Soft-body dynamics is a field of
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
that focuses on visually realistic physical
simulations of the motion and properties of
deformable objects (or ''soft bodies''). The applications are mostly in video games and films. Unlike in simulation of
rigid bodies, the shape of soft bodies can change, meaning that the relative distance of two points on the object is not fixed. While the relative distances of points are not fixed, the body is expected to retain its shape to some degree (unlike a
fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
). The scope of soft body dynamics is quite broad, including simulation of soft organic materials such as muscle, fat, hair and vegetation, as well as other deformable materials such as clothing and fabric. Generally, these methods only provide visually plausible emulations rather than accurate scientific/engineering simulations, though there is some crossover with scientific methods, particularly in the case of finite element simulations. Several
physics engines currently provide software for soft-body simulation.
Deformable solids
The simulation of volumetric solid soft bodies can be realised by using a variety of approaches.
Spring/mass models
In this approach, the body is modeled as a set of
point masses (nodes) connected by ideal weightless
elastic
Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics.
Elastic may also refer to:
Alternative name
* Rubber band, ring-shaped band of rubber used to hold objects togethe ...
springs obeying some variant of
Hooke's law
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of t ...
. The nodes may either derive from the edges of a two-dimensional
polygonal mesh representation of the surface of the object, or from a three-dimensional network of nodes and edges modeling the internal structure of the object (or even a one-dimensional system of links, if for example a rope or hair strand is being simulated). Additional springs between nodes can be added, or the force law of the springs modified, to achieve desired effects. Applying
Newton's second law
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in mo ...
to the point masses including the forces applied by the springs and any external forces (due to contact, gravity, air resistance, wind, and so on) gives a system of
differential equations
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
for the motion of the nodes, which is solved by standard numerical schemes for solving
ODEs. Rendering of a three-dimensional mass-spring lattice is often done using
free-form deformation In computer graphics, free-form deformation (FFD) is a geometric technique used to model simple deformations of rigid objects. It is based on the idea of enclosing an object within a cube or another hull object, and transforming the object within ...
,
in which the rendered mesh is embedded in the lattice and distorted to conform to the shape of the lattice as it evolves. Assuming all point masses equal to zero one can obtain the
Stretched grid method
The stretched grid method (SGM) is a numerical technique for finding approximate solutions of various mathematical and engineering problems that can be related to an elastic grid behavior.
In particular, meteorologists use the stretched grid meth ...
aimed at several engineering problems solution relative to the elastic grid behavior. These are sometimes known as
mass-spring-damper model
The mass-spring-damper model consists of discrete mass nodes distributed throughout an object and interconnected via a network of springs and dampers. This model is well-suited for modelling object with complex material properties such as nonlin ...
s. In pressurized soft bodies spring-mass model is combined with a pressure force based on the
ideal gas law
The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first s ...
.
Finite element simulation
This is a more physically accurate approach, which uses the widely used
finite element method to solve the
partial differential equations
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function.
The function is often thought of as an "unknown" to be solved for, similarly to ...
which govern the dynamics of an
elastic material. The body is modeled as a three-dimensional
elastic continuum by breaking it into a large number of solid elements which fit together, and solving for the
stresses and
strains in each element using a model of the material. The elements are typically tetrahedral, the nodes being the vertices of the tetrahedra (relatively simple methods exist to ''tetrahedralize'' a three dimensional region bounded by a polygon mesh into
tetrahedra
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all th ...
, similarly to how a two-dimensional
polygon
In geometry, a polygon () is a plane figure that is described by a finite number of straight line segments connected to form a closed '' polygonal chain'' (or ''polygonal circuit''). The bounded plane region, the bounding circuit, or the two ...
may be ''
triangulated'' into triangles). The strain (which measures the local deformation of the points of the material from their rest state) is quantified by the
strain tensor . The
stress (which measures the local forces per-unit area in all directions acting on the material) is quantified by the
Cauchy stress tensor . Given the current local strain, the local stress can be computed via the generalized form of
Hooke's law
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of t ...
:
where
is the "
elasticity tensor
In physics, Hooke's law is an empirical law which states that the force () needed to extend or compress a spring by some distance () scales linearly with respect to that distance—that is, where is a constant factor characteristic of ...
