View source for Soft-body dynamics

{{short description|Computer graphics simulation of deformable objects}}
'''Soft-body dynamics''' is a field of [[computer graphics]] that focuses on visually realistic physical [[computer simulation|simulations]] of the motion and properties of [[deformation (engineering)|deformable objects]] (or ''soft bodies'').<ref>{{cite web|citeseerx = 10.1.1.124.4664|title=Physically Based Deformable Models in Computer Graphics|year=2005|author=Nealen, Müller, Keiser, Boxerman & Carlson}}</ref> The applications are mostly in video games and films. Unlike in simulation of [[rigid body|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]]). 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.<ref>{{cite web|url=http://numerion-software.com|title=Numerion Software - Carbon Physics}}</ref><ref>
{{cite web|url=http://developer.nvidia.com/object/physx_features.html#soft|title=PhysX soft bodies|date=24 February 2014}}</ref><ref>{{cite web|url=http://www.pixelux.ch/content/view/31/45/|title=Pixelux's Digital Molecular Matter (DMM)|access-date=2010-03-07|archive-date=2009-09-21|archive-url=https://web.archive.org/web/20090921182741/http://www.pixelux.ch/content/view/31/45/|url-status=dead}}</ref><ref>{{cite web|url=http://www.havok.com/index.php?page=havok-cloth|title=Havok Cloth|access-date=2010-03-07|archive-date=2012-02-29|archive-url=https://web.archive.org/web/20120229205933/http://www.havok.com/index.php?page=havok-cloth|url-status=dead}}</ref><ref>
{{cite web|url=http://bulletphysics.org/wordpress/|title=Bullet Physics|date=21 March 2022 }}</ref><ref>
{{cite web|url=http://usa.autodesk.com/adsk/servlet/pc/index?siteID=123112&id=13583239#channels_Dynamics,%20Effects%20&amp;%20Simulation|title=Maya Nucleus}}</ref>
[[File:Blender3D HiddenTextSoftbody.ogv|400px|thumb|Softbody objects react to forces and are able to collide with other 3D objects. This example has been created with [[Blender (software)|Blender]].]]

== Deformable solids ==

The simulation of volumetric solid soft bodies<ref>{{Cite web|title = Doc:2.4/Manual/Physics/Soft Bodies 2.46 - BlenderWiki|url = http://wiki.blender.org/index.php/Doc:2.4/Manual/Physics/Soft_Bodies_2.46|website = wiki.blender.org|access-date = 2015-09-19|archive-date = 2015-11-04|archive-url = https://web.archive.org/web/20151104110134/http://wiki.blender.org/index.php/Doc:2.4/Manual/Physics/Soft_Bodies_2.46|url-status = dead}}</ref> can be realised by using a variety of approaches.

=== Spring/mass models ===

[[File:Two nodes as mass points connected by parallel circuit of spring and damper.svg|thumb|Two nodes as mass points connected by a parallel circuit of a spring and a damper]]

In this approach, the body is modeled as a set of [[point mass]]es (nodes) connected by ideal weightless [[deformation (engineering)#Elastic deformation|elastic]] [[spring (device)|springs]] obeying some variant of [[Hooke's law]]. 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]] 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]] for the motion of the nodes, which is solved by standard numerical schemes for solving [[ordinary differential equation|ODEs]].<ref>{{cite web|url=http://www.multires.caltech.edu/pubs/GI99.pdf|title=Interactive animation of structured deformable objects|year=1999|author=Desbrun, Schroder & Barr}}</ref> Rendering of a three-dimensional mass-spring lattice is often done using [[free-form deformation]],<ref name="ffd">{{cite web|url=http://tom.cs.byu.edu/~557/text/3dffd.pdf|title=Free-Form Deformation of Solid Geometric Models|year=1986|author=Sederberg & Parry}}</ref> 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]] aimed at several engineering problems solution relative to the elastic grid behavior. These are sometimes known as [[mass-spring-damper model]]s. In pressurized soft bodies<ref>{{cite web|url=http://www.ep.liu.se/ecp/010/007/ecp01007.pdf|title=A pressure model for soft body simulation|year=2003|author=Matyka & Ollila}}</ref> spring-mass model is combined with a pressure force based on the [[ideal gas law]].

