# American Institute of Mathematical Sciences

August & September  2019, 12(4&5): 1167-1178. doi: 10.3934/dcdss.2019080

## Three-dimensional computer simulation of twill woven fabric by using polynomial mathematical model

 1 Art School, Jinling Institute of Technology, Nanjing 211169, China 2 School of Fashion Art and Engineering, Beijing Institute of Fashion Technology, Beijing 100029, China 3 College of Textile and Clothing Engineering, Soochow University, Soochow, 215021, China 4 National Engineering Laboratory for Modern Silk, (NELMS) Soochow, 215123, China

* Corresponding author: Fang Qin

Received  September 2017 Revised  January 2018 Published  November 2018

This study was carried out to obtain visual simulations of twill woven fabrics on a computer screen using certain fabric characteristic. Based on the Peirce model, the polynomial curve fitting method is utilized to simulate the buckling configuration of twill weave yarns. Polynomial mathematical model was never used in constructing twill weave woven fabric structure in the past studies. In polynomial model, each point on yarn buckling track is calculated through the curvature, the radius of the warp and weft yarn, the geometric density, and the buckling curve height. Moreover, the twill weave structure is displayed through the arrangement of the warp and weft yarns. The polynomial mathematical model method was applied to convert the yarn path to a smooth curve and will be provided for three-dimensional computer simulation of satin weave fabric. Different twill weave is displayed by changing fabric parameters. In the VC++6.0 development environment, according to polynomial mathematical model, the three-dimensional simulation of twill fabric structure was given in details through the OpenGL graphics technology.

Citation: Fang Qin, Ying Jiang, Ping Gu. Three-dimensional computer simulation of twill woven fabric by using polynomial mathematical model. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1167-1178. doi: 10.3934/dcdss.2019080
##### References:
 [1] B. A. Barsky and D. P. Greenberg, Determining a set of b-spline control vertices to generate an interpolating surface, Computer Graphics & Image Processing, 14 (1980), 203-226.   Google Scholar [2] W. Gao and W. Wang, A tight neighborhood union condition on fractional (g, f, n', m)-critical deleted graphs, Colloquium Mathematicum, 149 (2017), 291-298.  doi: 10.4064/cm6959-8-2016.  Google Scholar [3] B. S. Jeon, Evaluation of the structural properties of plain fabrics woven from various fibers using peirce's model, Fibers & Polymers, 13 (2012), 130-134.   Google Scholar [4] F. Ji, R. Li and Y. Qiu, Three-dimensional garment simulation based on a mass-spring system, Textile Research Journal, 76 (2006), 12-17.   Google Scholar [5] W. Li, J. Qi, Z. Yu and D. Li, A social recommendation method based on trust propagation and singular value decomposition, Journal of Intelligent & Fuzzy Systems, 32 (2016), 1-10.   Google Scholar [6] S. Linbo and Q. Huayun, Performance of financial expenditure in china's basic science and math education: Panel data analysis based on ccr model and bbc model, Eurasia Journal of Mathematics Science and Technology Education, 13 (2017), 5217-5224.   Google Scholar [7] A. Moussa, D. Dupont, D. Steen and X. Zeng, Structure analysis and surface simulation of woven fabrics using fast fourier transform techniques, Journal of the Textile Institute Proceedings & Abstracts, 101 (2010), 556-570.   Google Scholar [8] S. Sha, G. Jiang, P. Ma and X. Li, 3-d dynamic behaviors simulation of weft knitted fabric based on particle system, Fibers & Polymers, 16 (2015), 1812-1817.   Google Scholar [9] R. B. Turan and G. Baser, Threea dimensional computer simulation of 2/2 twill woven fabric by using ba-splines, Journal of the Textile Institute Proceedings & Abstracts, 101 (2010), 870-881.   Google Scholar [10] L.-R. Wu, Zhuang-Wen Zhu, Cultivating innovative and entrepreneurial talent in the higher vocational automotive major with the "on-board educational factory" model, Eurasia Journal of Mathematics Science & Technology Education, 13. Google Scholar [11] X. Xu and F. Wang, A modeling method for complex system using hybrid method, Journal of Discrete Mathematical Sciences & Cryptography, 20 (2017), 239-254.   Google Scholar [12] F. Yamaguchi, A new curve fitting method using a crt computer display, Computer Graphics & Image Processing, 7 (1978), 425-437.   Google Scholar [13] J. Zhang, G. Baciu, J. Cameron and J. L. Hu, Particle pair system: An interlaced mass-spring system for real-time woven fabric simulation, Textile Research Journal, 82 (2012), 655-666.   Google Scholar [14] Y. B. Zhang, T. T. Ning and T. Xue, Autonomous learning ability training of college students in tianjin from the perspective of habitus and field, Journal of Discrete Mathematical Sciences & Cryptography, 20 (2017), 323-339.   Google Scholar [15] Q. Zhao, Computer simulation of reliability algorithm for wind-induced vibration response control of high structures, Journal of Discrete Mathematical Sciences & Cryptography, 20 (2017), 1519-1523.   Google Scholar

