Computer implemented method,system and computer program product for simulating the behavior of a woven fabric at yarn level

Abstract

Computer implemented method, system and computer program product for simulating the behavior of a woven fabric at yarn level. The method comprises. retrieving the layout of warp yarns (1), weft yarns (2) and yarn crossing nodes (3): describing each yarn crossing node (3) by a 3D position coordinate (x) and two sliding coordinates, warp sliding coordinate (u) and weft sliding coordinate (v) representing the sliding of warp (1) and weft (2) yarns; measuring forces on each yarn crossing node (3) based on a force model, the forces being measured on both the 3D position coordinate (x) and the sliding coordinates (u, v); calculating the movement of each yarn crossing node (3) using equations of motion derived using the Lagrange-Euler equations, and numerically integrated over time, wherein the equations of motion account for the mass density distributed uniformly along yarns, as well as the measured forces and boundary conditions.

Claims

1. A computer implemented method for simulating the behavior of a woven fabric at yarn level, the woven fabricassociated with a computer model comprising a layout of warp yarns, weft yarns and yarn crossing nodes, the method comprising: describing each yarn crossing node of the computer model of the woven fabric by a 3D position coordinate and two sliding coordinates, including a warp sliding coordinate and a weft sliding coordinate respectively representing a sliding of warp and weft yarns of the woven fabric; determining forces on each yarn crossing node based on a force model, the forces being determined on both the 3D position coordinate and the sliding coordinates of yarn crossing nodes; calculating the movement of each yarn crossing node at a plurality of time steps; and rendering by transforming the computer model of the woven fabric into a computer graphic representation of the woven fabric in a plurality of frames to simulate a behavior of the woven fabric.

2. The computerimplemented method of claim 1, wherein the computer model of the woven fabric further includes at least one of: a 2D pattern of the woven fabric, including panels and seam locations; a layout of warp yarns, weft yarns and yarn crossing nodes for each panel; a weave pattern of the woven fabric for each panel; a yarn density and width for each yarn type used in the woven fabric; and a mechanical parameter for each yarn type used in the woven fabric.

3. The computerimplemented method of claim 2, wherein the mechanical parameter includes at least one of an elastic modulus, a bending modulus, a shear contact modulus, a sliding friction coefficient, a damping-to-mass ratio, and a damping-to-elasticity ratio.

4. The computerimplemented method of claim 1, wherein the computer model of the woven fabric includes a slidingfriction coefficient of the yarns, and wherein the force model includes sliding friction forces by using the slidingfriction coefficient and the sliding coordinates.

5. The computerimplemented method of claim 1, wherein the computer model of the woven fabric further comprises a stiffness of the yarns, and wherein the force model includes contact between adjacent parallel yarns by usingthe sliding coordinates, the stiffness of the yarns and an inter-yarn distance being obtained from the layout of the yarns.

6. The computerimplemented method of claim 1, wherein the computer model of the woven fabric further comprises an elastic modulus of the yarns, and wherein the force model includes stretch forces.

7. The computer implemented method of claim 1, wherein the computer model of the woven fabric further comprises a bending modulus of the yarns, and wherein the force model includes bending forces.

8. The computerimplemented method of claim 1, wherein the computer model of the woven fabric further comprises a shear contact modulus of the yarns, and wherein the force model includes shear forces.

9. The computer implemented method of claim 1, wherein the force model uses inter-yarn normal compression at yarn crossings using normal components of stretch and bending forces.

10. A computer-based system for simulatingthe behavior of a woven fabric at yarn level, the woven fabric associated with a computer model comprising a layout of warp yarns, weft yarns and yarn crossing nodes, the system comprising: a non-transitory computer readable media comprising computer program instructions that when executed: (a) determine forces on each yarn crossing node based on a force model, the yarn crossing nodes associated with a 3D position coordinate and two sliding coordinates, including a warp sliding coordinate and a weft sliding coordinate respectively representing a sliding of the warp and the weft yarns of the woven fabric, the forces being determined on both the 3D position coordinate and the sliding coordinates of yarn crossing nodes, and (b) calculate the movement of each yarn crossing node at a plurality of time steps; a plurality of GPUs communicatively coupled to the non-transitory computer readable media, each GPU comprising a plurality of nodes for executingthe instructions substantially in parallel; and a graphics data controller configured to receive output from the GPUs and to render by using the calculated movement of each yarn crossing into a computer graphic representation of the woven fabric in a plurality of frames, each frame corresponding to at least one of the plurality of time steps, to simulate a behaviorof the woven fabric.

11. The computer-based system of claim 10, wherein the computer model of the woven fabric further includes at least one of: a 2D pattern of the woven fabric, including panels and seam locations; a layout of warp yarns, weft yarns and yarn crossing nodes for each panel; a weave pattern of the woven fabric for each panel; yarn densities and widths for all yarn types used in the woven fabric; or mechanical parameters for all yarn types used in the woven fabric.

12. The computer-based system of claim 11, wherein the mechanical parameters include at least one of an elastic modulus, a bending modulus, a shear contact modulus, a sliding friction coefficient, a damping-to-mass ratio, or a damping-to-elasticity ratio.

13. The computer-based system of claim 10, wherein the computer model of the woven fabric includes a sliding friction coefficient of the yarns, and wherein the force model includes sliding friction forces by using the sliding friction coefficient and the sliding coordinates.

14. The computer-based system of claim 10, wherein the computer model of the woven fabric further comprises a stiffness of the yarns, and wherein the force model includes contact between adjacent parallel yarns by usingthe sliding coordinates, the stiffness of the yarns and an inter-yarn distance obtained from the layout of the yarns.

