scholarly journals Magnetic field simulations using explicit time integration with higher order schemes

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Bernhard Kähne ◽  
Markus Clemens ◽  
Sebastian Schöps
Keyword(s):  
Time Integration ◽  
Schur Complement ◽  
Higher Order ◽  
Step Size ◽  
Time Step ◽  
Explicit Time Integration ◽  
Content Type ◽  
Time Step Size ◽  
Potential Formulation ◽  
Ode System

Purpose A transient magneto-quasistatic vector potential formulation involving nonlinear material is spatially discretized using the finite element method of first and second polynomial order. By applying a generalized Schur complement the resulting system of differential algebraic equations is reformulated into a system of ordinary differential equations (ODE). The ODE system is integrated in time by using explicit time integration schemes. The purpose of this paper is to investigate explicit time integration for eddy current problems with respect to the performance of the first-order explicit Euler scheme and the Runge-Kutta-Chebyshev (RKC) method of higher order. Design/methodology/approach The ODE system is integrated in time using the explicit Euler scheme, which is conditionally stable by a maximum time step size. To overcome this limit, an explicit multistage RKC time integration method of higher order is used to enlarge the maximum stable time step size. Both time integration methods are compared regarding the overall computational effort. Findings The numerical simulations show that a finer spatial discretization forces smaller time step sizes. In comparison to the explicit Euler time integration scheme, the multistage RKC method provides larger stable time step sizes to diminish the overall computation time. Originality/value The explicit time integration of the Schur complement vector potential formulation of eddy current problems is accelerated by a multistage RKC method.

Explicit time integration of eddy current problems using a selective matrix update strategy

2017 ◽  
Vol 36 (5) ◽  
pp. 1364-1371 ◽  
Author(s):  
Jennifer Susanne Dutiné ◽  
Markus Clemens ◽  
Sebastian Schöps
Keyword(s):  
Eddy Current ◽  
Time Integration ◽  
Schur Complement ◽  
Equation System ◽  
Differential Algebraic Equation ◽  
Time Step ◽  
Explicit Time Integration ◽  
Content Type ◽  
Ode System ◽  
Update Strategy

Purpose Discretizing the magnetic vector potential formulation of eddy current problems in space results in an infinitely stiff differential algebraic equation system that is integrated in time using implicit time integration methods. Applying a generalized Schur complement to the differential algebraic equation system yields an ordinary differential equation (ODE) system. This ODE system can be integrated in time using explicit time integration schemes by which the solution of high-dimensional nonlinear algebraic systems of equations is avoided. The purpose of this paper is to further investigate the explicit time integration of eddy current problems. Design/methodology/approach The resulting magnetoquasistatic Schur complement ODE system is integrated in time using the explicit Euler method taking into account the Courant–Friedrich–Levy (CFL) stability criterion. The maximum stable CFL time step can be rather small for magnetoquasistatic field problems owing to its proportionality to the smallest edge length in the mesh. Ferromagnetic materials require updating the reluctivity matrix in nonlinear material in every time step. Because of the small time-step size, it is proposed to only selectively update the reluctivity matrix, keeping it constant for as many time steps as possible. Findings Numerical simulations of the TEAM 10 benchmark problem show that the proposed selective update strategy decreases computation time while maintaining good accuracy for different dynamics of the source current excitation. Originality/value The explicit time integration of the Schur complement vector potential formulation of the eddy current problem is accelerated by updating the reluctivity matrix selectively. A strategy for this is proposed and investigated.

