Large Time Step Size Research Articles

A high‐resolution wetting and drying algorithm for free‐surface hydrodynamics

AbstractA new wetting and drying algorithm for numerical modeling free‐surface flows is proposed and analyzed. A well structured, mildly nonlinear system for the discrete water surface elevation is derived from the governing differential equations by requiring a correct mass balance in wet areas as well as in the region of transition from wet to dry and from dry to wet. Existence and uniqueness of the numerical solution, along with a convergence analysis of an iterative scheme for the mildly nonlinear system, is provided. The present algorithm is devised to use high‐resolution bathymetric data at subgrid level. The resulting model is quite efficient, does not require a threshold value for minimal water depth, does not produce un‐physical negative water depths and generates accurate results with relatively coarse mesh and large time step size. These features are illustrated on a severe test‐case with known analytical solution. Copyright © 2008 John Wiley & Sons, Ltd.

The Body-of-Revolution Hybrid Implicit–Explicit Finite-Difference Time-Domain Method With Large Time Step Size

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> A novel body-of-revolution finite-difference time-domain method (BOR-FDTD) based on the hybrid implicit–explicit (HIE) difference is presented in this paper. The time step in this method is independent of mode number <formula formulatype="inline"><tex>$m$</tex></formula>. It is only determined by the spatial increments <formula formulatype="inline"><tex>$\Delta r$</tex></formula> and <formula formulatype="inline"><tex>$\Delta z$</tex></formula>. Compared with the conventional BOR-FDTD method, this novel BOR-FDTD method has larger time step size, especially when the value of <formula formulatype="inline"><tex>$m$</tex></formula> is large. This novel BOR-FDTD method does not require the solution of tridiagonal matrices, and all the updates in this method are explicit. The stability condition of the proposed BOR-FDTD method is presented analytically, and the numerical performance is demonstrated through numerical examples. </para>

Real-time propagation of the reduced one-electron density matrix in atom-centered Gaussian orbitals: Application to absorption spectra of silicon clusters

We solve the time-dependent density functional theory equation by propagating the reduced one-electron density matrix in real-time domain. The efficiency of several standard solvers such as the short-iterative Krylov-subspace propagator, the low-order Magnus integration method with the matrix polynomial (MP) or Chebyshev matrix polynomial (CMP) expansion of the evolution operator, and Runge-Kutta algorithm are assessed. Fast methods for summing MP and CMP are implemented to speed the calculation of the matrix exponential. It is found that the exponential propagators can tolerate large time step size and retain the computational accuracy whereas the Krylov-subspace algorithm is a little inferior for a larger time step size compared with the second-order Magnus integration method with the MP/CMP expansion of the evolution operator in both weak and intense fields. As an application, we calculate the absorption spectra of hydrogen-passivated silicon nanoparticles Si(29)H(x). The popular hybrid and generalized gradient approximation exchange-correlation functionals are applied. We find that the experimental spectra can be reproduced by using B3LYP and that the silicon particles with sizes of 1 nm and the optical excitations at 3.7, 4.0, and 4.6 eV may consist of 29 Si atoms surrounded by 24 hydrogen atoms.

Numerical Simulation Using HIE-FDTD Method to Estimate Various Antennas With Fine Scale Structures

The hybrid implicit-explicit finite-difference time-domain (HIE-FDTD) method is employed to solve the problems of various antennas with fine scale structures. In this method, a larger time-step size than allowed by the CFL stability condition limitation can be set because the algorithm of this method is weakly conditionally stable. Consequently, an increase in computational efforts caused by fine cells due to thin slots or small substrate of antennas can be prevented. The results from the HIE-FDTD method agree well with results from the conventional finite-difference time-domain (FDTD) method, and the required CPU time for the HIE-FDTD method is much shorter than that for the FDTD method. Compared with the alternating-direction implicit FDTD (ADI-FDTD) method, the HIE-FDTD method has better accuracy and higher efficiency, which is demonstrated by numerical examples.

