Efficient Time Integration Research Articles

A ROBUST TIME-INTEGRATION ALGORITHM FOR SOLVING NONLINEAR DYNAMIC PROBLEMS WITH LARGE ROTATIONS AND DISPLACEMENTS

An efficient time-integration algorithm for nonlinear dynamic analysis of structures is presented. By adopting the temporal discretization for time finite element approximation, very large time steps can be used by the algorithm. With an accuracy of fourth order, this technique requires only displacements and velocities to be made available at the start of the current time step for integration in state space. Using the weighted momentum principle, the problem of discontinuity caused by impulsive loads is resolved after time-integration of the applied load in external momentum. Since no knowledge is required of acceleration at the current time step, the errors caused by estimation of acceleration by previous finite-difference methods are circumvented. Moreover, an iterative procedure is included for each time step, involving the three phases of predictor, corrector, and error-checking. The effectiveness and robustness of the proposed algorithm in solving nonlinear dynamic problems is demonstrated in the numerical examples.

An efficient backward Euler time-integration method for nonlinear dynamic analysis of structures

This paper presents an efficient time-integration method for obtaining reliable solutions of the transient nonlinear dynamic problems and of the stiff systems in structural engineering. This method employs the backward Euler formulae for evaluating both displacements and velocities of structures. It is a self-starting, two-step, second-order accurate algorithm with the same computational effort as the trapezoidal rule. The evaluations of the stability and accuracy of the proposed method are also given in this paper. With some numerical damping introduced, the proposed method remains stable in large deformation and long time range solutions even when the trapezoidal rule fails. Meanwhile, the proposed method has the following characteristics: (1) it is applicable to linear as well as general nonlinear analyses; (2) it does not involve additional variables (e.g. Lagrange multipliers) and artificial parameters; (3) it is a single-solver algorithm at the discrete time points with symmetric effective stiffness matrix and effective load vectors; and (4) it is easy to implement in an existing computational software. Some numerical results indicate that the proposed method is a powerful tool with some notable features for practical nonlinear dynamic analyses.

Higher Order Half-Explicit Time Integration of Eddy Current Problems Using Domain Substructuring

In this paper domain substructuring is adapted to the nonlinear transient eddy current problem: conductive and nonconductive domains are separately treated for a more efficient time integration. A matrix factorization of the linear (nonconductive) subproblem, e.g., air, is executed on beforehand and used throughout the simulation. For a general 3D problem these non-sparse factors must be replaced by sparse approximations or explicit time integration must be carried out to increase the efficiency of the solution process. This approach is validated using implicit and half-explicit time integration methods. Numerical results underline the feasibility and the possible gain obtained by higher order half-explicit methods.

Time Implicit High-Order Discontinuous Galerkin Method with Reduced Evaluation Cost

An efficient and robust time integration procedure for a high-order discontinuous Galerkin method is introduced for solving nonlinear second-order partial differential equations. The time discretization is based on an explicit formulation for the hyperbolic term and an implicit formulation for the parabolic term. The procedure uses an iterative algorithm with reduced evaluation cost. The size of the linear system to be solved is greatly reduced thanks to partial uncoupling in space between low-order and high-order degrees of freedom. Numerical examples are presented for the nonlinear convection-diffusion equation in one and two dimensions including steady and unsteady flow problems. The performance of the present method is investigated in terms of CPU time and compared to a fully implicit method. A von Neumann stability analysis is carried out in order to determine the stability and damping properties of the method. Besides a fairly reduced CPU effort, numerical results demonstrate better convergence properties of the present algorithm when compared to the fully implicit method.