=== Finite element simulation ===

This is a more physically accurate approach, which uses the widely used [[finite element method in structural mechanics|finite element method]] to solve the [[partial differential equations]] which govern the dynamics of an [[linear elasticity|elastic material]]. The body is modeled as a three-dimensional [[continuum mechanics|elastic continuum]] by breaking it into a large number of solid elements which fit together, and solving for the [[stress (mechanics)|stresses]] and [[deformation (mechanics)|strains]] in each element using a model of the material.<ref>{{cite web|url=http://www.graphics.uni-bielefeld.de/publications/sca08-dgfem.pdf|title=Flexible Simulation of Deformable Models Using Discontinuous Galerkin FEM|year=2008|author=Kaufmann, Martin, Botsch & Gross}}</ref> The elements are typically tetrahedral, the nodes being the vertices of the tetrahedra (relatively simple methods exist<ref>{{cite web|url=http://cg.informatik.uni-freiburg.de/publications/volumetricMeshesVMV2006.pdf|title=Robust Tetrahedral Meshing of Triangle Soups|year=2006|author=Spillmann, Wagner & Teschner}}</ref><ref>{{cite web|url=http://physbam.stanford.edu/~fedkiw/papers/stanford2003-12.pdf|title=Adaptive Physics Based Tetrahedral Mesh Generation Using Level Sets|year=2003|author=Bridson, Teran, Molino & Fedkiw}}</ref> to ''tetrahedralize'' a three dimensional region bounded by a polygon mesh into [[tetrahedra]], similarly to how a two-dimensional [[polygon]] may be ''[[polygon triangulation|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#Infinitesimal strain tensor|strain tensor]] <math>\boldsymbol{\epsilon}</math>. The [[stress (mechanics)|stress]] (which measures the local forces per-unit area in all directions acting on the material) is quantified by the [[Cauchy stress tensor]] <math>\boldsymbol{\sigma}</math>. Given the current local strain, the local stress can be computed via the generalized form of [[Hooke's law]]:
<math> 
\boldsymbol{\sigma} = \mathsf{C} \boldsymbol{\varepsilon} \,
</math>
where <math>\mathsf{C}</math> is the [[elasticity tensor]], which encodes the material properties (parametrized in linear elasticity for an isotropic material by the [[Poisson ratio]] and [[Young's modulus]]).

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]], 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.<ref name="pixelux">{{cite web|url=http://graphics.cs.berkeley.edu/papers/Parker-RTD-2009-08|title=Real-Time Deformation and Fracture in a Game Environment|year=2009|author=Parker & O'Brien}}</ref> Rendering is done via a form of [[free-form deformation]].<ref name="ffd" />

=== Energy minimization methods ===

This approach is motivated by [[calculus of variations|variational principles]] and the physics of surfaces, which dictate that a constrained surface will 
assume the shape which [[Minimum total potential energy principle|minimizes the total energy of deformation]] (analogous to a [[soap bubble]]). 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.<ref>{{cite web|url=http://www.cs.ucla.edu/~dt/papers/siggraph87/siggraph87.pdf|title=Elastically Deformable Models|year=1987|author=Terzopoulos, Platt, Barr & Fleischer}}</ref><ref>{{cite web|url=https://www.cs.columbia.edu/cg/pdfs/1166238667-bendingCAGD.pdf|title=Discrete Quadratic Curvature Energies|year=2007|author=Wardetzky, Bergou, Harmon, Zorin & Grinspun}}</ref>

=== Shape matching ===

In this scheme, penalty forces or constraints are applied to the model to drive it towards its original shape<ref name="shape">{{cite web|url=http://www.beosil.com/download/MeshlessDeformations_SIG05.pdf|title=Meshless Deformations Based on Shape Matching|year=2005|author= Müller, Heidelberger, Teschner & Gross}}</ref> (i.e. the material behaves as if it has [[shape memory alloy|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.<ref>{{cite web|url=http://graphics.ethz.ch/research/physics/fast_shape_matching.php|title=Fast Adaptive Shape Matching Deformations|year=2008|author=Steinemann, Otaduy & Gross|access-date=2010-03-08|archive-date=2011-01-10|archive-url=https://web.archive.org/web/20110110014217/http://graphics.ethz.ch/research/physics/fast_shape_matching.php|url-status=dead}}</ref>

