Dynamical simulation
Encyclopedia
Dynamical simulation, in computational physics
, is the simulation
of systems of objects that are free to move, usually in three dimensions according to Newton's laws of dynamics, or approximations thereto. Dynamical simulation is used in computer animation
to assist animators to produce realistic motion, in industrial design
(for example to simulate crashes as an early step in crash test
ing), and in video games. Body movement is calculated using time integration methods.
, a program called a Physics engine
is used to model the behaviors of objects in space. These engines allow simulation of the way bodies of many types are affected by a variety of physical stimuli. They are also used to create Dynamical simulations without having to know anything about physics. Physics engines are used throughout the video game and movie industry, but not all physics engines are alike; They are generally broken into real-time
and the high precision but these are not the only options. Most real-time physics engines are inaccurate and yield only the barest approximation of the real world, whereas most high-precision engines are far too slow for use in everyday applications.
To understand how these Physics engines are built, a basic understanding of physics is required. Physics engines are based on the actual behaviors of the world as described by Classical mechanics
. Engines do not typically account for Modern Mechanics (see Theory of relativity
and Quantum mechanics
) because most visualization deals with large bodies moving relatively slowly, but the most complicated engines perform calculations for Modern Mechanics as well as Classical. The models used in Dynamical simulations determine how accurate these simulations are.
This equation will allow us to fully model the behavior of particles, but this is not sufficient for most simulations because it does not account for the rotational motion of rigid bodies. This is the simplest model that can be used in a physics engine and was used extensively in early video games.
. Rigid body dynamics
deals with the motion of objects that cannot change shape, size, or mass but can change orientation and position.
To account for rotational energy and momentum, we must describe how force is applied to the object using a moment
, and account for the mass distribution of the object using a inertia tensor. We describe these complex interactions with an equation somewhat similar to the definition of force above:
where
is the central inertia tensor, 
is the angular velocity
vector, and
is the moment of the jth external force about the mass center
.
The inertia tensor describes the location of each particle of mass in a given object in relation to the object's center of mass. This allows us to determine how an object will rotate dependent on the forces applied to it. This angular motion is quantified by the angular velocity vector.
As long as we stay below relativistic speeds (see Relativistic dynamics
), this model will accurately simulate all relevant behavior. This method requires the Physics engine
to solve six ordinary differential equations at every instant we want to render, which is a simple task for modern computers.
The drawback to this model is that all the computation is on the front end, so it is still slower than we would like. The real usefulness is not apparent because it still relies on a system of non-linear differential equations. To alleviate this problem, we have to find a method that can remove the second term from the equation. This will allow us to integrate much more easily. The easiest way to do this is to assume a certain amount of symmetry.
scalar), which makes all of these equations simple:
These equations allow us to simulate the behavior of an object that can spin in a way very close to the method simulate motion without spin. This is a simple model but it is accurate enough to produce realistic output in real-time Dynamical simulations. It also allows a Physics engine
to focus on the changing forces and torques rather than varying inertia.
Computational physics
Computational physics is the study and implementation of numerical algorithms to solve problems in physics for which a quantitative theory already exists...
, is the simulation
Simulation
Simulation is the imitation of some real thing available, state of affairs, or process. The act of simulating something generally entails representing certain key characteristics or behaviours of a selected physical or abstract system....
of systems of objects that are free to move, usually in three dimensions according to Newton's laws of dynamics, or approximations thereto. Dynamical simulation is used in computer animation
Computer animation
Computer animation is the process used for generating animated images by using computer graphics. The more general term computer generated imagery encompasses both static scenes and dynamic images, while computer animation only refers to moving images....
to assist animators to produce realistic motion, in industrial design
Industrial design
Industrial design is the use of a combination of applied art and applied science to improve the aesthetics, ergonomics, and usability of a product, but it may also be used to improve the product's marketability and production...
(for example to simulate crashes as an early step in crash test
Crash test
A crash test is a form of destructive testing usually performed in order to ensure safe design standards in crashworthiness and crash compatibility for various modes of transportation or related systems and components.- Types :...
ing), and in video games. Body movement is calculated using time integration methods.