15. The computer-based system of claim 10, wherein the computer model of the woven fabric further comprises an elastic modulus of the yarns, and wherein the force model includes stretch forces.

16. The computer-based system of claim 10, wherein the computer model of the woven fabric further comprises a bending modulus of the yarns, and wherein the force model includes bending forces.

17. The computer-based system of claim 10, wherein the computer model of the woven fabric further comprises a shear contact modulus of the yarns, and wherein the force model includes shear forces.

18. The computer-based system of claim 10, wherein the force model uses inter-yarn normal compression at yarn crossings using normal components of stretch and bending forces.

19. A computer implemented method for displaying a simulation of a behavior of a woven fabric at yarn level, the method comprising: receiving a plurality of frames comprising a simulation of a behavior of a woven fabric represented with a computer model comprising a layout of warp yarns, weft yarns and yarn crossing nodes, each frame of the plurality of frames rendering a yarn geometry at a time step, the yarn geometry based on a description of each yarn crossing node of the computer model of the woven fabric by a 3D position coordinate and two sliding coordinates, including a warp sliding coordinate and a weft sliding coordinate respectively representing a sliding of warp and weft yarns of the woven fabric, the yarn geometry further based on determining forces on each yarn crossing node based on a force model, the forces determined on both the 3D position coordinate and the sliding coordinates of yarn crossing nodes, and further based on a calculated movement of each yarn crossing node at the time step; and displayingthe plurality of frames.

20. The computer implemented method of claim 19, wherein the computer model of the woven fabric further includes at least one of: a 2D pattern of the woven fabric, including panels and seam locations; a layout of warp yarns, weft yarns and yarn crossing nodes for each panel; a weave pattern of the woven fabric for each panel; yarn densities and widths for all yarn types used in the woven fabric; or mechanical parameters for all yarn types used in the woven fabric.

21. The computer implemented method of claim 20, wherein the mechanical parameters include at least one of an elastic modulus, a bending modulus, a shear contact modulus, a sliding friction coefficient, a damping-to-mass ratio, or a damping-to-elasticity ratio.

22. The computer implemented method of claim 19, wherein the computer model of the woven fabric includes a sliding friction coefficient of the yarns, and wherein the force model includes sliding friction forces by using the sliding friction coefficient and the sliding coordinates.

23. The computer implemented method of claim 19, wherein the computer model of the woven fabric further comprises a stiffness of the yarns, and wherein the force model includes contact between adjacent parallel yarns by usingthe sliding coordinates, the stiffness of the yarns and an inter-yarn distance obtained from the layout of the yarns.

24. The computerimplemented method of claim 19, wherein the computer model of the woven fabric further comprises an elastic modulus of the yarns, and wherein the force model includes stretch forces.

25. The computerimplemented method of claim 19, wherein the computer model of the woven fabric further comprises a bending modulus of the yarns, and wherein the force model includes bending forces.

26. The computer implemented method of claim 19, wherein the computer model of the woven fabric further comprises a shear contact modulus of the yarns, and wherein the force model includes shear forces.

27. The computer implemented method of claim 19, wherein the force model uses inter-yarn normal compression at yarn crossings using normal components of stretch and bending forces.

28. The computer implemented method of claim 19, wherein the displaying the plurality of frames comprises displaying a human body shape wearing the woven fabric.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) A series of drawings which aid in better understanding the invention and which are expressly related with an embodiment of said invention, presented as a non-limiting example thereof, are very briefly described below;

(2) FIGS. 1A, 1B and 1C show different weave patterns with increasing floats: plain, twill, and satin.

(3) FIG. 2 represents a graph relating the force with respect to shear angle for the three hanging sheet examples of FIG. 1.

(4) FIGS. 3A-3C shows the models of woven yarns used in the present invention: 3D volumetric yarns (FIG. 3A), interlaced rod segments with crimp (FIG. 3B), rod segments crossing at 5-DoF crossing nodes (FIG. 3C).

(5) FIG. 4A shows warp and weft yarns crossing at node q0, and the four adjacent yarn crossings.

(6) FIG. 4B shows the bending angle θ between two adjacent warp segments.

(7) FIG. 4C represents the forces producing normal compression at a crossing node.

(8) FIG. 4D represents the shear angle and shear jamming angle, between two adjacent warp and weft yarns.

(9) FIGS. 5A and 5B show shear friction effects, wherein the sample is stretched (FIG. 5A) and then relaxed (FIG. 5B), leaving a persistent wrinkle.

(10) FIG. 6 shows a table with parameter values used in the examples.

(11) FIG. 7 shows a table with average cost per time step (in milliseconds) for the examples, broken down by step.

(12) FIG. 8 shows an example of loose tank top with 2023 yarns and 350530 crossing nodes, showing large motions as well as small scale folds and wrinkles.

(13) FIG. 9 shows an example of long shirt with 3198 yarns and 559241 crossing nodes.

(14) FIG. 10 shows an example of a yarn-level simulation of a snag produced on a shirt with 2023 yarns and 350530 crossing nodes, by pulling on a seam node.

(15) FIGS. 11A-11C shows an example of a snag produced in the belly area of the loose tank top of FIG. 8. FIG. 11A shows a snag formed by pulling two yarns while blocking the outward motion of the cloth. FIG. 11B represents real snagging under similar conditions. FIG. 11C shows a close-up of the snag of FIG. 11A.

(16) FIG. 12 shows the loose tank top of FIG. 8 torn open by grabbing some nodes and pulling them apart. Yarns are detached and edges are frayed in the process.

