Implicit Algorithm Research Articles

On implementing the implicit discrete‐time super‐twisting observer on mechanical systems

AbstractIn this paper, an extension of an algorithm for the implicit discretization of a super‐twisting sliding mode observer is presented. Implicit and explicit discretization algorithms for homogeneous differentiators, where no physical model information is considered, are investigated in literature. This article studies the behavior when considering models of a rather general class of nonlinear systems. The discrete equations of the super‐twisting observer are reformulated as generalized equation and an algorithm for the step‐by‐step solution is given. The uniqueness of the derived algorithm is investigated with an equivalent variational inequality formulation which is derived for a class of nonlinear systems. Furthermore, a semi‐implicit predictor‐corrector discretization is presented which is an approximation method for the presented algorithms and allows an explicit implementation in practical applications. Accuracy properties under noise and sampling are given. The algorithm is applied on two mechanical example systems taken from practice.

A Three-Dimensional Elastoplastic Constitutive Model for Geomaterials

The Mohr-Coulomb (M-C) failure criterion has been a popular choice for geotechnical analysis because of its simplicity and ease of use. The fact that the M-C criterion disregards the intermediate principal stress’s impact is a significant drawback. As a result, the M-C criterion is only applied to materials under biaxial stress. This paper presents a three-dimensional version of the M-C criterion. The proposed criterion, called the Generalized Mohr-Coulomb (GMC) criterion, considers the intermediate principal stress’s effect, in addition to inheriting the original M-C criterion’s benefits. We obtained the conditions that the strength parameters must satisfy when the GMC criterion fulfills the π plane’s convexity. The GMC criterion can better describe geotechnical materials’ strengths under general stress conditions. Based on an implicit algorithm, the user material subroutine (UMAT) of the three-dimensional GMC model was developed in ABAQUS using the Fortran programming language. The established elastoplastic model’s validity and the program’s accuracy were examined using numerical simulation. Finally, a numerical simulation of a three-dimensional tunnel excavation under various working conditions was performed. The calculation results from the GMC model are precise and have some engineering-related practical significance.

Wasserstein Adversarially Regularized Graph Autoencoder

This paper introduces Wasserstein Adversarially Regularized Graph Autoencoder (WARGA), an implicit generative algorithm that directly regularizes the latent distribution of node embedding to a target distribution via the Wasserstein metric. To ensure the Lipschitz continuity, we propose two approaches: WARGA-WC that uses weight clipping method and WARGA-GP that uses gradient penalty method. The proposed models have been validated by link prediction and node clustering on real-world graphs with visualizations of node embeddings, in which WARGA generally outperforms other state-of-the-art models based on Kullback–Leibler (KL) divergence and typical adversarial framework.

An implicit, conservative and asymptotic-preserving electrostatic particle-in-cell algorithm for arbitrarily magnetized plasmas in uniform magnetic fields

We introduce a new electrostatic particle-in-cell algorithm capable of using large timesteps compared to particle gyro-period under a uniform external magnetic field. The algorithm extends earlier electrostatic fully implicit PIC implementations with a new asymptotic-preserving particle-push scheme that allows timesteps much larger than particle gyroperiods. In the large-timestep limit, the integrator preserves all particle drifts, while recovering the full orbit for small timesteps. The scheme allows for a seamless, efficient treatment of particles with coexisting magnetized and unmagnetized species, and conserves energy and charge exactly without spoiling implicit solver performance. We demonstrate by numerical experiment with several problems of variable species magnetization (diocotron instability, modified two-stream instability, and drift instability) that orders of magnitude wall-clock-time speedups vs. the standard fully implicit electrostatic PIC algorithm are possible without sacrificing solution accuracy.

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.

Demonstration of Pronghorn’s Subchannel Code Modeling of Liquid-Metal Reactors and Validation in Normal Operation Conditions and Blockage Scenarios

Pronghorn-SC is a subchannel code within the Multiphysics Object-Oriented Simulation Environment (MOOSE). Initially designed to simulate flows in water-cooled, square lattice, subchannel assemblies, Pronghorn-SC has been expanded to simulate liquid-metal-cooled flows in triangular lattices, hexagonal subchannel assemblies. For this purpose, the algorithm of Pronghorn-SC was adapted to solve the subchannel equations as they are applicable to a hexagonal wire-wrapped sodium-cooled fast reactor. Cheng–Todreas models for pressure drop and cross-flow models were adopted and a coolant heat conduction term was added. To solve these equations, an improved implicit algorithm was developed robust enough to deal with the numerical issues, associated with low flow and recirculation phenomena. To confirm the prediction capability of Pronghorn-SC, calculations and comparisons with available experimental data of 19- and 37-pin assemblies were performed, as well as other subchannel codes. Finally, a flow blockage modeling feature was added. This capability was validated for both water-cooled square sub-assemblies and sodium-cooled hexagonal sub-assemblies, using experimental data of partially and fully blocked cases.

