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.
Numerical analysis is the study of algorithms that use numerical approximation for the problems of mathematical analysis. It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics, numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living cells in medicine and biology.
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
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.
Mesh generation is the practice of creating a mesh, a subdivision of a continuous geometric space into discrete geometric and topological cells. Often these cells form a simplicial complex. Usually the cells partition the geometric input domain. Mesh cells are used as discrete local approximations of the larger domain. Meshes are created by computer algorithms, often with human guidance through a GUI, depending on the complexity of the domain and the type of mesh desired. A typical goal is to create a mesh that accurately captures the input domain geometry, with high-quality (well-shaped) cells, and without so many cells as to make subsequent calculations intractable. The mesh should also be fine in areas that are important for the subsequent calculations.
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.
Computational mechanics is the discipline concerned with the use of computational methods to study phenomena governed by the principles of mechanics. Before the emergence of computational science as a "third way" besides theoretical and experimental sciences, computational mechanics was widely considered to be a sub-discipline of applied mechanics. It is now considered to be a sub-discipline within computational science.
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.
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.
Traffic simulation or the simulation of transportation systems is the mathematical modeling of transportation systems through the application of computer software to better help plan, design, and operate transportation systems. Simulation of transportation systems started over forty years ago, and is an important area of discipline in traffic engineering and transportation planning today. Various national and local transportation agencies, academic institutions and consulting firms use simulation to aid in their management of transportation networks.
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.
20-sim is 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 widely 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.
Model order reduction (MOR) is a technique for reducing the computational complexity of mathematical models in numerical simulations. As such it is closely related to the concept of metamodeling, with applications in all areas of mathematical modelling.