Soft-body dynamics

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 masses (nodes) connected by ideal weightless elastic 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 ODEs.[9] Rendering of a three-dimensional mass-spring lattice is often done using free-form deformation,[10] 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 models. In pressurized soft bodies[11] spring-mass model is combined with a pressure force based on the ideal gas law.

This is a more physically accurate approach, which uses the widely used finite element method to solve the partial differential equations 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.[12] The elements are typically tetrahedral, the nodes being the vertices of the tetrahedra (relatively simple methods exist[13][14] to tetrahedralize a three dimensional region bounded by a polygon mesh into tetrahedra, similarly to how a two-dimensional polygon 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 ${\displaystyle {\boldsymbol {\epsilon }}}$. The stress (which measures the local forces per-unit area in all directions acting on the material) is quantified by the Cauchy stress tensor ${\displaystyle {\boldsymbol {\sigma }}}$. Given the current local strain, the local stress can be computed via the generalized form of Hooke's law: ${\displaystyle {\boldsymbol {\sigma }}={\mathsf {C}}{\boldsymbol {\varepsilon }}\,}$ where ${\displaystyle {\mathsf {C}}}$ 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.[15] Rendering is done via a form of free-form deformation.[10]

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). 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.[16][17]

In this scheme, penalty forces or constraints are applied to the model to drive it towards its original shape[18] (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.[19]

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.[20]

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 solvers (such as forward Euler integration), unless the timestep is made too small for interactive applications (since as is well known, explicit integrators are numerically unstable for sufficiently stiff systems). Therefore, implicit solvers must be used,[22] 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[23][24] is to use an explicit method with low stiffness, with ad hoc methods to avoid instability and excessive stretching (e.g. strain limiting corrections).

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),[25] 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 and 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.