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Liquid Diffusion Model that Accounts for a Variety of Dyeing ParametersYuki Morimoto, *Kyushu [email protected]
Masayuki Tanaka, The University of [email protected]
Reiji Tsuruno *[email protected]
Kiyoshi Tomimatsu *[email protected]
Outline of Dyeing = Overview of Our Method
Visual Characteristics of Dyeing on Woven Cloth
(a) thin spots, (b) bleeding ("nijimi"), (c) mottles, and (d) different shapes caused by the amount of dye.
Left:Batik simulation[1],
Right:Chinise ink simulation[2].
Cloth Model
Diffusion ModelDiffusion Equation with Variable D
D by Tortuosity and Porosity
Capacity of Dye Amount for Diffusion and Adsorption
Examples of Tie-dyeing(left) and Yuzen-dyeing(right).
① ② ③ ④
Some studies have been done in this area into watercolor painting and Chinese ink painting. While paper consists of
small fibers, the cloth consists of weaving wefts and warps. In cloth, there are some characteristics as shown as below.
Vmax
VVu
Vd
Diffusion cell⑥
Figure 1. Computer generated dye stains with various parameters.
This poster describes a method for simulating and visualizing dyeing based on weave patterns. Methods for simulating painting implements and drawing strokes are being advanced in the field of NPR. We describe the characteristics of the diffusion of pigments in cloth, and propose a
way to simulate these features. We define dye diffusion, diffusion coefficient, adsorption based on the dyeing theories. We describe a system to simulate a successive dyeing process.
(d)
Figure 3. The differences between dye stains on different weave patterns. (a) Plain weave, (b) diagonal weave, (c) satin weave.
Figure 5. Comparison of simulations with some adsorption models. (a), (b), (c) are calcurated with adsorption model (a” ), (b” ), (c” ) in ⑨. (a'), (b'), (c') show only the absorbed dye density of (a)(b)(c).
Our cloth model is two-layered cellular model.
①We first produce the weft layer and then the warp layer.
②Each layer is then divided into widths of threads before being subdivided into cells (cloth cell).
③The definition of a weave pattern is done by setting the smallest and basic pattern.
④Each cloth cell is divided into diffusion cells which are used to calculate dye diffusion.
Our dye diffusion model is based on Fick’ s second law which can be represented as diffusion
equation (eq 1). We discretized it to obtain (eq 2) which can consider the variable D.
Figure 2. Computer generated different shapes caused by the amount of dye.
[3] A. Fick. On liquid diffusion, 1855.
[1] B. S.-H. Diffusion/adsorption behaviour of reactive dyes in cellulose. Dyes and Pigments, 1997.
[1] B. Wyvill, K. van Overveld, and S. Carpendale. Rendering cracks in batik, 2004.
[2] Jintae Lee, Diffusion rendering of black ink paintings using new paper and ink models, April 2001.
[5] B. R. van den. Human exposure to soil contamination: a qualitative and quantitative analysis towards proposals for human toxicological intervention values (partly revised edition), 1994.
[2] N. Adabala and N. Magnenat-Thalmann. A procedural thread texture model, 2003. (This is a reference for generating cloth textures.)
Direction of the weft
Direction of the warp⑤τ3I :diffirent layerτ3II :two fibers are in the same layer, and are connected to each other perpendicularlyτ3III : fiber and gapτ3IV : gap and gapτ3V : fiber and fiber in the same layer
Result
Figure 4. The simulation of seikaiha along the time step.
333
1111111555
4
⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨⑨
Parameters from Figs. 1,3,4,5. r is a random value in (0.5, 1] for each yarn. ※In Fig. 4,5, the initial amount of dye is kept constant during the simulation.
The diffusion coefficient is calculated by eq 4 that is based on the Weisz-Zollinger model [1].
[Parameters] Tortuosity (the degree of twist): T, porosity: P, the dye concentration in the external solution
when equilibrium is achieved: φo , the diffusion coefficient in free water: Do, the molecular mass of dye:
M. 3 kinds of tortuosity in our method to calculate T (eq 3) :
(τ1) the twist of the thread,
(τ2) the position of a thread, including its orientation in the weave pattern,
(τ3) from the different orientations of fibers in neighboring diffusion cells.
⑤ τ3 has five different conditions which depend on properties such as the layer neighboring diffusion cells
are in and their porosity.
Our Method
⑦Red region in (a) is distribution of the dye, while the black region is the pressure distribution. (b) is the result with (a), (c) is a real dyed cloth showing the seikaiha pattern.
1. Prepare the cloth →Cloth model
2, Decide distribution of dye →Protecting & Dyeing Table
3, Add dye →Set the initial amount of dye
4, Dye diffusion →Diffusion model
5, Finish dye diffusion →Finish at a arbitral time
⑧a” is the case of b=1 in eq 6, b” is the case of b>1 in eq 6,c” is the case in eq 7.⑥ The total amount of dye in a diffusion cell V is defined by Vu (the dye capacity) and Vd (the
adsorbed dye capacity).
⑦ Vu is defined by the parameter B that denotes the volume ratio given by an imput image.
→Represent protection against dyeing (the resist paste using methods, tie-dyeing, batik and so on).
⑧ The maximum amount of dye adsorption (fixing dye) into fibers (Ad) is calculated each time step
by adsorption isotherm based on dyeing theories in our model. Also Vd is calculated by porosity.