(17) FIGS. 13A and 13B show a 100-yarn-per-inch plain-weave sheet (1 million crossing nodes). Small wrinkles appear during motion (FIG. 13A) until the sheet comes to rest exhibiting large draping wrinkles (FIG. 13B).

DESCRIPTION OF A PREFERRED EMBODIMENT OF THE INVENTION

(18) First, it will be described how to construct cloth models based on yarns. Next, it will be presented the key feature of our yarn-based cloth model: the discretization of yarn kinematics based on the positions of yarn crossing points and yarn sliding. Finally, the formulation of the equations of motion based on this discretization will be outlined.

(19) To construct garments at the yarn level, a tailoring approach is followed. The 2D pattern that forms a garment is taken as input, laying warp and weft yarns as orthogonal straight lines on each 2D panel independently, at an inter-yarn distance L. At each seam an additional yarn is placed, and weft and warp yarns are connected to seam yarns by sharing nodes. At cloth boundaries it may be chosen between adding seam yarns or letting yarn endings hang freely. 3D cloth models can be obtained automatically from commercial patterns [Berthouzoz et al. 2013], hence the present modeling process can also be easily automated.

(20) A float constitutes a gap between two yarns of the same type where the other yarn is not interlaced. Different weave patterns, such as plain weave (with no floats), twill, satin, etc. are obtained by varying the distribution of floats, thereby affecting the mechanics of the resulting fabric. FIGS. 1A, 1B and 1C show, respectively, different weave patterns with increasing floats; plain weave pattern (FIG. 1A), twill pattern (FIG. 1B) and satin pattern (FIG. 1C). More floats lead to lower shear resistance, hence cloth falls lower when pinned from two corners.

(21) To model the weave pattern, an arbitrary orientation for each panel is chosen, storing at each yarn crossing a flag indicating which yarn, warp or weft, is on top. This simple strategy allows modelling plain weave, twill, satin, and all other common weaves. FIG. 2 represents a graph relating the force (in N) with respect to shear angle (in radians) for the three hanging sheet examples of FIGS. 1A, 1B and 1C. Shear force, friction and jamming are clearly visible, as well as the different behaviors according to the weave pattern.

(22) In the kinematic representation the volume of yarns is ignored and all yarns are initialized flat on the same plane. However, for the purpose of force computation and rendering the volume of yarns is accounted for. FIGS. 3A, 3B and 3C shows the models of woven yarns (warp yarns 1 and weft yarns 2) used in the present invention. Crimp is the bending introduced in warp 1 and/or weft 2 yarns to allow interlacing, as shown in FIG. 3A. Crimp also produces compressive forces between interlaced yarns, and this compression allows the existence of friction forces that hold the fabric together. In the present implementation, crimp is applied to both weft 2 and warp 1 yarns, offsetting them by the yarn radius R in opposite directions, as shown in FIG. 3B. The present implementation could be extended to allow for anisotropic crimp. The 3D volumetric yarns shown in FIG. 3A of a piece of fabric are replaced by interlaced rod segments with crimp (FIG. 3B) for normal force computation, and by rod segments crossing at 5 degrees of freedom (5-DoF) crossing nodes 3 (FIG. 3C) for everything else. The volumetric appearance is restored at rendering.

(23) For ease of presentation, it is assumed that inter-yarn distance L and yarn radius R are the same for warp 1 and weft 2, but it is trivial to relax this assumption, and indeed the present implementation supports anisotropic cloth.

(24) In woven cloth, the vast majority of yarns are in contact at yarn crossings, so that it can generally assumed that such contacts are maintained throughout the simulation. The motion of cloth could be described as a constrained dynamics problem, with a node-based discretization of yarns, plus a large number of contact constraints that maintain a zero-distance between yarns at yarn crossings. But it was observed that, instead of detecting and resolving such contacts, it is utterly more efficient to choose a convenient discretization based on yarn crossings.

(25) FIG. 4A shows warp 1 and weft 2 yarns crossing at node q.sub.0, and the four adjacent yarn crossings (q.sub.1, q.sub.2, q.sub.3, q.sub.4). Warp 1 and weft 2 yarns are parameterized based on their undeformed arc length, u and v respectively. Hence, u is the undeformed length of the warp yarn (1) between the crossing point (3) and one end point of the yarn; and v is the undeformed length of the weft yarn (2) between the crossing point (3) and one end point of the yarn. Then, a yarn crossing is described by its 3D position, x, and the parametric coordinates of the warp 1 and weft 2 material points at the yarn crossing. The variation of u and v coordinates models, respectively, the siding of warp and weft yarns. A yarn crossing is considered as a 5-DoF node with 3 Lagrangian DoFs and 2 Eulerian DoFs. We denote the the coordinates of the i.sup.th yarn crossing node as q.sub.i=(x.sub.i,u.sub.i,v.sub.i).

(26) It is herewith proposed to discretize woven cloth with a combination of 5-DoF yarn-crossing nodes and regular 3-DoE nodes. A 5-DoF node is set at each yarn crossing, and regular 3-DoF nodes at the end-points of yarns. FIG. 4A shows a regular set-up with a yarn crossing node 3 and its four adjacent nodes.

(27) Given a warp segment [q.sub.0, q.sub.1] (and similarly for weft segments), positions are linearly interpolated according to the arc length u. Then, the 3D position of a point inside the segment is given by:

(28) $\begin{array}{cc}x\left(u\right)=\frac{u_1-u}{\Delta u}{x}_0+\frac{u-u_0}{\Delta u}{x}_1,& \left(1\right)\end{array}$
where Δu=u.sub.1−u.sub.0 is the rest length of the segment.

