Development of Simulation Software - from Simple ODE Modelling to Structural Dynamic Systems

Abstract

This contribution presents development and trends of simulation software, from the simple structures for ‘static’ explicit ODE models to modelling of structural dynamic systems with DAEs. Simulation emerged in the 1960’ in order to be able to analyse nonlinear dynamic system and to synthesize nonlinear control systems. Since that time simulation as problem solving tool has been developed towards the third pillar of science (beneath theory and experiment), and simultaneously simulation software has been developed further on. The paper first follows roots in the CSSL standard for simulation languages, from simple ODE modelling structures to discrete elements in ODE modelling, using the classical state space approach. Next, the extensions from explicit state space description to implicit model descriptions and their consequences for numerical algorithms and for structure of simulators are discussed, like DAE solvers and implicit model translation. Besides DAE modelling, state event description and state event handling has become a key feature for simulators – sketched by a state event classification and options for implementation. In the following, the last major steps of the development are presented: a-causal physical modelling, the new Modelica standard for ODE and DAE modelling, state chart and structural dynamic systems. Physical modelling and Modelica is outlined by examples, and for structural dynamic systems a new approach by means of internal and external events is presented – together comfortable state chart descriptions based on UML-RT. The last section reviews state-of-the-art simulators for availability of extended and structural features necessary for these last developments: DAE modelling, acausal physical modelling, state events, Modelica modelling, state chart modelling, structural decomposition for structural dynamic systems, and related features. At the end, a table summarises and compares the availability of structural approaches and features. CSSL STRUCTURE IN CONTINUOUS SIMULATION Simulation supported various developments in engineering and other areas, and simulation groups and societies were founded. One main effort of such groups was to standardise digital simulation programs and to work with a new basis: not any longer simulating the analog computer, but a self-standing structure for simulation systems. There were some unsuccessful attempts, but in 1968, the CSSL Standard became the milestone in the development: it unified the concepts and language structures of the available simulation programs, it defined a structure for the model, and it describes minimal features for a runtime environment. The CSSL standard suggests structures and features for a model frame and for an experimental frame. This distinction is based on Zeigler’s concept of a strict separation of these two frames. Model frame and experimental frame are the user interfaces for the heart of the simulation system, for the simulator kernel or simulation engine. A translator maps the model description of the model frame into state space notation, which is used by the simulation engine solving the system governing ODEs. This basic structure of a simulator is illustrated in Figure 1; an extended structure with service of discrete elements is given in Figure 3. In principle, in CSSL’s model frame, a system can be described in three different ways, as an interconnection of blocks, by mathematical expressions, and by conventional programming constructs as in FORTRAN or C. Mathematical basis is for the simulation engine is the state space description 0 0 ) ( ), , ), ( ), ( ( ) ( x t x p t t u t x f t x r r r r r r & r = = , which is used by the ODE solvers of the simulation engine. Any kind of textual model formulation, of graphical blocks or structured mathematical description or host languages constructs must be transformed to an internal state equation of the structure given above, so that the vector of derivatives ) , , , ( p t u x f r r r r can be calculated for a certain time instant ) , ), ( ), ( ( p t t u t x f f i i i i i r r r r r = . This vector of derivates is fed into an ODE solver in order to calculate a state update ) , . ( 1 h f x x i i i r r r Φ = + , h stepsize (all controlled by the simulation engine). Proceedings 22nd European Conference on Modelling and Simulation ©ECMS Loucas S. Louca, Yiorgos Chrysanthou, Zuzana Oplatkova, Khalid Al-Begain (Editors) ISBN: 978-0-9553018-5-8 / ISBN: 978-0-9553018-6-5 (CD) Essential is CSSL’s concept of SECTIONs or REGIONs, giving a certain structure to the model description. First, CSSL defines a set of operators like INTEG, which formulates parts of the state space description for the system governing ODEs. Other memory operators like DELAY for time delays, TABLE functions for generating (technical) tables, and transfer functions complete dynamic modelling parts. The dynamic model description builds up the DYNAMIC or DERIVATIVE section of the model description. Mapping the model description onto state space description, requires automatic sorting of the equations (blocks) to proper order of the calculation – an essential feature of the translator. Sometimes together with the state space equations we also meet parameter equations, parameter dependent initial values, and calculations with the terminal values (e.g. for cost functions in an optimisation). In principle, all this calculations could be done in the dynamic model description, but then they are calculated at each evaluation of the derivative vector of the ODE solver – although they have to be calculated only once. As example, we consider the model description for a pendulum. The well-known equations (length l, mass m, and damping coefficient d) and initial values, parameter and static relations and dependencies are given by ) ( 180 ) ( , , , 2 , 0 , , sin ) (