hence leading to discrete and movable mapping centers. Notably, the elastic zone concept is employed to produce a more realistic performance of nonlinearity during cyclic loading. These modifications formulate the novel model that can be utilized to enhance the ability in predicting the behavior of saturated clays. To improve the convergence of the model, an implicit integration scheme is used to solved the constitutive relationship through some nonlinear algebraic equations. The comparison of simulation results and published experimental data from one-way undrained cyclic and monotonic triaxial tests for saturated clays demonstrates the efficiency of this model under different consolidated conditions.

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.

Challenges in GPU-Accelerated Nonlinear Dynamic Analysis for Structural Systems

Many numerical simulation methods, such as finite-element analysis, were originally formulated to run serially or in parallel on central processing units (CPUs). However, computer engineering has seen a paradigm shift toward massive parallelism using graphics processing units (GPUs), which have become the default accelerators in many data-driven scientific disciplines outside of civil engineering. This state-of-the-art review highlights the challenges and practicalities of GPU-accelerating nonlinear dynamic analyses for civil structural problems. To demonstrate the feasibility of a fully GPU-accelerated finite-element analysis, a GPU-based program for linear-elastic dynamic analysis was implemented, where all stages of the analysis were ported to the GPU. Observed speedups were 115 times that of an equivalent CPU-driven analysis for 106 model degrees of freedom (dof). Importantly, the computational time for the assembly and update levels of the analyses were nearly independent of the number of dof. High-resolution simulations of complex structures can be computationally expensive, but these results and advances in other fields suggest that some levels of the finite-element analysis of civil structures could be accelerated using GPUs at increased model resolution with little increase in computational cost, demonstrating the potential for GPU-accelerated computing. However, compared to other GPU-accelerated finite-element analysis applications, the dynamic analysis of civil structures is subject to unique challenges that need to be addressed before GPU acceleration can be fully realized. Aspects of simulating the response of civil structures considering nonlinear response under extreme loading may not be immediately amenable to GPU acceleration; e.g., the use of many differing element formulations within a model, potential for inelastic response and varying degrees of nonlinearity across elements, and traditional reliance on implicit integration schemes with direct solvers. The shift to GPUs is part of a larger movement toward specialized hardware using fine-grained parallelism, and structural engineers need to address these challenges as these emerging technologies become more prevalent.

A 2D/3D implicit gradient-enhanced nonlocal meso-scale damage model for deformation and fracturing of brittle materials

An implicit gradient-enhanced nonlocal meso‑scale damage model is developed to investigate deformation and fracturing of brittle materials. In the model, an implicit gradient integration scheme is adopted to deal with mesh dependency and strain localization. The mechanical parameters are assigned randomly using a Weibull statistical distribution to reflect the material heterogeneity at the meso‑scale. A linear elastic damage constitutive law with residual strength is used to describe the stress-strain relationship at the mesoscale and the Mohr-Coulomb criterion as well as the tensile stress criterion are used to evaluate the damage of the elements in tension or shear. The local equivalent strain is replaced by a nonlocal description employing the non-local implicit gradient model. Benchmark tests are conducted to document the potential of the model in simulating crack initiation and propagation processes in brittle materials. The benchmark tests also demonstrate that the proposed non-local damage model can effectively achieve strain localization and eliminate mesh dependency to some extent.

Modeling the anisotropic behavior of highly orthotropic lithium-ion batteries polymer separators

Separators in commercial lithium-ion batteries are microporous polymeric films characterized by highly anisotropic mechanical behavior. An accurate characterization of their mechanical and fracture behavior is essential for optimal design, performance, and safety of lithium-ion batteries. In this work, the mechanical and fracture behavior of battery separators subjected to uniaxial tension and punch tests are characterized and discussed. An anisotropic continuum damage coupled elastic-hyperelastic-viscoplastic model is developed in the framework of the large deformation theory to capture the mechanical and fracture behavior of battery separators. A robust implicit integration scheme is developed then the model is implemented as a user-defined material subroutine (UMAT) in commercial finite-element software in order to solve boundary-value problems. The capability of the proposed model to predict the anisotropic mechanical behavior up to fracture for two different types of battery separators highlights the versatility of the proposed modeling approach. The model results showed a good agreement with the experimental data. Moreover, the model was able to predict the failure of the separators under different loading conditions.

Strength of reinforced concrete columns under special complex influences

A method for evaluating the strength, stiffness and stability of medium-flexible columns with a square cross-sectional shape and symmetrical reinforcement is proposed. Combined impacts from constant longitudinal force, horizontal impact with trapezoidal pulse shape and deterioration in mechanical properties of concrete from natural influences such as corrosion are taken into account. Various combinations of such influences are also considered. The computational model is represented by volumetric concrete elements rigidly connected to rebars. The dynamic transient process is based on an implicit integration scheme using a stepwise procedure. At each step, the nonlinear problem is solved by the Newton-Raphson method. The data characterizing the degree of danger of occurrence of local strength deterioration for column variants have been compared. The dependence of the value of longitudinal force on the value of transverse impact is found, provided that the load-carrying capacity of the column is ensured.

Co-rotational Reduced Order Kalman Filter Finite Element Method for the Real-time Implementation of Large-scale Structural Models

The real-time implementation of large-scale structural models without sacrificing their physical realism is challenging. This paper introduces a conceptual framework named the Co-rotational Reduced Order Kalman Filter Finite Element Method (CROKF-FEM) for the realistic linear elastic analysis of large-scale structures in real-time. The standard linear finite element method (FEM) is employed for high-precision deformation modeling, and element matrices are pre-computed in the space domain. To avoid the inflated elements of standard linear FEM due to large rotation, a co-rotational formulation is devised. Then, the system is discretized in the time domain using the Wilson- θ implicit integration scheme and a further state-space model is developed. Subsequently, a reduced-order model is constructed using the balanced truncation method. Finally, a Kalman filter algorithm is created on the reduced model for online estimation. This approach preserves all the benefits of traditional linear FEM with a similar deformation accuracy while achieving significant real-time performance.