(29) The velocity of a point inside the segment depends on the velocities of yarn crossing points, but also on yarn sliding, and it follows by differentiating equation (1):

(30) $\begin{array}{cc}\stackrel{.}{x}\left(u\right)=\frac{u_1-u}{\Delta u}\left({\stackrel{.}{x}}_0-{\stackrel{.}{u}}_{0}w\right)+\frac{u-u_0}{\Delta u}\left({\stackrel{.}{x}}_1-{\stackrel{.}{u}}_{1}w\right),\mathrm{where}w=\frac{x_1-x_0}{\Delta u}.& \left(2\right)\end{array}$

(31) By concatenating the coordinates of all yarn crossing, a vector of generalized coordinates q is defined. The equations of motion can then be derived using the Lagrange-Euler equations [Goldstein et al. 2001] The kinetic energy is T=½{dot over (q)}.sup.T M{dot over (q)}, with a generalized mass matrix M, V denotes the potential energy, and ∇ denotes the generalized gradient. Then, the Euler-Lagrange equations can be written as
M{umlaut over (q)}=∇T−∇V−{dot over (M)}{dot over (q)}.   (3)

(32) It is assumed that mass is distributed uniformly along yarns, with density ρ. Then, following the equation (2) of velocity for an arbitrary point in a warp segment, the kinetic energy of the segment [q.sub.0, q.sub.1] (and similarly for a weft segment) is:

(33) $\begin{array}{cc}{T}_{0,1}=\frac{1}{2}\left({\stackrel{.}{x}}_0^T{\stackrel{.}{u}}_0{\stackrel{.}{x}}_1^T{\stackrel{.}{u}}_1\right)M_{0,1}\left(\begin{array}{c}{\stackrel{.}{x}}_0\\ {\stackrel{.}{u}}_0\\ {\stackrel{.}{x}}_1\\ {\stackrel{.}{u}}_1\end{array}\right),\mathrm{with}& \left(4\right)\\ M_{0,1}=\frac{1}{6}\mathrm{\rho \Delta }u\left(\begin{array}{cccc}2I_3& -2w& I_3& -w\\ -2w^T& 2w^{T}w& -w^T& w^{T}w\\ I_3& -w& 2I_3& -2w\\ -w^T& w^{T}w& -2w^T& 2w^{T}w\end{array}\right)& \left(5\right)\end{array}$

(34) The potential energy V includes multiple terms, such as gravity and conservative internal forces. Gravity is defined, e.g., for the warp segment [q.sub.0, q.sub.1] as

(35) $\begin{array}{cc}{V}_{0,1}=\mathrm{\rho \Delta }{\mathrm{ug}}^T\frac{x_0+x_1}{2}& \left(6\right)\end{array}$

(36) The formulation of internal forces is discussed now in detail. In addition to the conservative forces derived from energy potentials, it is also incorporated other force terms directly to the right-hand side of the Euler-Lagrange equations (3), such as friction and contact forces. It is also incorporated damping through the Rayleigh damping model which uses a damping-to-mass ratio and a damping-to-elasticity ratio as parameters, with mass and stiffness-proportional terms controlled by parameters αand β respectively.

(37) For the force model it is considered two types of internal forces in woven cloth. The forces due to the deformation of individual yarns include stretch and bending forces. Yarn torsion is not considered, as its effect is minimal on cloth. Next, it will be described internal forces due to contact between interlaced yarns, which include normal compression sliding friction, shear, and contact between parallel yarns.

(38) Conservative forces are described in a concise manner using energy potentials. In the general case, these potentials will produce forces on both the yarn crossing points and the sliding coordinates. In addition, the application of numerical integration requires the computation of force Jacobians, including mixed-terms relating crossing points and sliding coordinates.

(39) To model stretch, the approach by Spillmann et al. [2007] is followed, defining a stretch energy that is quadratic in the strain along the yarn's centerline. With the present discretization, stretch strain is constant on each yarn segment. For the warp segment [q.sub.0, q.sub.1], it is simply ε=∥w|−1. Then, the stretch energy of the segment for a stiffness k.sub.s can be computed as:
V.sub.0,1=½k.sub.sΔu(∥w∥−1).sup.2   (7)
where k.sub.s=YπR.sup.2, and Y is the elastic modulus. Yarns of woven cloth are often close to inextensible, which requires the use of a high elastic modulus. An alternative would be to enforce inextensibility through constraints and Lagrange multipliers. However, a solver for implicit integration has been designed for the present implementation, which will be later described in detail and which allows the efficient simulation of stiff yarns.

(40) For model bending, a discrete differential geometry approach is taken, defining bending energies based on discrete curvatures at yarn crossings, separately for warp and weft yarns. There are several possible definitions of discrete curvature at yarn crossings [Sullivan 2008], but it is berewith defined it simply as the angle between yarn segments. This curvature is transformed into a curvature density dividing it by the arc length between segment centers. For the warp yarn 1 in FIG. 4B, given an angle θ between segments [q.sub.2, q.sub.0] and [q.sub.0, q.sub.1], curvature density at node q.sub.0 is defined as

(41) $k=\frac{2\theta }{u_1-u_2}.$
It is defined bending energy density with stiffness k.sub.b that is quadratic in curvature. Integrating it over the half-segments adjacent to q.sub.0 results in a discrete bending energy

(42) $\begin{array}{cc}{V}_{2,0,1}=k_b\frac{{\theta }^2}{u_1-u_2},& \left(8\right)\end{array}$
where k.sub.b=BπR.sup.2, and B is the bending modulus. The expression could turn numerically unstable if yarn crossings became arbitrary close. However, this does not happen in practice due to the contact model between parallel yarns later described. Bergou et al. [2008]) choose a different discrete curvature metric, based on the tangent of the angle between segments. The resulting energy grows to infinity if a yarn bends completely, and this also creating excessive resistance to bending in practice. Yet another option is to use a discrete curvature metric based on the sine of the half-angle between segments, but this metric produces a non-convex bending energy.