Hybrid state-space time integration of rotating beams

An efficient time integration algorithm for the dynamic equations of flexible beams in a rotating frame of reference is presented. The equations of motion are formulated in a hybrid state-space format in terms of local displacements and local components of the absolute velocity. With inspiration from Hamiltonian mechanics, where displacement and momentum have similar roles, both sets of state-space variables are interpolated between nodes by the same shape functions, leading to a general format where all inertia effects are represented via the classic constant mass matrix, while effects of the system rotation enter via global operations with the angular velocity vector. The algorithm is based on an integrated form of the equations of motion with energy and momentum conserving properties, if a kinematically consistent non-linear formulation is used. A consistent monotonic scheme for algorithmic energy dissipation in terms of local displacements and velocities, typical of structural vibrations, is developed and implemented in the form of forward weighting of appropriate mean value terms in the algorithm. The algorithm is implemented for a beam theory with consistent quadratic non-linear kinematics, valid for moderate finite rotations. The equations of this non-linear beam theory are generated in explicit form by extension of the constitutive stiffness and the geometric stiffness of a linear beam theory at the element level. The performance of the algorithm is illustrated by numerical examples.

Implementation of feedback in smoothed particle hydrodynamics: towards concordance of methods

We perform simulations of feedback from supernovae and black holes with smoothed particle hydrodynamics (SPH). Such strong perturbations are inaccurately handled with standard time integration schemes, leading to poor energy conservation, a problem that is commonly overlooked. We show for the first time that, in the absence of radiative cooling, concordance of thermal and kinetic feedback are achieved when using an accurate time integration. In order to preserve the concordance of feedback methods when using a more efficient time integration scheme - as for instance the hierarchical time-step scheme - we implement a modified version of the time-step limiter proposed by Saitoh & Makino (2009). We apply the limiter to general test cases, and first show that this scheme violates energy conservation up to almost four orders of magnitude when energy is injected at random times. To tackle this issue, we find necessary, not only to ensure a fast information propagation, but also to enforce a prompt response of the system to the energy perturbation. The method proposed here to handle strong feedback events enables us to achieve energy conservation at percent level in all tests, even if all the available energy is injected into only one particle. We argue that concordance of feedback methods can be achieved in numerical simulations only if the time integration scheme preserve a high energy conservation level.

CDM Model and its Application to Numerical Simulation on Rotating Band’s Engraving Process

A coupled constitutive model of viscoplasticity and ductile damage for penetration problems has been deduced. The model proposed by Lemaitre was based on the continuum damage mechanics. In the model, large strain, high strain-rate, thermal softening and damage evolution were taken into account. At the meanwhile, Von Mises yield criterion, Johnson-Cook hardening model and Johnson-Cook fracture strain model were adopted. The model was implemented in the explicit finite element code Abaqus\Explicit through the Vumat subroutine by using an efficient explicit time integration algorithm. Based on the model, the rotating band’s engraving process was simulated, at the same time the influence of the stress state on damage evolution and the thermal softening effect was discussed. An effective way to simulate the fire process of gun was afforded.

The differential quadrature solution of nonlinear reaction-diffusion and wave equations using several time-integration schemes

Three different time-integration schemes, namely the finite difference method (FDM) with a relaxation parameter, the least-squares method (LSM) and the finite element method (FEM), are applied to the differential quadrature (DQM) solution of one-dimensional nonlinear reaction–diffusion and wave equations. In the solution procedure, the space derivatives are discretized using DQM, which may also be used without the need of boundary conditions. The aim of the paper is to find computationally more efficient time-integration algorithm by comparison. For the nonlinear reaction–diffusion equation, which is a parabolic type, FEM is found to be the method of choice in terms of accuracy and computational cost. For the nonlinear wave equation, it is seen that both the FDM and LSM are better than FEM. Therefore, LSM can be preferred as a direct method for the nonlinear wave equation. Copyright © 2009 John Wiley & Sons, Ltd.

