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.