=== 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) [[skeletal animation|matrix-palette skinning]] to generate a surface mesh for rendering. This is the approach used for deformable objects in [[Havok (software)#Havok Cloth and Destruction|Havok Destruction]].<ref>{{cite web|url=http://www.havok.com/index.php?page=havok-destruction|title=Havok Destruction|access-date=2010-03-07|archive-date=2010-02-21|archive-url=https://web.archive.org/web/20100221035711/http://www.havok.com/index.php?page=havok-destruction|url-status=dead}}</ref>

== 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]] on the [[yarn]] level can be ignored (though modeling cloth on the yarn level has been tried).<ref>{{cite web|url=http://www.graphics.cornell.edu/~jmkaldor/sg08_knityarns_sm.pdf|title=Simulating Knitted Cloth at the Yarn Level|year=2008|author=Kaldor, James & Marschner}}</ref> Via [[rendering (computer graphics)|rendering]] effects, this can produce a visually plausible emulation of [[textiles]] and [[clothing]], 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 [[character animation|animated characters]].

Cloth simulators are generally based on [[soft body dynamics#Mass-spring models|mass-spring models]], but a distinction must be made between force-based and position-based solvers.

=== Force-based cloth ===

The [[soft body dynamics#Spring/mass models|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 [[ordinary differential equation|ODE]] solvers. To create high resolution cloth with a realistic stiffness is not possible however with simple [[explicit and implicit methods|explicit]] solvers (such as forward [[Euler integration]]), unless the timestep is made too small for interactive applications (since as is well known{{Citation needed |date=September 2021}}, [[numerical ordinary differential equations#The Euler method|explicit]] integrators are numerically unstable for sufficiently [[stiff equation|stiff]] systems). Therefore, [[explicit and implicit methods|implicit solvers]] must be used,<ref>{{cite web|url=https://www.cs.cmu.edu/~baraff/papers/sig98.pdf|title=Large Steps in Cloth Simulation|year=1998|author=Baraff & Witkin}}</ref> requiring solution of a large [[sparse matrix]] system (via e.g. the [[conjugate gradient method]]), which itself may also be difficult to achieve at interactive frame rates. An alternative<ref>{{cite web|citeseerx = 10.1.1.89.9232|title=Collision and self-collision handling in cloth model dedicated to design garments|year=1997|author=Provot}}</ref><ref>{{cite web|url=http://graphics.stanford.edu/papers/cloth-sig02/cloth.pdf|title=Robust Treatment of Collisions, Contact and Friction for Cloth Animation|year=2002|author=Bridson, Fedkiw & Anderson}}</ref> 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 [[ordinary differential equation|ODEs]], many real-time cloth simulators (notably [[PhysX]], [[Havok (software)#Havok Cloth and Destruction|Havok Cloth]], and [[Autodesk Maya#Dynamics and simulation|Maya nCloth]]) use ''position based dynamics'' (PBD),<ref>{{cite web|url=http://www.matthiasmueller.info/publications/posBasedDyn.pdf|title=Position Based Dynamics|year=2006|author=Müller, Heidelberger, Hennix & Ratcliff}}</ref> 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 [[conservation of linear momentum|linear]] and [[conservation of angular momentum|angular momentum]] 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 ==
{{main article|Collision detection}}

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]] has presented a method which uses a global topological analysis of mesh intersections in configuration space to detect and resolve self-interpenetration of cloth.<ref>{{cite web|url=http://graphics.pixar.com/library/UntanglingCloth/paper.pdf|title=Untangling Cloth|year=2003|author=Baraff, Witkin & Kass}}</ref> 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 <math>O[n^2]</math> primitive collisions. Approaches used include:

