Adaptive Time Step Control Research Articles

Parameter identification of vibratory conveying systems including statistical part behavior

This work presents a complete workflow for the parameter identification of an MBS model representing a vibratory conveying system. First, the MBS model built within the multibody dynamics tool HOTINT is shown. Essential components, such as contact modeling and adaptive time step control, are explained in detail. The model includes a considerable amount of parameters that need to be identified. Altough, this model is fully deterministic, the complex contact dynamics can cause major effect on results by minor parameter variations. Thus, parameter variations usually show statistical variance, which is characteristic for real conveying processes too. For parameter identification it is important to detect parameters that significantly correlate with results. A method that allows to distinguish significant and non-significant parameters is presented. The application to the multibody system shows that stiffness, restitution and sliding friction of the contact model are the essential parameters. Next, the parameter identification is discussed with particular attention to multimodal distributions, that can occur in conveying systems. Furthermore, the integration of statistics into the deterministic simulation model is discussed. In order to demonstrate the applicability of the proposed methods, a parameter identification is performed based on measurement data of a single part on a real conveying system in the development site of STIWA Automation GmbH. Finally, the validity of the fully parametrized simulation model is shown by investigating geometric adjustments of the conveyor. In order to further verify the identified set of parameters, a conveying process with multiple parts is analyzed and compared in measurement and simulation.

High fidelity transient solver in STREAM based on multigroup coarse-mesh finite difference method

This study incorporates a high-fidelity transient analysis solver based on multigroup CMFD in the MOC code STREAM. Transport modeling with heterogeneous geometries of the reactor core increases computational cost in terms of memory and time, whereas the multigroup CMFD reduces the computational cost. The reactor condition does not change at every time step, which is a vital point for the utilization of CMFD. CMFD correction factors are updated from the transport solution whenever the reactor core condition changes, and the simulation continues until the end. The transport solution is adjusted once CMFD achieves the solution. The flux-weighted method is used for rod decusping to update the partially inserted control rod cell material, which maintains the solution's stability. A smaller time-step size is needed to obtain an accurate solution, which increases the computational cost. The adaptive step-size control algorithm is robust for controlling the time step size. This algorithm is based on local errors and has the potential capability to accept or reject the solution. Several numerical problems are selected to analyze the performance and numerical accuracy of parallel computing, rod decusping, and adaptive time step control. Lastly, a typical pressurized LWR was chosen to study the rod-ejection accident.

EVALUATION OF DISSIPATION ENERGY OF ISOTROPIC CONTINUUM DAMAGE MECHANICS MODEL WITH ADAPTIVE TIME-STEP CONTROL APPROACH

This paper proposes an algorithm to evaluate dissipation energy of an isotropic continuum damage mechanics model with an adaptive time-step control approach. The algorithm takes place at each integration point under the scope of finite element analysis. The total amount of the dissipation energy of one structure can be used to verify the conservation law of energy, where the summation of the elastic strain energy and the dissipation energy should be equal to the total work by external force. However, the paper shows that this condition may not be valid when the unstable crack propagation occurs. When the crack propagation is unstable, the development of damage can be driven by stored strain energy without external load. The strains at the beginning and ending time within one increment at the damaged elements will give inaccurate dissipation energy by the classic trapezoidal integration scheme. In addition, this paper proposes a new damage evolution shape function with C1 continuity of the strain-stress curve under the uniaxial tension test, which is used to demonstrate the numerical procedure of the adaptive algorithm.

ESPSim: An Efficient Scalable Power Grid Simulator Based on Parallel Algebraic Multigrid

Fast verification for the extremely large-scale power grid is demanding as CMOS technology advances consistently. In this work, we propose ESPSim, an efficient scalable power grid simulator based on a parallel smoothed aggregation-based algebraic multigrid technique. ESPSim has the ability to do fast DC and transient analysis through MPI and adaptive timestep control mechanism. Thanks to the smoother applied on the prolongation operator, ESPSim copes well with the convergence rate on extremely large-scale power grid transient analysis. Extensive experiments are conducted with a variety of serial/parallel solvers. The runtime of ESPSim is linear with case size. With 16 processors, 1,000 timesteps transient analysis of 63.4M nodes can be completed in 22.1 min. Over 22× speedup compared to the well-known direct solver Cholmod is observed.