Interactions between adaptive time-integrators and adaptive meshing in a monolithic FEM solver

2019 ◽  
Vol 29 (7) ◽  
pp. 2297-2323 ◽  
Author(s):  
Etienne Muller ◽  
Dominique Pelletier ◽  
André Garon
Keyword(s):  
Time Integration ◽  
Adaptive Meshing ◽  
Step Size ◽  
Time Step ◽  
Content Type ◽  
Time Step Size ◽  
Time Integrators ◽  
Grid Convergence ◽  
Adaptive Time ◽  
Grid Interpolation

Purpose This paper aims to focus on characterization of interactions between hp-adaptive time-integrators based on backward differentiation formulas (BDF) and adaptive meshing based on Zhu and Zienkiewicz error estimation approach. If mesh adaptation only occurs at user-supplied times and results in a completely new mesh, it is necessary to stop the time-integration at these same times. In these conditions, one challenge is to find an efficient and reliable way to restart the time-integration. The authors investigate what impact grid-to-grid interpolation errors have on the relaunch of the computation. Design/methodology/approach Two restart strategies of the time-integrator were used: one based on resetting the time-step size h and time-integrator order p to default values (used in the initial startup phase), and another designed to restart with the time-step size h and order p used by the solver prior to remeshing. The authors also investigate the benefits of quadratically interpolate the solution on the new mesh. Both restart strategies were used to solve laminar incompressible Navier–Stokes and the Unsteady Reynolds Averaged Naviers-Stokes (URANS) equations. Findings The adaptive features of our time-integrators are excellent tools to quantify errors arising from the data transfer between two grids. The second restart strategy proved to be advantageous only if a quadratic grid-to-grid interpolation is used. Results for turbulent flows also proved that some precautions must be taken to ensure grid convergence at any time of the simulation. Mesh adaptation, if poorly performed, can indeed lead to losing grid convergence in critical regions of the flow. Originality/value This study exhibits the benefits and difficulty of assessing both spatial error estimates and local error estimates to enhance the efficiency of unsteady computations.

Newmark-Beta Method in Discrete Elastic Rods Algorithm to Avoid Energy Dissipation

2019 ◽  
Vol 86 (8) ◽  
Author(s):  
Weicheng Huang ◽  
Mohammad Khalid Jawed
Keyword(s):  
Energy Dissipation ◽  
Time Integration ◽  
Integration Scheme ◽  
Integration Step ◽  
Elastic Rods ◽  
Computationally Efficient ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Time Integration Scheme

Discrete elastic rods (DER) algorithm presents a computationally efficient means of simulating the geometrically nonlinear dynamics of elastic rods. However, it can suffer from artificial energy loss during the time integration step. Our approach extends the existing DER technique by using a different time integration scheme—we consider a second-order, implicit Newmark-beta method to avoid energy dissipation. This treatment shows better convergence with time step size, specially when the damping forces are negligible and the structure undergoes vibratory motion. Two demonstrations—a cantilever beam and a helical rod hanging under gravity—are used to show the effectiveness of the modified discrete elastic rods simulator.

scholarly journals Accuracy Analysis of a Spectral Element Atmospheric Model Using a Fully Implicit Solution Framework

2010 ◽  
Vol 138 (8) ◽  
pp. 3333-3341 ◽  
Author(s):  
Katherine J. Evans ◽  
Mark A. Taylor ◽  
John B. Drake
Keyword(s):  
Shallow Water Equation ◽  
Time Integration ◽  
Atmospheric Model ◽  
Test Cases ◽  
Implicit Methods ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Fully Implicit ◽  
Stability Limits

Abstract A fully implicit (FI) time integration method has been implemented into a spectral finite-element shallow-water equation model on a sphere, and it is compared to existing fully explicit leapfrog and semi-implicit methods for a suite of test cases. This experiment is designed to determine the time step sizes that minimize simulation time while maintaining sufficient accuracy for these problems. For test cases without an analytical solution from which to compare, it is demonstrated that time step sizes 30–60 times larger than the gravity wave stability limits and 6–20 times larger than the advective-scale stability limits are possible using the FI method without a loss in accuracy, depending on the problem being solved. For a steady-state test case, the FI method produces error within machine accuracy limits as with existing methods, but using an arbitrarily large time step size.

