Multirate Integration Research Articles

Spline-oriented inter/extrapolation-based multirate schemes of higher order

Multirate integration uses different time step sizes for different components of the solution based on the respective transient behavior. For inter/extrapolation-based multirate schemes, we construct a new subclass of schemes by using clamped cubic splines to obtain multirate schemes up to order 4. Numerical results for a n -mass-oscillator demonstrate that 4th order of convergence can be achieved for this class of schemes.

Reduced order multirate schemes in industrial circuit simulation

In this paper the industrial application of Reduced Order Multirate (ROMR) schemes is presented. This paper contains the mathematical foundations of the ROMR schemes and elaborates on the construction of these schemes using specific Model Order Reduction (MOR) techniques. Especially the Maximum Entropy Snapshot Sampling method for generating a reduced basis and reduction by Gauß–Newton with Approximated Tensors (GNAT). This basis generation method is also used for generating the basis for the gappy hyper-reduction method used for nonlinear function evaluation. For the multirate integration part, a Backward Differentation Formula approach to integration is used in conjunction with a coupled-slowest-first multirate approach. After introducing the numerical approach to industrial circuit simulation validation experiments are performed. First a simple academic model is used, and then an industrial test case is simulated as presented by STMicroelectronics. A significant speedup in simulation time is achieved whilst accuracy and convergence is kept.

Reduced order multirate schemes for coupled differential-algebraic systems

In the context of time-domain simulation of integrated circuits, one often encounters large systems of coupled differential-algebraic equations. Simulation costs of these systems can become prohibitively large as the number of components keeps increasing. In an effort to reduce these simulation costs a twofold approach is presented in this paper. We combine maximum entropy snapshot sampling method and a nonlinear model order reduction technique, with multirate time integration. The obtained model order reduction basis is applied using the Gauß-Newton method with approximated tensors reduction. This reduction framework is then integrated using a coupled-slowest-first multirate integration scheme. The convergence of this combined method is verified numerically. Lastly it is shown that the new method results in a reduction of the computational effort without significant loss of accuracy.

Multirate Solver with Speed and Gap Control

Computer simulation of vehicular traffic on the real road network can be used to solve a whole range of relevant and practical problems. The microscopic approach and tens of thousands of vehicles used in simulation lead to large systems of ordinary differential equations. The vehicle dynamics can vary significantly. As a result, the corresponding systems of differential equations have a certain feature, i.e., the rate of change in unknown vector components, which in this case are the speeds of vehicles and the distances (gaps) between them, varies significantly. In this paper, we propose a multirate numerical integration scheme, in which an individual microstep is used for each component of the vector of unknowns within each macrostep. The values of the steps are determined using the obtained local error estimate of the given numerical scheme. The corresponding time-stepping strategy is obtained both for vehicle speeds and for distances between vehicles. Moreover, the multirate solver’s local error for gaps is estimated one order of accuracy higher than for speeds because drivers estimate the gap primarily rather than the speed. The developed numerical method shows a significant computational speedup in comparison with the corresponding single-rate method.

A high-fidelity wave-to-wire model for wave energy converters

Abstract Mathematical models incorporating all the necessary components of wave energy converters (WECs) from ocean waves to the electricity grid, known as wave-to-wire (W2W) models, are vital in the development of wave energy technologies. Ideally, precise W2W models should include all the relevant nonlinear dynamics, constraints and energy losses. This paper presents a balanced W2W model that incorporates high-fidelity models for each conversion system, and can accommodate different types of WECs, hydraulic power take-off (PTO) topologies, electric generators and grid connections. The models of the different conversion stages presented herein are efficiently implemented in the W2W model using a multi-rate integration scheme that reduces the computational requirements by a factor of 10. Two W2W models, i.e. one with the constant-pressure hydraulic PTO configuration and one with the variable-pressure configuration, are compared in this paper. Results show that a higher PTO efficiency (30% higher for the constant-pressure configuration) does not necessarily imply a higher electricity generation (2% higher for the variable-pressure configuration), which reinforces the need for high-fidelity W2W models for the design of successful WECs.

Variational multirate integration in multi‐body dynamics

AbstractFor systems that contain slow and fast dynamics, variational multirate integration schemes are used. These schemes split the system into parts which are simulated using two time grids consisting of micro and macro nodes. This formulation can be extended for multi‐body systems. The rigid multi‐body system is described by the so called director formulation and constraints describing the joints connecting the bodies. With the Lagrange multiplier method, the constraints are introduced into the equations of motion. A way to implement the null space method into the variational multirate framework is shown and the influence on the number of unknowns is investigated. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Efficient Computation of State Derivatives for Multi-Rate Integration of Object-Oriented Models

Abstract Multi-rate integration algorithms offer huge advantages in terms of simulation time savings when large, loosely coupled systems or systems with mixed fast and slow dynamics are considered. Such algorithms require the computation of different sub-sets of the state derivative vector at each time step. This paper presents a method to efficiently compute these sub-sets, starting from a continuous-time, object-oriented dynamic model. The method is demonstrated on a simple test case.