A unified method to simulate electrodeposition and galvanic corrosion using the peridynamic differential operator

This study presents a unified method to model electrodeposition and galvanic corrosion. The governing equations and boundary conditions are recast in their nonlocal form by using the peridynamic differential operator. Electric potential and mass transport equations are solved simultaneously in ANSYS through implicit algorithms for computational efficiency and complex geometry. Deposition and corrosion fronts are established by applying the concept of deposition fraction and examining the concentration values. Element matrices associated with peridynamic interactions are constructed through MATRIX27 elements native to ANSYS. The damping is modeled using combin14 element and the resulting equations are solved in an implicit fashion. Accuracy of this unified method is established by comparison with the published experimental measurements and simulations concerning simple and irregular geometries.

Numerical Analysis of Unsteady Heat Transfer in the Chamber in the Piston Hybrid Compressor with Regenerative Heat Exchange

The adjoint thermodynamic and heat exchange processes in the new class piston compressor with regenerative heat exchange are considered in the paper. The implicit tridiagonal matrix algorithm is implemented to study the unsteady thermal conductivity in the cylinder–piston group. After the formulation of proper initial and boundary conditions, the optimal relationship between temporal and spatial discretization steps is determined. Two different time steps are used in the numerical solution of the two-way coupled fast thermodynamic and slow heat exchange models. The relationship between those time steps is determined as well. The conducted numerical experiments allow the analysis of the dynamics of heat exchange in the cylinder–piston group, temperature variation in different parts of the cylinder–piston group, the impact of the heat transfer processes on isothermal and adiabatic efficiency, the impact of heat exchange dynamics on the thermodynamic cycle, as well as other thermodynamic and energetic effects.

Complex Envelope Weaker HIE With Higher Order PML Algorithm for FDTD Simulation

Weaker hybrid implicit–explicit (WHIE) algorithm with complex envelope (CE) method is proposed for structures with fine details along a single direction in bandpass problems. Higher order perfectly matched layer (PML) formulation is proposed for absorption improvement in open regions. By introducing alternating-direction implicit (ADI) discretized form and adding proper disturbance terms, it can obtain relaxed stability condition under the same Courant–Friedrichs–Levy (CFL) numbers (CFLNs) compared with the other implicit algorithms resulting in the improvement of efficiency. Furthermore, time step can also be enlarged according to the bandpass lowpass-sampling theorem which improves the efficiency in bandpass circumstances.

Implicit Algorithm of the SBSP-R Model for Predicting the Non-Unique Critical State of Soils

The non-unique critical state represents the distance between the critical state line (CSL) and the isotropic consolidation line (ICL) that significantly varies with stress paths and particle size distribution of soils. A structural bounding surface plasticity model with spacing ratio r (SBSP-R model) was implemented using an explicit algorithm. However, the explicit algorithm did not well capture the non-unique critical state of soils with a large spacing ratio r, which prevented the soil mechanics research on non-unique critical state via finite element analysis. To overcome the limitation, the implicit algorithm of the SBSP-R model is formulated, and it mainly includes elastic prediction and plastic correction. The plastic correction is realized using the Newton–Simpson scheme with a controlling equation set related to consistency condition, plastic flow, hardening parameter, structural bounding surface, plastic modulus, and mapping rule. Case studies indicate that the implicit algorithm of the SBSP-R model is right and stable in predicting non-unique critical states. Comparisons between predicted and tested results indicate that the implicit algorithm of the SBSP-R model not only captures the critical state, stress-strain, and stress paths of various soils but also shows higher computational accuracy and efficiency compared with the previous explicit algorithm. These results indicate that the formulated implicit algorithm of the SBSP-R model is an alternative approach to the previous explicit algorithm.

A method of fully implicit coupled analyses for thermoset resin subjected to cure based on variationally consistent formulation for finite thermo-viscoelasticity

A variationally consistent formulation is presented to devise a method of fully implicit analysis for predicting thermo-mechanically coupled behavior of thermosetting resin subjected to cure in finite strain theory. To characterize the finite thermo-viscoelasticity coupled with cure kinetics, the dual dissipation potential (DDP) is originally defined in the thermodynamically consistent formulation, which begins with the definition of stored energy. The governing equations are derived as stationary conditions of the relevant total energy rate potential containing internal energy, and viscous and curing DDPs, and then discretized in time and space in order. In the temporal discretization process within a time interval, we utilize the function form of the equilibrium temperature obtained from the Legendre–Fenchel transformation of internal energy. The obtained time discretized equations are spatially discretized by means of the standard finite element method. The present formulation results in a fully implicit algorithm with separate systems of governing equations that can be solved in a monolithic manner. Two representative numerical examples are presented to demonstrate that, while allowing relatively large time increments, the devised analysis method provides a more realistic prediction than the previous method formulated within the framework of small strain theory.