show all references

##### References:
 [1] B. A. Barsky and D. P. Greenberg, Determining a set of b-spline control vertices to generate an interpolating surface, Computer Graphics & Image Processing, 14 (1980), 203-226.   Google Scholar [2] W. Gao and W. Wang, A tight neighborhood union condition on fractional (g, f, n', m)-critical deleted graphs, Colloquium Mathematicum, 149 (2017), 291-298.  doi: 10.4064/cm6959-8-2016.  Google Scholar [3] B. S. Jeon, Evaluation of the structural properties of plain fabrics woven from various fibers using peirce's model, Fibers & Polymers, 13 (2012), 130-134.   Google Scholar [4] F. Ji, R. Li and Y. Qiu, Three-dimensional garment simulation based on a mass-spring system, Textile Research Journal, 76 (2006), 12-17.   Google Scholar [5] W. Li, J. Qi, Z. Yu and D. Li, A social recommendation method based on trust propagation and singular value decomposition, Journal of Intelligent & Fuzzy Systems, 32 (2016), 1-10.   Google Scholar [6] S. Linbo and Q. Huayun, Performance of financial expenditure in china's basic science and math education: Panel data analysis based on ccr model and bbc model, Eurasia Journal of Mathematics Science and Technology Education, 13 (2017), 5217-5224.   Google Scholar [7] A. Moussa, D. Dupont, D. Steen and X. Zeng, Structure analysis and surface simulation of woven fabrics using fast fourier transform techniques, Journal of the Textile Institute Proceedings & Abstracts, 101 (2010), 556-570.   Google Scholar [8] S. Sha, G. Jiang, P. Ma and X. Li, 3-d dynamic behaviors simulation of weft knitted fabric based on particle system, Fibers & Polymers, 16 (2015), 1812-1817.   Google Scholar [9] R. B. Turan and G. Baser, Threea dimensional computer simulation of 2/2 twill woven fabric by using ba-splines, Journal of the Textile Institute Proceedings & Abstracts, 101 (2010), 870-881.   Google Scholar [10] L.-R. Wu, Zhuang-Wen Zhu, Cultivating innovative and entrepreneurial talent in the higher vocational automotive major with the "on-board educational factory" model, Eurasia Journal of Mathematics Science & Technology Education, 13. Google Scholar [11] X. Xu and F. Wang, A modeling method for complex system using hybrid method, Journal of Discrete Mathematical Sciences & Cryptography, 20 (2017), 239-254.   Google Scholar [12] F. Yamaguchi, A new curve fitting method using a crt computer display, Computer Graphics & Image Processing, 7 (1978), 425-437.   Google Scholar [13] J. Zhang, G. Baciu, J. Cameron and J. L. Hu, Particle pair system: An interlaced mass-spring system for real-time woven fabric simulation, Textile Research Journal, 82 (2012), 655-666.   Google Scholar [14] Y. B. Zhang, T. T. Ning and T. Xue, Autonomous learning ability training of college students in tianjin from the perspective of habitus and field, Journal of Discrete Mathematical Sciences & Cryptography, 20 (2017), 323-339.   Google Scholar [15] Q. Zhao, Computer simulation of reliability algorithm for wind-induced vibration response control of high structures, Journal of Discrete Mathematical Sciences & Cryptography, 20 (2017), 1519-1523.   Google Scholar
$\frac{3\ 2}{2\ 3}$ weave diagram
A unit curve and the corresponding coordinate of $\frac{3\ 2}{2\ 3}$ twill weft yarn
A unit curve of twill weft yarn
The segmentation diagram of a unit curve of twill weft yarn
Schematic diagram of a unit of plain fabric weft yarn
Coordinate system of the weft yarn
A unit curve and the corresponding coordinate of the $\frac{3\ 2}{2\ 3}$ twill weave warp yarn
A unit curve of the twill weave warp yarn
The segmentation of a unit curve of the twill weave warp yarn
Segment of the plain warp yarn
The coordinate system of the warp yarn
The curved surface of twill weave weft yarn
The curved surface of twill weave warp yarn
Yarn model and 3D image of $\frac32$ left twill fabric structure
