To demonstrate how the spring mass system reacts to the stretching and wrinkling when a fabric surface is driven by an external force, we presented several simulation examples.
The first example is a square fabric surface driven by a hyperbolic velocity field. Fig-ure 7 is the top view of the wrinkling of the edges when the velocity at the center is larger. The spring model does not conserve the total area exactly, but the restoring force preserves the total area approximately. The second example is the reaction of a round cloth to an almost point velocity, or a hanging force. Even though only a few points are driven by the external velocity field, the entire fabric surface feels the dragging force by the moving points through the spring system. Collision is handled by functions in the front tracking library. Figure 8 shows the simulation of a table cloth of draping under the gravitational field.
The main motivation of studying the spring mass model for fabric surface is to study
Figure 10: Cross parachute inflation for wind tunnel experiment. The initial shape is flat cross, the diameter is 1.27 m, the parachute has 20 suspension lines which are 1.27 m each. The simulation starts from a fully folded state and ended when the canopy is opened. The fabric parameters are the same as being used in Figure 9.
the parachute dynamics for the air delivery system. In this application, the fabric surface is an immersed surface in the incompressible fluid solver. We couple the PDE solver for the Navier-Stokes equation with the ODE solver for the point mass system through the impulse method. The physical interaction between the parachute canopy and the fluid is though the exchange of the normal impulse of the mass points in the fabric surface.
Figure 9 shows the inflation of the G-11 cargo parachute and Figure 10 shows the inflation of the cross parachute. The validation study of the parachute system is presented in a different paper [21].
5 Conclusions
We use the front tracking data structure and functionalities to model the dynamic motion of fabric material. Our objective is to use this model for the computational study of the air delivery system such as the parachute system. We established the computational platform by using the spring mass system. We considered two spring systems, the linear system and the fabric system, the latter has no bending energy and is a suitable model for fabric material. For the linear system (Model-I), we have proved, through the Levy-Desplanques Theorem and the Gershgorin circle theorem, that all the eigen values of the coefficient matrix are imaginary and therefore the motion is pure oscillatory, and there
exists an upper bound|µ| ≤√
2Mk/m, where M is the maximum number of neighbors a spring mass point can have.
The nonlinear spring model is more difficult to analyze. But we found that the force along the direction to the neighbors of a vertex is the same as in the linear model. Nu-merically, we have showed that indeed, the motion of a spring mass point is oscillatory along the tangential direction of the fabric surface. The motion in the direction normal to the surface is not oscillatory in general. Fourier analysis of the tangential motion on an arbitrary sample point showed that the frequency of the oscillation is bounded by
√Mk/m/2π, where M≈7.
For the oscillatory motion, the first order Euler forward scheme for the ODE system is increasing in amplitude. Higher order Runge-Kutta scheme is a much efficient way to solve the equations. Our computation showed that the fourth order Runge-Kutta scheme with µmax∆t≤0.1 gives very stable and accurate solution to the spring mass ODE system.
There are still two open problems. The first is that although the analysis of Model-I through the Gershgorin circle theorem gives an upper bound of the eigen frequency as ωc≤√
2Mk/m, our numerical tests suggest that this upper bound is not a sharp bound.
The minimum bound appears to be ωc≤√
Mk/m. The second is that we still need the analytical proof of the upper bound for the nonlinear system of Model-II.
6 Acknowledgement
The authors would like to acknowledge many discussions with Xiangmin Jiao and Keh-Ming Shyue. Joungdong Kim and Xiaolin Li are supported in part by the US Army Re-search Office under the ARO grant award W911NF0910306. Xiaolin Li would like to thank the Department of Mathematics, National Taiwan University and to acknowledge the generous support from National Science Council of The Republic of China, Grant NSC 101-2811-M-002-006 on his sabbatical visit during which this work is accomplished.
References
[1] R. Fedkiw A. Selle, M. Lentine. A mass spring model for hair simulation. ACM Transactions on Graphics (Proceedings of the ACM SIGGRAPH’08 conference), 27, 2008.
[2] M. Aono, D. Breen, and M. Wozny. Fitting a woven cloth model to a curved surface: map-ping algorithms. Computer-Aided Design, 26(4):278–292, April 1994.
[3] M. Aono, P. Denti, D. Breen, and M. Wozny. Fitting a woven cloth model to a curved surface:
dart insertion. IEEE Computer Graphics and Applications, 16(5):60–70, September 1996.
[4] U. Ascher and E. Boxerman. On the modified conjugate gradient method in cloth simulation.
The Visual Computer, 19(7–8):523–531, December 2003.
[5] D. Baraff and A. Witkin. Large steps in cloth simulation. In Proceedings of ACM SIGGRAPH 98, pages 43–54. ACM Press, 1998.
