基于双弯曲质点弹簧系统的Waterbomb铺嵌快速模拟
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摘要
通过对Waterbomb铺嵌模型的结构分析,提出双弯曲质点弹簧系统并对其进行模拟。根据与棱相邻的三角形面片是否对称,将Waterbomb铺嵌模型的非边界棱(边)归为两类,建立两类弯曲弹簧,设置不同的弯曲弹性系数,该质点弹簧系统被称为双弯曲质点弹簧系统。然后运用隐式欧拉法和块坐标下降法求解最小化问题,得到下一时刻物体运动变形的速度和位置,实现对Waterbomb铺嵌模型快速、稳定和准确的模拟。实验证明,在误差允许范围内,该算法速度快于牛顿法。相比单弯曲(基本)系统,双弯曲质点弹簧系统提升了模拟稳定性、真实性,并减小了误差。
According to the analysis of waterbomb tessellation model structure,bi-bending mass-spring system is proposed to simulate waterbomb tessellation.According to the symmetry of the triangular facets adjacent to the edge,the non boundary edges of the waterbomb tessellation model are classified into two classes,and two kinds of bending springs are established with different bending elasticity coefficients.This mass-spring system is called the bi-bending mass-spring system.Then,we use the implicit Euler method and solve the minimization problem by block coordinate descent method(also known as alternating optimization)to get the velocity and position of the movement and deformation of the object at the next time.This paper realizes the fast,stable and accurate simulation of waterbomb tessellation model.Within the allowable error range,the algorithm is faster than Newton's method.Compared with the single bending(basic)system,the bi-bending mass-spring system further improves the stability and realism of the simulation and reduces the error.
According to the analysis of waterbomb tessellation model structure,bi-bending mass-spring system is proposed to simulate waterbomb tessellation.According to the symmetry of the triangular facets adjacent to the edge,the non boundary edges of the waterbomb tessellation model are classified into two classes,and two kinds of bending springs are established with different bending elasticity coefficients.This mass-spring system is called the bi-bending mass-spring system.Then,we use the implicit Euler method and solve the minimization problem by block coordinate descent method(also known as alternating optimization)to get the velocity and position of the movement and deformation of the object at the next time.This paper realizes the fast,stable and accurate simulation of waterbomb tessellation model.Within the allowable error range,the algorithm is faster than Newton's method.Compared with the single bending(basic)system,the bi-bending mass-spring system further improves the stability and realism of the simulation and reduces the error.
引文
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[4]TACHI T.Freeform variations of origami[J].Journal for Geometry and Graphics,2010,14(2):203-215.
[5]LU S,ZHAO C,GUO X,et al.Bi-layered mass-spring model for leaf motions[J].Journal of System Simulation,2009,21(14):4343-4388.
[6]刘雪梅,毛磊,李运华,等.耦合无网格伽辽金与质点弹簧实现软组织形变仿真[J].计算机辅助设计与图形学学报,2013,25(1):1-6.
[7]陈瑛,吴明珠,胡新荣.一种改进的质点-弹簧织物建模与仿真[J].华南师范大学学报:自然科学版,2013,45(5):43-46.
[8]LIU T,BARGTEIL A W,O'BRIEN JAMES F,et al.Fast simulation of mass-spring systems[J].ACM Transactions on Graphics,2013,32(6):803-812.
[9]王崴,周诚,杨云,等.基于改进弹簧-质点模型的柔性绳索仿真[J].计算机辅助设计与图形学学报,2015,27(11):2230-2236.
[10]王舒珍,陈国栋,李琦.基于改进的质点-弹簧模型的软组织形变仿真研究[J].通化师范学院学报,2017(1):64-69.
[11]TERZOPOULOS D,PLATT J,BARR A,et al.Elastically deformable models[C].Conference on Computer Graphics&Interactive Tec,1987:205-214.
[12]GARG A,GRINSPUN E,WARDETZKY M,et al.Cubic shells[C].Siggraph/Eurographics Symposium on Computer Animation,2007:91-98.
[13]BOUAZIZ S,MARTIN S,LIU T,et al.Projective dynamics:fusing constraint projections for fast simulation[J].ACM Transactions on Graphics,2014,33(4):1-11.
[14]ZHANG X,LE X,WU Z,et al.Data-driven bending elasticity design by shell thickness[J].Computer Graphics Forum,2016,35(5):157-166.
[15]LIU T,BOUAZIZ S,KAVAN L,et al.Quasi-Newton methods for real-time simulation of hyperelastic meterials[J].ACM Transactions on Graphics,2017,36(3):1-16.
[16]BURGOON R,WOOD Z J,GRINSPUN E.Discrete shells origami[C].Proceedings of the 21st International Conference on Computers and Their Applications,2006:180-187.
[17]TACHI T.Origanizing polyhedral surfaces[J].IEEE Transactions on Visualization and Computer Graphics,2010,16(2):298-311.
[18]TACHI T.Designing freeform origami tessellations by generalizing Resch's patterns[C].Proceedings of ASME 2013 IDETC/CIE,2013:1-10.
[19]TACHI T,OBREBSKI J B,TARCZEWSKI R.Interactive form-finding of elastic origami[C].Proceedings of the International Association for Shell and Spatial Structures(IASS)Symposium,2013:23-27.
[20]DAVIS E,DEMAINE E D,DEMAINE M.L,et al.Reconstructing David Huffman's origami tessellations[C].Proceedings of the ASME2013 International Design Engineering Technical Conferences&Computersand Information in Engineering Conference,2013:1-10.