Direct Numerical Simulation of the Interaction of an Ultra Short-Pulsed Intense Laser With a H2+ Molecule

2008 ◽  
Author(s):  
Y.-M. Lee ◽  
J.-S. Wu ◽  
T.-F. Jiang ◽  
Y.-S. Chen
Keyword(s):  
Domain Decomposition Method ◽  
Time Integration ◽  
Three Dimensional ◽  
Time Algorithm ◽  
Intense Laser ◽  
Angle Of Incidence ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
H2 Molecule

In this paper, interactions of a linearly polarized ultra short-pulsed intense laser with a single H2+ molecule at various angles of incidence are studied by directly solving the time-dependent three-dimensional Schrodinger equation (TDSE), assuming Born-Oppenheimer approximation. An explicit stagger-time algorithm is employed for time integration of the TDSE, in which the real and imaginary parts of the wave function are defined at alternative times, while a cell-centered finite-volume method is utilized for spatial discretization of the TDSE on Cartesian grids. The TDSE solver is then parallelized using domain decomposition method on distributed memory machines by applying a multi-level graph-partitioning technique. The solver is applied to simulate laser-molecular interaction with test conditions including: laser intensity of 0.5*1014 W/cm2, wavelength of 800 nm, three pulses in time, angle of incidence of 0–90° and inter-nuclear distance of 2 a.u.. Simulation conditions include 4 million hexahedral cells, 90 a.u. long in z direction, and time-step size of 0.005 a.u.. Ionization rates, harmonic spectra and instantaneous distribution of electron densities are then obtained from the solution of the TDSE. Future possible extension of the present method is also outlined at the end of this paper.

Modeling the damped dynamic behavior of a flexible pendulum

2019 ◽  
Vol 54 (2) ◽  
pp. 116-129 ◽  
Author(s):  
Roberto Ortega ◽  
Geraldine Farías ◽  
Marcela Cruchaga ◽  
Matías Rivero ◽  
Mariano Vázquez ◽  
...  
Keyword(s):  
High Speed ◽  
Time Integration ◽  
Integration Scheme ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Rayleigh Damping ◽  
Integration Schemes ◽  
Time Integration Schemes ◽  
Damping Model

The focus of this work is on the computational modeling of a pendulum made of a hyperelastic material and the corresponding experimental validation with the aim of contributing to the study of a material commonly used in seismic absorber devices. From the proposed dynamics experiment, the motion of the pendulum is recorded using a high-speed camera. The evolution of the pendulum’s positions is recovered using a capturing motion technique by tracking markers. The simulation of the problem is developed in the framework of a parallel multi-physics code. Particular emphasis is placed on the analysis of the Newmark integration scheme and the use of Rayleigh damping model. In particular, the time step size effect is analyzed. A strong time step size dependency is obtained for dissipative time integration schemes, while the Rayleigh damping formulation without time integration dissipation shows time step–independent results when convergence is achieved.

Picard-Newton Iterative Method with Time Step Control for Multimaterial Non-Equilibrium Radiation Diffusion Problem

2011 ◽  
Vol 10 (4) ◽  
pp. 844-866 ◽  
Author(s):  
Jingyan Yue ◽  
Guangwei Yuan
Keyword(s):  
Iterative Method ◽  
Time Integration ◽  
Step Size ◽  
Time Step ◽  
Radiation Diffusion ◽  
Reaction Diffusion Systems ◽  
Time Step Size ◽  
Step Control ◽  
Equilibrium Radiation ◽  
Non Equilibrium

AbstractFor a new nonlinear iterative method named as Picard-Newton (P-N) iterative method for the solution of the time-dependent reaction-diffusion systems, which arise in non-equilibrium radiation diffusion applications, two time step control methods are investigated and a study of temporal accuracy of a first order time integration is presented. The non-equilibrium radiation diffusion problems with flux limiter are considered, which appends pesky complexity and nonlinearity to the diffusion coefficient. Numerical results are presented to demonstrate that compared with Picard method, for a desired accuracy, significant increase in solution efficiency can be obtained by Picard-Newton method with the suitable time step size selection.