Facilitating the Adoption of Unstructured High-Order Methods Amongst a Wider Community of Fluid Dynamicists

Theoretical studies and numerical experiments suggest that unstructured high-order methods can provide solutions to otherwise intractable fluid flow problems within complex ge- ometries. However, it remains the case that existing high-order schemes are generally less robust and more complex to implement than their low-order counterparts. These issues, in conjunction with difficulties generating high-order meshes, have limited the adoption of high-order techniques in both academia (where the use of low-order schemes remains widespread) and industry (where the use of low-order schemes is ubiquitous). In this short review, issues that have hitherto pre- vented the use of high-order methods amongst a non-specialist community are identified, and cur- rent efforts to overcome these issues are discussed. Attention is focused on four areas, namely the generation of unstructured high-order meshes, the development of simple and efficient time integration schemes, the development of robust and accurate shock capturing algorithms, and fi- nally the development of high-order methods that are intuitive and simple to implement. With regards to this final area, particular attention is focused on the recently proposed flux reconstruc- tion approach, which allows various well known high-order schemes (such as nodal discontinuous Galerkin methods and spectral difference methods) to be cast within a single unifying framework. It should be noted that due to the experience of the authors the review is directed somewhat to- wards aerodynamic applications and compressible flow. However, many of the discussions have a wider applicability. Moreover, the tone of the review is intended to be generally accessible, such that an extended scientific community can gain insight into factors currently pacing the adoption of unstructured high-order methods.

Numerical resolution of linear evolution multidimensional problems of second order in time

AbstractWe present a new class of efficient time integrators for solving linear evolution multidimensional problems of second‐order in time named Fractional Step Runge‐Kutta‐Nyström methods (FSRKN). We show that these methods, combined with suitable spliting of the space differential operator and adequate space discretizations provide important advantages from the computational point of view, mainly parallelization facilities and reduction of computational complexity. In this article, we study in detail the consistency of such methods and we introduce an extension of the concept of R‐stability for Runge‐Kutta‐Nyström methods. We also present some numerical experiments showing the unconditional convergence of a third order method of this class applied to resolve one Initial Boundary Value Problem of second order in time. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 597–620, 2012

Implicit methods for viscous incompressible flows

Numerical methods and simulation tools for incompressible flows have been advanced largely as a subset of the computational fluid dynamics (CFD) discipline. Especially within the aerospace community, simulation of compressible flows has driven most of the development of computational algorithms and tools. This is due to the high level of accuracy desired for predicting aerodynamic performance of flight vehicles. Conversely, low-speed incompressible flow encountered in a wide range of fluid engineering problems has not typically required the same level of numerical accuracy. This practice of tolerating relatively low-fidelity solutions in engineering applications for incompressible flow has changed. As the design of flow devices becomes more sophisticated, a narrower margin of error is required. Accurate and robust CFD tools have become increasingly important in fluid engineering for incompressible and low-speed flow. Accuracy depends not only on numerical methods but also on flow physics and geometry modeling. For high-accuracy solutions, geometry modeling has to be very inclusive to capture the elliptic nature of incompressible flow resulting in large grid sizes. Therefore, in this article, implicit schemes or efficient time integration schemes for incompressible flow are reviewed from a CFD tool development point of view. Extension of the efficient solution procedures to arbitrary Mach number flows through a unified time-derivative preconditioning approach is also discussed. The unified implicit solution procedure is capable of solving low-speed compressible flows, transonic, as well as supersonic flows accurately and efficiently. Test cases demonstrating Mach-independent convergence are presented.

High order finite volume methods on wavelet-adapted grids with local time-stepping on multicore architectures for the simulation of shock-bubble interactions

We present a space–time adaptive solver for single- and multi-phase compressible flows that couples average interpolating wavelets with high-order finite volume schemes. The solver introduces the concept of wavelet blocks, handles large jumps in resolution and employs local time-stepping for efficient time integration. We demonstrate that the inherently sequential wavelet-based adaptivity can be implemented efficiently in multicore computer architectures using task-based parallelism and introducing the concept of wavelet blocks. We validate our computational method on a number of benchmark problems and we present simulations of shock-bubble interaction at different Mach numbers, demonstrating the accuracy and computational performance of the method.