(43) Woven cloth is held together by inter-yarn friction, and admissible friction forces are a function of inter-yarn normal compression at yarn crossings. The present yarn discretization ignores the relative motion between warp 1 and weft 2 yarns along their normal direction, hence normal compression cannot be modeled as an elastic potential. Instead, it is berewith proposed a quasi-static approximation that captures the desired friction effects. In essence, the compression force is estimated by averaging the normal components of warp and weft forces, depicted in FIG. 4C, and then this compression can be used to model friction and shear forces [Page and Wang 2000].

(44) The detailed computation is as follows. At each yarn crossing 3, a best-fitting plane is computed using the positions of the node and its four adjacent nodes. As normal direction n is chosen the normal of the plane pointing from the warp yarn 1 toward the weft yarn 2. Crimp is applied by offsetting the positions of warp and waft points in the normal direction by the yarn radii (FIG. 3B), and bending forces are recomputed. At each yarn crossing, the compression force is estimated by summing the normal components of stretch F.sub.s and bending F.sub.b forces (in FIG. 4C superscripts + and − denote positive arid negative yarn directions), and averaging the resulting forces for warp and weft directions, i.e.,
F.sub.n=½n.sup.T(F.sub.s(u)+F.sub.b(U)−F.sub.s(v)−F.sub.b(v)0   (9)

(45) If the compression force is negative, the yarns are considered to be separating, and the force is clamped to zero. It would be possible to extend this model to handle adhesion.

(46) Note that we account only for stretch and bending forces. If the fabric is stretched, then compression is dominated by stretch. However, when it is not stretched, then it is dominated by bending. For a flat cloth, it is crucial to account for the misalignments produced by crimp, otherwise friction forces cannot hold the yarns in place, and this is why bending forces are recomputed after offsetting warp and weft points.

(47) At each yarn crossing, friction forces that try to prevent sliding between warp 1 and weft 2 yarns are also computed. Inter-yarn friction is modeled using a penalty-based approximation of the Coulomb model, similar to the one of Vamane and Nakamura [2006]. The present discretization based on yarn crossings simplifies greatly the formulation of friction, and a simple spring on each sliding coordinate produces effective results.

(48) Given the yarn crossing q.sub.0, we set an anchor position u.sub.0 on the warp yarn 1, and similarly for the weft yarn 2. The anchor position is initialized as the warp sliding u.sub.0 at the crossing. Friction is modeled as a zero-rest-length viscoelastic spring between the anchor position and the actual warp coordinate.

(49) The Coulomb model sets a limit μF.sub.n on the elastic component of the friction force, where μ is the sliding friction coefficient and F.sub.n is the inter-yarn compression as computed in equation (9) above. If the limit is not reached, the contact is in stick mode, and the force is defined by the spring. If the limit is exceeded, the contact is in slip mode, and the force is given by the Coulomb limit. In summary, the warp friction force is computed as:

(50) $\begin{array}{cc}{F}_{u_0}=\{\begin{array}{cc}-k_f\left(u_0-{\stackrel{\_}{u}}_0\right)-d_f{\stackrel{.}{u}}_0,& \mathrm{if}\mathrm{stick}\\ -\mathrm{sign}\left(u_0-{\stackrel{\_}{u}}_0\right)\mu F_n-d_f{\stackrel{.}{u}}_0,& \mathrm{if}\mathrm{slip}.\end{array}& \left(11\right)\end{array}$

(51) In addition, in slip mode the anchor position is maintained at a constant distance from the warp coordinate, such that the resulting spring force equals the Coulomb limit.

(52) At yarn crossings 3, adjacent warp 1 and weft 2 yarns rotate on top of each other as a function of the shear angle, as shown in FIG. 4D. This rotation produces two effects; yarn compression and contact friction. In addition, at interlaced crossings yarns suffer jamming as they collide.

(53) To capture these effects, for every pair of warp and weft segments at a yarn crossing, an angular friction force and an elastic potential that depend on the shear angle are modeled. Let us consider, for example, the warp segment [q.sub.0, q.sub.1] and the weft segment [q.sub.0, q.sub.3] in FIG. 4D. A shear energy density given by the shear rotation

(54) $\varphi -\frac{\pi }{2}$
is defined, integrating it over the two half-segments incident in q.sub.0. In this integration, it was found that it is sufficient to use the default inter-yarn distance L. This approximation has little effect in practice and it eliminates the need to compute shear forces and their Jacobians for sliding coordinates. The resulting shear energy with stiffness k.sub.x is

(55) $\begin{array}{cc}{V}_{0,1,3}=\frac{1}{2}{k}_x{L\left(\varphi -\frac{\pi }{2}\right)}^2& \left(11\right)\end{array}$
where k.sub.xSR.sup.2, and S is the contact shear modulus.

