First order explicit time integration scheme with large time steps for parabolic problems using irregular grids

We present a family of fast explicit time integration schemes of first, second and third order accuracy for parabolic problems in mechanics solved via standard numerical methods that have considerable higher computational efficiency versus existing explicit methods of the same order. The derivation of the new explicit schemes is inspired on the finite increment calculus (FIC) procedure used for obtaining stabilized numerical schemes in fluid and solid mechanics. The new (so‐called) explicit FIC‐Time (EFT) schemes allow considerable larger time steps than the standard first order forward Euler (FE) scheme and the second and third order Adams–Bashforth schemes. The comparison with Runge–Kutta schemes also favors the FIC‐Time schemes in terms of the limit time step size (for second order schemes) and the total number of matrix‐vector multiplications per time step (for second and third order schemes). The new first order explicit schemes have a faster convergence to steady‐state than the FE scheme. The accuracy and efficiency of the new EFT schemes are verified in examples of application to the transient heat conduction equation using the finite element method.

Explicit time integration scheme with large time steps for first order transient problems using finite increment calculus

Three-dimensional transient heat conduction problems in FGMs via IG-DRBEM

Explicit/implicit multi-time step co-computations for blast analyses on a reinforced concrete frame structure

Recently, a number of explicit time integration schemes have been proposed. They have been shown to perform better for mostly linear problems. However, there is no work to compare the performance of various explicit time integration schemes for nonlinear problems. Hence, the objective of the present work is to fill this gap. In the paper, a number of recently proposed explicit time integration schemes are analyzed for their performance when applied to nonlinear problems. In particular, a multiple degree-of-freedom nonlinear problem and a single degree-of-freedom nonlinear adhesive contact problem are solved using different schemes for different time steps. The error in energy for different time steps for each scheme is compared. The computational cost associated with each scheme is also compared. It is shown that still the classical central difference scheme performs equally well as compared to all the recently proposed explicit schemes and takes least amount of computational time.

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.

Analyzing three-dimensional transient heat conduction problems with the dimension splitting reproducing kernel particle method

The dimension splitting interpolating element-free Galerkin method for solving three-dimensional transient heat conduction problems

A combined approach of the radial integration boundary-element method (RIBEM) and the precise algorithm in the time domain is presented for solving three-dimensional transient heat conduction problems with variable thermal conductivity. First, by expanding physical quantities at discrete time intervals, the recursive formulation of the governing equation is derived. Then, the recursive equation is solved by the RIBEM, and a self-adaptive check technique is carried out to estimate how many expansion terms are needed in a time step. Finally, three numerical examples show that the present approach can obtain very stable and accurate results for different time-step size.

IG-DRBEM of three-dimensional transient heat conduction problems

A meshless analysis of three-dimensional transient heat conduction problems

The interpolating element-free Galerkin method for three-dimensional transient heat conduction problems

Adaptive conservative time integration for transport in fractured porous media

PurposeDiscretizing 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/approachThe 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.FindingsNumerical 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/valueThe 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.

Implicit/explicit multi-time step co-computations for predicting reinforced concrete structure response under earthquake loading