* [[bounding volume hierarchy|Bounding volume hierarchies]] ([[axis-aligned bounding box|AABB]] trees,<ref>{{cite web|url=http://knight.temple.edu/~lakaemper/courses/cis350_2004/etc/jgt98deform.pdf|title=Efficient Collision Detection of Complex Deformable Models using AABB Trees|year=1998|author=Van Den Bergen}}</ref> [[oriented bounding box|OBB]] trees, sphere trees)
* Grids, either uniform<ref>{{cite web|url=http://www.beosil.com/download/CollisionDetectionHashing_VMV03.pdf|title=Optimized Spatial Hashing for Collision Detection of Deformable Objects|year=2003|author=Teschner, Heidelberger, Müller, Pomeranets & Gross}}</ref> (using [[hash table|hashing]] for memory efficiency) or hierarchical (e.g. [[Octree]], [[kd-tree]])
* Coherence-exploiting schemes, such as [[sweep and prune]] with insertion sort, or tree-tree collisions with front tracking.
* Hybrid methods involving a combination of various of these schemes, e.g. a coarse AABB tree plus sweep-and-prune with coherence between colliding leaves.

== Other applications ==

Other effects which may be simulated via the methods of soft-body dynamics are:

* ''Destructible'' materials: [[fracture]] of brittle solids, [[cutting]]<ref>{{cite web|url=http://graphics.ethz.ch/~deniss/SCA06/SCA06_steinemann.pdf|title=Fast Arbitrary Splitting of Deforming Objects|year=2006|author=Steinemann, Otaduy & Gross}}</ref> of soft bodies, and [[tearing]] of cloth. The [[soft body dynamics#Finite element simulation|finite element method]] is especially suited to modelling fracture<ref name="pixelux"/> as it includes a realistic model of the distribution of internal stresses in the material, which physically is what determines when fracture occurs, according to [[fracture mechanics]].
* [[plasticity (physics)|Plasticity]]<ref name="shape"/> (permanent deformation) and [[melting]]<ref>{{cite web|url=http://www.inf.ethz.ch/personal/giesen/tch/AGCGsem_SS05/mueller.pdf|title=Point Based Animation of Elastic, Plastic and Melting Objects|year=2004|author=Müller, Keiser, Nealen, Pauly, Gross & Alexa}}</ref>
* Simulated hair,<ref name="sellehair">{{cite web|url=http://physbam.stanford.edu/~aselle/papers/9/|title=A Mass Spring Model for Hair Simulation|year=2008|author=Selle, Lentine & Fedkiw|access-date=2010-03-07|archive-date=2009-12-14|archive-url=https://web.archive.org/web/20091214040532/http://physbam.stanford.edu/~aselle/papers/9/|url-status=dead}}</ref> fur, and feathers
* Simulated organs for biomedical applications<ref>{{cite web|url=http://www.scientificamerican.com/article.cfm?id=virtual-surgery-computer-graphic|title=When Will Virtual Surgery Make the Cut?|year=2007|publisher=[[Scientific American]]}}</ref>

[[fluid simulation|Simulating fluids in the context of computer graphics]] would not normally be considered soft-body dynamics, which is usually restricted to mean simulation of materials which have a tendency to retain their shape and form. In contrast, a [[fluid]] assumes the shape of whatever vessel contains it, as the particles are bound together by relatively weak forces.