The regularization continuation method with an adaptive time step control for linearly constrained optimization problems

This paper considers the regularization continuation method and the trust-region updating strategy for the optimization problem with linear equality constraints. The proposed method utilizes the linear conservation law of the regularization continuation method such that it does not need to compute the correction step for preserving the feasibility other than the previous continuation methods and the quasi-Newton updating formulas for the linearly constrained optimization problem. Moreover, the new method uses the special limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) formula as the preconditioning technique to improve its computational efficiency in the well-posed phase, and it uses the inverse of the regularized two-sided projection of the Lagrangian Hessian as the pre-conditioner to improve its robustness. Numerical results also show that the new method is more robust and faster than the traditional optimization method such as the alternating direction method of multipliers (ADMM), the sequential quadratic programming (SQP) method (the built-in subroutine fmincon.m of the MATLAB2020a environment), and the recent continuation method (Ptctr). The computational time of the new method is about 1/3 of that of SQP (fmincon.m). Finally, the global convergence analysis of the new method is also given.

Numerical simulation of mid-deep geothermal system based on dynamic mesh adaptive finite element

Numerical simulation is an important tool in the study of geothermal science and even the development and utilization of geothermal resources. Aiming at the development of middle and deep geothermal resources, this paper takes the numerical simulation of U-shaped heat exchange well as the research object. Based on Fourier’s law and heat conduction equation, the dynamic mesh adaptive finite element method is used to simulate the heat transfer process. The results show that the fixed mesh non-adaptive finite element method lacks a feasible error control mechanism, and the adaptive high-order finite element method with dynamic mesh and adaptive time step control can effectively simulate the process of geothermal energy transmission.

Accelerating numerical simulation of continuous-time Boolean satisfiability solver using discrete gradient

To explore the design of analog computing devices, modeling the problem-solving process as a continuous-time dynamical system is important. Ercsey-Ravasz and Toroczkai [Nature Physics 7, 966 (2011)] proposed such a model for solving the Boolean satisfiability (SAT) problem. This system consists of a gradient system that minimizes the potential function reduced from the SAT problem and a system for achieving a temporal variation of the potential function to avoid the problem of non-solution local minima. Although the ability of the system to find a solution to the SAT problem is demonstrated, its large simulation cost hinders theoretical research towards the physical realization and limits its utility on digital computers. This is due to the necessity of small time steps to maintain the numerical stability of the simulation. In this study, we propose a fast and stable numerical simulation algorithm for this solver using the discrete gradient method to allow a larger time step. We also propose an adaptive time step control method for this system. The proposed algorithm achieves a faster simulation by a factor of approximately 100, compared to conventional methods. Although taking a large time step degrades the accuracy, we found that it does not necessarily degrade the performance as a SAT solver; this indicates the new utility of the discrete gradient apart from the conventional studies of numerical simulation algorithms that pursue accuracy as well as efficiency.

Adaptive time-step control for a monolithic multirate scheme coupling the heat and wave equation

We study the dynamics of a parabolic and a hyperbolic equation coupled on a common interface. We develop time-stepping schemes that can use different time-step sizes for each of the subproblems. The problem is formulated in a strongly coupled (monolithic) space-time framework. Coupling two different step sizes monolithically gives rise to large algebraic systems of equations. There, multiple states of the subproblems must be solved at once. For efficiently solving these algebraic systems, we inherit ideas from the partitioned regime. Therefore we present two decoupling methods, namely a partitioned relaxation scheme and a shooting method. Furthermore, we develop an a posteriori error estimator serving as a mean for an adaptive time-stepping procedure. The goal is to optimally balance the time-step sizes of the two subproblems. The error estimator is based on the dual weighted residual method and relies on the space-time Galerkin formulation of the coupled problem. As an example, we take a linear set-up with the heat equation coupled to the wave equation. We formulate the problem in a monolithic manner using the space-time framework. In numerical test cases, we demonstrate the efficiency of the solution process and we also validate the accuracy of the a posteriori error estimator and its use for controlling the time-step sizes.

Variational inequality transport model on the sphere by the active-set reduced-space algorithm

The simulation of the transport problem on the sphere is crucial in the numerical modeling of the transport of trace constituents in atmospheric models. One major issue in the numerical simulation is the ability of the solver to obtain accurate constraint-preserving solutions, i.e., ensuring the predicted solution to stay within the physical range. In this paper, we develop and study a variational inequality (VI) based optimization methodology for constructing a new transport model that naturally satisfies this restriction by removing over- and under-shoots of the solution. With the use of the cubed-sphere mesh, we present an explicit-first-step, single-diagonal coefficient, diagonally implicit Runge–Kutta (ESDIRK) method with an adaptive time step control strategy for the fully implicit temporal integration of the variational inequality problem. And then the resulting nonlinear system arising at each time step is solved by using some nonlinear and linear fast solver technologies, including a variant of inexact Newton methods, i.e., the active-set reduced-space (ASRS) method, and the domain decomposition based preconditioners with a novel analytical Jacobian matrix. A set of tracer transport test cases based on the variational inequality model is presented to demonstrate the efficiency and robustness of the proposed method.

