Time Step Reduction Research Articles

Flux limiting embedded boundary technique for electromagnetic FDTD

A general approach for incorporating embedded boundaries into an electromagnetic finite difference time domain (FDTD) code is presented. This algorithm is shown to satisfy Gauss’s law and enforces no magnetic monopoles while maintaining a globally second-order result (first-order at physical boundaries), with no added time-step restriction. Theoretically predicted superior results are shown with an 11% time-step reduction from the Courant stability limit. This is achieved through a physics-based flux limiting scheme near physical boundaries. Stability, local truncation error and energy conservation analysis are also provided.

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.

A divergence‐free interpolation scheme for the immersed boundary method

AbstractThe immersed boundary approach for the modeling of complex geometries in incompressible flows is examined critically from the perspective of satisfying boundary conditions and mass conservation. It is shown that the system of discretized equations for mass and momentum can be inconsistent, if the velocity is used in defining the force density to satisfy the boundary conditions. As a result, the velocity is generally not divergence free and the pressure at locations in the vicinity of the immersed boundary is not physical. However, the use of the pseudo‐velocities in defining the force density, as frequently done when the governing equations are solved using a fractional step or projection method, combined with the use of the specified velocity on the immersed boundary, is shown to result in a consistent set of equations which allows a divergence‐free velocity but, depending on the time step, is shown to have the undesirable effects of inaccurately satisfying the boundary conditions and allowing a significant permeability of the immersed boundary. If the time step is reduced sufficiently, the boundary conditions on the immersed boundary can be satisfied. However, this entails an unacceptable increase in computational expense. Two new methods that satisfy the boundary conditions and allow a divergence‐free velocity while avoiding the increased computational expense are presented and shown to be second‐order accurate in space. The first new method is based on local time step reduction. This method is suitable for problems where the immersed boundary does not move. For these problems, the first new method is shown to be closely related to the second new method. The second new method uses an optimization scheme to minimize the deviation from the interpolation stencil used to represent the immersed boundary while ensuring a divergence‐free velocity. This method performs well for all problems, including those where the immersed boundary moves relative to the grid. Additional results include showing that the force density that is added to satisfy the boundary conditions at the immersed boundary is unbounded as the time step is reduced and that the pressure in the vicinity of the immersed boundary is unphysical, being strongly a function of the time step. A method of computing the total force on an immersed boundary which takes into account the specifics of the numerical solver used in the iterative process and correctly computes the total force irrespective of the residual level is also presented. Copyright © 2007 John Wiley & Sons, Ltd.

Conformal FDTD-methods to avoid time step reduction with and without cell enlargement

During the last decades there have been considerable efforts to develop accurate and yet simple conformal methods for modelling curved boundaries within the finite difference time domain (FDTD) algorithm. In an earlier publication we proposed the uniformly stable conformal (USC) approach as a general three-dimensional extension of FDTD without the need to reduce the maximum stable time step. The main idea of USC is the usage of virtually enlarged cells near to the boundary, leading to an increased implementation effort. In this paper we review the USC method and introduce a new simple and accurate conformal scheme which does not use such enlarged cells. This simplified conformal (SC) scheme has the same number of operations and algorithmic logic as the standard “staircase” method, and thus is easily realizable in existing FDTD codes. Like USC, it leads to accurate results without time step reduction, showing a nearly second order convergence in practice. The method is verified and compared to other approaches by means of several numerical 2D and 3D examples.

A New 3-D Conformal PEC FDTD Scheme With User-Defined Geometric Precision and Derived Stability Criterion

A new conformal finite-difference time-domain (CFDTD) updating scheme for metallic surfaces nonaligned in the grid is presented in this paper. In contrast to existing conformal models, the new model can be formulated with the original Yee FDTD update equation. Therefore, the proposed scheme can be easily added in standard FDTD codes even if the codes are already parallelized or hardware-accelerated. In addition, based on the commonly used conventional stability criterion, a derivation of the stability is presented and based on the conformal geometric information, a time step reduction formula is presented. The time step reduction is used as a user-defined parameter to tradeoff speed versus accuracy. The achievable geometric precision is optimized to a given time step. Therefore, even with the conventional time step (no reduction) the presented scheme profits from the conformal discretization. To show the performance and the robustness of the proposed scheme canonical validations and two real world applications were investigated. A broadband low profile (circular) antenna was successfully simulated showing the benefit of the conformal FDTD method compared to the conventional scheme. Furthermore, a CAD based mobile phone was conformally discretized and successfully simulated showing that the proposed scheme is highly suited for the simulation of advanced engineering problems.