(56) Normal compression increases the resistance to shear, and we model this effect by making the shear stiffness a function of the compression force, i.e., k.sub.x(F.sub.n). Moreover, if either the warp or weft yarn segment is interlaced, shear jamming is also considered, modeled according to the following beuristics. We define the shear jamming angle .sub.j as the angle at which the end-points of the warp and weft segments with radius R touch each other, i.e.,

(57) 0${\varphi }_j=2\arcsin \left(\frac{R}{L}\right),$
We model jamming as a strong nonlinearity in the shear stiffness, leaving it as a constant for shear angles above the jamming angle, and making it grow cubically for smaller angles.

(58) Shear friction can be modeled using an angular spring between the current shear angle and an anchor angle ϕ, following the same approach as for sliding friction previously described. We apply shear friction force only to the position of yarn crossing nodes, and it can be computed for each of the three nodes q.sub.1 in the example in FIG. 4D as

(59) $\begin{array}{cc}{F}_{x_i}=\{\begin{array}{cc}{-k_{f\varphi }\left(\varphi -\stackrel{\_}{\varphi }\right)\hspace{1em}\hspace{1em}\frac{\partial \varphi }{\hspace{1em}\partial x_i}\hspace{1em}}^T,& \mathrm{if}\mathrm{stick}\\ {-\mathrm{sign}\left(\varphi -\stackrel{\_}{\varphi }\right){\mu }_{\varphi }{F}_n\frac{\partial \varphi }{\partial x_i}\hspace{1em}}^T,& \mathrm{if}\mathrm{slip}\end{array}.& \left(12\right)\end{array}$

(60) One of the visual effects of internal friction is the creation of persistent wrinkles, as demonstrated by Miguel and collaborators [2013]. FIG. 5A shows a small fabric sample that is first stretched and then relaxed (FIG. 5B), leaving a persistent wrinkle along the stretch direction due to shear friction.

(61) Contact between adjacent parallel yarns can be easily modeled by adding a penalty energy if two yarn crossings get too close. We define the distance threshold d as four times the yarn radius if there is an interlaced yarn between the two crossings, and as twice the radius if the two yarns form a float. Given, for example, the weft yarns passing through q.sub.0 and q.sub.1 in FIG. 4A, we define an energy density based on the distance between crossing points, and integrate this density over the weft half-segments incident on both nodes. Assuming that yarns are practically inextensibie, the distance between crossing points can be approximated as the difference between warp sliding coordinates. And same as for shear, it was found that it was sufficient to integrate the energy density using the default inter-yarn distance L, thus eliminating the need to compute complex coupling forces with weft sliding coordinates. The resulting penalty energy with stiffness k.sub.c is:

(62) $V_{0,1}=\{\begin{array}{cc}\frac{1}{2}{k}_c{L\left(u_1-u_0<d\right)}^2,& \mathrm{if}{u}_1-u_0<d\\ 0,& \mathrm{if}{u}_1-u_0\ge d\end{array}.$

(63) Contact between interlaced yarns is handled implicitly by the present discretization, and contact between adjacent parallel yarns is easily handled as described above. On the other hand, contact with other objects as well as long-range self-collisions require explicit collision processing. Existing methods to detect and resolve collisions can be used. As a summary, a thin volume around the cloth is defined, which allows to compute penetration depth and implement collision response through penalty energies.

(64) To detect contact with volumetric objects, distance fields are used. In the examples later commented, only rigid or articulated objects have been used, hence it was sufficient to compute the distance field once as preprocessing. Given an object O, at every time step we query every yarn node x against the distance field of O, and define a collision if the distance to O is smaller than γ (in the examples, γ is 4 times the yarn radius R). The collision information is formed by the crossing point x, the closest point p on the surface of O, and a contact normal n. The normal at p has been used as contact normal, although other options are possible.

(65) To detect self-collisions, small volumetric elements on the surface of the cloth are defined, and the yarn nodes are queried against these volumetric elements following the approach of Teschner et al. [2003]. Two triangles are formed with the 4 nodes defined by every two pairs of adjacent warp and weft yarns, protruding the triangles by a distance γ in the directions of and opposite the normal at each crossing point to form each volumetric element (the estimation of normal was previously discussed). All nodes are queried against the protruded triangles, hashing AABBs of the protruded triangles on a regular grid for culling [Teschner et al. 2003]. If a point x is inside a protruded triangle, a collision is defined, finding the projected point p on the surface, and computing a contact normal n by interpolating the normals of the triangle's nodes.

(66) For collision response, both with external objects or in self-collisions, a penalty force on the colliding point x is defined, with penalty distance n.sup.T(p−x)+γ, and direction n. In the self-collision case, we also distribute the opposite force to the nodes that define the triangle, by using as weights the barycentric coordinates of p in the triangle. In addition to penalty response, Coulomb friction approximated through clamped springs is applied [Yamane and Nakamura 2006].

(67) The obvious limitation of penalty-based response on thin objects is the chance of suffering pop-through problems. In the examples given below, pop-through was prevented by adding a damping term to the penalty response. A more robust solution would be to use a barrier method [Harmon et al. 2009].

(68) The equations of motion (3) are integrated using backward Euler implicit integration [Baraff and Witkin 1998] with Newton's method and adaptive time stepping to ensure convergence. In practice, most solve steps require only one Newton iteration, and we allow up to five before having the time step. The simulation cost is dominated by the solution to linear systems of the type Av=b, where v={dot over (q)} is the vector of generalized velocities, and the system matrix is defined as

(69) $A=M-\Delta t\frac{\partial F}{\partial v}-\Delta t^2\frac{\partial F}{\partial q}.$
With the present discretization based on yarn crossings. A is formed by blocks of size 5×5.