Adaptive time-step control for modal methods to integrate the neutron diffusion equation

The solution of the time-dependent neutron diffusion equation can be approximated using quasi-static methods that factorise the neutronic flux as the product of a time dependent function times a shape function that depends both on space and time. A generalization of this technique is the updated modal method. This strategy assumes that the neutron flux can be decomposed into a sum of amplitudes multiplied by some shape functions. These functions, known as modes, come from the solution of the eigenvalue problems associated with the static neutron diffusion equation that are being updated along the transient. In previous works, the time step used to update the modes is set to a fixed value and this implies the need of using small time-steps to obtain accurate results and, consequently, a high computational cost. In this work, we propose the use of an adaptive control time-step that reduces automatically the time-step when the algorithm detects large errors and increases this value when it is not necessary to use small steps. Several strategies to compute the modes updating time step are proposed and their performance is tested for different transients in benchmark reactors with rectangular and hexagonal geometry.

Sensitivity Analysis of Time-Averaged Quantities of Chaotic Systems

Sensitivity analysis of chaotic dynamics in resolved turbulence simulations is still considered a major technical challenge, and existing methods are cost-prohibitive for practical applications. Accurate and efficient sensitivity analysis of such responses is necessary for successful application of design optimization to systems that exhibit chaotic dynamics. This paper presents a fundamentally new approach for sensitivity analysis of time-averaged quantities of chaotic systems. This includes the development of a stabilized time integrator with adaptive time-step control to maintain stability of the sensitivity calculations. Therefore, the approach maintains the initial-value formulation of the problem and avoids costly boundary-value space-time formulation in the least-square shadowing approaches. Both direct- and adjoint-sensitivity analyses approaches are discussed. The method is then applied to the Lorenz system and Kuramoto–Sivashinsky equation, where the sensitivity of time-averaged quantities is computed. The results show that the method provides a robust computational approach with effectively no extra cost other than to compute the sensitivity using a standard implicit time integrator for linear systems.

A hybrid stochastic method with adaptive time step control for reaction–diffusion systems

Randomness often plays an important role in the spatial and temporal dynamics of biological systems. General stochastic simulation methods may lead to excessive computational cost for a system in which a large number of molecules involved. Therefore, multi-scale hybrid simulation methods become important for stochastic simulations. Here we build a spatially hybrid method which couples two approaches: discrete stochastic simulation and continuous stochastic differential equations. In our method, the locations of the interfaces between the two approaches are changing according to the distribution of molecules in a one-dimensional domain. To balance the accuracy and efficiency, the time step of the numerical method for the continuous stochastic differential equations is adapted to the dynamics of the molecules near the adaptive interfaces. The simulation results for a linear system and two nonlinear biological systems in different one-dimensional domains demonstrate the effectiveness and advantage of our new hybrid method with the adaptive time step control.

A staggered procedure for fluid–object interaction with free surfaces, large rotations and driven by adaptive time stepping

The coupling between a rigid body under large rotations and incompressible fluids is investigated within the arbitrary Lagrangian–Eulerian framework. We use here a staggered type of coupling with a predictor/corrector approach for the forces applied to the rigid body. Adaptive time stepping based on feedback control theory imposing a CFL condition on the mesh is investigated. The coupling scheme is first tested on a case illustrating vortex-induced vibrations around a rotating plate. We show the advantages of using the residual-based variational multiscale method for the fluid in the present context. Also, the time-step control and the role of the parameters introduced for the predictor/corrector approach are illustrated using the same test case. A reduced model FPSO ship is then studied, comparing its pitch decay with experimental results. A complex wave–rigid body interaction calculation is finally presented. Results demonstrated the robustness of the predictor/corrector staggered approach with adaptive time-step control for simulating complex interactions of a rigid body under large rotations and free-surface flows.

An adaptive time step control scheme for the transient diffusion equation

The stability and accuracy of an adaptive time step control scheme are analyzed for the transient diffusion equation. This scheme is based on the commonly-implemented backward difference discretization of the diffusion equation and recommends optimal time steps based on constraints applied to estimates of the local truncation error. Methods are derived for both error estimation and error control, each of which potentially impacts the stability of the scheme and the global accuracy of the solution. Asymptotic stability and convergence of the recommended time steps are investigated theoretically and demonstrated numerically to identify optimal realizations of the method. This adaptive time stepping scheme requires no solution evaluations or operator inversions beyond those already performed in the adaption-free solution and requires no modifications to the numerical solution algorithm. As such, this adaptivity scheme can be easily implemented in virtually any reactor physics simulation code based on a backward difference discretization of transient neutronics.