Physics Engines
In Computer scienceComputer science
Computer science or computing science is the study of the theoretical foundations of information and computation and of practical techniques for their implementation and application in computer systems...
, a program called a Physics engine
Physics engine
A physics engine is computer software that provides an approximate simulation of certain physical systems, such as rigid body dynamics , soft body dynamics, and fluid dynamics, of use in the domains of computer graphics, video games and film. Their main uses are in video games , in which case the...
is used to model the behaviors of objects in space. These engines allow simulation of the way bodies of many types are affected by a variety of physical stimuli. They are also used to create Dynamical simulations without having to know anything about physics. Physics engines are used throughout the video game and movie industry, but not all physics engines are alike; They are generally broken into real-time
Real-time computing
In computer science, real-time computing , or reactive computing, is the study of hardware and software systems that are subject to a "real-time constraint"— e.g. operational deadlines from event to system response. Real-time programs must guarantee response within strict time constraints...
and the high precision but these are not the only options. Most real-time physics engines are inaccurate and yield only the barest approximation of the real world, whereas most high-precision engines are far too slow for use in everyday applications.
To understand how these Physics engines are built, a basic understanding of physics is required. Physics engines are based on the actual behaviors of the world as described by Classical mechanics
Classical mechanics
In physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces...
. Engines do not typically account for Modern Mechanics (see Theory of relativity
Theory of relativity
The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity. However, the word relativity is sometimes used in reference to Galilean invariance....
and Quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
) because most visualization deals with large bodies moving relatively slowly, but the most complicated engines perform calculations for Modern Mechanics as well as Classical. The models used in Dynamical simulations determine how accurate these simulations are.
Particle Model
The first model which may be used in Physics engines governs the motion of infinitesimal objects with finite mass called “particles.” This equation, called Newton’s Second law (see Newton's laws) or the definition of force, is the fundamental behavior governing all motion:
This equation will allow us to fully model the behavior of particles, but this is not sufficient for most simulations because it does not account for the rotational motion of rigid bodies. This is the simplest model that can be used in a physics engine and was used extensively in early video games.
Inertial Model
Bodies in the real world deform as forces are applied to them, so we call them “soft,” but often the deformation is negligibly small compared to the motion, and it is very complicated to model, so most physics engines ignore deformation. A body that is assumed to be non-deformable are called rigid bodyRigid body
In physics, a rigid body is an idealization of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant in time regardless of external forces exerted on it...
. Rigid body dynamics
Rigid body dynamics
In physics, rigid body dynamics is the study of the motion of rigid bodies. Unlike particles, which move only in three degrees of freedom , rigid bodies occupy space and have geometrical properties, such as a center of mass, moments of inertia, etc., that characterize motion in six degrees of...
deals with the motion of objects that cannot change shape, size, or mass but can change orientation and position.
To account for rotational energy and momentum, we must describe how force is applied to the object using a moment
Moment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...
, and account for the mass distribution of the object using a inertia tensor. We describe these complex interactions with an equation somewhat similar to the definition of force above:
where
is the central inertia tensor, is the angular velocityAngular velocity
In physics, the angular velocity is a vector quantity which specifies the angular speed of an object and the axis about which the object is rotating. The SI unit of angular velocity is radians per second, although it may be measured in other units such as degrees per second, revolutions per...
vector, and
is the moment of the jth external force about the mass centerCenter of mass
In physics, the center of mass or barycenter of a system is the average location of all of its mass. In the case of a rigid body, the position of the center of mass is fixed in relation to the body...
.
The inertia tensor describes the location of each particle of mass in a given object in relation to the object's center of mass. This allows us to determine how an object will rotate dependent on the forces applied to it. This angular motion is quantified by the angular velocity vector.
As long as we stay below relativistic speeds (see Relativistic dynamics
Relativistic dynamics
Relativistic dynamics refers to a combination of relativistic and quantum concepts to describe the relationships between the motion and properties of a relativistic system and the forces acting on the system...
