Efficient Parallel-in-Time Solution of Time-Periodic Problems Using a Multi-Harmonic Coarse Grid Correction

This paper presents a highly-parallelizable parallel-in-time algorithm for efficient solution of nonlinear time-periodic problems. It is based on the time-periodic extension of the Parareal method, known to accelerate sequential computations via parallelization on the fine grid. The proposed approach reduces the complexity of the periodic Parareal solution by introducing a simplified Newton algorithm, which allows an additional parallelization on the coarse grid. In particular, at each Newton iteration a multi-harmonic correction is performed, which converts the block-cyclic periodic system in the time domain into a block-diagonal system in the frequency domain, thereby solving for each frequency component in parallel. The convergence analysis of the method is discussed for a one-dimensional model problem. The introduced algorithm and several existing solution approaches are compared via their application to the eddy current problem for both linear and nonlinear models of a coaxial cable. Performance of the considered methods is also illustrated for a three-dimensional transformer model

An efficient steady-state analysis of the eddy current problem using a parallel-in-time algorithm

This paper introduces a parallel-in-time algorithm for efficient steady-state solution of the eddy current problem. Its main idea is based on the application of the well-known multi-harmonic (or harmonic balance) approach as the coarse solver within the periodic parallel-in-time framework. A frequency domain representation allows for the separate calculation of each harmonic component in parallel and therefore accelerates the solution of the time-periodic system. The presented approach is verified for a nonlinear coaxial cable model

A New Parareal Algorithm for Time-Periodic Problems with Discontinuous Inputs

The Parareal algorithm, which is related to multiple shooting, was introduced for solving evolution problems in a time-parallel manner. The algorithm was then extended to solve time-periodic problems. We are interested here in time-periodic systems which are forced with quickly-switching discontinuous inputs. Such situations occur, e.g., in power engineering when electric devices are excited with a pulse-width-modulated signal. In order to solve those problems numerically we consider a recently introduced modified Parareal method with reduced coarse dynamics. Its main idea is to use a low-frequency smooth input for the coarse problem, which can be obtained, e.g., from Fourier analysis. Based on this approach, we present and analyze a new Parareal algorithm for time-periodic problems with highly-oscillatory discontinuous sources. We illustrate the performance of the method via its application to the simulation of an induction machine

Accelerated Steady-State Torque Computation for Induction Machines using Parallel-In-Time Algorithms

This paper focuses on efficient steady-state computations of induction machines. In particular, the periodic Parareal algorithm with initial-value coarse problem (PP-IC) is considered for acceleration of classical time-stepping simulations via non-intrusive parallelization in time domain, i.e., existing implementations can be reused. Superiority of this parallel-in-time method is in its direct applicability to time-periodic problems, compared to, e.g, the standard Parareal method, which only solves an initial-value problem, starting from a prescribed initial value. PP-IC is exploited here to obtain the steady state of several operating points of an induction motor, developed by Robert Bosch GmbH. Numerical experiments show that acceleration up to several dozens of times can be obtained, depending on availability of parallel processing units. Comparison of PP-IC with existing time-periodic explicit error correction method highlights better robustness and efficiency of the considered time-parallel approach

Parareal for index two differential algebraic equations

This article proposes modifications of the Parareal algorithm for its application to higher index differential algebraic equations (DAEs). It is based on the idea of applying the algorithm to only the differential components of the equation and the computation of corresponding consistent initial conditions later on. For differential algebraic equations with a special structure as, e.g. given in flux-charge modified nodal analysis, it is shown that the usage of the implicit Euler method as a time integrator suffices for the Parareal algorithm to converge. Both versions of the Parareal method are applied to numerical examples of nonlinear index 2 differential algebraic equations

A New Parareal Algorithm for Problems with Discontinuous Sources

The Parareal algorithm allows to solve evolution problems exploiting parallelization in time. Its convergence and stability have been proved under the assumption of regular (smooth) inputs. We present and analyze here a new Parareal algorithm for ordinary differential equations which involve discontinuous right-hand sides. Such situations occur in various applications, e.g., when an electric device is supplied with a pulse-width-modulated signal. Our new Parareal algorithm uses a smooth input for the coarse problem with reduced dynamics. We derive error estimates that show how the input reduction influences the overall convergence rate of the algorithm. We support our theoretical results by numerical experiments, and also test our new Parareal algorithm in an eddy current simulation of an induction machine

Parallel-in-Time Simulation of Electromagnetic Energy Converters

Computer-aided simulations are widely used in industry, as they allow to optimize the design and to understand the life cycle of engineering products, before their physical prototypes are constructed. Such simulations must be typically performed in the time domain and are especially then time consuming, when long time intervals have to be computed, e.g., until the steady state. Parallel-in-time methods such as the Parareal algorithm are powerful candidates for an acceleration of these development stages due to their capability to distribute the workload among multiple processing units. This dissertation develops and analyzes novel efficient Parareal-based approaches, particularly suitable for applications in electrical engineering such as pulse-width modulated (PWM) power converters, electric motors or transformers. The main contributions of this thesis are the following. First, a multirate Parareal method is proposed for parallel-in-time solution of systems excited with PWM signals. The idea of the approach is to solve a surrogate model with a smooth excitation on the coarse level, while on the fine level the original discontinuous PWM excitation is used. Convergence analysis gives an error estimate in terms of the deviation of the coarse input form the PWM signal. Numerical study for an RL-circuit model is in agreement with the theoretical derivations. An extension of the method to time-periodic problems is proposed and analyzed for a linear model problem. The multirate Parareal-based methods are applied to a buck converter and a four-pole induction machine. Second, time parallelization with Parareal is incorporated into an industrial simulation tool and used for the design of an electric vehicle drive. In contrast to many other methods Parareal is not limited to particular operating points or motor configurations and can employ already existing solvers due to its non-intrusiveness. By means of a periodic Parareal method and 80 cores, the steady state of the motor can be obtained up to 28 times faster compared to the sequential calculation. This is a great aid to industry as it speeds up the design workflow significantly. Such a good performance of Parareal for induction machine simulations is justified also based on an eigenvalue analysis of two circuit schemes in this thesis. Third, a parallel-in-time algorithm for time-periodic problems based on a multi-harmonic coarse grid correction is presented. It introduces an additional parallelization on the coarse level due to a Newton-based linearization with a block-cyclic Jacobian matrix, followed by a frequency-domain transformation. Convergence analysis is performed for a model problem and confirmed by a numerical study. Application to a nonlinear coaxial cable model and a nonlinear transformer model yields acceleration of the sequential computations up to factors of 175 when exploiting 20 cores. Finally, this thesis develops a Parareal-based approach for time-periodic problems with unknown period as, e.g., autonomous evolution systems. The method is tested on a Colpitts oscillator model