Subcycling Algorithm Research Articles

A reacting multiphase computational flow model for 2,3-butanediol synthesis in industrial-scale bioreactors

The microbe, Zymomonas mobilis, can efficiently convert sugars to 2,3 butanediol (BDO), an important intermediate for downstream chemical products, only in a well-controlled microaerated environment. However, controlling oxygen distribution in industrial-scale bioreactors is challenging, and at-scale BDO production is hard to optimize using traditional engineering methods. This study takes a step towards addressing this problem through a computational model for reacting multiphase flows in large-scale bioreactors. A phenomenological metabolic model was first developed and validated against experiments, and then it was coupled to a multiphase computational fluid dynamics (CFD) solver using a subcycling algorithm for simulating long conversion times ( ∼ 30 h). Large-scale ( ∼ 500 m3) bubble column simulations using this coupled approach demonstrated a 25% improvement in BDO yield with low sparging rates (0.002 m/s) and low height-to-diameter ratio (0.875) compared to a baseline bubble column (35 m height, 5 m diameter) with a superficial gas velocity of 0.06 m/s.

An inner-outer subcycling algorithm for parallel cardiac electrophysiology simulations.

This paper explores cardiac electrophysiological simulations of the monodomain equations and introduces a novel subcycling time integration algorithm to exploit the structure of the ionic model. The aim of this work is to improve upon the efficiency of parallel cardiac monodomain simulations by using our subcycling algorithm in the computation of the ionic model to handle the local sharp changes of the solution. This will reduce the turnaround time for the simulation of basic cardiac electrical function on both idealized and patient-specific geometry. Numerical experiments show that the proposed approach is accurate and also has close to linear parallel scalability on a computer with more than 1000 processor cores. Ultimately, the reduction in simulation time can be beneficial in clinical applications, where multiple simulations are often required to tune a model to match clinical measurements.

The Role of Radiation in Common Envelope Evolution

The common envelope phase in binary star systems is simulated using the 3-D moving-mesh hydrodynamic code MANGA. The radiation hydrodynamics module in MANGA is modified using a sub-cycling algorithm to skip unnecessary calculations. The sub-cycling as well as other optimizations reduce computation time by a factor of 3. The generation of appropriate initial conditions for radiation hydrodynamic simulations is discussed.

An improvement of rigid bodies contact for particle-based non-smooth walls modeling

Problems of fluid–structure interaction with free-surface flow and multiple bodies are highly nonlinear phenomena, which is challenging for computational modeling and simulation. Particle-based methods have been proven to provide a substantial potential for simulation of free-surface fluid flows and their interactions with multiple solids that are often encountered in fluvial and coastal mechanics fields, due to their Lagrangian tracking scheme and meshless nature. When contact between solids occurs, a numerical modeling to detect it and prevent penetration between bodies is required to avoid numerical inconsistencies. The objective of this work is to propose a solid–solid contact model that relays on contact mechanics theories and, by an effective way to determine the normal direction of contact as well as the distance between bodies faces, eliminates the numerical instabilities induced by non-smooth modeling of bodies surfaces in the particle-based methods. The model adopts a penalty-based method that uses a nonlinear spring and dashpot concept, and thus reproducing the macroscopic properties of the multiple bodies interactions. To avoid very small time step for all the domain, a dynamic sub-cycling algorithm for rigid bodies contact is adopted. First, the present fluid solver, based on the moving particle semi-implicit method, is validated by hydrodynamic and fluid–structure interaction problems. Then, analytical and numerical results are compared, demonstrating the improvements achieved by the proposed model. Finally, the present model is validated via 3D dam breaking events, in which rigid bodies interact with each other or with fixed walls.

Adaptive multi-time-domain subcycling for crystal plasticity FE modeling of discrete twin evolution

Crystal plasticity finite element (CPFE) models that accounts for discrete micro-twin nucleation-propagation have been recently developed for studying complex deformation behavior of hexagonal close-packed (HCP) materials (Cheng and Ghosh in Int J Plast 67:148–170, 2015, J Mech Phys Solids 99:512–538, 2016). A major difficulty with conducting high fidelity, image-based CPFE simulations of polycrystalline microstructures with explicit twin formation is the prohibitively high demands on computing time. High strain localization within fast propagating twin bands requires very fine simulation time steps and leads to enormous computational cost. To mitigate this shortcoming and improve the simulation efficiency, this paper proposes a multi-time-domain subcycling algorithm. It is based on adaptive partitioning of the evolving computational domain into twinned and untwinned domains. Based on the local deformation-rate, the algorithm accelerates simulations by adopting different time steps for each sub-domain. The sub-domains are coupled back after coarse time increments using a predictor-corrector algorithm at the interface. The subcycling-augmented CPFEM is validated with a comprehensive set of numerical tests. Significant speed-up is observed with this novel algorithm without any loss of accuracy that is advantageous for predicting twinning in polycrystalline microstructures.

