Dynamic simulation (or dynamic system simulation) is the use of a computer program to model the time-varying behavior of a dynamical system. The systems are typically described by ordinary differential equations or partial differential equations. A simulation run solves the state-equation system to find the behavior of the state variables over a specified period of time. The equation is solved through numerical integration methods to produce the transient behavior of the state variables. Simulation of dynamic systems predicts the values of model-system state variables, as they are determined by the past state values. This relationship is found by creating a model of the system. [1]
Simulation models are commonly obtained from discrete-time approximations of continuous-time mathematical models. [2] As mathematical models incorporate real-world constraints, like gear backlash and rebound from a hard stop, equations become nonlinear. This requires numerical methods to solve the equations. [3]
A numerical simulation is done by stepping through a time interval and calculating the integral of the derivatives through numerical integration. Some methods use a fixed step through the interval, and others use an adaptive step that can shrink or grow automatically to maintain an acceptable error tolerance. Some methods can use different time steps in different parts of the simulation model.
There are two types of system models to be simulated: difference-equation models, and differential-equation models. Classical physics is usually based on differential equation models. This is why most old simulation programs are simply differential equation solvers and delegate solving difference-equations to “procedural program segments.”Some dynamic systems are modeled with differential equations that can only be presented in an implicit form. These differential-algebraic-equation systems require special mathematical methods for simulation. [2]
Some complex systems’ behavior can be quite sensitive to initial conditions, which could lead to large errors from the correct values. To avoid these possible errors, a rigorous approach can be applied, where an algorithm is found which can compute the value up to any desired precision. For example, the constant e is a computable number because there is an algorithm that is able to produce the constant up to any given precision. [4]
The first applications of computer simulations for dynamic systems was in the aerospace industry. [5] Commercial uses of dynamic simulation are many and range from nuclear power, steam turbines, 6 degrees of freedom vehicle modeling, electric motors, econometric models, biological systems, robot arms, mass-spring-damper systems, hydraulic systems, and drug dose migration through the human body to name a few. These models can often be run in real time to give a virtual response close to the actual system. This is useful in process control and mechatronic systems for tuning the automatic control systems before they are connected to the real system, or for human training before they control the real system. Simulation is also used in computer games and animation and can be accelerated by using a physics engine, the technology used in many powerful computer graphics software programs, like 3ds Max, Maya, Lightwave, and many others to simulate physical characteristics. In computer animation, things like hair, cloth, liquid, fire, and particles can be easily modeled, while the human animator animates simpler objects. Computer-based dynamic animation was first used at a very simple level in the 1989 Pixar short film Knick Knack to move the fake snow in the snowglobe and pebbles in a fish tank.
This animation was made with a software system dynamics, with a 3D modeler. The calculated values are associated with parameters of the rod and crank. In this example the crank is driving, we vary both the speed of rotation, its radius, and the length of the rod, the piston follows.
Computer simulation is the process of mathematical modelling, performed on a computer, which is designed to predict the behaviour of, or the outcome of, a real-world or physical system. The reliability of some mathematical models can be determined by comparing their results to the real-world outcomes they aim to predict. Computer simulations have become a useful tool for the mathematical modeling of many natural systems in physics, astrophysics, climatology, chemistry, biology and manufacturing, as well as human systems in economics, psychology, social science, health care and engineering. Simulation of a system is represented as the running of the system's model. It can be used to explore and gain new insights into new technology and to estimate the performance of systems too complex for analytical solutions.
A hybrid system is a dynamical system that exhibits both continuous and discrete dynamic behavior – a system that can both flow and jump. Often, the term "hybrid dynamical system" is used, to distinguish over hybrid systems such as those that combine neural nets and fuzzy logic, or electrical and mechanical drivelines. A hybrid system has the benefit of encompassing a larger class of systems within its structure, allowing for more flexibility in modeling dynamic phenomena.
Computational science, also known as scientific computing, technical computing or scientific computation (SC), is a division of science that uses advanced computing capabilities to understand and solve complex physical problems. This includes
Modelica is an object-oriented, declarative, multi-domain modeling language for component-oriented modeling of complex systems, e.g., systems containing mechanical, electrical, electronic, hydraulic, thermal, control, electric power or process-oriented subcomponents. The free Modelica language is developed by the non-profit Modelica Association. The Modelica Association also develops the free Modelica Standard Library that contains about 1400 generic model components and 1200 functions in various domains, as of version 4.0.0.
Numerical methods for partial differential equations is the branch of numerical analysis that studies the numerical solution of partial differential equations (PDEs).
Finite-difference time-domain (FDTD) or Yee's method is a numerical analysis technique used for modeling computational electrodynamics. Since it is a time-domain method, FDTD solutions can cover a wide frequency range with a single simulation run, and treat nonlinear material properties in a natural way.
Computational electromagnetics (CEM), computational electrodynamics or electromagnetic modeling is the process of modeling the interaction of electromagnetic fields with physical objects and the environment using computers.
EcosimPro is a simulation tool developed by Empresarios Agrupados A.I.E for modelling simple and complex physical processes that can be expressed in terms of Differential algebraic equations or Ordinary differential equations and Discrete event simulation.
MapleSim is a Modelica-based, multi-domain modeling and simulation tool developed by Maplesoft. MapleSim generates model equations, runs simulations, and performs analyses using the symbolic and numeric mathematical engine of Maple. Models are created by dragging-and-dropping components from a library into a central workspace, resulting in a model that represents the physical system in a graphical form. Maplesoft began development of MapleSim partly in response to a request from Toyota to produce physical modeling tools to aid in their new model-based development process.
SimulationX is a CAE software application running on Microsoft Windows for the physical simulation of technical systems. It is developed and sold by ESI Group.
Continuous Simulation refers to simulation approaches where a system is modeled with the help of variables that change continuously according to a set of differential equations.
Ecolego is a simulation software tool that is used for creating dynamic models and performing deterministic and probabilistic simulations. It is also used for conducting risk assessments of complex dynamic systems evolving over time.
The Functional Mock-up Interface defines a standardized interface to be used in computer simulations to develop complex cyber-physical systems.
20-sim is a commercial modeling and simulation program for multi-domain dynamic systems, which is developed by Controllab. With 20-sim, models can be entered as equations, block diagrams, bond graphs and physical components. 20-sim is used for modeling complex multi-domain systems and for the development of control systems.
In mathematics a partial differential algebraic equation (PDAE) set is an incomplete system of partial differential equations that is closed with a set of algebraic equations.
Simcenter Amesim is a commercial simulation software for the modeling and analysis of multi-domain systems. It is part of systems engineering domain and falls into the mechatronic engineering field.
System-level simulation (SLS) is a collection of practical methods used in the field of systems engineering, in order to simulate, with a computer, the global behavior of large cyber-physical systems.
OpenModelica is a free and open source environment based on the Modelica modeling language for modeling, simulating, optimizing and analyzing complex dynamic systems. This software is actively developed by Open Source Modelica Consortium, a non-profit, non-governmental organization. The Open Source Modelica Consortium is run as a project of RISE SICS East AB in collaboration with Linköping University.