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An energy based model for the flattening of woven fabrics
Journal of Materials Processing Technology 107 (2000) 312±318
An energy based model for the ¯attening of woven fabrics J. McCartney*, B.K. Hinds, B.L. Seow, D. Gong School of Mechanical and Manufacturing Engineering, The Queen's University of Belfast, Belfast, Northern Ireland BT9 5AH UK
Abstract Applications such as garment manufacture and composite structure fabrication require a two dimensional (2D) woven material to assume a three dimensional (3D) shape. The speci®cation of the process is usually initiated by de®ning the 3D surface. Hence, the problem arises of determining the best 2D pattern. The problem is made more complex by the anistropic nature of woven fabrics which are often used as raw material. Such materials display a variation in mechanical properties with respect to the woven structure. This paper presents a model for determining the optimum 2D pattern for a speci®ed 3D surface where optimality is determined in terms of minimising the energy distribution required to force the 2D pattern to assume the 3D shape. The 3D surface speci®cation is assumed to consist of a polygonal mesh. The model allows af®ne transformations to be applied to the weave structure which can be unique for each polygon in the mesh. Important considerations in the modelling process include the following: 1. The degree to which the speci®ed 3D surface departs from a developable surface. 2. The energy components used to model the woven structure and their sensitivity to weave direction. Essentially, these stem from tensile strain in each direction of the weave and shear strain. 3. The prediction of weave geometry as it reacts to the energy distribution being applied. The model is demonstrated by applying it to a relatively simple pyramidal 3D shape. Energy values are optimised to produce a pattern that requires the minimum overall energy to be applied to the 2D pattern in order for it to assume the 3D shape. This 2D pattern is sensitive to the orientation of the woven structure with predictions being made of how the woven structure will behave in 3D. # 2000 Elsevier Science B.V. All rights reserved. Keywords: Flattening of woven fabrics; Woven fabric element; Af®ne transformation
1. Introduction In many industries, it is required to manufacture three dimensional (3D) structures or surfaces from two dimensional (2D) raw material. It is common for the design function to originate a speci®cation of the 3D shape and then for a technical department to convert this speci®cation into the requisite 2D pattern. Such situations range from ship hull production to garment manufacture. The problem is complicated by the fact that the 3D shape speci®cation might be of a generalised form for which an exact 2D pattern may not exist. In this context, `exact' means that the 2D pattern can assume the 3D shape by bending only of the raw material. Such 3D surfaces are described as being developable [1]. Situations where a non-developable surface have been speci®ed imply that that some amount of distortion has to be applied to the 2D pattern in order for it to assume the 3D shape. There will exist many 2D patterns for a *
Corresponding author. Tel.: 28902-74127; fax: 28906-61729. E-mail address: [email protected] (J. McCartney).
given 3D surface, each requiring different distortion pro®les in order to deform the 2D pattern in a unique way to obtain the 3D shape. In such circumstances, some criterion must be speci®ed to determine the optimum 2D pattern. Speci®cally, this paper examines the situation where the raw material is of a woven construction. This is the case with the design of many garments and composite structures. The major problem here is that the material displays anistropic behaviour regarding its ability to distort relative to the orientation of the weave. The model described below attempts to obtain a minimum energy solution to ®nding a 2D pattern for a non-developable surface, assuming that the raw material is of woven construction. 2. 3D Surface representation It is assumed that the speci®cation of the 3D surface will be in the form of a polygon list that contains pointers to the vertices used to de®ne each triangular polygon. This is an
0924-0136/00/$ ± see front matter # 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 6 9 4 - 4
J. McCartney et al. / Journal of Materials Processing Technology 107 (2000) 312±318
313
Fig. 2. Initial 2D ¯attening of hexahedron.
