Abstract In this paper, a symplectic structure-preserved algorithm is presented to solve Hamiltonian dynamic model of ultra- flexible cloth simulation with high computation stability. Our method can preserve the conserved quantity of a Hamiltonian, which enables a long-time stable simulation of ultra-flexible cloth. Firstly, the dynamic equation of ultra-flexible cloth simulation is transferred into Hamiltonian system which is slightly perturbed from the original one, but with generalized structure preservability. Secondly, semi-implicit symplecticRunge-Kutta and Euler algorithms are constructed, and able to be converted into explicit algorithms for the separable dynamic models. Thirdly, in order to show the advantages, the presented algorithms are utilized to solve a conservative system which is the primary ultra-flexible cloth model unit. The results show that the presented algorithms can preserve the system energy constant and can give the exact results even at large time-step, however the ordinary non-symplectic explicit methodsexhabit large error with the increasing of time-step. Finally, the presented algorithms are adopted to simulate a large-areaultra-flexible cloth to validate the computation capability and stability. The method employs the symplectial features and analytically integrates the force for better stability and accuracy while keeping the integration scheme is still explicit. Experiment results show that our symplectic schemes are more powerful for integrating Hamiltonian systems than non-symplectic methods. Our method is a common scheme for physically based system to simultaneously maintain real-time and long-time simulation.It has been implemented in the scene building platform- World Max Studio.

References

[1] Bridson R., Marino S., and Fedkiw R., Simulation of Clothing with Folds and Wrinkles, in Proceedings of the ACM SIGGRAPH/Eurographics Symposium on Computer Animation, San Diego, pp. 28-36, 2003.

[2] Chen S., Human Visual Perception-based Image Quality Analyzer for Assessment of Contrast, The International Arab Journal of Information Technology, vol. 13, no. 2, pp. 238-245, 2016.

[3] Chen Y., He B., and Yu X., Research on Garment Wrinkle Synthetic Method Based on Mass-Spring Model, Journal of Electroanalytical Chemistry, vol. 621, no. 2, pp. 121-123, 2015.

[4] Choi K. and Ko H., Stable but Responsive Cloth, in Proceedings of the 29th Annual Conference on Computer Graphics and Interactive Techniques, San Antonio, pp. 604- 611, 2005.

[5] Hauth M., Etzmuss O., and Strasser W., Analysis of Numerical Methods for the Simulation of Deformable Models, The Visual Computer, vol. 19, no. 7-8, pp. 581-600, 2003.

[6] Huang Y., Yin Z., and Xiong Y., Modeling and Computation for the High-Speed Rotating Flexible Structure, Journal of Vibration and Acoustics, vol. 130, no. 4, pp. 2727-2747, 2008.

[7] Huang Y., Deng Z., and Yao L., An improved Symplectic Precise Integration Method for Analysis of the Rotating Rigid-flexible Coupled System, Journal of Sound and Vibration, vol. 299, no. 1-2, pp. 229-246, 2007.

[8] Huang Y., Deng Z., Xiong Y., High-order Model and Slide Mode Control for Rotating Flexible Smart Structure, Mechanism and Machine Theory, vol. 43, no. 8, pp. 1038-1054, 2008.

[9] Huang Y., Deng Z., and Li W., Sliding Mode Control based on Neural Network for the Vibration Reduction of Flexible Structures, Structural Engineering and Mechanics, vol. 26, no. 4, pp. 377-392, 2007.

[10] Huang Y., Deng Z., and Yao L., Dynamic Analysis of a Rotating Rigid-flexible Coupled Smart Structure with Large Deformations, Applied Mathematics and Mechanics, vol. 28, no. 10, pp. 1349-1360, 2007.

[11] Liu L., Wang R., Su Z., Luo X., and Gao C., Mesh-based Anisotropic Cloth Deformation for Virtual Fitting, Multimedia Tools and Applications, vol. 71, no. 2, pp. 411-433, 2014.

[12] Magnenat N. and Volino P., From Early Draping to Haute Couture Models: 20 Years of Research, The Visual Computer, vol. 21, no. 8- 10, pp. 506-519, 2005.

[13] M ller M., Heidelberger B., Hennix M., and Ratcliff J., Position based Dynamics, Journal of Visual Communication and Image Representation, vol. 18, no. 2, pp. 109-118, 2007.

[14] Nealen A., M ller M., Keiser R., Bxerman E., and Carlson M., Physically based Deformable Models in Computer Graphics, Wiley Online Library, 2006.

[15] Qin Y., Deng Z., and Hu W., Structure- Preserving Algorithm for Steady-State Solution to the Infinite Dimensional Hamilton System, Applied Mathematics and Mechanics, vol. 35, no. 1, pp. 22-28, 2014.

[16] Selle A., Lentine M., and Fedkiw R., A mass Spring Model for Hair Simulation, ACM Transactions on Graphics, vol. 27, no. 3, pp. 15- 19, 2008.

[17] Stam J., Towards a Unified Dynamical Solver for Computer Graphics, Computer Graphics Forum, vol. 25, no. 3, pp. 232-242, 2006.

[18] Stern A. and Desbrun M., Discrete geometric Mechanics for Variational Time Integrators, in Proceedings of ACM SIGGRAPH Courses, Boston, pp. 75-80, 2006.

[19] Stern A. and Grinspun E., Implicit-explicit Variational Integration of Highly Oscillatory Problems, Multiscale Modeling and Simulation, vol. 7, no. 4, pp. 1779-1794, 2009.

[20] Thomaszewski B., Pabsts S., and Straber W., Continuum-based Strain Limiting, Computer Graphics Forum, vol. 28, no. 2, pp. 569-576, 2009. Explicitly Symplectic Algorithm for Long-time Simulation of Ultra-flexible Cloth 1075

[21] Volino P. and Magnenat-Thalmann N., Accurate Garment Prototyping and Simulation, Computer-Aided Design and Applications, vol. 2, no. 5, pp. 645-654, 2005.

[22] Volino P. and Magnenat-Thalmann N., Comparing Efficiency of Integration Methods for Cloth Simulation, in Proceedings of Computer Graphics International, Hong Kong, pp. 265-272, 2001.

[23] Wang H., O'Brien J., and Ramamoorthi R., Multi-Resolution Isotropic Strain Limiting, ACM Transactions on Graphics, vol. 29, no. 6, pp. 156, 2010. Xiao-hui Tan, born in 1977, Ph.D., a lecturer of Capital NormalUniversity, Beijing, China.Her main research interest is computer graphics and virtual reality. Zhou Ming-quan, born in 1954, professor at Beijing Normal University. His research interests include computer graphics, virtual reality, and the digital protection of culture heritage. Prof. He is a senior member of the China Computer Federation (CCF) and China Society of Image and Graphics. YaChun Fan, born in 1978, Ph.D., assistance professor. Her main research interest is omputer graphics and virtual reality. Wang Xuesong, born in 1975, Ph.D., senior engineer of Beijing Normal University, Beijing, China. His main research interest is Virtual Reality and Intelligent Internet Data An Wu Zhongke, born in 1963, professor in College of information science and technology, Beijing Normal University (BNU), China. Prof. WU s current research interests include computer graphics, animation virtual reality, geometric modeling, volume graphics and medical imaging.