== Software supporting soft body physics ==

=== Simulation engines ===
{| class="wikitable"
! Engine
! Website
! License
! Description
|-
| [[Bullet (software)|Bullet]]
| http://bulletphysics.org
| {{Free|[[zlib license]]}}
|-
| Carbon
| http://numerion-software.com
| {{Proprietary}}
|
|-
| [[CryEngine 3]]
| http://mycryengine.com
| {{Nonfree}}
|
|-
| [[Digital Molecular Matter]]
| http://pixelux.com
| {{Proprietary}}
|
|-
| [[Havok (software)#Havok Cloth and Destruction|Havok Cloth]]
| http://havok.com
| {{Nonfree}}
|
|-
|[[Houdini (software)|Houdini]]
|https://www.sidefx.com/products/houdini/<nowiki/>| https://www.sidefx.com/products/houdini/
| {{Proprietary}}
|Procedural generation & VFX software - soft-body FEM, cloth simulation (Vellum)
|-
| [[Autodesk Maya|Maya nCloth]]
| http://autodesk.com/maya
| {{Nonfree}}
|
|-
| [[OpenCloth]]
| https://github.com/mmmovania/opencloth
| {{dunno}}
| A collection of source codes implementing cloth simulation algorithms as well as soft body dynamics in OpenGL.
|-
| OpenTissue
| http://opentissue.org
| {{Free|[[zlib license]]}}
|
|-
| [[Physics Abstraction Layer]]
| http://www.adrianboeing.com/pal/index.html
| {{Free|[[3-clause BSD license]]}}
| Uniform API, supports multiple physics engines.
|-
| [[PhysX]]
| http://developer.nvidia.com/physx
| {{Nonfree}}
|
|-
| [[Phyz]]
| http://phyz.ath.cx
| {{Free|Public domain}}
|
|-
| [[Simulation Open Framework Architecture]]
| http://www.sofa-framework.org/
| {{Free|[[GNU Lesser General Public License]] v2.1+}}
|
|-
| [[Source Engine]]
| https://developer.valvesoftware.com
| {{Proprietary}}
| Supports the use of simulated hair and cloth physics.<ref>{{cite web |title=$jigglebone |url=https://developer.valvesoftware.com/wiki/$jigglebone |website=developer.valvesoftware |access-date=4 August 2022}}</ref>
|-
| [[Step (software)|Step]]
| http://edu.kde.org/step/
| {{Free|[[GNU General Public License]] v2}}
|
|-
| [[SyFlex]]
| http://syflex.biz
| {{Nonfree}}
|
|-
| [[Torque (game engine)|Torque]]
| https://torque3d.org/torque3d/
| {{Free|[[MIT License]]}}
|-
| [[Unreal Engine]]
| https://unrealengine.com
| {{Proprietary}}
|
|-
| [[Vega FEM]]
| https://viterbi-web.usc.edu/~jbarbic/vega/
| {{Free|[[3-clause BSD license]]}}
|
|-
| [[Blender (software)]]
| https://www.blender.org/
| {{Free|[[GNU General Public License]] v2}}
|
|-
| Ziva VFX
| http://zivadynamics.com
| {{Proprietary}}
| Character simulation software - FEM muscles, fat, skin and cloth
|}

=== Games ===
{| class="wikitable"
! Game
! Website
! License
! Description
|-
| [[BeamNG.drive]]
| http://beamng.com
| {{Nonfree}}
| Commercial sandbox-esque game based on soft-body vehicle physics. Stimulates damage to Vehicles And their functions
|-
| Crash Effects Inc.
| https://www.crasheffects.com/
| {{Nonfree}}
|Vehicle construction / first person crash test simulator. Soft-body physics are implemented for superstructures, some vehicle components use simple damage modeling.
|-
| [[Rigs of Rods]]
| https://rigsofrods.github.io
| {{Free|[[GNU General Public License]] v3}}
| Free and open source vehicle simulator.
|-
| [[Wreckfest]]
| http://nextcargame.com
| {{Nonfree}}
|Derby racing game. Soft-body physics are implemented for superstructures, but parts and components (e.g. the engine) use simple damage modeling.
|}

== See also ==
* [[Deformable body]]
* [[Dynamical simulation]]
* [[Rigid body dynamics]]
* [[Strength of materials]]
* [[Cloth modeling]]
* [[Breast physics]]

== References ==
{{Reflist}}

== External links ==
* [http://graphics.cs.cmu.edu/courses/15-869/ "The Animation of Natural Phenomena", CMU course on physically based animation, including deformable bodies]
* [https://www.youtube.com/watch?v=gbXCGpuJI7w Soft body dynamics video example]
* [https://web.archive.org/web/20110313153718/http://vizproto.prism.asu.edu/classes/sp03/wyman_g/Soft%20Body%20Dynamics.htm Introductory article]
* [https://web.archive.org/web/20100111035201/http://www.teknikus.dk/tj/gdc2001.htm Article by Thomas Jakobsen which explains the basics of the PBD method]

[[Category:Classical mechanics]]
[[Category:3D computer graphics]]
[[Category:Video game development]]
[[Category:Computer physics engines]]
[[Category:Articles containing video clips]]