Parallel-in-time simulation of an electrical machine using MGRIT

A parallel-in-time algorithm for variable step multistep methods

In this paper we develop a two-grid convergence theory for the parallel-in-time scheme known as multigrid reduction in time (MGRIT), as it is implemented in the open-source package [XBraid: Parallel Multigrid in Time, http://llnl.gov/casc/xbraid]. MGRIT is a scalable and multilevel approach to parallel-in-time simulations that nonintrusively uses existing time-stepping schemes, and in a specific two-level setting it is equivalent to the widely known parareal algorithm. The goal of this paper is twofold. First, we present a two-level MGRIT convergence analysis for linear problems where the spatial discretization matrix can be diagonalized, and then apply this analysis to our two basic model problems, the heat equation and the advection equation. One important assumption is that the coarse and fine time-grid propagators can be diagaonalized by the same set of eigenvectors, which is often the case when the same spatial discretization operator is used on the coarse and fine time grids. In many cases, the MGRI...

SummaryThis paper presents some recent advances for parallel‐in‐time methods applied to linear elasticity. With recent computer architecture changes leading to stagnant clock speeds, but ever increasing numbers of cores, future speedups will be available through increased concurrency. Thus, sequential algorithms, such as time stepping, will suffer a bottleneck. This paper explores multigrid reduction in time (MGRIT) for an important application area, linear elasticity. Previously, efforts at parallel‐in‐time for elasticity have experienced difficulties, for example, the beating phenomenon. As a result, practical parallel‐in‐time algorithms for this application area currently do not exist. This paper proposes some solutions made possible by MGRIT (e.g., slow temporal coarsening and FCF‐relaxation) and, more importantly, a different formulation of the problem that is more amenable to parallel‐in‐time methods. Using a recently developed convergence theory for MGRIT and Parareal, we show that the changed formulation of the problem avoids the instability issues and allows the reduction of the error using two temporal grids. We then extend our approach to the multilevel case, where we demonstrate how slow temporal coarsening improves convergence. The paper ends with supporting numerical results showing a practical algorithm enjoying speedup benefits over the sequential algorithm.

Use of fuzzy logic to increase stability in control volume-based facility modeling

Conservation of power in time-stepping finite-element (FE) simulation of electrical machines is studied. We propose a method for accurately obtaining the instantaneous time derivative of the FE solution, from which the instantaneous eddy-current losses and the rate-of-change of the magnetic field energy are calculated. The method is shown to be consistent with different time-integration schemes, unlike the typically used backward-difference (BWD) approximation, which is only accurate if the BWD method is also used for the time integration. We first formulate the FE equations for a locked-rotor induction machine as a differential-algebraic equation (DAE) system. An approach called the collocation method is then used to derive the BWD, trapezoidal (TR), and implicit midpoint integration rules in order to show how these methods approximate the solution in time. We then differentiate the constraint equations of the DAE to form a system from which the time derivative of the solution can be solved. The obtained derivative is shown to satisfy the power balance exactly in the collocation points. In case of the TR rule, the losses calculated with the proposed method are shown to be less sensitive to the time-step length than ones obtained with the BWD approximation for the time derivatives. The collocation approach also allows studying the power balance continuously during the time step.

The need for parallelism in the time dimension is being driven by changes in computer architectures, where performance increases are now provided through greater concurrency, not faster clock speeds. This creates a bottleneck for sequential time marching schemes because they lack parallelism in the time dimension. Multigrid reduction in time (MGRIT) is an iterative procedure that allows for temporal parallelism by utilizing multigrid reduction techniques and a multilevel hierarchy of coarse time grids. MGRIT has been shown to be effective for linear problems, with speedups of up to 50 times. The goal of this work is the efficient solution of nonlinear problems with MGRIT, where efficiency is defined as achieving similar performance when compared to an equivalent linear problem. The benchmark nonlinear problem is the $p$-Laplacian, where p=4 corresponds to a well-known nonlinear diffusion equation and $p=2$ corresponds to the standard linear diffusion operator, our benchmark linear problem. The key difficu...