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.

Using the two-way nesting technique AGRIF with MARS3D V11.2 to improve hydrodynamics and estimate environmental indicators

Abstract. In the ocean, mesoscale or submesoscale structures and coastal processes are associated with fine scales. The simulation of such features thus requires the hydrodynamic equations to be solved at high-resolution (from a few hundred meters down to a few tens of meters). Therefore, local mesh refinement is a primary issue for regional and coastal modeling. The AGRIF (adaptive grid refinement in Fortran) library is committed to tackling such a challenge for structured grids. It has been implemented in MARS3D (Model for Application at Regional Scale), a semi-implicit, free-surface numerical model developed by Ifremer (the French Research Institute for Exploitation of the Sea) for coastal environmental research and studies. As its time scheme uses an alternating-direction implicit (ADI) algorithm, the two-way nesting implementation differs from the one in explicit models. The present paper describes the specifics of the AGRIF introduction and how the nesting preserves some essential properties (mass, momentum and tracer conservations) along with the induced constraints (bathymetric coherence between grids and increase in computation cost). The use and the performance of this new tool are detailed over two configurations that illustrate the wide range of scales and resolutions typically targeted by coastal applications. The first one is based on multiple high-resolution (500 m) grids that pave the coastal ocean over thousands of kilometers, allowing a continuum between the regional and coastal scales. The second application is more local and has a finer resolution (50 m). It targets a recurrent question for semi-enclosed bays, i.e., the renewal time indicator. Throughout these configurations, the paper intends to compare the two-way nesting method with the traditional one-way approach. It highlights how the MARS3D-AGRIF tool proves to be an efficient way to both improve the physical hydrodynamics and unravel ecological challenges.

Parallel fully implicit chemical potential-based modeling of unconventional shale gas reservoirs

The design of novel mathematical models and state-of-the-art simulators for unconventional shale gas reservoirs plays an increasingly important role in the current energy mix of the world’s growing energy demand. In this paper, we present a robust and scalable framework, which is based on the chemical potential-based modeling and the fully implicit method (FIM), to model and simulate this highly nonlinear flow transport problem on a large scale. In the proposed approach, a thermodynamically consistent mathematical model of gas flow in shale formations, which employs the gas density and the chemical potential gradient, is developed to satisfy the second law of thermodynamics and meanwhile guarantee the energy dissipation property. And then the family of fully implicit finite element algorithms is utilized to accurately capture the complicated flow physics behind the transport process in shale media on high resolution grids. In particular, our approach further enhances the numerical formulation by proposing the family of Newton–Krylov methods for efficiently computing, and the parallel implementation of the simulator is achieved by using the domain decomposition technique. Numerical experiments are presented to demonstrate the robustness and parallel scalability of the solution strategies for several interesting shale gas flow problems in two or three dimensions. With the proposed parallel method, large-scale reservoir simulation can be obtained on the Shaheen-II supercomputer with up to 40960 processors, which enables to achieve a good strong scalability by saving more than 90 percent of computing time, when the problem size is enlarged to hundreds of millions of degrees of freedom.

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.

Three-dimensional time-domain particle-in-cell calculations of impedances and centroid deflections in a linear-accelerator cell

In linear induction accelerators, high-brightness electron beams are accelerated by voltages applied at gaps at discrete locations along the accelerator. At high currents, the beam itself induces wakefields in the accelerating gaps which can then feedback and distort the beam. The coupling of the gap-cavity modes and the beam can be characterized by frequency-dependent quantities known as the parallel and perpendicular gap impedances. Assessing the effects of instabilities resulting from these interactions requires accurate knowledge of these quantities. In this paper, we describe how a 3D finite-difference time-domain particle-in-cell code can be used to calculate both parallel and perpendicular gap impedances. We also demonstrate good agreement between full particle-in-cell simulations and results from a beam transport code making use of gap impedances to model beam centroid deflections. In practice, ferrite materials are often used in accelerating gaps to dampen cavity modes and reduce the severity of the resulting instabilities. We describe an implicit recursive convolution algorithm used to model the linear response of dispersive ferrite materials.

