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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.

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.

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.

The explicit time integration of the Schur complement vector potential formulation of eddy current problems is accelerated by a multistage RKC method.

The numerical treatment of coupled field interaction problems frequently uses mixed time integration methods. These methods permit different time integration methods (implicit, explicit) and/or different timesteps to be used simultaneously in different parts of the mesh. This paper summarizes the various mixed time integration methods and provides a unified presentation. Computer implementation of the generalized scheme is provided through a 1D linear structural dynamics program (GEMIX). Two common examples illustrate the use of GEMIX program.

The purpose of this study, which is designed for the implementation of models in the implicit finite element framework, is to propose a robust, stable and efficient explicit integration algorithm for rate-independent elasto-plastic constitutive models.

The proposed automatic substepping algorithm is founded on an explicit integration scheme. The estimation of the maximal subincrement size is based on the stability analysis.

In contrast to other explicit substepping schemes, the algorithm is self-correcting by definition and generates no cumulative drift. Although the integration proceeds with maximal possible subincrements, high level of accuracy is attained. Algorithmic tangent stiffness is calculated in explicit form and optionally no analytical second-order derivatives are needed.

The algorithm is convenient for elasto-plastic constitutive models, described with an algebraic constraint and a set of differential equations. This covers a large family of materials in the field of metal plasticity, damage mechanics, etc. However, it cannot be directly used for a general material model, because the presented algorithm is convenient for solving a set of equations of a particular type.

The estimation of the maximal stable subincrement size is computationally cheap. All expressions in the algorithm are in explicit form, thus the implementation is simple and straightforward. The overall performance of the approach (i.e. accuracy, time consumption) is fully comparable with a default (built-in) ABAQUS/Standard algorithm.

The estimated maximal subincrement size enables the algorithm to be stable by definition. Subincrements are much larger than those in conventional substepping algorithms. No error control, error correction or local iterations are required even in the case of large increments.

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.

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.

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.

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.

Describes simulations of impact forging processes. Uses the explicit time integration finite element method, which is based on direct time integration of equation of motion, to compute the deformation of the workpiece and the dies. Uses the program developed to simulate the copper blow test performed on a 350,000J counter‐blow hammer. The calculated result reveals a good agreement in the final deformed configurations between the experiment and the explicit simulation. In order to compare this with the explicit method, the implicit time integration rigid‐plastic finite element program considering the inertia effect is also applied to the copper blow test simulation. As a result of the copper blow test simulation using the explicit program and the implicit program, finds that the calculated results have good agreements in available plastic deformation energy, forging load and equivalent plastic strain distribution. Finally, applies the developed program to simulations of multi‐blow forging processes. Presents three major findings from the multi‐blow forging simulations: (1) the continuous analysis technique used for the multi‐blow forging simulations works well; (2) the blow efficiency and the forging load generated by blow operations can be analysed efficiently and simulated results coincide with previous experimental and analytical ones; (3) the geometrical configuration of the workpiece is closely related to blow efficiency.

Two kinds of time integration methods; the dynamic explicit method and the static implicit method, have been compared, especially with emphasis on the shell formulations and the stress integration methods. Two methods have been applied to the benchmark problem named the S‐rail stamping process, provided by NUMISHEET’96 committee, in order to compare their numerical results with each other and with the average values of the experiments as well. Based on the comparisons, it is shown that both time integration methods can be successfully applied to industrial sheet metal stamping simulations. In detail, the static implicit method is advantageous over the dynamic explicit method in terms of accuracy, while the latter is known to be more efficient than the former in terms of computation efficiency.

The purpose of this paper is to propose a new explicit time integration algorithm for solution to the linear and non-linear finite element equations of structural dynamic and wave propagation problems.

The algorithm is completely explicit so that no linear equation system requires solving, if the mass matrix of the finite element equation is diagonal and whether the damping matrix does or not. The algorithm is a single-step method that has the simple starting and is applicable to the analysis with the variable time step size. The algorithm is second-order accurate and conditionally stable. Its numerical stability, dissipation and dispersion are analyzed for the dynamic single-degree-of-freedom equation. The stability of the multi-degrees-of-freedom non-proportional damping system can be evaluated directly by the stability theory on ordinary differential equation.