A Validation of FEM3MP with Joint Urban 2003 Data

Abstract Under the sponsorship of the U.S. Department of Energy and U.S. Department of Homeland Security, a computational fluid dynamics (CFD) model for simulating airflow and dispersion of chemical/biological agents released in urban areas has recently been developed. This model, the Finite Element Model in 3-Dimensions and Massively Parallelized (FEM3MP), is based on solving the three-dimensional, time-dependent Navier–Stokes equations with appropriate physics submodels on massively parallel computer platforms. It employs finite-element discretization for effective treatment of complex geometries and a semi-implicit projection scheme for efficient time integration. A simplified CFD approach, using both explicitly resolved and virtual buildings, was implemented to improve further the model’s efficiency. Results from our model are continuously being verified against measured data from wind-tunnel and field studies. Herein, this model is further evaluated using observed data from intensive operation periods (IOP) 3 and 9 of the Joint Urban 2003 field study conducted in Oklahoma City, Oklahoma, in July 2003. The model simulations of wind and concentration fields in the near and intermediate regions, as well as profiles of wind speed, wind direction, friction velocity, and turbulent kinetic energy (TKE) in the urban wake region, are generally consistent with and compared reasonably well to field observations. In addition, this model was able to reproduce the observed split plume of IOP 3 and the end vortices along Park Avenue in IOP 9. The dispersion results and TKE profiles at the crane station indicate that the effects of convective mixing are relatively important for the daytime release of IOP 3 but that the stable effects are relatively unimportant for the nighttime release of IOP 9. Results of this study also suggest that the simplified CFD approach implemented in FEM3MP can be a cost-effective tool for simulating urban dispersion problems.

Spin-dynamics simulations of theXYvector Blume-Emery-Griffiths model in three dimensions

We present extensive Monte Carlo spin-dynamics simulations of the vector Blume-Emery-Griffiths model with three-dimensional spins on a simple cubic lattice and periodic boundary conditions. For a range of Hamiltonian parameters, this model reproduces the phase diagram topology of the one experimentally observed for bulk mixtures of 3He-4He . Highly efficient decomposition time integration algorithms have been used to obtain the evolution of the equations of motion from which we have computed the dynamic structure factor S(q,omega) close to critical and tricritical points. The dispersion relation and the corresponding dynamic exponents have been obtained in the critical and multicritical transitions. We find that the dynamic critical exponents for in-plane and out-of-plane fluctuations are the same at the tricritical point but are in fact different along the critical line.

ICONE15-10488 DEVELOPMENT OF MULTISCALE LES CODE MISTRAL AND ITS APPLICATION TO A T-JUNCTION PIPE FLOW

The necessity of multiscale numerical model is explained first to mimic the energy cascade of turbulence in the introduction of this paper. Then, the outline of a multiscale large-eddy simulation code MISTRAL is explained, including the basic equations, turbulence models, numerical formulation and numerical techniques employed. In MISTRAL, both velocity and pressure are split into the large-scale and small-scale resolvable components, and unresolvable components. Each resolvable component is approximated with large-scale and small-scale finite elements. Variational method is used not only to yield the spatially discretized equation system but also to close the turbulence terms resulted from unresolvable components. With such a formulation, multiscale turbulent viscosities can be given and the adequate magnitudes of them may be estimated by the balance of turbulent kinetic energy and model dissipation. The multiscale mass matrix lumping technique is used for an efficient time integration in MISTRAL. As an example of the application, an isothermal flow in a T-junction pipe is analyzed with MISTRAL code. For comparison, numerical results obtained by a conventional LES code are also presented. The 1st and 2nd moments of turbulent velocities as well as the power spectra are computed with these numerical results, to discuss the superiority of MISTRAL in predicting statistical quantities.

A note on time integrators in water-wave simulations

The importance of a suitable temporal integrator for fully nonlinear simulations of surface gravity waves is emphasized. Via numerical examples, it is demonstrated that constant-step procedures are inefficient. This relates to the practice of energy-conserving symplectic integration, assuming constant time steps, and is compared to direct numerical simulations using Runge–Kutta integrators with variable time-step control. It is concluded that the latter with automatic variable time-step control is the more efficient, and should be applied. The practice and efficiency of a stabilization procedure for the time-step selection is described and illustrated. An important point of the method is that the linear part of the prognostic equations is integrated analytically, which means that this part is obtained to machine presicion for any (large) time step (time interval). The evolution and instabilities of highly nonlinear water waves in three dimensions are exemplified through an accurate and efficient time-integration procedure.

