A Consistent Bending Model for Cloth Simulation with Corotational Subdivision Finite Elements
Wrinkles and folds play an important role in the appearance of real textiles. The way in which they form depends
mainly on the bending properties of the specific material type. Existing approaches fail to reliably reproduce
characteristic behaviour like folding and buckling for different material types or resolutions. It is therefore crucial
for the realistic simulation of cloth to model bending energy in a physically accurate and consistent way. In this
paper we present a new method based on a corotational formulation of subdivision finite elements. Due to the
non-local nature of the employed subdivision basis functions a C1-continuous displacement field can be defined.
In this way, it is possible to use the governing equations of thin shell analysis leading to physically accurate
bending behaviour. Using a corotated strain tensor allows the large displacement analysis of cloth while retaining
a linear system of equations. Hence, known convergence properties and computational efficiency are preserved
while convincing and detailed folding behaviour is obtained in the simualtion.
Throughout the last two decades there has been a lot of interest
in cloth simulation and animation. A complete discussion
of the relevant work is beyond the scope of this paper
and the reader is therefore referred to the textbook by House
and Breen [HB00], Volino and Thalmann [VMT00] or the
overview compiled by Ng and Grimsdale [NG96] and Volino
et al. [VCMT05]. In the following, we classify the previous
work relevant to the presented approach into different categories.
FEM Finite element methods have not yet seen much attention
in cloth simulation – at least not in computer graphics.
While we only mention the most relevant work here an
extensive list can be found in [HB00]. Most of the existing
FE-approaches are based on the geometrically exact thin
shell formulation presented by Simo et al. [SF89]. Eischen et
al. [EDC96] depart from the fully nonlinear theory and apply
it to cloth simulation using quadrilateral, curvilinear elements.
Because of the buckling behaviour of cloth which can
lead to divergence in the algorithm an adaptive arc-length
control is used. Etzmuss et al. [EKS03] presented a linear
FE-approach based on a plane-stress assumption. Bending is
treated separately from in-plane deformation while a corotational
strain formulation is used to account for arbitrary rigid
body transformations. Cirak et al. [COS00] use the formulation
of [SF89] to derive equilibrium equations. They introduce
a new kind of element based on subdivision basis functions.
Unlike former FE-formulations C1-continuity is ensured
by the nonlocal nature of the element shape functions
while retaining linearity in the displacements. The method
was later extended to the finite deformation range using nonlinear
theory [CO01,GCSO99].
Corotational Formulation The extraction of the rotational
part from the displacement field used in [EKS03] was first
addressed by Müller et al [MDM+02] who used a warping
heuristics. However, due to the inaccuracy of this method
occuring ghost forces have to be treated separately. A more
precise method was presented by Hauth et al. [HS04] who
used the polar decomposition of the deformation gradient.
While the latter work proposes an iterative solution for the
3D case, Etzmuss et al. [EKS03] take a direct way for the
simpler 2D problem. In our approach, the rotation field is
extracted in a similar way. This allows stable treatment of
arbitrarily large rigid body transformations and strict separation
of curvature and membrane strains.
Bending Models Most of the existing cloth simulation techniques
use an angular expression to model bending energy
or forces. Breen et al. [BHW94] use the linear beam theory
relating bending moment to curvature. Curvature is approximated
by fitting a circle to the three points defined by
two incident edges. A biphasic expression in terms of the
enclosed angle is then used for approximation. Volino et
al. [VCMT95] use a similar approach but rely on the dihedral
angle formed by two neighbouring triangles. Choi
and Ko [CK02] propose a bending model simultaneously
accounting for compression and buckling. Specific assumptions
on the post-buckling state and associated energy lead
to the derivation of bending forces. Bridson et al [BMF03]
identify an independent bending mode where the requirement
is to not affect rigid body motions and in-plane deformations.
Thus, they derive directions and relative magnitudes
for the bending forces of a basic bending element
consisting of two neighbouring triangles. Grinspun et
al. [GH+03] use a discrete mean curvature approximation
for a sound definition of bending energy for flexible shelllike
objects. Because the necessary gradient computation is
intricate the use of automatic differentiation is suggested.
A method to use thin shell dynamics with point sampled
surfaces for efficient animation was recently proposed by
Wicke et al. [WSG05]. This approach is particularly useful
for scenes where the topology changes due to cutting or tearing.
However, the method is limited to explicit time integration
and the accuracy of the computational framework (e.g.
boundedness of the solution) is not clear.
In most of the above mentioned approaches, the treatment
of bending is physically motivated but not accurate. This
means that realistic material mapping and resolution independence
cannot be expected. Since implicit time integration
is mandatory for stable and efficient cloth simulation
(see [BW98]) the approach by Wicke et al. is not an option.
The methods by Bridson et al. and Grinspun et al. (which
have in fact some aspects in common) are more promising.
An implicit treatment of bending, however remains difficult
as second order derivatives of the bending energy are then
required. For this reason Bridson et al. use explicit integration
for their directly derived bending forces. This works
well with small enough time steps and materials without
too strong a resistance to bending deformation. However,
this method is not unconditionally stable anymore. On the
contrary, Grinspun et al. treat bending implicitly and compute
the required derivatives using automatic differentiation.
Their method works well for inextensible rigid shells where
the bending energy is largely predominant. For physically
accurate cloth simulation, however, significant in-plane deformations
can also occur and the interplay between bending
and membrane energy needs to be treated consistently.
Furthermore, their method is not founded on continuum mechanics.
Therefore curvature and membrane strains are not
explicitly available and modelling anisotropic materials becomes
difficult. Additionally, independence of discretisation
and convergence with higher resolution to the continuous
equivalent at the same time is not given.