Deformable solids Soft-body dynamics

1 deformable solids

1.1 spring/mass models
1.2 finite element simulation
1.3 energy minimization methods
1.4 shape matching
1.5 rigid-body based deformation

deformable solids

the simulation of volumetric solid soft bodies can realised using variety of approaches.

spring/mass models

two nodes mass points connected parallel circuit of spring , damper.

in approach, body modeled set of point masses (nodes) connected ideal weightless elastic springs obeying variant of hooke s law. nodes may either derive edges of two-dimensional polygonal mesh representation of surface of object, or three-dimensional network of nodes , edges modeling internal structure of object (or one-dimensional system of links, if example rope or hair strand being simulated). additional springs between nodes can added, or force law of springs modified, achieve desired effects. applying newton s second law point masses including forces applied springs , external forces (due contact, gravity, air resistance, wind, , on) gives system of differential equations motion of nodes, solved standard numerical schemes solving odes. rendering of three-dimensional mass-spring lattice done using free-form deformation, in rendered mesh embedded in lattice , distorted conform shape of lattice evolves. assuming point masses equal 0 1 can obtain stretched grid method aimed @ several engineering problems solution relative elastic grid behavior. these known mass-spring-damper models.

finite element simulation

this more physically accurate approach, uses used finite element method solve partial differential equations govern dynamics of elastic material. body modeled three-dimensional elastic continuum breaking large number of solid elements fit together, , solving stresses , strains in each element using model of material. elements typically tetrahedral, nodes being vertices of tetrahedra (relatively simple methods exist tetrahedralize 3 dimensional region bounded polygon mesh tetrahedra, how two-dimensional polygon may triangulated triangles). strain (which measures local deformation of points of material rest state) quantified strain tensor




ϵ



{\displaystyle {\boldsymbol {\epsilon }}}

. stress (which measures local forces per-unit area in directions acting on material) quantified cauchy stress tensor




σ



{\displaystyle {\boldsymbol {\sigma }}}

. given current local strain, local stress can computed via generalized form of hooke s law:




σ

=


c



ε

 
,


{\displaystyle {\boldsymbol {\sigma }}={\mathsf {c}}{\boldsymbol {\varepsilon }}\ ,}







c




{\displaystyle {\mathsf {c}}}

elasticity tensor encodes material properties (parametrized in linear elasticity isotropic material poisson ratio , young s modulus).

the equation of motion of element nodes obtained integrating stress field on each element , relating this, via newton s second law, node accelerations.

pixelux (developers of digital molecular matter system) use finite-element-based approach soft bodies, using tetrahedral mesh , converting stress tensor directly node forces. rendering done via form of free-form deformation.

energy minimization methods

this approach motivated variational principles , physics of surfaces, dictate constrained surface assume shape minimizes total energy of deformation (analogous soap bubble). expressing energy of surface in terms of local deformation (the energy due combination of stretching , bending), local force on surface given differentiating energy respect position, yielding equation of motion can solved in standard ways.

shape matching

in scheme, penalty forces or constraints applied model drive towards original shape (i.e. material behaves if has shape memory). conserve momentum rotation of body must estimated properly, example via polar decomposition. approximate finite element simulation, shape matching can applied 3 dimensional lattices , multiple shape matching constraints blended.

rigid-body based deformation

deformation can handled traditional rigid-body physics engine, modeling soft-body motion using network of multiple rigid bodies connected constraints, , using (for example) matrix-palette skinning generate surface mesh rendering. approach used deformable objects in havok destruction.