Reply to 'Comment on "Enlarged cells for the conformal FDTD method to avoid the time step reduction"'

For original article by T. Xiao and Q.H. Liu see ibid., vol.14, no.12, p.551-3, Dec. 2004. For comments by R. Schuhmann, I.A. Zagorodnov and T. Weiland see ibid., vol.16, no.1, p.55, Jan. 2006.

Comment on "Enlarged cells for the conformal FDTD method to avoid the time step reduction"

For original article by T. Xiao and Q.H. Liu see ibid., vol.14, no.12, p.551-3, Dec. 2004

The force/work differencing of exceptional points in the discrete, compatible formulation of Lagrangian hydrodynamics

This study presents the force and mass discretization of exceptional points in the compatible formulation of Lagrangian hydrodynamics. It concludes a series of papers that develop various aspects of the theoretical exposition and the operational implementation of this numerical algorithm. Exceptional points are grid points at the termination of lines internal to the computational domain, and where boundary conditions are therefore not applied. These points occur naturally in most applications in order to ameliorate spatial grid anisotropy, and the consequent timestep reduction, that will otherwise arise for grids with highly tapered regions or a center of convergence. They have their velocity enslaved to that of neighboring points in order to prevent large excursions of the numerical solution about them. How this problem is treated is given herein for the aforementioned numerical algorithm such that its salient conservation properties are retained. In doing so the subtle aspects of this algorithm that are due to the interleaving of spatial contours that occur with the use of a spatially-staggered-grid mesh are illuminated. These contours are utilized to define both forces and the work done by them, and are the central construct of this type of finite-volume differencing. Additionally, difficulties that occur due to uncertainties in the specification of the artificial viscosity are explored, and point to the need for further research in this area.

Enlarged cells for the conformal FDTD method to avoid the time step reduction

Recently, the conformal finite-difference time-domain (CFDTD) method has emerged as an efficient FDTD method with a higher order accuracy than the conventional FDTD methods that are degraded by staircasing errors. The only obvious point to further improve on the CFDTD method is its requirement for a smaller time step increment due to the existence of small irregular cells near the boundary. In this letter, an enlarged cell technique is introduced to ensure the stability of the CFDTD method without the time step reduction. The introduction of the enlarged cells therefore makes the CFDTD method much more efficient and suffers from a smaller dispersion error, as shown in several two-dimensional examples.

Two-dimensional response of crushable polyurethane foam to low velocity impact

An experimental and numerical study of the two-dimensional response of crushable foam to low velocity impact is undertaken. Rigid polyurethane foam blocks are subjected to normal impact by gravity-driven impactors of different geometries, at velocities ranging from 2 to 4 m/s. The impactors comprise a rectangular block, a wedge-tipped block and a cylinder. Quantities measured during impact are the impactor deceleration, velocity and displacement, and the energy dissipated. The effects of impact velocity and geometry on the deformation and energy absorbed are studied. A two-dimensional numerical model is proposed to simulate the gross deformation induced in the impact process. It employs a lumped mass approach and is formulated in terms of finite deformation. Appropriate equations of motion, stress–strain relations, failure criteria and failure patterns are developed. Results generated by this model exhibit good correlation with experiments, thus substantiating its validity. The proposed model demonstrates advantages over traditional finite element approaches, in that it accommodates severe deformation and extensive structural failure without the problems of excessive mesh distortion and untenable time step reduction which accompany finite element simulations.