Limitations of computing time savings in variational multirate integrations

AbstractFor systems that contain slow and fast dynamics, variational multirate integration schemes are used. These schemes split the system into parts which are simulated using a time grid consisting of micro and macro nodes. This leads to computing time savings, however not unlimited, for a certain number of micro steps per macro step the computing time is minimal. To find a relation between this minimum computing time and the number of variables in the system, the computing time for the Fermi‐Pasta‐Ulam problem (FPU) is measured for different numbers of masses and different numbers of micro steps. In addition, the numerical convergence of the variational multirate integration is shown for the FPU. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

High-order algorithms for compressible reacting flow with complex chemistry

In this paper we describe a numerical algorithm for integrating the multicomponent, reacting, compressible Navier–Stokes equations, targeted for direct numerical simulation of combustion phenomena. The algorithm addresses two shortcomings of previous methods. First, it incorporates an eighth-order narrow stencil approximation of diffusive terms that reduces the communication compared to existing methods and removes the need to use a filtering algorithm to remove Nyquist frequency oscillations that are not damped with traditional approaches. The methodology also incorporates a multirate temporal integration strategy that provides an efficient mechanism for treating chemical mechanisms that are stiff relative to fluid dynamical time-scales. The overall methodology is eighth order in space with options for fourth order to eighth order in time. The implementation uses a hybrid programming model designed for effective utilisation of many-core architectures. We present numerical results demonstrating the convergence properties of the algorithm with realistic chemical kinetics and illustrating its performance characteristics. We also present a validation example showing that the algorithm matches detailed results obtained with an established low Mach number solver.

Computing time investigations for variational multirate integration

AbstractConsider a mechanical system that contains slow and fast dynamics. Let it be possible, to split the potential energy into a slow and a fast potential and the configuration vector into slow and fast variables. For such systems, multirate schemes simulate the different parts using different time steps with the goal to save computing time. For the proposed multirate scheme, a time grid consisting of micro and macro nodes is used and the integrator is derived from a discrete variational principle. Variational integrators conserve properties like symplecticity and momentum maps and have good energy behavior. To solve the resulting system of coupled nonlinear equations, a Newton‐Raphson iteration with an analytical Jacobian is used. It is demonstrated that the multirate approach leads to less computing time compared to singlerate simulation by means of three example systems, the Fermi‐Pasta‐Ulam problem, a triple spherical pendulum and a simple atomistic model, where the latter two are subject to constraints. Computing times are compared for different numbers of micro and macro nodes for dynamic simulations during a certain time interval. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Posteriori analysis of a multirate numerical method for ordinary differential equations

In this paper, we analyze a multirate time integration method for systems of ordinary differential equations that present significantly different scales within the components of the model. The main purpose of this paper is to present a hybrid a priori – a posteriori error analysis that accounts for the effects of projections between the coarse and fine scale discretizations, the use of only a finite number of iterations in the iterative solution of the discrete equations, the numerical error arising in the solution of each component, and the effects on stability arising from the multirate solution. The hybrid estimate has the form of a computable a posteriori leading order expression and a provably-higher order a priori expression. We support this estimate by an a priori convergence analysis. We present several examples illustrating the accuracy of multirate integration schemes and the accuracy of the a posteriori estimate.

Variational multirate integration of constrained dynamics

AbstractMechanical systems with dynamics on varying time scales, in particular those including highly oscillatory motion, impose challenging questions for numerical integration schemes. Tiny step sizes are required to guarantee a stable integration of the fast frequencies. However, for the simulation of the slow dynamics, integration with a larger time step is accurate enough. Small time steps increase integration times unnecessarily, especially for costly function evaluations. For systems comprising fast and slow dynamics, multirate methods integrate the slow part of the system with a relatively large step size while the fast part is integrated with a small time step. Main challenges are the identification of fast and slow parts (e.g. by separating the energy or by distinguishing sets of variables), the synchronisation of their dynamics and in particular the treatment of mixed parts that often appear when fast and slow dynamics are coupled by constraints. In this contribution, a multirate integrator is derived in closed form via a discrete variational principle on a time grid consisting of macro and micro time nodes. Variational integrators (based on a discrete version of Hamilton's principle) lead to symplectic and momentum preserving integration schemes that also exhibit good energy behavior. The resulting multirate variational integrator has the same preservation properties. An example demonstrates the performance of the multirate integrator for constrained multibody dynamics. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Towards a theory of continuous flow models

We present the continuous flow system specification formalism (CFSS). This formalism provides a digital computer approach to the representation of continuous systems based on a broad interpretation of the sampling concept. Continuous flow models permit to represent systems with both time-varying and multirate sampling. The equivalent dynamic system of a CFSS model is defined. Based on the CFSS formalism a multirate integration method is developed.