Accurate and stable time stepping in ice sheet modeling

In this paper we introduce adaptive time step control for simulation of the evolution of ice sheets. The discretization error in the approximations is estimated using “Milne's device” by comparing the result from two different methods in a predictor–corrector pair. Using a predictor–corrector pair the expensive part of the procedure, the solution of the velocity and pressure equations, is performed only once per time step and an estimate of the local error is easily obtained. The stability of the numerical solution is maintained and the accuracy is controlled by keeping the local error below a given threshold using PI-control. Depending on the threshold, the time step Δt is bound by stability requirements or accuracy requirements. Our method takes a shorter Δt than an implicit method but with less work in each time step and the solver is simpler. The method is analyzed theoretically with respect to stability and applied to the simulation of a 2D ice slab and a 3D circular ice sheet. The stability bounds in the experiments are explained by and agree well with the theoretical results.

Adaptive Strategies in the Leader Propagation Model for Lightning Shielding Failure Evaluation: Implementation and Applications

Lightning shielding failure accounts for the highest proportion of lightning accidents for transmission systems above 500 kV. The leader propagation model (LPM) was found to be effective in the analysis of lightning shielding failure of high-voltage transmission lines, especially for the ultrahigh voltage (UHV) transmission systems. However, it was generally inefficient to calculate the shielding failure rate by LPM, since the progress of lightning development must be simulated repeatedly, which is time-consuming, and no adaptive implementation of LPM has been reported. In this paper, adaptive strategies to accelerate the computation of shielding failure analysis by LPM were proposed, which include the adaptive time step control according to the electric field at the conductor surface, non-uniform lightning current sampling based on the lightning current amplitude probability distribution, and the adaptive lightning location sampling based on recursive adaptive Simpson’s method. The analysis results were validated by the field data of the UHV transmission line in Japan, and the efficiency of the adaptive strategies was shown by the shielding failure analysis of a typical 500 kV ac transmission line.

Efficient Moving Mesh Methods for $Q$-Tensor Models of Nematic Liquid Crystals

This paper describes a robust and efficient numerical scheme for solving the system of six coupled partial differential equations which arises when using $Q$-tensor theory to model the behavior of a nematic liquid crystal cell under the influence of an applied electric field. The key novel feature is the use of a full moving mesh partial differential equation approach to generate an adaptive mesh which accurately resolves important solution features. This includes the use of a new monitor function based on a local measure of biaxiality. In addition, adaptive time-step control is used to ensure the accurate predicting of the switching time, which is often critical in the design of liquid crystal cells. We illustrate the behavior of the method on a one-dimensional time-dependent problem in a Pi-cell geometry which admits two topologically different equilibrium states, modeling the order reconstruction which occurs on the application of an electric field. Our numerical results show that, in addition to achieving optimal rates of convergence in space and time, we obtain higher levels of solution accuracy and a considerable improvement in computational efficiency compared to other moving mesh methods used previously for liquid crystal problems.

Adaptive time step control for higher order variational time discretizations applied to convection–diffusion–reaction equations

Higher order variational time stepping schemes allow an efficient post-processing for computing a higher order solution. This paper presents an adaptive algorithm whose time step control utilizes the post-processed solution. The algorithm is applied to convection-dominated convection–diffusion–reaction equations. It is shown that the length of the time step properly reflects the dynamics of the solution. With respect to the performance (accuracy, efficiency), the variational time stepping schemes are compared with an adaptive Crank–Nicolson scheme, whose time step control relies on comparing two solutions computed with schemes of the same order.

A temporally adaptive hybridized discontinuous Galerkin method for time-dependent compressible flows

The potential of the hybridized discontinuous Galerkin (HDG) method has been recognized for the computation of stationary flows. Extending the method to time-dependent problems can, e.g., be done by backward differentiation formula (BDF) or diagonally implicit Runge–Kutta (DIRK) methods. In this work, we investigate the use of embedded DIRK methods in an HDG solver, including the use of adaptive time-step control. Numerical results demonstrate the performance of the method for both linear and nonlinear (systems of) time-dependent convection–diffusion equations.

시간차분 유한요소법을 이용한 대용량 삼상 변압기의 정상상태 해석에 관한 연구

This paper presents the fast steady state analysis using time-stepping finite element method for a high power three-phase transformer. The high power transformer spends huge computational cost of the time-stepping finite element method. It is because that the high power transformer requires a lot of time to reach steady state by its large inductance component. In order to reduce computational cost, in this paper, the adaptive time-step control algorithm combined with the embedded 2nd 4th singly diagonally implicit Runge-Kutta method and the analysis strategy using variation of the winding resistance are studied, and their numerical results are compared with those from the typical time-stepping finite element method.