Parallel-in-time methods for partial differential equations (PDEs) have been the subject of intense development over recent decades, particularly for diffusion-dominated problems. It has been widely reported in the literature, however, that many of these methods perform quite poorly for advection-dominated problems. Here we analyze the particular iterative parallel-in-time algorithm of multigrid reduction-in-time (MGRIT) for discretizations of constant-wave-speed linear advection problems. We focus on common method-of-lines discretizations that employ upwind finite differences in space and Runge-Kutta methods in time. Using a convergence framework we developed in previous work, we prove for a subclass of these discretizations that, if using the standard approach of rediscretizing the fine-grid problem on the coarse grid, robust MGRIT convergence with respect to CFL number and coarsening factor is not possible. This poor convergence and non-robustness is caused, at least in part, by an inadequate coarse-grid correction for smooth Fourier modes in space-time known as characteristic components. We propose an alternative coarse-grid operator that provides a better correction of these modes. This coarse-grid operator is related to previous work and uses a semi-Lagrangian discretization combined with an implicitly treated truncation error correction. Theory and numerical experiments show the proposed coarse-grid operator yields fast MGRIT convergence for many of the method-of-lines discretizations considered, including for both implicit and explicit discretizations of high order. Parallel results demonstrate speed-up over sequential time-stepping.

SummaryAlthough convergence of the Parareal and multigrid‐reduction‐in‐time (MGRIT) parallel‐in‐time algorithms is well studied, results on their optimality is limited. Appealing to recently derived tight bounds of two‐level Parareal and MGRIT convergence, this article proves (or disproves) hx‐ and ht‐independent convergence of two‐level Parareal and MGRIT, for linear problems of the form , where is symmetric positive definite and Runge‐Kutta time integration is used. The theory presented in this article also encompasses analysis of some modified Parareal algorithms, such as the θ‐Parareal method, and shows that not all Runge‐Kutta schemes are equal from the perspective of parallel‐in‐time. Some schemes, particularly L‐stable methods, offer significantly better convergence than others as they are guaranteed to converge rapidly at both limits of small and large htξ, where ξ denotes an eigenvalue of and ht time‐step size. On the other hand, some schemes do not obtain h‐optimal convergence, and two‐level convergence is restricted to certain regimes. In certain cases, an factor change in time step ht or coarsening factor k can be the difference between convergence factors ρ≈0.02 and divergence! The analysis is extended to skew‐symmetric operators as well, which cannot obtain h‐independent convergence and, in fact, will generally not converge for a sufficiently large number of time steps. Numerical results confirm the analysis in practice and emphasize the importance of a priori analysis in choosing an effective coarse‐grid scheme and coarsening factor. A Mathematica notebook to perform a priori two‐grid analysis is available at https://github.com/XBraid/xbraid‐convergence‐est.

Parareal and multigrid reduction in time (MGRiT) are two of the most popular parallel-in-time methods. The basic idea is to treat time integration in a parallel context by using a multigrid method ...

In dislocation dynamics simulations, strain hardening simulations require integrating stiff systems of ordinary differential equations in time with expensive force calculations, discontinuous topological events and rapidly changing problem size. Current solvers in use often result in small time steps and long simulation times. Faster solvers may help dislocation dynamics simulations accumulate plastic strains at strain rates comparable to experimental observations. This paper investigates the viability of high-order implicit time integrators and robust nonlinear solvers to reduce simulation run times while maintaining the accuracy of the computed solution. In particular, implicit Runge–Kutta time integrators are explored as a way of providing greater accuracy over a larger time step than is typically done with the standard second-order trapezoidal method. In addition, both accelerated fixed point and Newton's method are investigated to provide fast and effective solves for the nonlinear systems that must be resolved within each time step. Results show that integrators of third order are the most effective, while accelerated fixed point and Newton's method both improve solver performance over the standard fixed point method used for the solution of the nonlinear systems.