The influence of material model on the neck development for aluminium alloy sheet

The neck development for bore expansion problem is analyzed. Static implicit calculations are carried out for the A2219 Aluminium alloy sheet using ABAQUS/Standard. The equations to give the conditions of the onset of the localized necking of sheet metal based on the general bifurcation theory are presented for plane stress. Since accurate calculation of stresses and strains are the basis for the necking prediction, both associated and non-associated flow rule are adopted in the material model in which Hill’s quadratic functions are used. User defined subroutine UMAT is developed using a fully implicit integration scheme. The results of necking prediction show that necking occurrence delays for the Hill solid using both AFR and NAFR based model compared with Mises solid and necking occurrence slightly delays for the Hill solid using AFR based model compared with Hill solid using NAFR based model. Initial necking occurs when equivalent plastic strain is respectively 0.33, 0.39 and 0.348 for the Mises solid and Hill solids using AFR and NAFR based models. Theses indicate that the material model has significant influence on the accuracy of necking prediction.

Steady‐state analysis of discrete‐time nonsingular terminal sliding mode control systems based on Euler method

AbstractIn this article, the steady‐state behaviors of discrete‐time nonsingular terminal sliding mode (NTSM) control systems are investigated for the explicit and implicit Euler integration schemes. It is demonstrated that implicit Euler method can guarantee the global convergence of both system states and sliding variable for discrete‐time NTSM control systems, whereas explicit Euler approximation does not. For explicit discrete‐time NTSM, the range of quasi sliding mode band is deduced on the basis of the reachability condition of discrete SM, showing that discretized NTSM can achieve higher control accuracy than the discrete‐time linear sliding mode. By analyzing the distribution features of steady‐state points, the upper bound for steady states is obtained to estimate chattering amplitude, which establishes the influence relationship of sampling time, switching gain and movement characteristics on the bound of steady states. Simulations are presented to validate the theoretical results.

Physics-Based Self-Learning Recurrent Neural Network enhanced time integration scheme for computing viscoplastic structural finite element response

In the current study, we present an application to the class of deep learning known as the Physics Informed Neural Networks (PINNs), more specifically we develop a new implicit material integration scheme based on Recurrent Neural Network (RNN). In this study, we refer to it as the Recurrent Enhanced Implicit Integration Scheme (REIIS). We illustrate how to incorporate the constitutive relations into REIIS and explore its application to Lemaitre–Chaboche viscoplastic model. Whilst common applications in this class of deep learning utilize feed-forward neural networks (FFNN) for training, we propose a combination of the Long Term Short Memory (LSTM) network with FFNNs for an accurate representation of the state variables. Furthermore, with the present strategy, we combine the data-driven approach of replacing the material behavior with NN along with its ability to learn continuously through physical laws. To validate the approach, we test the results of the framework with the numerical solution obtained with Finite Element Method (FEM) for Reissner–Mindlin plate elements at different strain rates. We further observe that obeying physical constraints leads to improved robustness in the learning of REIIS and broadens the scope of integrating the NNs in FEM.

A finite element-peridynamic combined multiscale analysis strategy based on implicit integration scheme

Although peridynamics (PD) method can address discontinuity issues existing in civil engineering, it is much more computationally expensive than the conventional finite element (FE) method because of inherent shortcomings such as small grid spacing and explicit integration scheme. In this regard, a multiscale analysis method reformulated in an implicit integration scheme, named FE-PD method, is proposed, which integrates the PD method with the FE method to take the advantages of discontinuity solving capacity of the PD method and high computing efficiency of the FE method. This study commences with the introduction of the bond-based PD theory. Subsequently, the framework of the FE-PD method together with two newly developed interface elements is presented. Finally, how to implement the FE-PD method into an open-source FE software, OpenSees, is illustrated, and the proposed FE-PD method is verified by a benchmark problem involving an l-shaped concrete plate sustaining a vertical loading, as well as a 2D and a 3D cantilever beam sustaining uniaxial tensile and shear loadings, respectively. Besides, the FE-PD method is employed to simulate the bond-slip behavior of the rebar embedded in a solid concrete specimen. The investigation results confirm that the FE-PD method not only consumes relatively much less (e.g., about 70%) computing resources compared with the PD method but also well captures the crack propagation of concrete and the bond-slip behavior of the rebar embedded in a quasi-brittle concrete specimen.

The role of time integration in energy conservation in Smoothed Particle Hydrodynamics fluid dynamics simulations

The choice of a time integration scheme is a crucial aspect of any transient fluid simulation, and Smoothed-Particle Hydrodynamics (SPH) is no exception. The influence of the time integration scheme on energy balance is here addressed. To do so, explicit expressions allowing to compute the deviations from the energy balance, induced by the time integration scheme, are provided. These expressions, computed a posteriori, are valid for different integration methods. Besides, a new formulation that improves energy conservation by enhancing stability, based on an implicit integration scheme, is proposed. Such formulation is tested with the simulation of a two-dimensional non-viscous impact of two jets, with no artificial dissipation terms. To the best of our knowledge, this is the first stable simulation of a non-dissipative system with a weakly-compressible SPH method. A viscous case, the Taylor–Green vortex, has also been simulated. Results show that an implicit time integration scheme also behaves better in a viscous context.