Research progress and performance evaluations for self-starting single-solve explicit time integrators

PurposeA 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/approachThe 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.FindingsThe 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/valueThe explicit time integration of the Schur complement vector potential formulation of eddy current problems is accelerated by a multistage RKC method.

A nonconformal hybrid finite difference time domain (FDTD)/finite element time domain (FETD) method was previously introduced, which implemented the hybridization through a buffer zone. Although this method has been demonstrated to be accurate and long-time stable, further efforts are still desirable to remove the buffer zone and to implement an implicit–explicit time integration from the perspective of practical applications. In this paper, a novel hybrid method is proposed, which not only successfully eliminates the necessity of the buffer zone without compromising the featured advantage (e.g., nonconformal mesh) but also effectively applies an implicit–explicit time integration scheme to improve the computational efficiency. Furthermore, the new method extends the hybridization to a broader level by incorporating the spectral element time domain (SETD) method based on the discontinuous Galerkin and domain decomposition techniques, resulting in a more general hybrid FDTD/SETD/FETD framework. The framework employs the explicit leapfrog time integration for the FDTD region while it employs the implicit Crank–Nicolson time integration for the FETD region. For the SETD region, either the implicit or explicit time integration can be employed, depending on the mesh sizes in it. When the implicit region becomes large, it can be further split into multiple subdomains to reduce computational complexity. Numerical examples are included to demonstrate the performance of the proposed hybrid method, which is accurate, long-time stable and more efficient than the hybrid method with a buffer zone.

This study presents the impact of the difference between the implicit and explicit time integration methods on a steady turbulent flow field. In contrast to the explicit time integration method, the implicit time integration method may produce significant kinetic energy conservation error because the widely used spatial difference method for discretizing the governing equations is explicit with respect to time. In this study, the second-order Crank-Nicolson method is used as the implicit time integration method, and the fourth-order Runge-Kutta, second-order Runge-Kutta and second-order Adams-Bashforth methods are used as explicit time integration methods. In the present study, both isotropic and anisotropic steady turbulent fields are analyzed with two values of the Reynolds number. The turbulent kinetic energy in the steady turbulent field is hardly affected by the kinetic energy conservation error. The rms values of static pressure fluctuation are significantly sensitive to the kinetic energy conservation error. These results are examined by varying the time increment value. These results are also discussed by visualizing the large scale turbulent vortex structure.

Dispersion-corrected explicit integration of the wave equation

The use of sandwich plates has gained significant popularity in the construction and machinery industries due to their exceptional stiffness-to-mass and strength-to-mass ratios. Among various structural types, I-core sandwich plates (T-joints) and corrugated-core sandwich plates (K-joints) are widely utilized. The welding method employed significantly impacts joint performance and overall structural characteristics. This review paper examines recent research on these commonly used sandwich plates and their joint performance, encompassing preparation methods, performance evaluation, structure optimization, and overall research progress. Furthermore, the influence of different preparation methods on the overall performance of sandwich plates is also addressed.

In computational structural dynamics, particularly in the presence of nonsmooth behavior, the choice of the time-step and the time integrator has a critical impact on the feasibility of the simulation. Furthermore, in some cases, as in the case of a bridge crane under seismic loading, multiple time-scales coexist in the same problem. In that case, the use of multi-time scale methods is suitable. Here, we propose a new explicit–implicit heterogeneous asynchronous time integrator (HATI) for nonsmooth transient dynamics with frictionless unilateral contacts and impacts. Furthermore, we present a new explicit time integrator for contact/impact problems where the contact constraints are enforced using a Lagrange multiplier method. In other words, the aim of this paper consists in using an explicit time integrator with a fine time scale in the contact area for reproducing high frequency phenomena, while an implicit time integrator is adopted in the other parts in order to reproduce much low frequency phenomena and to optimize the CPU time. In a first step, the explicit time integrator is tested on a one-dimensional example and compared to Moreau-Jean’s event-capturing schemes. The explicit algorithm is found to be very accurate and the scheme has generally a higher order of convergence than Moreau-Jean’s schemes and provides also an excellent energy behavior. Then, the two time scales explicit–implicit HATI is applied to the numerical example of a bridge crane under seismic loading. The results are validated in comparison to a fine scale full explicit computation. The energy dissipated in the implicit–explicit interface is well controlled and the computational time is lower than a full-explicit simulation.

A rotation-free thin shell quadrilateral

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.

Efficient sublaminate-scale impact damage modelling with higher-order elements in explicit integration

An approach that increases the critical time step in explicit time integration is presented. By derivation as a variational integrator, symplecticity and momentum preservation are ensured. The linear response is assumed to be dominant and is integrated implicitly, while the nonlinear forces are explicitly integrated. MOLLY filters the high‐frequent portions of the nonlinear forces and increases the critical time step. An SDOF example and a structural example using modal reduction verify the improved stability. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Selective mass scaling and critical time-step estimate for explicit dynamics analyses with solid-shell elements

This short but systematic work demonstrates a link between Chebyshev's theorem and the explicit integration in cosmological time t and conformal time η of the Friedmann equations in all dimensions and with an arbitrary cosmological constant Λ. More precisely, it is shown that for spatially flat universes an explicit integration in t may always be carried out, and that, in the non-flat situation and when Λ is zero and the ratio w of the pressure and energy density in the barotropic equation of state of the perfect-fluid universe is rational, an explicit integration may be carried out if and only if the dimension n of space and w obey some specific relations among an infinite family. The situation for explicit integration in η is complementary to that in t. More precisely, it is shown in the flat-universe case with Λ ≠ 0 that an explicit integration in η can be carried out if and only if w and n obey similar relations among a well-defined family which we specify, and that, when Λ = 0, an explicit integration can always be carried out whether the space is flat, closed, or open. We also show that our method may be used to study more realistic cosmological situations when the equation of state is nonlinear.

Implicit-explicit-compact methods for advection diffusion reaction equations

EULER SOLUTIONS FOR FLOW PAST AN AIRFOIL FROM SUBSONIC TO HIGH SUPERSONIC MACH NUMBERS

SummaryTo improve the computational efficiency of the numerical manifold method for discontinuous deformation simulations, a spatial‐domain coupled explicit‐implicit time integration algorithm is proposed. A subdomain partition algorithm based on a super manifold element is developed for the numerical manifold method to simulate dynamic motions of blocky rock mass. In different subdomains, explicit or implicit time integration method is employed respectively based on its contact and motion status. These subdomains interact through assembling the corresponding explicit or implicit time integration‐based matrices of different rock blocks. The computational efficiency of the discontinuity system under dynamic loading is improved by partially diagonalizing the global matrices. Two verification examples of a sliding block along an inclined plane under a horizontal acceleration excitation and a multiblock system acted on by dynamic forces are studied to examine the accuracy of the proposed numerical method, respectively. A highly fractured rock mass situated on an inclined slope subjected to seismic excitations is then studied to show the computational efficiency of the developed algorithm. The simulated results are in good agreement with those from the versions using purely implicit or explicit time integration algorithm for the numerical manifold method. The computational efficiency is shown to be higher using the proposed algorithm, which demonstrates its potential for application in dynamic analysis of highly fractured rock masses.