" which encodes the material properties (parametrized in linear elasticity for an isotropic material by the
Poisson ratio and
Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied ...
).
The equation of motion of the element nodes is obtained by integrating the stress field over each element and relating this, via
Newton's second law
Newton's laws of motion are three basic laws of classical mechanics that describe the relationship between the motion of an object and the forces acting on it. These laws can be paraphrased as follows:
# A body remains at rest, or in mo ...
, to the node accelerations.
Pixelux (developers of the
Digital Molecular Matter system) use a finite-element-based approach for their soft bodies, using a tetrahedral mesh and converting the stress tensor directly into node forces.
Rendering is done via a form of
free-form deformation In computer graphics, free-form deformation (FFD) is a geometric technique used to model simple deformations of rigid objects. It is based on the idea of enclosing an object within a cube or another hull object, and transforming the object within ...
.
Energy minimization methods
This approach is motivated by
variational principles and the physics of surfaces, which dictate that a constrained surface will
assume the shape which
minimizes the total energy of deformation (analogous to a
soap bubble
A soap bubble is an extremely thin film of soap or detergent and water enclosing air that forms a hollow sphere with an iridescent surface. Soap bubbles usually last for only a few seconds before bursting, either on their own or on contact wi ...
). Expressing the energy of a surface in terms of its local deformation (the energy is due to a combination of stretching and bending), the local force on the surface is given by differentiating the energy with respect to position, yielding an equation of motion which can be solved in the standard ways.
Shape matching
In this scheme, penalty forces or constraints are applied to the model to drive it towards its original shape
(i.e. the material behaves as if it has
shape memory). To conserve momentum the rotation of the body must be estimated properly, for example via
polar decomposition. To approximate finite element simulation, shape matching can be applied to three dimensional lattices and multiple shape matching constraints blended.
Rigid-body based deformation
Deformation can also be handled by a traditional rigid-body
physics engine, modeling the soft-body motion using a network of multiple rigid bodies connected by constraints, and using (for example)
matrix-palette skinning to generate a surface mesh for rendering. This is the approach used for deformable objects in
Havok Destruction.
Cloth simulation
In the context of computer graphics, ''
cloth simulation'' refers to the simulation of soft bodies in the form of two dimensional continuum elastic membranes, that is, for this purpose, the actual structure of real
cloth
Textile is an umbrella term that includes various fiber-based materials, including fibers, yarns, filaments, threads, different fabric types, etc. At first, the word "textiles" only referred to woven fabrics. However, weaving is not the ...
on the
yarn
Yarn is a long continuous length of interlocked fibres, used in sewing, crocheting, knitting, weaving, embroidery, ropemaking, and the production of textiles. Thread is a type of yarn intended for sewing by hand or machine. Modern manu ...
level can be ignored (though modeling cloth on the yarn level has been tried). Via
rendering effects, this can produce a visually plausible emulation of
textiles
Textile is an umbrella term that includes various fiber-based materials, including fibers, yarns, filaments, threads, different fabric types, etc. At first, the word "textiles" only referred to woven fabrics. However, weaving is not the ...
and
clothing
Clothing (also known as clothes, apparel, and attire) are items worn on the body. Typically, clothing is made of fabrics or textiles, but over time it has included garments made from animal skin and other thin sheets of materials and natura ...
, used in a variety of contexts in video games, animation, and film. It can also be used to simulate two dimensional sheets of materials other than textiles, such as deformable metal panels or vegetation. In video games it is often used to enhance the realism of clothed
animated characters.
Cloth simulators are generally based on
mass-spring models, but a distinction must be made between force-based and position-based solvers.
Force-based cloth
The
mass-spring model (obtained from a
polygonal mesh representation of the cloth) determines the internal spring forces acting on the nodes at each timestep (in combination with gravity and applied forces). Newton's second law gives equations of motion which can be solved via standard
ODE solvers. To create high resolution cloth with a realistic stiffness is not possible however with simple
explicit
Explicit refers to something that is specific, clear, or detailed. It can also mean:
* Explicit knowledge
Explicit knowledge (also expressive knowledge) is knowledge that can be readily articulated, codified, stored and accessed. It can be expres ...
solvers (such as forward
Euler integration
In mathematics and computational science, the Euler method (also called forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit ...