A New Method to Avoid Reduction of Time Step in the CP-FDTD Method

A new CP-WCS FDTD method is developed to avoid the time step reduction in the conformal path FDTD method. The stability condition in this method is not related with the shape and size of the distorted cell, and the time step size in this method is only determined by the space discretization perpendicular to the curved metal surface. The CP-WCS FDTD method has the same accuracy as that for the CP FDTD method, and due to large time step size can be used, the CP-WCS FDTD has higher computation efficiency, which is demonstrated by numerical examples.

Accurate evaluation of divided differences for polynomial interpolation of exponential propagators

In this paper, we propose an approach to the computation of more accurate divided differences for the interpolation in the Newton form of the matrix exponential propagator φ(hA)v, φ (z) = (ez − 1)/z. In this way, it is possible to approximate φ (hA)v with larger time step size h than with traditionally computed divided differences, as confirmed by numerical examples. The technique can be also extended to “higher” order φk functions, k≥0.

An iterative stabilized CNBS–CG scheme for incompressible non-isothermal non-Newtonian fluid flow

An iterative stabilized fractional step scheme abbreviated as I-CNBS–CG is developed, in which the Crank–Nicolson method based split (CNBS) scheme and the characteristic-Galerkin (CG) method are, respectively, used to discretize and solve the non-Newtonian momentum–mass conservation equations and the energy conservation equation in consideration of their convective character. Owing to introduction of an iterative procedure into the scheme the stability of the proposed scheme in time domain is greatly enhanced and much larger time step sizes are allowed to be used than those limited in existing explicit and semi-implicit ones. The proposed I-CNBS–CG scheme particularly suits to numerically model the non-isothermal non-Newtonian fluid flows with moderate or high viscosity and low thermal conductivity, such as molten polymer flow process in a mould cavity. Numerical experiments with the power-law fluid model demonstrate the improved performances of the proposed scheme.

A fractional step ELLAM approach to high-dimensional convection–diffusion problems with forward particle tracking

In this paper, a fractional step method combined with an Eulerian–Lagrangian localized adjoint method (ELLAM) is proposed to solve high-dimensional convection–diffusion problems. The method reduces high-dimensional problems to a series of uncoupled one-dimensional problems in each time step interval, in which one-dimensional ELLAM is used to solve the one-dimensional splitting equations. The approach takes the attractive advantages of the ELLAM method and the fractional step technique. It reduces computational complexities, large memory requirements, and long computation durations due to the application of the splitting technique. It reduces temporal errors and generates accurate numerical solutions even if large time and coarse spatial step sizes are used in computation. It effectively eliminates non-physical oscillation or excessive numerical dispersion and treats boundary conditions well and in a natural way. Numerical experiments show the efficient performance of the approach.

Efficient numerical methods for simulation of high-frequency active devices

We present two new numerical approaches for physical modeling of high-frequency semiconductor devices using filter-bank transforms and the alternating-direction implicit finite-difference time-domain method. In the first proposed approach, a preconditioner based on the filter-bank and wavelet transforms is used to facilitate the iterative solution of Poisson's equation and the other semiconductor equations discretized using implicit schemes. The second approach solves Maxwell's equations which, in conjunction with the semiconductor equations, describe the complete behavior of high-frequency active devices, with larger time-step size. These approaches lead to the significant reduction of the full-wave simulation time. For the first time, we can reach over 95% reduction in the simulation time by using these two techniques while maintaining the same degree of accuracy achieved using the conventional approach

Meshfree iterative stabilized Taylor–Galerkin and characteristic-based split (CBS) algorithms for incompressible N–S equations

This paper presents two stabilized schemes of the fractional step algorithm using characteristic-based split and Taylor–Galerkin like based split respectively in the meshfree context. An iterative procedure which makes the convective term satisfy the momentum conservation equation in an implicit sense is introduced into the two schemes to allow much larger time step sizes. A modified version of finite increment calculus process for incompressibility stabilization is developed to avoid the calculation of high order spatial derivatives. Numerical experiments demonstrate the capability and stability of the proposed schemes of the meshfree-based fractional step method in the simulation of incompressible flows.