Crystal plasticity finite element modeling of discrete twin evolution in polycrystalline magnesium

This paper develops an advanced, image-based crystal plasticity finite element (CPFE) model, for predicting explicit twin formation and associated heterogeneous deformation in single crystal and polycrystalline microstructures of hexagonal close-packed or hcp materials, such as magnesium. Twin formation is responsible for premature failure of many hcp materials. The physics of nucleation, propagation and growth of explicit twins are considered in the CPFE formulation. The twin nucleation model is based on dissociation of sessile dislocations into stable twin loops, while propagation is assumed by atoms shearing on twin planes and shuffling to reduce the thermal activation energy barrier. The explicit twin evolution model however has intrinsic issues of low computational efficiency. Very fine simulation time steps with enormous computation costs are required to simulate the fast propagating twin bands and associated strain localization. To improve the computational efficiency, a multi-time scale subcycling algorithm is developed. It decomposes the computational domain into sub-domains of localized twins requiring very fine time-steps and complementary domains of relatively low resolution. Each sub-domain updates the stress and the deformation-dependent variables in different rates, followed by a coupling at the end of every coarse time step to satisfy global equilibrium. A 6-fold increase in computing speed is obtained for a polycrystalline Mg microstructure simulation in this paper. CPFE simulations of high purity Mg microstructures are compared with experiments with very good agreement in stress-strain response as well as heterogeneous twin formation with strain localization.

Efficient time integration in dislocation dynamics

The efficiencies of one implicit and three explicit time integrators have been compared in line dislocation dynamics simulations using two test cases: a collapsing loop and a Frank–Read (FR) source with a jog. The time-step size and computational efficiency of the explicit integrators is shown to become severely limited due to the presence of so-called stiff modes, which include the oscillatory zig-zag motion of discretization nodes and orientation fluctuations of the jog. In the stability-limited regime dictated by these stiff modes, the implicit integrator shows superior efficiency when using a Jacobian that only accounts for short-range interactions due to elasticity and line tension. However, when a stable dislocation dipole forms during a jogged FR source simulation, even the implicit integrator suffers a substantial drop in the time-step size. To restore computational efficiency, a time-step subcycling algorithm is tested, in which the nodes involved in the dipole are integrated over multiple smaller, local time steps, while the remaining nodes take a single larger, global time step. The time-step subcycling method leads to substantial efficiency gain when combined with either an implicit or an explicit integrator.

Study on sub-cycling algorithm for flexible multi-body system: stability analysis and numerical examples

Numerical stability is an important issue for any integral procedure. Since sub-cycling algorithm has been presented by Belytschko et al. (Comput Methods Appl Mech Eng 17/18: 259–275, 1979), various kinds of these integral procedures were developed in later 20 years and their stability were widely studied. However, on how to apply the sub-cycling to flexible multi-body dynamics (FMD) is still a lack of investigation up to now. A particular sub-cycling algorithm for the FMD based on the central difference method was introduced in detail in part I (Miao et al. in Comp Mech doi: 10.1007/s00466-007-0183-9) of this paper. Adopting an integral approximation operator method, stability of the presented algorithm is transformed to a generalized eigenvalue problem in the paper and is discussed by solving the problem later. Numerical examples are performed to verify the availability and efficiency of the algorithm further.

Study on sub-cycling algorithm for flexible multi-body system—integral theory and implementation flow chart

A sub-cycling integration algorithm (or named multi-time-steps integration algorithm), which has been successfully applied to FEM dynamical analysis, was firstly presented by Belytschko et al. (Comput Methods Appl Mech Eng 17/18:259–275, 1979). However, the problem of how to apply this type of algorithm to flexible multi-body dynamics (FMD) problems still lacks investigation up to now. Similar to the region-partitioning method used in FEM, this paper presents a central-difference-based sub-cycling integral method by decomposing the variables of an FMD equation into several groups and adopting different integral step sizes to each group of the variables. Based on the condensed form of an FMD equation, a group of common update formulae and a sub-step update formula, which constitute the sub-cycling together, are established in the paper. Furthermore, an implementation flowchart of the sub-cycling is presented. Stability of the sub-cycling will be analyzed and numerical examples will be performed to verify availability and precision of the sub-cycling in part II of the paper.