 [1] A. Naga, Z. Zhang. The polynomial-preserving recovery for higher order finite element methods in 2D and 3D. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 769-798. doi: 10.3934/dcdsb.2005.5.769 [2] Guoliang Ju, Can Chen, Rongliang Chen, Jingzhi Li, Kaitai Li, Shaohui Zhang. Numerical simulation for 3D flow in flow channel of aeroengine turbine fan based on dimension splitting method. Electronic Research Archive, 2020, 28 (2) : 837-851. doi: 10.3934/era.2020043 [3] Tiago de Carvalho, Rodrigo Donizete Euzébio, Jaume Llibre, Durval José Tonon. Detecting periodic orbits in some 3D chaotic quadratic polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 1-11. doi: 10.3934/dcdsb.2016.21.1 [4] Yong Zhou. Remarks on regularities for the 3D MHD equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (5) : 881-886. doi: 10.3934/dcds.2005.12.881 [5] Hyeong-Ohk Bae, Bum Ja Jin. Estimates of the wake for the 3D Oseen equations. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 1-18. doi: 10.3934/dcdsb.2008.10.1 [6] Indranil SenGupta, Weisheng Jiang, Bo Sun, Maria Christina Mariani. Superradiance problem in a 3D annular domain. Conference Publications, 2011, 2011 (Special) : 1309-1318. doi: 10.3934/proc.2011.2011.1309 [7] Giovanny Guerrero, José Antonio Langa, Antonio Suárez. Biodiversity and vulnerability in a 3D mutualistic system. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4107-4126. doi: 10.3934/dcds.2014.34.4107 [8] Sadek Gala. A new regularity criterion for the 3D MHD equations in $R^3$. Communications on Pure & Applied Analysis, 2012, 11 (3) : 973-980. doi: 10.3934/cpaa.2012.11.973 [9] Jiahong Wu. Regularity results for weak solutions of the 3D MHD equations. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 543-556. doi: 10.3934/dcds.2004.10.543 [10] Gabriel Deugoue. Approximation of the trajectory attractor of the 3D MHD System. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2119-2144. doi: 10.3934/cpaa.2013.12.2119 [11] Alp Eden, Varga K. Kalantarov. 3D convective Cahn--Hilliard equation. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1075-1086. doi: 10.3934/cpaa.2007.6.1075 [12] Chongsheng Cao. Sufficient conditions for the regularity to the 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1141-1151. doi: 10.3934/dcds.2010.26.1141 [13] Ning Ju. The global attractor for the solutions to the 3D viscous primitive equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 159-179. doi: 10.3934/dcds.2007.17.159 [14] Tomás Caraballo, Antonio M. Márquez-Durán, José Real. Pullback and forward attractors for a 3D LANS$-\alpha$ model with delay. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 559-578. doi: 10.3934/dcds.2006.15.559 [15] Jianqing Chen. Best constant of 3D Anisotropic Sobolev inequality and its applications. Communications on Pure & Applied Analysis, 2010, 9 (3) : 655-666. doi: 10.3934/cpaa.2010.9.655 [16] Ming Lu, Yi Du, Zheng-An Yao. Blow-up phenomena for the 3D compressible MHD equations. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1835-1855. doi: 10.3934/dcds.2012.32.1835 [17] Felipe Linares, Jean-Claude Saut. The Cauchy problem for the 3D Zakharov-Kuznetsov equation. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 547-565. doi: 10.3934/dcds.2009.24.547 [18] Rafel Prohens, Antonio E. Teruel. Canard trajectories in 3D piecewise linear systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4595-4611. doi: 10.3934/dcds.2013.33.4595 [19] Makram Hamouda, Chang-Yeol Jung, Roger Temam. Asymptotic analysis for the 3D primitive equations in a channel. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 401-422. doi: 10.3934/dcdss.2013.6.401 [20] Ning Ju. The finite dimensional global attractor for the 3D viscous Primitive Equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7001-7020. doi: 10.3934/dcds.2016104

2018 Impact Factor: 0.545