General approach for Nth-order dispersive material

In some recent publications, formulations of a time-domain beam propagation method (TD-BPM) were developed to incorporate dispersive media with complex permittivity described by a single first-order pole or a modified version thereof. However, for many emerging optical applications, procedures are needed for general linear dispersive media with dispersions described by rational functions, such as a combination of multiple second-order poles. The authors present an efficient frequency-dependent full-band TD-BPM formulation for the analysis of general linear dispersive media. In this method, the complex time-domain convolution of the dispersive media is evaluated efficiently via a Z-transform. The formulation is validated by simulating dispersive media with complex permittivity described by single and multiple second-order poles at microwave and optical frequencies. It is shown that the proposed formulation can accurately simulate broadband electromagnetic responses using a much larger time-step size than those required by other conventional numerical techniques. Extension of this formulation to the treatment of any general material, such as dispersive non-linear material, is straightforward.

A Moment Method for Evaporation Using a Logarithmic Size Distribution with the Smallest Size

A moment method of the log-normal distribution with the smallest size is applied to evaporation by newly introducing correction factors obtained from the error function. In this article, the improved moment method is tested for the evaporation, and is compared with the exact solutions calculated by the CIP semi-Lagrangian (CIP-SL) method. In small particle regime, the size distributions and the time histories of the total number and volume per unit volume are reproduced by the moment method for the evaporation near the smallest size. In large particle regime, however, the differences between the moment and exact solutions are larger with time, after the exact distribution spreads to the smallest size. This new moment solver can reproduce the evaporation near the smallest size even when large time step size is given, and thus is expected to be used for the parameterization of the evaporation of small particles in aerosol-transport model.

Analysis of Joint Clearances in Multibody Systems

A methodology for the study of typical smooth joint clearances in multibody systems is presented. The proposed approach takes advantage of the analytical definition of the material surfaces defining the clearance, resulting in a formulation where the gap does not play a central role, as it happens in standard contact models. The contact forces are formulated in conserving form, such that the balance of total energy during the intermittent contact is exactly established in the discrete time integration scheme. Some numerical applications are presented, showing that the proposed methodology is very stable in long-term simulations with relatively large time step sizes. Therefore, it appears to be promising in terms of efficiency and robustness for the numerical analysis of real joints with clearances.

An iterative stabilized fractional step algorithm for numerical solution of incompressible N-S equations

SUMMARY Stabilized fractional step algorithm has been widely employed for numerical solution of incompressible Navier-Stokes equations. However, smaller time step sizes are required to use for existing explicit and semi-implicit versions of the algorithm due to their fully or partially explicit nature particularly for highly viscousow problems. The purpose of this paper is to present two modied versions of the fractional step algorithm using characteristic based split and Taylor-Galerkin like based split. The proposed modied versions of the algorithm are based on introducing an iterative procedure into the algorithm and allow much larger time step sizes than those required to the preceding ones. A numerical study of stability at acceptable convergence rate and accuracy as well as capability in circumventing the restriction imposed by the LBB condition for the proposed iterative versions of the algorithm is carried out with the plane Poisseuilleow problem under dierent Reynolds numbers ranging from low to high viscosities. Numerical experiments in the plane Poisseuilleow and the lid- driven cavityow problems demonstrate the improved performance of the proposed versions of the algorithm, which are further applied to numerical simulation of the polymer injection moulding process. Copyright ? 2005 John Wiley & Sons, Ltd.

A split operator approach to reactive transport with the forward particle tracking Eulerian Lagrangian localized adjoint method

A forward particle tracking Eulerian Lagrangian localized adjoint method (ELLAM) is applied to the multicomponent reactive transport problem using a split operator approach. Two split operator algorithms are compared, the Strang algorithm and the sequential non-iterative algorithm (SNIA). The reaction equations are integrated using a coupled predictor corrector algorithm with adaptive time stepping. Reaction time steps are adjusted at the inflow boundary to reflect the actual time of transport inside the solution domain. Results show that split operator ELLAM formulations are competitive with direct or fully coupled ELLAM solutions for reactive transport problems. The SNIA algorithm is more accurate than the Strang splitting algorithm when large time steps are used. The reaction algorithm employed dominates computational effort in runs with large time step sizes. To illustrate the use of the method in practical problems, the model is fitted to aerobic aniline degradation data from laboratory scale column experiments. Model inversion is achieved using non-linear regression with a shuffled complex evolution optimization algorithm and parameter uncertainty is assessed using a Bayesian uncertainty analysis procedure.

