During the last decade, a big progress has been achieved in the analysis and numerical treatment of Initial Value Problems (IVPs) in Differential Algebraic Equations (DAEs) and Ordinary Differential Equations (ODEs). In spite of the rich variety of results available in the literature, there are still many specific problems that require special attention. Two of such, which are considered in this work, are the optimization of order of accuracy and reduction of cost of functional evaluations of Explicit Runge - Kutta (ERK) methods.

Traditionally, the maximum attainable order p of an s-stage ERK method for advancing the solution of an IVP is such that
p(s)
 4
 
In 1999, Goeken presented an s-stage ERK Method of order p(s)=s +1,s>2.
However, this work focuses on the design of a new approximation algorithm that reduces the cost of functional evaluations and yet increases the attainable order higher 
U n and Jonhson [94]; and the classical ERK methods. The order p of 
the new scheme called Multiderivative Explicit Runge-Kutta (MERK) Methods is such that p(s) 2. The stability, convergence and implementation for the optimization of IVPs in DAEs and ODEs systems are also considered.

Numerical solution of initial value problems in differential - algebraic equations

Digital twin can be defined as a virtual representation of a physical asset enabled through data and simulators for real-time prediction, optimization, monitoring, controlling, and improved decision making. Recent advances in computational pipelines, multiphysics solvers, artificial intelligence, big data cybernetics, data processing and management tools bring the promise of digital twins and their impact on society closer to reality. Digital twinning is now an important and emerging trend in many applications. Also referred to as a computational megamodel, device shadow, mirrored system, avatar or a synchronized virtual prototype, there can be no doubt that a digital twin plays a transformative role not only in how we design and operate cyber-physical intelligent systems, but also in how we advance the modularity of multi-disciplinary systems to tackle fundamental barriers not addressed by the current, evolutionary modeling practices. In this work, we review the recent status of methodologies and techniques related to the construction of digital twins mostly from a modeling perspective. Our aim is to provide a detailed coverage of the current challenges and enabling technologies along with recommendations and reflections for various stakeholders.

/pdf/digital-twin-values-challenges-and-enablers-from-a-modeling-3hpoelaogn.pdf

Digital Twin: Values, Challenges and Enablers From a Modeling Perspective

The Functional Mockup Interface (FMI) is a tool independent standard for the exchange of dynamic models and for co-simulation. The development of FMI was initiated and organized by Daimler AG within the ITEA2 project MODELISAR. The primary goal is to support the exchange of simulation models between suppliers and OEMs even if a large variety of different tools are used. The FMI was developed in a close collaboration between simulation tool vendors and research institutes. In this article an overview about FMI is given and technical details about the solution are discussed.

https://www.ep.liu.se/ecp/063/013/ecp11063013.pdf

The Functional Mockup Interface for Tool independent Exchange of Simulation Models

The Functional Mockup Interface (FMI) is a tool independent standard for the exchange of dynamic models and for co simulation. The first version, FMI 1.0, was published in 2010. Already more than 30 tools support FMI 1.0. In this paper an overview about the recently published version 2.0 of FMI is given that combines the formerly separated interfaces for Model Exchange and Co-Simulation in one standard. Based on the experience on using FMI 1.0, many small details have been improved and new features ease the usability and increase the performance especially for larger models. Additionally, a free FMI compliance checker will become soon available and FMI models from different tools are made available on the web to simplify testing.

/pdf/functional-mockup-interface-2-0-the-standard-for-tool-4cimolga2r.pdf

Functional Mockup Interface 2.0: The Standard for Tool independent Exchange of Simulation Models

All you need to know about model predictive control for buildings

Co-Simulation is a general approach to simulate coupled technical systems. In a master-slave concept the slaves simulate sub-problems whereas the master is responsible for both coordinating the overall simulation as well as transferring data. To unify the interface between master and slave the FMI for CoSimulation was developed. Using FMI a master was implemented with simple and advanced algorithms which can be applied depending on the properties of the involved slave simulators. The master was tested amongst others by coupling with SimulationX.

/pdf/master-for-co-simulation-using-fmi-5a4k16ut56.pdf

Master for Co-Simulation Using FMI

Complex multi-disciplinary models in system dynamics are typically composed of subsystems. This modular structure of the model reflects the modular structure of complex engineering systems. In industrial applications, the individual subsystems are often modeled separately in different mono-disciplinary simulation tools. The Functional Mock-Up Interface (FMI) provides an interface standard for coupling physical models from different domains and addresses problems like export and import of model components in industrial simulation tools (FMI for Model Exchange) and the standardization of co-simulation interfaces in nonlinear system dynamics (FMI for Co-Simulation), see [8]. In November 2011, the third b -version of FMI for Model Exchange and Co-Simulation v2.0 was released [13] that supports advanced numerical techniques in co-simulation like higher order extrapolation and interpolation of subsystem inputs, step size control including step rejection and Jacobian based linearly implicit stabilization techniques. Well known industrial simulation tools for applied dynamics support Version 1.0 of this standard and plan to support the forthcoming Version 2.0 in the near future, see the “Tools” tab of website [8] for up-to-date information. The renewed interest in algorithmic and numerical aspects of co-simulation inspired some new investigations on error estimation and stabilization techniques in FMI for Model Exchange and Co-Simulation v2.0 compatible co-simulation environments. The present paper extends recent results from [3] on reliable error estimation and communication step size control in the framework of FMI for Model Exchange and Co-Simulation v2.0. Based on a strict mathematical analysis, we study the asymptotic behaviour of the local error and two error estimates that may be used to adapt the communication step size automatically to the changing solution behaviour during time integration. These theoretical results are illustrated by numerical tests for a (linear) quarter car model and provide a basis for future investigations with more complex cou

https://www.ep.liu.se/ecp/076/020/ecp12076020.pdf

Co-simulation with communication step size control in an FMI compatible master algorithm

Functional Digital Mock-up (FDMU) and Functional Mock-up Interface (FMI) are two keywords arising in the last years in simulation technology. In this paper, we would like to show that both principles, aiming at a comprehensive investigation of heterogeneous systems, e.g. from mechatronics, are not necessarily competing with each other but may be combined to benefit from the ideas behind. The approaches are based on different ideas and cover different aspects of the interaction of modern simulation tools. For that reason different constraints have to be considered, which do not make things easier. Both principles have advantages and disadvantages. However, by combining both ways, a powerful framework for handling a broad variety of simulation tasks can be formed. In the paper, a possible approach for integrating both technologies will be shown.

/pdf/functional-digital-mock-up-and-the-functional-mock-up-341h94q2c9.pdf

Functional Digital Mock-up and the Functional Mock-up Interface - Two Complementary Approaches for a Comprehensive Investigation of Heterogeneous Systems

The new operator homotopy (...) was introduced in Modelica 3.2 to improve the solution of difficult initialization problems. The background and motivation for this approach is discussed and it is demonstrated how to apply it for mechanical, electrical and fluid systems. Furthermore, it is shown at hand of several examples how an inappropriate formulation might lead to ill-posed problems.

/pdf/robust-initialization-of-differential-algebraic-equations-4n9cnvbk9e.pdf

Christoph Clauß

Papers

The Functional Mockup Interface for Tool independent Exchange of Simulation Models

Master for Co-Simulation Using FMI

Co-simulation with communication step size control in an FMI compatible master algorithm

Functional Digital Mock-up and the Functional Mock-up Interface - Two Complementary Approaches for a Comprehensive Investigation of Heterogeneous Systems

Robust Initialization of Differential-Algebraic Equations Using Homotopy