Scaling of Constraints and Augmented Lagrangian Formulations in Multibody Dynamics Simulations

2009 ◽  
Vol 4 (2) ◽  
Author(s):  
Olivier A. Bauchau ◽  
Alexander Epple ◽  
Carlo L. Bottasso
Keyword(s):  
Physical Properties ◽  
Augmented Lagrangian ◽  
Equations Of Motion ◽  
Time Integration ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Integration Schemes

This paper addresses practical issues associated with the numerical enforcement of constraints in flexible multibody systems, which are characterized by index-3 differential algebraic equations (DAEs). The need to scale the equations of motion is emphasized; in the proposed approach, they are scaled based on simple physical arguments, and an augmented Lagrangian term is added to the formulation. Time discretization followed by a linearization of the resulting equations leads to a Jacobian matrix that is independent of the time step size, h; hence, the condition number of the Jacobian and error propagation are both O(h0): the numerical solution of index-3 DAEs behaves as in the case of regular ordinary differential equations (ODEs). Since the scaling factor depends on the physical properties of the system, the proposed scaling decreases the dependency of this Jacobian on physical properties, further improving the numerical conditioning of the resulting linearized equations. Because the scaling of the equations is performed before the time and space discretizations, its benefits are reaped for all time integration schemes. The augmented Lagrangian term is shown to be indispensable if the solution of the linearized system of equations is to be performed without pivoting, a requirement for the efficient solution of the sparse system of linear equations. Finally, a number of numerical examples demonstrate the efficiency of the proposed approach to scaling.

Unconditional Stability in Numerical Time Integration Methods

1973 ◽  
Vol 40 (2) ◽  
pp. 417-421 ◽  
Author(s):  
R. D. Krieg
Keyword(s):  
Time Integration ◽  
Step Size ◽  
Time Step ◽  
Central Difference ◽  
Integration Methods ◽  
Time Step Size ◽  
Time Integration Method ◽  
Time Integration Methods ◽  
Numerical Time Integration ◽  
Difference Time

Methods of numerical time integration of the equation M¯q¨ + K¯q = f are examined in this paper. A particular class of explicit time integration methods is defined and this class is searched for an unconditionally stable method. The class is found to contain no such method and, furthermore, is found to contain no method with a larger stable time step size than that characterized by the simple central difference time integration method.

scholarly journals Implicit–explicit (IMEX) Runge–Kutta methods for non-hydrostatic atmospheric models

2018 ◽  
Vol 11 (4) ◽  
pp. 1497-1515 ◽  
Author(s):  
David J. Gardner ◽  
Jorge E. Guerra ◽  
François P. Hamon ◽  
Daniel R. Reynolds ◽  
Paul A. Ullrich ◽  
...  
Keyword(s):  
Acoustic Waves ◽  
Time Integration ◽  
Coupled System ◽  
Atmospheric Model ◽  
Step Size ◽  
Time Step ◽  
Solution Quality ◽  
Time Step Size ◽  
Runge Kutta ◽  
The Impact

Abstract. The efficient simulation of non-hydrostatic atmospheric dynamics requires time integration methods capable of overcoming the explicit stability constraints on time step size arising from acoustic waves. In this work, we investigate various implicit–explicit (IMEX) additive Runge–Kutta (ARK) methods for evolving acoustic waves implicitly to enable larger time step sizes in a global non-hydrostatic atmospheric model. The IMEX formulations considered include horizontally explicit – vertically implicit (HEVI) approaches as well as splittings that treat some horizontal dynamics implicitly. In each case, the impact of solving nonlinear systems in each implicit ARK stage in a linearly implicit fashion is also explored.The accuracy and efficiency of the IMEX splittings, ARK methods, and solver options are evaluated on a gravity wave and baroclinic wave test case. HEVI splittings that treat some vertical dynamics explicitly do not show a benefit in solution quality or run time over the most implicit HEVI formulation. While splittings that implicitly evolve some horizontal dynamics increase the maximum stable step size of a method, the gains are insufficient to overcome the additional cost of solving a globally coupled system. Solving implicit stage systems in a linearly implicit manner limits the solver cost but this is offset by a reduction in step size to achieve the desired accuracy for some methods. Overall, the third-order ARS343 and ARK324 methods performed the best, followed by the second-order ARS232 and ARK232 methods.

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