IUTAM Symposium on Solver-Coupling and Co-Simulation

Non-iterative or weak-coupling is the most applicable scheme for the co-simulation of interacting subsystems, where subsystems are solved independently with data exchange at restricted time instants. This contribution analyzes the continuous co-simulation from a different, a system-oriented, point of view and three coupling challenges are identified: co-simulation discretization error, sampling and discontinuities introduced. Introduction of smoothing filters can be interpreted as an additional co-simulation discretization error and affects the entire system behavior in general. However, energy-preservation-based considerations has proven to improve co-simulation performance, enabling filter applications according to the communication step-size, where mitigated frequency parts are added by the recently proposed correction schemes. This way, numerical stiffness is relaxed by an energy preserving mapping of high frequencies into low frequency ranges, based on Parseval’s identity. The proposed approaches are demonstrated along a theoretical as well as an industrial co-simulation example.

Numerical stability is a key aspect in co-simulation of physical systems. Decoupling a system into independent sub-models will introduce time delays on interface variables. By utilizing physical time delays for decoupling, affecting the numerical stability can be avoided. This requires interpolation, to allow solvers to request input variables for the time slot where they are needed. The FMI for co-simulation standard does not support fine-grained interpolation using interpolation tables. Here, various modifications to the FMI standard are suggested for improved handling of interpolation. Mechanical and thermodynamic models are used to demonstrate the need for interpolation, as well as to provide an industrial context. It is shown that the suggested improvements are able to stabilize the otherwise unstable connections.

We consider prediction strategies in a parallel coupling scheme for modular co-simulation: local extrapolation and a linear-implicit stabilization technique based on model information. That is, concerning local extrapolation, instead of using data points at the macro time points for generating the extrapolation polynomial (as it is done in the conventional global case), we use local data points only within the last macro time step. The linear-implicit stabilization technique predicts coupling quantities based on model information in terms of Jacobian matrices by performing a linear-implicit Euler step forward in time. We introduce and discuss these two prediction strategies and analyze their numerical properties, stability and accuracy, based on a simple test model.

In order to couple several simulation models, the corresponding software tools can be interconnected by means of a co-simulation. The inputs and outputs of the models depend on each other and have to be updated during the time integration process of the numerical solvers. Since the tools can only communicate at discrete macro-time points, the model inputs are mostly approximated, e.g., by using polynomial interpolation and extrapolation techniques. As a drawback of classical extrapolation methods, discontinuities occur at the macro-time points. This can slow down the solvers and reduces the efficiency of the co-simulation. The current paper considers continuous approximation techniques of \(C^0\), \(C^1\) and \(C^2\) type which are capable to overcome the discontinuity issues. The approaches are analyzed regarding numerical stability, global error and performance. To show the benefit of the continuity, the methods are implemented in a master-slave co-simulation and a comparison with the classical discontinuous approach is done. The \(C^2\)-continuous approach mostly outperforms the methods of lower continuity. The \(C^0\)-continuous method fails due to a limitation of the error order. With a here-presented enhancement the order drop of the \(C^0\)-continuous method can be avoided.

Co-simulation promotes the idea that domain specific simulation tools should cooperate in order to simulate the inter-domain interactions that are often observed in complex systems. To get trustworthy results, it is important that this technique preserves the stability properties of the original system. In this paper, we show how to preserve stability for adaptive co-simulation schemes, which offer fine grained control over the performance/accuracy of the co-simulation. To this end, we apply the joint spectral radius theory to certify that an adaptive co-simulation scheme is stable, and, if that is not possible, we use recent results in this field to create a trace of decisions that lead to instability. With this trace, it is possible to adjust the adaptive co-simulation in order to make it stable. Our approach is limited by the fact that computing the joint spectral radius is NP-Hard and undecidable in general. Nevertheless, we successfully applied our results to the co-simulation of a double mass-spring-damper system.

We introduce the SNiMoWrapper, an FMI-compatible software tool which enables the integration of models with an integrated, adapted solver in the form of a co-simulation FMU into simulation tools by conducting the co-simulation and hiding its details from the simulator. We describe the used algorithm in detail, give a short proof for the order of convergence of the SNiMoWrapper, show results for its application to an academic test example and describe an industrial proof-of-concept application.

Coupled finite element analysis is expected in several fields of research and development, where necessity of coupled analysis of not only two phenomena but also three or more phenomena is increasing. The demand for multiscale and multiphysics analyses is also increasing. A coupled analysis approach that combines an in-house code and a general-purpose commercial finite element analysis code using a message passing interface is very promising for these analyses. To examine the practical usefulness and effectiveness, this approach was applied to the triply coupled and multiscale analyses of a resistance spot welding problem and to the dynamic coupled analysis of an ultra sonic piezoelectric motor. The approach is effective to perform various types of coupled finite element analyses of multiphysics and multiscale problems.