Efficient and robust constitutive integrators for single-crystal plasticity modeling

Small-scale deformation phenomena such as subgrain formation, development of texture, and grain boundary sliding require simulations with a high degree of spatial resolution. When we consider finite-element simulation of metal deformation, this equates to thousands or hundreds of thousands of finite elements. Simulations of the dynamic deformations of metal samples require elastic–plastic constitutive updates of the material behavior to be performed over a small time step between updates, as dictated by the Courant condition. Further, numerical integration of physically-based equations is inherently sensitive to the step in time taken; they return different predictions as the time step is reduced, eventually approaching a stationary solution. Depending on the deformation conditions, this converged time step becomes short (10 −9 s or less). If an implicit constitutive update is applied to this class of simulation, the benefit of the implicit update (i.e., the ability to evaluate over a relatively large time step) is negated, and the integration is prohibitively slow. The present work recasts an implicit update algorithm into an explicit form, for which each update step is five to six times faster, and the compute time required for a plastic update approaches that needed for a fully-elastic update. For dynamic loading conditions, the explicit model is found to perform an entire simulation up to 50 times faster than the implicit model. The performance of the explicit model is enhanced by adding a subcycling algorithm to the explicit model, by which the maximum time step between constitutive updates is increased an order of magnitude. These model improvements do not significantly change the predictions of the model from the implicit form, and provide overall computation times significantly faster than the implicit form over finite-element meshes. These modifications are also applied to polycrystals via Taylor averaging, where we also see improved model performance.

The subcycled Newmark algorithm

The popular Newmark algorithm, used for implicit direct integration of structural dynamics, is extended by means of a nodal partition to permit use of different timesteps in different regions of a structural model. The algorithm developed has as a special case an explicit-explicit subcycling algorithm previously reported by Belytschko, Yen and Mullen. That algorithm has been shown, in the absence of damping or other energy dissipation, to exhibit instability over narrow timestep ranges that become narrower as the number of degrees of freedom increases, making them unlikely to be encountered in practice. The present algorithm avoids such instabilities in the case of a one to two timestep ratio (two subcycles), achieving unconditional stability in an exponential sense for a linear problem. However, with three or more subcycles, the trapezoidal rule exhibits stability that becomes conditional, falling towards that of the central difference method as the number of subcycles increases. Instabilities over narrow timestep ranges, that become narrower as the model size increases, also appear with three or more subcycles. However by moving the partition between timesteps one row of elements into the region suitable for integration with the larger timestep these the unstable timestep ranges become extremely narrow, even in simple systems with a few degrees of freedom. As well, accuracy is improved. Use of a version of the Newmark algorithm that dissipates high frequencies minimises or eliminates these narrow bands of instability. Viscous damping is also shown to remove these instabilities, at the expense of having more effect on the low frequency response.

Analysis and implementation of a new constant acceleration subcycling algorithm

An algorithm for explicit integration of structural dynamics problems with multiple time steps is proposed that averages accelerations to obtain subcycle states at a nodal interface between regions integrated with different time steps. With integer time step ratios, the resulting subcycle updates at the interface sum to give the same effect as a central difference update over a major cycle. The algorithm is shown to have good accuracy, and stability properties in linear elastic analysis similar to those of constant velocity subcycling algorithms. The implementation of a generalised form of the algorithm with non-integer time step ratios is presented. (C) 1997 by John Wiley & Sons, Ltd.

Convergence and stability analyses of multi-time step algorithm for parabolic systems

An analysis of the consistency and convergence of a subcycling algorithm together with the stability conditions for parabolic systems of both element and nodal partitions are provided. It is shown that the algorithms only attain a first-order rate-of-convergence for both element and nodal partitions in the sense of the total cycle update. The stability conditions for general integration parameters in an element partition are also given for the first time.

SECOND-ORDER-ACCURATE EXPLICIT SUBCYCLING ALGORITHM FOR UNSTEADY HEAT CONDUCTION

An explicit subcycling algorithm that attains second-order temporal accuracy is presented for solving the semidiscrete equations of unsteady heat conduction. The algorithm is based on a nodal partitioning of an explicit, single-step method in which the second time derivatives of temperature are computed. All nodal temperatures are updated at each time step using an extension of the variable explicit time integration method developed by Mizu-kami. A time step restriction is derived that provides a sufficient condition for stability. Numerical results are presented that verify the improved accuracy of the subcycling scheme compared to the first-order accuracy obtained by other known explicit subcycling methods.

Electron sub-cycling in particle simulation of plasma

A straightforward modification which reduces by half the computational cost of standard particle-in-cell algorithms for simulation of plasmas is described. The saving is obtained by integrating only the electrons through a number of time steps (sub-cycles) in order to resolve their evolution, while integrating the much slower ions only once per cycle, i.e., to match the time step of each species to their characteristic frequencies. A dispersion relation is derived which describes the numerical instabilities expected by sampling frequency arguments. Simulations support the broad features of the analytical results, viz., the maximum growth rate and domain of the instability, and its stabilization by the addition of weak damping. An implicit sub-cycling algorithm is suggested which may provide further saving while avoiding a limitation of implicit algorithms described elsewhere.