Linearly implicit time stepping methods for numerical weather prediction

The efficient time integration of the dynamic core equations for numerical weather prediction (NWP) remains a key challenge. One of the most popular methods is currently provided by implementations of the semi-implicit semi-Lagrangian (SISL) method, originally proposed by Robert (J. Meteorol. Soc. Jpn., 1982). Practical implementations of the SISL method are, however, not without certain shortcomings with regard to accuracy, conservation properties and stability. Based on recent work by Gottwald, Frank and Reich (LNCSE, Springer, 2002), Frank, Reich, Staniforth, White and Wood (Atm. Sci. Lett., 2005) and Wood, Staniforth and Reich (Atm. Sci. Lett., 2006) we propose an alternative semi-Lagrangian implementation based on a set of regularized equations and the popular Stormer–Verlet time stepping method in the context of the shallow-water equations (SWEs). Ultimately, the goal is to develop practical implementations for the 3D Euler equations that overcome some or all shortcomings of current SISL implementations.

Efficient time integration of the Boltzmann-Poisson system applied to semiconductor device simulation

This paper presents new efficient methods for performing the time integration of finite-difference WENO approximations of the Boltzmann-Poisson system applied to semiconductor device simulation. The developed methods are based on local time-step schemes for hyperbolic conservation laws which permit the use of different time increments at different positions in space. A strategy to dynamically adapt the space-time grid according to the actual stability criteria imposed by the CFL-condition is proposed. The resulting numerical schemes are used to simulate the electron transport in a n+−n−n+ silicon diode and in a silicon MESFET. Several numerical tests and comparisons with computations performed with TVD Runge-Kutta type algorithms prove the efficiency of the presented time integration methods.

Development and evaluation of a micro–macro algorithm for the simulation of polymer flow

A micro–macro algorithm for the calculation of polymer flow is developed and numerically evaluated. The system being solved consists of the momentum and mass conservation equations from continuum mechanics coupled with a microscopic-based rheological model for polymer stress. Standard finite element techniques are used to solve the conservation equations for velocity and pressure, while stochastic simulation techniques are used to compute polymer stress from the simulated polymer dynamics in the rheological model. The rheological model considered combines aspects of reptation, network and continuum models. Two types of spatial approximation are considered for the configuration fields defining the dynamics in the model: piecewise constant and piecewise linear. The micro–macro algorithm is evaluated by simulating the abrupt planar die entry flow of a polyisobutylene solution described in the literature. The computed velocity and stress fields are found to be essentially independent of mesh size and ensemble size, while there is some dependence of the results on the order of spatial approximation to the configuration fields close to the die entry. Comparison with experimental data shows that the piecewise linear approximation leads to better predictions of the centerline first normal stress difference. Finally, the computational time associated with the piecewise constant spatial approximation is found to be about 2.5 times lower than that associated with the piecewise linear approximation. This is the result of the more efficient time integration scheme that is possible with the former type of approximation due to the pointwise incompressibility guaranteed by the choice of velocity–pressure finite element.

Efficient corrector iteration for DAE time integration in multibody dynamics

Efficient time integration is a key issue in computational multibody dynamics. Implicit time integration methods for stiff systems and constrained systems require the solution of a system of nonlinear equations in each time step. The nonlinear equations are solved iteratively by Newton type methods that are tailored to the structure of the equations of motion in multibody dynamics. In the present paper we discuss classical and recent methods for reducing the numerical effort in the application to multibody systems that are modelled in joint coordinates. The methods have been implemented in an industrial multibody system simulation package. Results of numerical tests for two benchmark problems from vehicle dynamics are presented.