Modeling and Simulation of Dynamic Structure Heterogeneous Flow Systems

The author presents the Heterogeneous Flow System Specification (HFSS), a formalism suited to represent combined (discrete/continuous) systems. The new formalism is based on the concept of sampling that is described in general terms to include multirate sampling systems. The HFSS formalism permits the construction of continuous/discrete input/output models in a modular and hierarchical manner. Hierarchical systems have a dynamic structure nature that can be changed over time. To show an application of the HFSS formalism, the author presents a new formalism to numerically solve differential equations with discontinuities using multirate integration methods. Applications of combined and dynamic structure systems are presented.

Developments in Multirating for Coupled Systems

AbstractIn technologically computer‐aided design, the demand for a refined modelling yields the numerical simulation of coupled systems. Often the dynamics of these systems can be described as initial value problems of ordinary differential equations, which are characterised by largely varying time constants. A natural approach is to use multirate integration schemes which integrate the subsystems by their inherent step size. The realisation of the couplings of these subsystems, however done, is the crucial point.To begin, we give a short survey on multirate behaviour and its possible numerical exploitation in one‐step‐schemes. Based on the conception of generalised multirate, we classify multirate methods by the means, which compute the coupling terms for the internal stages: On one hand, extra/interpolation, on the other hand, the incremental formulation, which yields a genuine one‐step‐method. For the latter approach, we discuss the multirate W‐method as an example and give finally test results for a multirate version of Prothero‐Robinson's equation and the Inverter‐Chain.

Flexible Multibody Dynamic Analysis Using Multirate Integration Method

A Nordsick form opf the multirate integration scheme has been proposed for flexible multibody dynamic systems. It is assumed that vibrational modal coordinates in the equations of motion are treated as fast variables, whereas the relative joint coordinates are treated as slow variables. In the multirate integration, the fast variables are integrated with small step-size, and the slow variables are integrated with larger step-size. The proposed multirate integration method is based on the Adams-Bashforth-Moulton predictor-corrector method and implemented in the Nordsieck vector form. The Nordsieck form of multrate integration method provides effective step-size control and at the same time, inherits the efficiency from the Adams integration method. Simulations of a flexible gun and turret system of the military tank have been carried out to show the effectiveness and efficiency of the proposed method.

BEYOND SPICE, A REVIEW OF MODERN ANALOG CIRCUIT SIMULATION TECHNIQUES

We present a review of modern analog simulation techniques based on time- and frequency-domain algorithms. For time-domain techniques, important topics such as circuit decomposition, relaxation methods, latency, multirate integration, continuation methods, parallel algorithms, and finite difference time-domain methods are discussed. Frequency-domain simulation techniques included are harmonic balance, harmonic relaxation, harmonic-Newton, spectral balance, methods for quasiperiodic circuits, and device modeling for frequency domain simulators. Also included are examples of modern simulators.

Simulation of modular mechatronic systems with application to vehicle dynamics

The example of articulated vehicles is used to introduce a new simulation concept. Modules found by functional decomposition of an engineering system can be encapsulated by an input-output representation. They are then interconnected by the system inputs and outputs, or, in the case of mechanical systems, by constraints using joints. It is shown that the resulting mechanical DAE structure can effciently be solved by a projection method. A multirate integration is proposed for efficient simulation of systems with different time scales. The simulatorNEWMOS is introduced supporting all the features described. Simulation runs show that the multirate scheme is essential to achieve a stable integration of the articulated vehicles problem.

Efficient Dynamic Simulation of a Quadruped Using a Decoupled Tree-Structure Approach

Efficient simulation of a dynamically stable, fast-moving quadruped vehicle has been undertaken at The Ohio State University. Individual leg-link inertial properties, espe cially important at high speeds, are incorporated into the simulation. Also incorporated are the ground contact con siderations of compliance, friction, and impulsive impact forces. Basic efficiency is gained by decoupling of the closed-chain system into a tree-structured, open chain through introduction of spring/damper systems at the ground. The ensuing tree-structure dynamics are solved by developing and applying an extended form of the effi cient O(N) open-chain algorithm of Brandl et al. (1986). Additional speedup in the computations is gained through application of multirate integration and parallelization. One second of real-time simulation on a single Intel iPSC/ 2 Hypercube node takes 66.7 s. Use of multirate integra tion improves four-node parallel performance, giving a total speedup factor of more than 3.

CONNECTING WAVEFORM RELAXATION CONVERGENCE PROPERTIES TO THE A‐ STABILITY OF MULTIRATE INTEGRATION METHODS

Application of the waveform relaxation algorithm to the differential‐algebraic equations generated by problems in circuit and semiconductor device simulation have demonstrated that the method often contracts uniformly in time. In addition, instabilities in the underlying multirate integration method have not been observed. In this paper, it is proved that multirate A‐stability and waveform relaxation uniform contractivity are connected, and use the result to show that the first and second‐order backward‐difference based multirate methods are A‐stable when applied to block diagonally‐dominant problems.