# 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 and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1167-1178. doi: 10.3934/dcdss.2019080
##### References:
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##### 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [12] F. Yamaguchi, A new curve fitting method using a crt computer display, Computer Graphics & Image Processing, 7 (1978), 425-437. [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. [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. [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.
$\frac{3\ 2}{2\ 3}$ 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
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