(70) The linear system is solved using the conjugate residual (CR) method. It is more robust than the conjugate gradient (CG) method for matrices that are close to semi-definite, as it computes the least-squares solution to the linear problem, at the expense of slightly higher cost and memory consumption.

(71) An advantage of the present yarn-level cloth model is the strong regularity of the system matrix A, which allows a highly efficient implementation of numerical integration on GPUs, similar to GPU cloth solvers for regular triangle meshes [Tang et al. 2013]. The computation of internal forces, their Jacobians, and the solution to the linear system are parallelized on the GPU, but executing collision detection on the CPU. All in all, the bottleneck of the solver is the sparse matrix-vector multiplication needed on each iteration of PCR.

(72) This product has therefore been optimized in multiple ways, exploiting the regularity of yarn-level cloth and our yarn-crossing discretization. Due to regularity of the weave pattern, the internal force of a node yields non-zero Jacobians w.r.t. 13 nodes (excluding collisions, each node interacts with 12 neighbors plus itself). Then, the system matrix A is split as the sum of a regular matrix A, and the remaining tail matrix A.sub.t, where A.sub.r contains the 13 regular 5×5 blocks per node, and A.sub.t contains other blocks resulting from collisions. At amounts to only 14% of the product cost even with simple COO storage. The coefficients of A.sub.r are stored in a dense matrix in column-major order, with one row and 325=13×25 columns per node (2:53 KB per node in double precision). The indices of A.sub.r require a much smaller matrix, with one row and only 13 columns per node. Products involving A.sub.r are parallelized on a per-node basis, and column-major storage of the coefficients provides extremely efficient coalescent access to the data. The parallelization strategy mimics the one of the ELL and HYB matrix formats in the Cusp CUDA library [Bell and Garland 2012], but obtaining more than a 40% speed-up over Cusp in sparse matrix-vactor products thanks to node-level parallelization.

(73) Overall, it is achieved a 16× to 24× speedup over a multithreaded CPU implementation, depending mainly on the number of nodes (higher speedup for a higher number of nodes) and the number of collisions (lower speedup for a higher number of collisions, since collisions are treated on the CPU).

(74) For off-line rendering of the results, the yarn geometry is transformed into a volumetric representation at the microfiber level, and use the volumetric path tracer Mitsuba [Jakob 2010]. It accounts for the anisotropic scattering of microfibers using a microflakes model. For each yarn, a polyline with offsetted node positions is generated to account for crimp. The polyline yarns are smoothed using Catmull-Rom splines, and then a modified version of the Lumislice method is used [Chen et al. 2003; Lopez-Moreno et al. 2014] to define be volumetric representation of the yarn geometry to be passed to the Mitsube renderer. Each smoothed yarn is composed of thousands of twisted microfibers, and slices representing the microfiber density distribution are set perpendicularly along the thread segment at regular steps and incremental rotations. The density is computed by intersecting the absolute position of each texel at each site with a 3D texture volume. This is done in a fragment shader with asynchronous 3D texel store calls. The present implementation is based on OpenGL shaders and instanced geometry, and up to 8M slices are processed in less than 100 ms on a standard desktop machine. The tangents of microfibers are also stored in a 3D texture, computing differentials of texel positions at the current and previous slices, which differ mainly by the rotation along the yarn axis. Due to the asynchronous nature of texel calls in the shader, the previous slice is not accessible, hence the local differentials are precomputed and passed to each slice as a texture. All in all, the density and orientation of yarns at the microfiber level are stored in volumetric textures (3 GB per frame in the examples).

(75) Visual and numerical large-scale examples for several woven cloth simulation scenarios are now described. All the examples were executed on a 3.4 GHz Quad-core Intel Core i7-3770 CPU with 32 GB of memory, with an NVIDIA Titan Black graphics card with 6 GB of memory. Collision detection is parallelized on the CPU, while the solution to dynamics is parallelized on the GPU as previously discussed. All the simulations were executed with a time step of 1 ms. The parameter values used in these examples are listed in FIG. 6. Representative timings are summarized in FIG. 7.

(76) The examples are the following: Loose tank top: A male mannequin was dressed with a loose tank top made of 2023 yarns and 350530 crossing nodes, one seam on each side and one seam on each shoulder (see FIG. 8). Yarn density is one yarn per millimeter (25 yarns per inch). The mannequin performs highly dynamic karate motions. The simulation shows large-scale motion and folds resolved at the yarn level, combined with fine-scale effects. Cloth dynamics and contact resolution are robust even under such challenging motions. Despite the highly dynamic motion of the mannequin, the simulation is robust. Long shirt: A shirt with sleeves was designed to dress a dancing female mannequin (see FIG. 9). The shirt is made of 3199 yarns and 559241 crossing nodes, with seams on the sides of the body, the shoulders, the sleeve-body junctions and along the sleeves. Yarn density is one yarn per millimeter (25 yarns per inch). Compared to the loose tank top, this simulation shows a higher complexity due to a higher crossing node count and the additional dynamics and contact mechanics of the sleeves. Using a yarn-level model inherently produces high resolution cloth dynamics, as shown by the small-scale wrinkles throughout the sleeves. Snags: These examples show how extreme deformations of cloth produce highly complex plastic deformations at the yarn level, as well as the influence of local yarn dynamics on the global shape of the fabric. A snag in the loose tank top was produced, by pinching a node on the side seam and pulling it outwards very fast (see FIG. 10). The deformation due to pulling generates a small hole: the warp yarn being pulled pushes the weft yarns away, in a clear example of yarn sliding and yarn contact dynamics. In addition, the snag is transferred all across the shirt, showing fine wrinkles as the complex effect of yarn sliding and friction. Such a plastic effect can only be achieved by simulating fabric at the yarn level with yarn-yarn interaction. Fine-scale deformations showing yarn sliding and thin wrinkles are combined with large-scale motion of the shirt.