), this model will accurately simulate all relevant behavior. This method requires the Physics engine
Physics engine
A physics engine is computer software that provides an approximate simulation of certain physical systems, such as rigid body dynamics , soft body dynamics, and fluid dynamics, of use in the domains of computer graphics, video games and film. Their main uses are in video games , in which case the...
to solve six ordinary differential equations at every instant we want to render, which is a simple task for modern computers.
Euler Model
The inertial model is much more complex than we typically need but it is the most simple to use. In this model, we do not need to change our forces or constrain our system. However, if we make a few intelligent changes to our system, simulation will become much easier, and our calculation time will decrease. The first constraint will be to put each torque in terms of the principal axes. This makes each torque much more difficult to program, but it simplifies our equations significantly. When we apply this constraint, we diagonalize the moment of inertia tensor, which simplifies our three equations into a special set of equations called Euler's equations. These equations describe all rotational momentum in terms of the principal axes:- The N terms are applied torques about the principal axes
- The I terms are the principal moments of inertia
- The
terms are angular velocities about the principal axes
The drawback to this model is that all the computation is on the front end, so it is still slower than we would like. The real usefulness is not apparent because it still relies on a system of non-linear differential equations. To alleviate this problem, we have to find a method that can remove the second term from the equation. This will allow us to integrate much more easily. The easiest way to do this is to assume a certain amount of symmetry.
Symmetric/Torque Model
The two types of symmetric objects that will simplify Euler's equations are “symmetric tops” and “symmetric spheres.” The first assumes one degree of symmetry, this makes two of the I terms equal. These objects, like cylinders and tops, can be expressed with one very simple equation and two slightly simpler equations. This does not do us much good, because with one more symmetry we can get a large jump in speed with almost no change in appearance. The symmetric sphere makes all of the I terms equal (the Moment of inertiaMoment of inertia
In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...
scalar), which makes all of these equations simple:
- The N terms are applied torques about the principal axes
- The
terms are angular velocities about the principal axes - The I term is the scalar Moment of inertiaMoment of inertiaIn classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...
:
- where
-
- V is the volume region of the object,
- r is the distance from the axis of rotation,
- m is mass,
- v is volume,
- ρ is the pointwise densityDensityThe mass density or density of a material is defined as its mass per unit volume. The symbol most often used for density is ρ . In some cases , density is also defined as its weight per unit volume; although, this quantity is more properly called specific weight...
function of the object, - x, y, z are the Cartesian coordinates.
-
These equations allow us to simulate the behavior of an object that can spin in a way very close to the method simulate motion without spin. This is a simple model but it is accurate enough to produce realistic output in real-time Dynamical simulations. It also allows a Physics engine
Physics engine
A physics engine is computer software that provides an approximate simulation of certain physical systems, such as rigid body dynamics , soft body dynamics, and fluid dynamics, of use in the domains of computer graphics, video games and film. Their main uses are in video games , in which case the...
to focus on the changing forces and torques rather than varying inertia.
See also
- Bounding volumeBounding volumeIn computer graphics and computational geometry, a bounding volume for a set of objects is a closed volume that completely contains the union of the objects in the set. Bounding volumes are used to improve the efficiency of geometrical operations by using simple volumes to contain more complex...
- Collision detectionCollision detectionCollision detection typically refers to the computational problem of detecting the intersection of two or more objects. While the topic is most often associated with its use in video games and other physical simulations, it also has applications in robotics...
- Euler's equations (rigid body dynamics)
- Moment of inertiaMoment of inertiaIn classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation...
- Physics Abstraction Layer
- Physics enginePhysics engineA physics engine is computer software that provides an approximate simulation of certain physical systems, such as rigid body dynamics , soft body dynamics, and fluid dynamics, of use in the domains of computer graphics, video games and film. Their main uses are in video games , in which case the...
- Rigid body dynamicsRigid body dynamicsIn physics, rigid body dynamics is the study of the motion of rigid bodies. Unlike particles, which move only in three degrees of freedom , rigid bodies occupy space and have geometrical properties, such as a center of mass, moments of inertia, etc., that characterize motion in six degrees of...