ABAQUS and ANSYS Implementations of the Peridynamics-Based Finite Element Method (PeriFEM) for Brittle Fractures

In this study, we propose the first unified implementation strategy for peridynamics in commercial finite element method (FEM) software packages based on their application programming interface using the peridynamics-based finite element method (PeriFEM). Using ANSYS and ABAQUS as examples, we present the numerical results and implementation details of PeriFEM in commercial FEM software. PeriFEM is a reformulation of the traditional FEM for solving peridynamic equations numerically. It is considered that the non-local features of peridynamics yet possesses the same computational framework as the traditional FEM. Therefore, this implementation benefits from the consistent computational frameworks of both PeriFEM and the traditional FEM. An implicit algorithm is used for both ANSYS and ABAQUS; however, different convergence criteria are adopted owing to their unique features. In ANSYS, APDL enables users to conveniently obtain broken-bond information from UPFs; thus, the convergence criterion is chosen as no new broken bond. In ABAQUS, obtaining broken-bond information is not convenient for users; thus, the default convergence criterion is used in ABAQUS. The codes integrated into ANSYS and ABAQUS are both verified through benchmark examples, and the computational convergence and costs are compared. The results show that, for some specific examples, ABAQUS is more efficient, whereas the convergence criterion adopted in ANSYS is more robust. Finally, 3D examples are presented to demonstrate the ability of the proposed approach to deal with complex engineering problems.

Numerical study on the effect of airflow on powder mixing in a container blender

Powder mixing is critical in many industries. Despite the wide variety of available mixers, the container blender is favored in industries due to easy manufacturing and convenient operation. As fine powders are frequently encountered in practical mixing, the presence of air during mixing may significantly impact the process. However, a scientific understanding of air–particle interactions in powder mixing has not been established so far. From a physical view, the air drag force on particles might be significant when the gas velocity is high and the particles are fine. Therefore, this novel study numerically investigates the effects of particle size and air presence on powder mixing under typical conditions, such that the relationship between particle–fluid dynamics and mixing performance is clarified for the first time. In the calculation, our advanced computational fluid dynamics–discrete element method, namely, the flexible Eulerian–Lagrangian method with an implicit algorithm, is utilized. To examine the effect of particle size on powder mixing, the coarse-grained discrete element method is employed for fine particle systems. Through the advanced numerical framework, the effect of airflow on powder mixing can be discussed in depth. The numerical results show that airflow accelerates the mixing of fine particles under the investigated rotation speeds. Based on the calculation results of the particle kinetics and fluid velocity distribution, it is clarified that the accelerated mixing results from the fine particle movement under air entrainment. How the movement of fine particles is developed is also elucidated by the continuously circulating gas flow and fluid drag force. Thus, this study provides a new understanding of the effects of airflow on powder mixing, which has not been scientifically clarified in previous studies.

Three-dimensional X-FEM modeling of crack coalescence phenomena in the Smart CutTM technology

A new 3D X-FEM approach is proposed to model crack coalescence phenomena in the Smart Cut TM technology. An adaptive mesh is developed based on a 3D fractal mesh allowing a significant gain in computing time compared to regular meshes. In order to calculate stress intensity factors (SIFs), the term corresponding to internal pressure, that is normal loading on crack faces has been determined, added to the interaction integral and the results of SIFs in mode I have been compared to their analytical values with mean error less than 1%. An implicit algorithm is implemented to predict the evolution of the internal pressure for growing cracks of any shape; a constitutive law in pressure correlated with the equation of state of ideal gases has therefore been used. Compared to the analytical solution, numerically evaluated pressure is accurate with maximum errors less than 1% for relevant convergence tolerance. Finally, the coalescence of two cracks is taken into account in the model. Qualitative local stress analysis makes it possible to explain some phenomena experimentally observed in the Smart Cut TM such as neighboring effects of growing cracks, the outgrowth of nearby cracks or influence of the presence of concave areas at a front on growth. • A 3D X-FEM model has been proposed to model crack coalescence phenomena under variable pressure (Smart Cut TM process). • The interaction integral has been adapted to take into account the internal pressure when computing 3D stress intensity factors. • Prediction of the internal pressure of propagating cracks whatever are their shapes. • Analysis of neighboring effects by stress analysis on a two penny-shape crack coalescence model.

A three stages transition into chaos for mixed convection nanofluid inside a differentially-heated square ventilated cavity

In this paper, we present a numerical study of an unsteady Al2O3 water nanofluid mixed convection in a differentially-heated square ventilated cavity. The routes from steady state to chaotic mixed convection state have been also studied for a constant Reynolds Number (Re), Re = 50 and Richardson number (Ri) ranging from Ri = 40 to Ri = 150. The governing equations are solved using an implicit finite-volume scheme and TDMA algorithm. Results are presented through streamlines, isotherms, mean temperature, average Nusselt number, pressure drop, etc. It is observed that the average Nusselt number and the mean cavity temperature values are all the higher as the Ri number is great. The pressure drop slightly decreases as Ri number increases. The fully chaotic flow regime is reached after three stages. The first stage is similar to the Ruelle-Takens-Newhouse scenario. At the end of this stage, apparition of periodicity interrupted the development of a full chaos initiating the second stage of the route to chaos where another series of three supercritical Hopf type bifurcations and a period doubling bifurcation takes place. The third stage starts by the emergence of intermittencies which interrupts the apparition of a full chaotic flow. A complete chaos is achieved at the end of this third stage.