Co-simulation methods can be used advantageously not only in the field of multidisciplinary simulations, but also to parallelize large monodisciplinary dynamical models. This paper focuses on the reduction of computation time that can be achieved in the simulation of multibody systems by partitioning a monolithic model into a variable number of coupled subsystems. The connection between the subsystems can be described in various ways. In this work, different subsystems are coupled by nonlinear constitutive equations (applied-force coupling approach). Exchange of coupling information takes only place at distinct macro-time points. The essential point is that the subsystems are integrated independently of each other between the macro-time points. If a Jacobi-type co-simulation scheme is used, all subsystems can be solved in parallel.

For coupling different solvers, explicit co-simulation approaches are frequently applied, especially when proprietary software tools have to be coupled without full solver access. Explicit solver coupling approaches are usually much simpler to implement than implicit methods. A major drawback of explicit coupling approaches is their reduced numerical stability behavior. Applying classical explicit co-simulation techniques, the coupling variables are approximated by extrapolation/interpolation polynomials in order to carry out the subsystem integration. Typically, Lagrange polynomials are applied for generating the approximation polynomials, using the coupling variables at the macro-time points as sampling points. In this manuscript, an explicit coupling approach is presented, which shows an improved numerical stability behavior. The key idea of the proposed method is to apply polynomial approximation to predict the acceleration variables of the coupling bodies. By integrating the predicted accelerations, one obtains predicted state variables for the coupling bodies, which are used to predict the coupling variables by making use of the constitutive equations of the coupling element. Compared to classical coupling techniques, where the coupling variables are directly approximated, the approach presented here based on approximated accelerations exhibits a significantly improved numerical stability behavior. Moreover, the numerical error is markedly reduced. The co-simulation method is introduced for flexible multibody systems. However, the proposed approach can generally be applied to couple arbitrary mechanical systems. Also, the coupling technique may be used to couple non-mechanical systems, e.g. electrical or hydraulic systems.

The secondary flow inside a round tube undergoing motion perpendicular to its longitudinal axis is considered. The motion of the fluid is modelled by a discretization of the secondary flow and is simulated separately from the motion of the tube, which is modelled as a flexible multibody system. The interaction between the two problems is taken into account by means of co-simulation. The Navier–Stokes equations are simplified and linearized, and then discretized with a spectral expansion in the circumferential direction and with finite elements in the radial direction. A tube rotating with a constant angular velocity and a tube undergoing a harmonic vibration are considered as examples. The main objective of this study is to show that the proposed co-simulation is actually feasible for this problem.

In this paper, an approach for controlling the communication-step size in connection with explicit co-simulation methods is suggested. In the framework of the proposed communication-step size controller, each subsystem integration is carried out with two different explicit co-simulation methods. By comparing the variables for both integrations, an error estimator for the local error can be constructed. Making use of the estimated local error, a step-size controller for the communication step-size can be implemented. Examples are presented demonstrating the applicability and accuracy of the proposed communication-step size controller.

Co-simulation schemes are designed to couple subsystems during the integration process. Therefore, any complex or multi-physics system can be split into subsystems in its mathematical representation, and re-coupled using a co-simulation scheme. Dealing separately with each subsystem, its own characteristics and specifically its own solver is the purpose of this decoupling/re-coupling mechanism. Before making a choice between all the existing solver-coupling schemes for a complex mechanical system, it is interesting to know which one is the most efficient. Therefore, this paper studies the performance of the Jacobi and Gauß–Seidel methods using one-step integration schemes applied on a double harmonic oscillator. However, since most of the mechanical joints generate elastic forces, the study concerns applied-force schemes only.

In order to simulate wind turbines under different load scenarios, the computational model should take into account the aerodynamics of the rotor, the flexibility of tower, foundation and soil, transient operational phases and, first and foremost, the interaction of all these aspects. Whenever vibrations are emitted to soil, they induce waves traveling through the ground. Here, the main focus lies on the Soil-Structure-Interaction (SSI) effects on the dynamic behavior of operating wind turbines. The wind turbine with its foundation and the surrounding soil is modeled by a coupled Finite Element Method/Scaled Boundary Finite Element Method approach.

We consider a differential-algebraic co-simulation approach to couple flexible structures to a rigid multibody system. That is, the coupling is realized using an algebraic constraint equation. The spatial discretization of flexible structures introduces an algebraic loop in the data exchange of subsystems. We investigate how this influences the stability and how modifying the discretization can help to stabilize the co-simulation.