The performance of the proposed algorithm is demonstrated by several numerical examples including the linear single-degree-of-freedom problem, non-linear two-degree-of-freedom problem, wave propagation problem in two-dimensional layer and seismic elastoplastic analysis of high-rise structure.

A new single-step second-order accurate explicit time integration algorithm is proposed to solve the linear and non-linear dynamic finite element equations. The algorithm has advantages on the numerical stability and accuracy over the existing modified central difference method and Chung-Lee method though the theory and numerical analyses.

The purpose of this paper is to present new implementation aspects of unified explicit time integration algorithms, called the explicit GS4-II family of algorithms, of a second-order time accuracy in all the unknowns (e.g. positions, velocities, and accelerations) with particular attention to the moving-particle simulation (MPS) method for solving the incompressible fluids with free surfaces.

In the present paper, the explicit GS4-II family of algorithms is implemented in the MPS method in the following two different approaches: a direct explicit formulation with the use of the weak incompressibility equation involving the (modified) speed of sound; and a predictor-corrector explicit formulation. The first approach basically follows the concept of the explicit MPS method, presented in the literature, and the latter approach employs a similar concept used in, for example, a fractional-step method in computational fluid dynamics.

Illustrative numerical examples demonstrate that any scheme within the proposed algorithmic framework captures the physics with the necessary second-order time accuracy and stability.

The new algorithmic framework extended with the GS4-II family encompasses a multitude of pastand new schemes and offers a general purpose and unified implementation.

The distinct element method as proposed by Cundall and Strack uses the computationally efficient, explicit, central difference time integration scheme. A limitation of this scheme is that it is only conditionally stable, so small time steps must be used. Some researchers have proposed using an implicit time integration scheme to avoid the stability issues arising from the explicit time integrator typically used in these simulations. However, these schemes are computationally expensive and can require a significant number of iterations to form the stiffness matrix that is compatible with the contact state at the end of each time step. In this paper, a new, simple approach for calculating the critical time increment in explicit discrete element simulations is proposed. Using this approach, it is shown that the critical time increment is a function of the current contact conditions. Considering both two‐ and three‐dimensional scenarios, the proposed refined estimates of the critical time step indicate that the earlier recommendations contained in the literature can be unconservative, in that they often overestimate the actual critical time step. A three‐dimensional simulation of a problem with a known analytical solution illustrates the potential for erroneous results to be obtained from discrete element simulations, if the time‐increment exceeds the critical time step for stable analysis.

In the present work a rigid‐plastic finite element formulation using a dynamic explicit time integration scheme is proposed for numerical analysis of sheet metal forming processes. The rigid‐plastic finite element method, based on membrane elements, has long been employed as a useful numerical technique for the analysis of sheet metal forming because of its time effectiveness. The explicit scheme, in general, is based on the elastic‐plastic modelling of material requiring large computation time. The resort to rigid‐plastic modelling would improve the computational efficiency, but this involves new points of consideration such as zero energy mode instability. A damping scheme is proposed in order to achieve a stable solution procedure in dynamic sheet forming problems. In order to improve the drawbacks of the conventional membrane elements, BEAM (abbreviated from Bending Energy Augmented Membrane) elements, are employed. Rotational damping and spring about the drilling direction are introduced to prevent a zero energy mode. The lumping scheme is employed for the diagonal mass matrix and linearizing dynamic formulation. A contact scheme is developed by combining the skew boundary condition and a direct trial‐and‐error method. Computations are carried out for analysis of complicated sheet metal forming processes such as forming of an oilpan and a front fender. The numerical results of explicit analysis are compared with the implicit results, with good agreement, and it is shown that the explicit scheme requires much shorter computational times, especially when the problem becomes more complicated. It is thus shown that the proposed dynamic explicit rigid‐plastic finite element enables an effective computation for complicated sheet metal processes.