An important application of explicit integration methods is to find the solution of a set of ordinary differential equations which is obtained using the finite element method. In many engineering applications, the finite element mesh consists of areas where large deformations are expected and areas where only small deformations are expected. In the areas where large deformations are expected, a fine mesh is necessary, while in the other areas a coarse grid is sufficient. The Courant criterion states that the time step must be smaller than the characteristic length of the element divided by the dilatational wave speed. Due to this Courant criterion, in the fine grid a small time step is necessary to ensure a stable solution. In the coarse grid, a much larger time step is allowed for the integration. If subcycling is applied, different time steps are allowed in the same mesh. This means that the smaller elements can be integrated with smaller time steps than the larger elements. This will lead to a decrease of the number of element calculations and hence to a decrease of the computer time.

For poroelastic media, the existence of a slow P-wave mode, next to the standard fast P and S waves, hinders efficient numerical implementations to propagate poroelastic waves through arbitrary seismic models. The slow P-wave speed can be an order of magnitude smaller than the fast P-wave speed. Hence, a stable and accurate simulation that can capture the slow P wave requires a fine grid and a small time step, which increases the overall computation cost greatly. To decrease the computation cost, we propose a poroelastic finite-difference simulation method that combines a discontinuous curvilinear collocated-grid method with a nonuniform time step Runge-Kutta (NUTS-RK) scheme. The fine grid and small time step are only used for areas near interfaces, where the contribution of the slow P wave is nonnegligible. The NUTS-RK scheme is derived from a Taylor expansion and it can circumvent the need for interpolations or extrapolations otherwise required by communications between different time levels. The accuracy and efficiency of the proposed method are verified by numerical tests. Compared with the curvilinear collocated-grid finite-difference method that uses a globally uniform space grid as well as a uniform time step, the proposed method requires fewer computing resources and can reduce the computing time greatly.

This study aims to present the discrete models and the methodology to implement real‐time simulations of electric machines using a low‐cost digital signal processor (DSP). The DC machine and the three‐phase induction machine are modelled in real‐time using a Texas Instruments DSP TMS28379D, where the discrete models are implemented using C language. A minimum time‐step of 1 µs can be achieved for the DC machine and 1.5 µs for the inductions machine in the experimental hardware. To validate the described models and show their precision, they are compared with commercial computational models from PSIM®. In addition, closed‐loop speed control strategies are applied to the real‐time DSP experimental models, showing perfect concordance with the machine theory. For the DC machine, a speed control strategy with an inner current control loop is applied and for the induction machine, a field‐oriented control for the speed control. The proposed real‐time simulation hardware has a great potential for low‐budget research and educational purposes since it can replace a real machine setup for a very low price, with great accuracy, variable parameters and free from risks, such as accidents or equipment damage. Furthermore, it uses cheap hardware with free software and a high‐level programing language.

Time adaptive conservative finite volume method

SUMMARYWe present a new unconditionally positivity‐preserving (PP) implicit time integration method for the DG scheme applied to shallow water flows. This novel time discretization enhances the currently used PP DG schemes, because in the majority of previous work, explicit time stepping is implemented to deal with wetting and drying. However, for explicit time integration, linear stability requires very small time steps. Especially for locally refined grids, the stiff system resulting from space discretization makes implicit or partially implicit time stepping absolutely necessary. As implicit schemes require a lot of computational time solving large systems of nonlinear equations, a much larger time step is necessary to beat explicit time stepping in terms of CPU time. Unfortunately, the current PP implicit schemes are subject to time step restrictions due to a so‐called strong stability preserving constraint. In this work, we hence give a novel approach to positivity preservation including its theoretical background. The new technique is based on the so‐called Patankar trick and guarantees non‐negativity of the water height for any time step size while still preserving conservativity. In the DG context, we prove consistency of the discretization as well as a truncation error of the third order away from the wet–dry transition. Because of the proposed modification, the implicit scheme can take full advantage of larger time steps and is able to beat explicit time stepping in terms of CPU time. The performance and accuracy of this new method are studied for several classical test cases. Copyright © 2014 John Wiley & Sons, Ltd.