[6] R. Bargmann. Real time cloth simulation. Master’s thesis, Ecoly Polytechnique F´ed´erale de Lausanne (EPFL) / Eidgen ¨ossische Technische Hoschschule Z ¨urich (ETHZ), Lausanne / Zurich, Switzerland, 2003.
[7] H. Bez, A. Bricis, and J. Ascough. A collision detection method with applications in CAD systems for the apparel industry. Computer-Aided Design, 28(1):27–32, 1996.
[8] W. Bo, X. Liu, J. Glimm, and X. Li. Primary breakup of a high speed liquid jet. ASME Journal of Fluids Engineering, submitted, 2010.
[9] D. Breen. A particle-based model for simulating the draping behavior of woven cloth. PhD thesis, Rensselaer Polytechnic Institute, 1993.
[10] D. Breen, D. House, and M. Wozny. Predicting the drape of woven cloth using interacting particles. In Proceedings of ACM SIGGRAPH 94, pages 365–372. ACM Press, 1994.
[11] R. Bridson, S. Marino, and R. Fedkiw. Simulation of clothing with folds and wrinkles. In Proceedings of ACM SIGGRAPH/Eurographics Symposium on Computer Animation (SCA 2003), pages 28–36. ACM Press, 2003.
[12] Robert Bridson, Ronald Fedkiw, and John Anderson. Robust treatment of collisions, contact and friction for cloth animation. ACM Transactions on Graphics, 21:594–603, 2002.
[13] J. C. Butcher. The numerical analysis of ordinary differential equations : Runge-Kutta and general linear methods. 1987.
[14] M. Carignan, Y. Yang, N. Magnenat-Thalmann, and D. Thalmann. Dressing animated syn-thetic actors with complex deformable clothes. In Computer Graphics (Proceedings of ACM SIGGRAPH 92), pages 99–104. ACM Press, 1992.
[15] K.-J. Choi and H.-S. Ko. Stable but responsive cloth. ACM Transactions on Graphics, 21:604–
611, 2002.
[16] Jian Du, Brian Fix, James Glimm, Xicheng Jia, Xiaolin Li, Yunhua Li, and Lingling Wu. A simple package for front tracking. J. Comput. Phys., 213:613–628, 2006.
[17] B. Eberhardt, M. Meißner, and W. Straßer. Knit fabrics. In D. House and D. Breen, editors, Cloth Modeling and Animation, pages 123–144. A.K. Peters, 2000.
[18] B. Eberhardt, A. Weber, and W. Straßer. A fast, flexible, particle-system model for cloth draping. IEEE Computer Graphics and Applications, 16(5):52–59, September 1996.
[19] E. George, J. Glimm, X. L. Li, Y. H. Li, and X. F. Liu. The influence of scale-breaking phe-nomena on turbulent mixing rates. Phys. Rev. E, 73:016304, 2006.
[20] A. H. Barr J. C. Platt. Constraints methods for flexible models. In SIGGRAPH ’88 Proceedings of the 15th annual conference on Computer graphics and interactive techniques, pages 279–288, 1988.
[21] J.-D. Kim, Y. Li, and X.-L. Li. Simulation of parachute FSI using the front tracking method.
Journal of Fluids and Structures, 37:101–119, 2013.
[22] X. Provot. Deformation constraints in a mass-spring model to describe rigid cloth behavior.
In Proceedings of Graphics Interface (GI 1995), pages 147–154. Canadian Computer-Human Communications Society, 1995.
[23] R. Samulyak, T. Lu, P. Parks, J. Glimm, and X. Li. Simulation of pellet ablation for tokamak fuelling with itaps front tracking. Journal of Physics: Conf. Series, 125:012081, 2008.
[24] D. Terzopoulos and K. Fleischer. Deformable models. The Visual Computer, 4(6):306–331, December 1988.
[25] D. Terzopoulos and K. Fleischer. Modeling inelastic deformation: viscoelasticity, plasticity, fracture. In Computer Graphics (Proceedings of ACM SIGGRAPH 88), pages 269–278. ACM Press, July 1988.
[26] D. Terzopoulos, J. Platt, A. Barr, and K. Fleischer. Elastically deformable models. In Computer Graphics (Proceedings of ACM SIGGRAPH 87), pages 205–214. ACM Press, July 1987.
[27] D. Terzopoulos and K. Waters. Analysis and synthesis of facial image sequences using phys-ical and anatomphys-ical models. IEEE Transactions on Pattern Analysis and Machine Intelligence
(TPAMI), 15:569–579, 1993.
[28] P. Volino, M. Courchesne, and N. Magnenat-Thalmann. Versatile and efficient techniques for simulating cloth and other deformable objects. In Proceedings of ACM SIGGRAPH 95, pages 137–144. ACM Press, 1995.