A novel continuum-based fast projection scheme is proposed for cloth simulation. Cloth geometry is described by NURBS, and the dynamic response is modeled by a displacement-only Kirchhoff-Love shell element formulated directly on NURBS geometry. The fast projection method, which solves strain limiting as a constrained Lagrange problem, is extended to the continuum version. Numerical examples are studied to demonstrate the performance of the current scheme. The proposed approach can be applied to grids of arbitrary topology and can eliminate unrealistic over-stretching efficiently if compared to spring-based methodologies.

Cloth simulation has many practical applications, such as computer-aided garment design, character animation, and electronic e-commerce. Terzopoulos et al. [

The garment industry is now beginning to use virtual simulation for prototyping [

Another challenge of cloth simulation is how to efficiently enforce realistic strain on cloth. One of the characteristics of fabric is that the bending stiffness is far lower than the in-plane stiffness. A consequence of this property is that in-plane deformation of practical cloth in most cases is negligible if compared to the out-of-plane deformation. However, using high physical in-plane stiffness introduces significant difficulty in simulation. For explicit methods, higher in-plane stiffness requires smaller time increments. In Barraf et al. [

To overcome this undesired side effect, Provot [

Isogeometric analysis has been proposed to bridge computer-aided design (CAD) and analysis seamlessly [

The rest of this paper is organized as follows.

Kirchhoff-Love shell theory assumes the following:

The normal to the undeformed middle surface remains straight and perpendicular to the deformed middle surface.

The transverse normal stress is small compared with other normal stress components and may be neglected.

The thickness of the shell is small compared to the other dimensions.

The displacements of any given point on the shell are small in comparison to the thickness.

These assumptions are a good approximation for fabrics in which the energetic contribution from transverse shear is negligibly small compared to the bending and in-plane energy. Hence, the kinetics are completely characterized by the surface strain and curvature, which are determined by the surface geometry. For numerical computation, the theory can lead to a displacement-only formulation, which does not involve rotational degrees of freedom, thus increasing the efficiency of the method. A Kirchhoff-Love shell element is not commonly used in traditional finite-element analysis because constructing a

In the present NURBS Kirchkoff-Love shell element, the primary unknowns are the displacements of the control points. No rotational degree-of-freedoms are introduced.

The NURBS formulation below follows that of Kiendl et al. [

Here, the

The unit normal

With respect to the convected basis vectors, the surface deformation tensor

It is convenient to use a local ortho-normal basis to perform the element computations presented later. To this end, we introduce a pair of orthonormal bases

Derivatives of a basis function

In the current configuration, the bases (

With respect to the physical basis, the Green-Lagrangian strain assumes the form

The local bases

Cloth response is typically inelastic, exhibiting anisotropic properties and a small to moderate amount of hysteresis [

For the bending model, we employ a nonlinear bending function [

External forces acting on a piece of cloth normally include a body force

In the NURBS representation,

Similarly, from the definition of curvature, we can derive

In the above,

Substituting

For low-speed air drag, we assume that

Our strain-limiting scheme is independent of time integration. Here, we use the velocity Verlet scheme that was first applied to cloth simulation by [

Predict average velocity and candidate configuration at

Compute

The fast projection method begins with the constrained Lagrange problem. For the general coordinates

The Euler-Lagrange equation is:

Thus, we have,

Supposing that we check the constraint condition at the end of a time step,

Now, we use the splitter:

Predict candidate configuration

This sub-step can be replaced by any time-integration scheme.

2. Correct the candidate configuration by

The second term of

Linearizing

Solve

Correct

The iteration exits when the

We note that

Substituting

Substituting

The iterative fast projection process flow is as follows:

Solve

Correct

The iteration exits when the

For the spring-mass method, the length of each spring is a constraint,

For the continuum-based approach, the first task is to select sampling points. We tried to select the Gaussian points as sampling points, but that does not work well. Because the number of cells are close to the number of control points, and if there are three or more constraint conditions on each cell, the model will be locked. Thus,

The constraint condition

The constraint energy term

However, we found a checkerboard pattern in the values of

A piece of cloth in the x-y plane is subject to constraints at two corners and will swing under gravity in the z direction. The cloth is represented by a second-order NURBS patch with 100 control points. The bending parameters are

The simulation results of different Young’s moduli at

Case (a) | Case (b) | Case (c) | |
---|---|---|---|

CPU time (s/frame) | 0.13 | 0.325 | 0.054 |

Maximum |
0.015 | 0.0828 | 2.24 |

Maximum |
0.013 | 0.0512 | 2.20 |

Average |
0.081 | ||

Average |
0.111 |

The draping of a soft armor was simulated in this example. The armor is represented by a second-order NURBS patch with 1,616 control points. The initial configuration is obtained by virtual try-on simulation and is shown in

To more clearly show the results, the simulation results of upper- and lower-body armor at

Case (a) | Case (b) | Case (c) | |
---|---|---|---|

CPU time (s/frame) | 5.69 | 25.25 | 3.47 |

Maximum |
0.097 | 0.41 | |

Maximum |
0.018 | 0.075 | 0.34 |

Average |
0.037 | ||

Average |
0.043 |

This example simulates the draping process of a skirt. The initial configuration of the skirt is obtained by a try-on simulation and is shown in

The garment shape at

Case (a) | Case (b) | Case (c) | |
---|---|---|---|

CPU time (s/frame) | 7.4 | 20.4 | 8.2 |

Maximum |
0.0093 | 0.26 | 0.64 |

Maximum |
0.012 | 0.13 | 0.65 |

Average |
0.073 | ||

Average |
0.10 |

Finally, we conducted a standard patch test to check the stress obtained from the fast projection method. A 1 m-by−1 m square was fixed on the top edge and we applied 1 N/m of uniform force on the bottom edge. As the problem itself is based on dynamic analysis, we modeled the static patch test by applying a large damping factor and waiting until the vibration was damped out.

In this work, a rotational-free Kirchhoff-Love shell-based isogeometric analysis was outlined for cloth simulation. To overcome the numerical burden caused by high in-plane stiffness, a continuum version of the fast projection method was applied. The highlights of this work are the following:

Compared with spring-based models, the constraint directions of strain limiting are independent of grid lines. This implies that the fast projection method can be applied to a grid with arbitrary topology.

Examples show that the present scheme can eliminate unrealistic over-stretching efficiently, while the stress field remains reliable.

The trimmed NURBS patches are used directly for geometry, indicating seamless application of CAD and analysis.

Future work will focus on developments of efficient numerical methods to handle localized features, e.g., wrinkles, to facilitate a “real-time” simulation of cloth.