(77) A second snag is produced in the belly area of the loose tank top by pulling a crossing node and fixing the four neighboring nodes that are not on the yarns being pulled (see FIGS. 11A, 11B and 11C). This setup tries to mimic the pulling of a yarn while locally blocking the outward motion of the fabric with the hand. The cloth wrinkles forming a cross shape, showing another familiar snagging pattern. Tearing: Simulating the tearing of cloth using the yarn-level model is straightforward, as the complex and visually rich behavior of frayed edges and loose yarns come naturally with yarn-level dynamics. Fracture was implemented simply by splitting yarns when a threshold of stretch stress is exceeded, followed by a relaxation step to allow a correct stress relief and avoid erratic crack propagation. More sophisticated approaches could be used, such as separation tensors [O'Brien and Hodgins 1999] and local relaxation substeps [Pfaff et al. 2014]. Node resampling is frequently triggered during fracture and highly plastic behaviors due to yarn pullouts and sliding past the end of a yarn.

(78) The loose tank top was torn by pinching two sets of crossing notes in the torso area and pulling them apart in opposite directions, creating a vertical fracture path and a diamond-shaped opening as shown in FIG. 12. Individual yarns detach from the edges of the crack, and either hang or stretch the edges across the opening. These loose yarns and the resulting frayed edges are commonly seen in the tearing of many types of fabric. More subtle plastic deformations can be observed around the crack, mainly due to yarn sliding. Weaving patterns: The yarn-level model allows for easy configuration and simulation of different weave patterns. As previously mentioned, configuring the fabric for a specific weave pattern is just a matter of setting a flag for each node that specifies which yarn is on top. Weave patterns directly affect the global and local behavior of cloth, mainly due to the different number of floats. Shear, for instance, is greatly influenced by the number of crossings and floats in the fabric. The visual aspect of the cloth also changes according to the pattern.

(79) Three 25×25 cm cloth sheets (see FIGS. 1A, 1B and 1C) were simulated by hanging them from two corners. Yarn density is one yarn per millimeter (25 yarns per inch). The three sheets are exactly the same except for the weaving pattern, where the first one is plain weave (FIG. 1A), the second one is twill (FIG. 1B) and the third one is satin (FIG. 1G). FIGS. 1A, 1B and 1C shows a still of each sheet after two seconds of simulation. The sheets exhibit clearly distinctive behaviors, from FIG. 1A to FIG. 1C the wrinkles move to the bottom, the bottom edge of the fabric falls lower, and the top edge shows higher curvature. These effects are due to lower shear stiffness for weaves with more floats, with is the expected result reality. Lower shear stiffness results in better drape quality. The visual appearance is also different between the three stills. In the top part of each sheet, the “see-through” effect due to stretch reveals the different weaving structures of the cloth. It can also be observed how the twill woven sample exhibits its characteristic diagonal pattern.

(80) The three sheets are put through a shear frame test and measured the overall shear through time. Results are plotted in FIG. 2., showing force-angle plots for each weave pattern. The plots exhibit hysteresis due to friction and nonlinearity due to jamming as observed on real fabrics [Miguel et al. 2012], as well as the influence of the weave pattern. Again, weave patterns with more floats are less resistant to shear, as expected.

(81) A fourth cloth sheet using plain weave was simulated, but this time with 4 yarns per millimeter (100 yarns per inch). Given the sheet's size, this yarn density translates into 1 million crossing nodes. This example, shown FIGS. 13A and 13B, shows how the model can handle very high yarn densities found in common woven fabrics such as bed linen. Small wrinkles appear during motion (FIG. 13A) until the sheet comes to rest exhibiting large draping wrinkles (FIG. 13B). As per textile nomenclature, 100 yarns per inch is equivalent to a thread count of 200 .

(82) Hence, the present invention is an efficient method to simulate woven cloth at the yarn level. The key novelty is a discretization of yarn crossings that resolves yarn-yarn contact implicitly and represents inter-yarn sliding efficiently. Effects such as inter-yarn friction, shear, and contact are also captured with simple force models. This yarn level model enables the simulation of effects such as tearing with frayed edges, plasticity due to snags, or nonlinear behavior due to fine-scale friction.

(83) One of the advantages of yarn-level models is the possibility to replicate with high fidelity the nonlinear mechanics of real cloth. This requires estimating the parameters of the model from force-deformation measurements of real cloth. The fitting results could be compared to those of nonlinear cloth models.

(84) The model approximates the compression between crossing yarns as a function of stretch and bending forces. Another possibility would be to incorporate compression as an extra degree of freedom, and add a compression potential to the system energy. Stretch forces are currently modeled using a stretch potential, but another possibility would be to consider yarns to be inextensible, and account for the compression produced during stretch due to crimping.

(85) Even though the examples are limited to orthogonal weave patterns, the discretization is general and could be applied to arbitrary settings with interlaced yarns. One simple extension would be to handle triaxial weaving.

(86) The implementation makes use of penetration-depth queries and penalty-based collision response. To ensure robustness of contact handling, stiff penalty energies must be used and the amount of motion per time step must be limited. Robustness could be improved using continuous collision detection and constraint-based response, although contact handling might then become the bottleneck.