), unless the timestep is made too small for interactive applications (since as is well known,
explicit
Explicit refers to something that is specific, clear, or detailed. It can also mean:
* Explicit knowledge
Explicit knowledge (also expressive knowledge) is knowledge that can be readily articulated, codified, stored and accessed. It can be expres ...
integrators are numerically unstable for sufficiently
stiff systems). Therefore,
implicit solvers must be used, requiring solution of a large
sparse matrix system (via e.g. the
conjugate gradient method
In mathematics, the conjugate gradient method is an algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a c ...
), which itself may also be difficult to achieve at interactive frame rates. An alternative is to use an explicit method with low stiffness, with ''ad hoc'' methods to avoid instability and excessive stretching (e.g. strain limiting corrections).
Position-based dynamics
To avoid needing to do an expensive implicit solution of a system of
ODEs, many real-time cloth simulators (notably
PhysX,
Havok Cloth, and
Maya nCloth) use ''position based dynamics'' (PBD), an approach based on constraint relaxation. The mass-spring model is converted into a system of constraints, which demands that the distance between the connected nodes be equal to the initial distance. This system is solved sequentially and iteratively, by directly moving nodes to satisfy each constraint, until sufficiently stiff cloth is obtained. This is similar to a
Gauss-Seidel solution of the implicit matrix system for the mass-spring model. Care must be taken though to solve the constraints in the same sequence each timestep, to avoid spurious oscillations, and to make sure that the constraints do not violate
linear
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
and
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
conservation. Additional position constraints can be applied, for example to keep the nodes within desired regions of space (sufficiently close to an animated model for example), or to maintain the body's overall shape via shape matching.
Collision detection for deformable objects
Realistic interaction of simulated soft objects with their environment may be important for obtaining visually realistic results. Cloth self-intersection is important in some applications for acceptably realistic simulated garments. This is challenging to achieve at interactive frame rates, particularly in the case of detecting and resolving self collisions and mutual collisions between two or more deformable objects.
Collision detection may be ''discrete/a posteriori'' (meaning objects are advanced in time through a pre-determined interval, and then any penetrations detected and resolved), or ''continuous/a priori'' (objects are advanced only until a collision occurs, and the collision is handled before proceeding). The former is easier to implement and faster, but leads to failure to detect collisions (or detection of spurious collisions) if objects move fast enough. Real-time systems generally have to use discrete collision detection, with other ''ad hoc'' ways to avoid failing to detect collisions.
Detection of collisions between cloth and environmental objects with a well defined "inside" is straightforward since the system can detect unambiguously whether the cloth mesh vertices and faces are intersecting the body and resolve them accordingly. If a well defined "inside" does not exist (e.g. in the case of collision with a mesh which does not form a closed boundary), an "inside" may be constructed via extrusion. Mutual- or self-collisions of soft bodies defined by tetrahedra is straightforward, since it reduces to detection of collisions between solid tetrahedra.
However, detection of collisions between two polygonal cloths (or collision of a cloth with itself) via discrete collision detection is much more difficult, since there is no unambiguous way to locally detect after a timestep whether a cloth node which has penetrated is on the "wrong" side or not. Solutions involve either using the history of the cloth motion to determine if an intersection event has occurred, or doing a global analysis of the cloth state to detect and resolve self-intersections.
Pixar
Pixar Animation Studios (commonly known as Pixar () and stylized as P I X A R) is an American computer animation studio known for its critically and commercially successful computer animated feature films. It is based in Emeryville, Californ ...
has presented a method which uses a global topological analysis of mesh intersections in configuration space to detect and resolve self-interpenetration of cloth. Currently, this is generally too computationally expensive for real-time cloth systems.
To do collision detection efficiently, primitives which are certainly not colliding must be identified as soon as possible and discarded from consideration to avoid wasting time.
To do this, some form of
spatial subdivision scheme is essential, to avoid a brute force test of