Symplectic Semi-Lagrangian Schemes for Computational Fluid Dynamics

The symplectic semi-Lagrangian (SSL) method was developed in order to improve the preservation of the physical quantities of the semi-Lagrangian method. In this method, symplectic integrators are adopted in searching for the upstream point in the semi-Lagrangian method. The purpose of this paper is to verify the validity of the SSL method for a simple passive scalar advection (PSA) problem and for a more practical fluid problem, two-dimensional incompressible viscous flow. From the computed results, it is confirmed that the SSL can take a large time-step size over Courant number 2 without losing computational accuracy.

On the use of large time steps with an energy momentum conserving algorithm for non-linear hypoelastic constitutive models

This paper presents an extension of the energy momentum conserving algorithm, developed by the authors for hypoelastic constitutive models. For such a material, contrarily to hyperelastic models, no potential can be defined, and thus the conservation of the energy is ensured only if the elastic work of deformation can be restored by the scheme. In a previous paper, we proposed a new expression of internal forces at the finite element level which is shown to verify this property. We also demonstrated that the work of plastic deformation is positive and consistent with the material model. In this paper, the second order terms that were neglected in the previous work are now taken into account. Several numerical applications are presented to demonstrate the necessity of taking these terms into account once large time step sizes are used. The limitations of the Newmark algorithm in the non-linear range are also illustrated.

An iterative stabilized fractional step algorithm for finite element analysis in saturated soil dynamics

The discretization of the u –p model for saturated porous media results in the semi-discrete system of mixed type in displacements and pressures. The Babuska–Brezzi condition or the simpler patch test proposed by Zienkiewicz and Taylor precludes the use of elements with the equal low order of interpolation for u and p, unless special stabilization techniques, such as the fractional step algorithms, are used. The purpose of this paper is to present a modified version of the fractional step algorithm which allows much larger time step sizes than those for the existing ones. The method is based on introducing an iteration algorithm. The numerical experiments demonstrate the effectiveness and improved performance of the proposed modified version of the fractional step algorithm.

Rate‐dependent projection operators for frictional contact constraints

AbstractWe develop rate‐dependent regularization approaches for three‐dimensional frictional contact constraints based on the Kelvin and Maxwell viscoelastic constitutive models. With the present regularization schemes, we aim to provide a basis to better model friction and to stabilize the contact analysis while keeping the contact model as simple as possible. The key feature of the regularization approaches, implemented using an implicit time integrator, is that one can recover in the limit the widely used rate‐independent elastoplastic regularization framework without encountering numerical difficulties. Intermediate contact tractions are defined in terms of the relative displacement, the relative velocity, and the regularization parameters. The projection operators operate on the intermediate tractions and yield contact tractions that satisfy all the discretized contact constraints. The use of projection operators allows a systematic implementation of the present regularization schemes. Through numerical simulations, we observed that the Maxwell‐type regularization effectively avoids convergence problems, even for relatively large time step sizes, while the Kelvin‐type regularization does not. Copyright © 2003 John Wiley &amp; Sons, Ltd.

Sub-timing in fluid flow and transport simulations

A sub-time stepping method is described for computational fluid dynamics problems that utilize implicit-type time marching procedures to resolve transients. In this method, small time-step sizes are used in portions of a domain where interest and activity is high, with larger time-step sizes being applied in other locations. The sub-time step is an integral portion of a larger time step––i.e., multiple sub-time steps over a sub-timed part of a domain add up to the time interval of the full time step used over the remainder of the domain. The technique is particularly suitable for extensive simulations where large portions of a domain are temporally over-discretized. The principles underlying implementation of the implicit sub-timing procedure, the computational effort in relation to conventional implicit time-stepping methods, and an analysis of the effects of sub-timing on the matrix structure are presented. Feasibility and applicability of the implicit sub-timing method is demonstrated through three proof-of-the-concept example problems. These examples include implementation of implicit sub-timing to one-dimensional as well as two-dimensional flow and solute transport problems.