An effective optimization-based algorithm for job shop scheduling with fixed-size transfer lots

Abstract Effective scheduling of production lots is of great importance for manufacturing medium to high-volume products that require significant setup times. Compared to traditional entire-lot production, lot splitting techniques divide a production lot into multiple smaller sublots so that each sublot can be “transferred” from one stage of operation to the next as soon as it has been completed. “Transfer lots,” therefore, significantly reduce lead times and lower work-in-process (WIP) inventory. The mathematical modeling, analysis, and control of transfer lots, however, is extremely difficult. This paper presents a novel integer programming formulation with separable structure for scheduling job shops with fixed-size transfer lots. A solution methodology based on a synergistic combination of Lagrangian relaxation, backward dynamic programming (BDP), and heuristics is developed. Through explicit modeling of lot dynamics, transfer lots are handled on standard machines, machines with setups, and machines requiring all transfer lots within a production lot to be processed simultaneously. With “substates” and the derivation of DP functional equations considering transfer lot dynamics, the standard BDP is extended to solve the lot-level subproblems. The recently developed “time step reduction technique” is also used for increased efficiency. It implicitly establishes two time scales to reduce computational requirements without much loss of modeling accuracy and scheduling performance, thus enabling resolution of long-horizon problems within controllable computational requirements. The method has been implemented using object-oriented programming language C++, and numerical tests show that high-quality schedules involving transfer lots are efficiently generated to achieve on-time delivery of products with low WIP inventory.

Large-scale crystal plasticity computations of microstructural failure modes

A computational scheme is introduced for the integration of rate-dependent multiple-slip crystal plasticity constitutive relations. Fundamental issues of accuracy, stability, and stiffness that are intrinsically related to the evolution of microstructural failure modes in metallic crystals are addressed. An adaptive finite-element methodology is introduced to classify these characteristics. A nonlinear initial value system is derived to update the plastic deformation-rate tensor. An explicit method is used in non-stiff domains, where accuracy is required. If a time-step reduction is due to stability, a harbinger of numerical stiffness, the algorithm is automatically switched to an A-stable method. A stiffness ratio is defined to measure the eigenvalue dispersion of the system. The adaptability of the proposed algorithm for the solution of a class of inelastic constitutive relations is illustrated by investigating the influence of high angle grain boundary orientations on failure in face-centered cubic (f.c.c.) bicrystals. The effects of grain boundary misorientation, dislocation densities, strain hardening, and geometrical softening on failure evolution are investigated. This study underscores the importance of understanding the origin of numerical instabilities, such that these instabilities are not mistaken for inherent material instabilities.

An accurate and stable algorithm for high strain-rate finite strain plasticity

A computational scheme is introduced for the integration of rate-dependent constitutive relations for the high strain-rate finite inelastic deformation and failure of metallic materials. Fundamental issues of accuracy, stability, and stiffness that are intrinsically related to the evolution of dynamic inelastic deformation and failure modes, such as shear-band formation, are addressed. Due to this evolution, accuracy, and stability characteristics of the constitutive relations change throughout the integration domain. Stiffness, a stability problem, may occur due to widely dispersed solution components that result from material instabilities. An adaptive methodology is introduced to classify these characteristics, and appropriate numerical methods were used. The current deformation configuration was updated by obtaining the current total deformation and the plastic deformation-rate tensors. The total deformation-rate tensor was obtained by an explicit finite element method. A nonlinear initial value problem is derived to update the plastic deformation-rate tensor. A combination of a fifth-order accurate explicit Runge-Kutta variable-step method and the A-stable backward Euler method were used to integrate this initial value system. The explicit method was used in nonstiff domains, where accuracy is required. If a time-step reduction was due to stability, a harbinger of numerical stiffness, the algorithm was automatically switched to the A-stable method. To distinguish a time-step reduction due to stability from one due to accuracy, a stiffness ratio is defined to measure the eigenvalue dispersion of the initial-value system. The adaptability of the proposed algorithm to a wide class of inelastic constitutive relations is illustrated by applications to crystal plasticity. This study underscores the importance of understanding the origin of numerical instabilities, such that these instabilities are not mistaken for inherent material instabilities.