Fig. 1. Example hexahedron surface.
approximation process usually since the original design may have been of a continuous form. As a simple example, consider the hexahedron shown in Fig. 1. A list speci®cation of this is given in Tables 1 and 2. 3. 2D Pattern representation Now consider the problem of constructing a one-piece pattern for the hexahedron in Fig. 1. There is no pattern that will take up this shape without some form of distortion. Let the 2D axes system for such patterns be represented by (x0 , y0 ). From classical treatment of surface geometry [2], this shape can be considered as an approximation to an elliptical surface. As such, it will display an angular defect of 41.218 when neighbouring triangles are sequentially laid on the 2D plane (Fig. 2). If a one-piece pattern is required, then some Table 1 3D Surface node list Node
1 2 3 4 5 6 0
distortion has to take place. As an initial attempt, force the two locations (600 ) for node 6 to a mean position (60 ). The resultant pattern will have triangles P02 , P03 , P04 and P05 remaining undistorted but triangles P01 and P06 will have some degree of distortion introduced. The one-piece pattern that results can now be superimposed onto an underlying woven structure. This structure can be speci®ed by the following parameters Weft increment, wfinc . Warp increment, wpinc . Weft angle, wa . In effect, these parameters constitute a weft and warp axes system (wf ; wp ) for the pattern. Assuming that the (wf ; wp ) origin is located at the same position as the (x0 , y0 ) origin, then Appendix A provides a transformation which can be used to provide (wf ; wp ) co-ordinates from (x0 , y0 ) co-ordinates. If the following values are assumed for the warp and weft geometry, wfinc 1:0;
wpinc 1:0;
wa 50
Table 3 can be compiled. This speci®cation only has to contain the new co-ordinates for each node since the polygon composition is the same as Tables 1 and 2.
3D co-ordinates X
Y
Z
5.00 ÿ5.00 ÿ10.00 ÿ5.00 5.00 10.00 0.00
8.67 8.67 0.00 ÿ8.67 ÿ8.67 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 5.00
4. Deformation of a woven fabric element At this stage a 2D pattern has been precisely speci®ed. This is displayed in Fig. 3 where alternate triangles are shaded lighter and darker to highlight the triangular Table 3 Initial pattern node co-ordinates Node
Table 2 3D Surface polygon speci®cation
2D pattern x
Polygon
Node pointers
P1 P2 P3 P4 P5 P6
6, 1, 2, 3, 4, 5,
1, 2, 3, 4, 5, 6,
0 0 0 0 0 0
10 20 30 40 50 60 00
0
3.13 ÿ6.71 ÿ11.18 ÿ6.71 3.13 10.46 0.00
2D weft and warp y
0
10.73 8.94 0.00 ÿ8.94 ÿ10.73 0.00 0.00
wx
wy
9.80 1.58 ÿ7.91 ÿ11.07 ÿ5.38 7.40 0.00
5.38 11.07 7.91 ÿ1.58 ÿ9.80 ÿ7.40 0.00
314
J. McCartney et al. / Journal of Materials Processing Technology 107 (2000) 312±318
Fig. 5. (a) Af®ne transformation; (b) af®ne transformation for woven element in triangle P1.
Fig. 3. Hexahedron 2D pattern on weft and warp axes.
composition. For this pattern to assume the 3D shape of the hexahedron, strain must be applied consistent with the geometric distortions implied above. i.e. that only triangles P01 and P06 are distorted. The approach adopted here is to calculate the energy required to perform such distortions on an individual triangle basis. The process starts by displaying individual triangles on a unifying axes system (u, v) for comparing the shapes of the two forms that exist for each triangle. First the 2D undeformed triangle is positioned so that one of its vertices is coincident with the (u, v) origin with the weft direction aligned with the u direction. Then the 3D deformed triangle is superimposed on the same axes system using the same positioning conditions. Note that the deformed 3D triangle must maintain the same woven structure composition as the 2D in terms of the total number of woven elements and how the weft and warp directions intersect with the edges of the triangle. This process is depicted in Fig. 4 for triangle P1. The 2D triangle (P01 ) is A0 B0 C0 and the deformed triangle on the 3D surface (P1) is ABC. This unifying process enables the speci®cation of the geometric transformation that a representative rectangular woven element must undergo in order for the complete triangle A0 B0 C0 to deform to ABC. The total energy required for the distortion can be calculated by determining the energy to distort the representative weave element and then applying the same transformation to the triangle as a whole. The distortion of the representative woven element can be represented by an af®ne transformation [3]. The general
Fig. 4. Unifying (u,v) axes system for triangles P1 (ABC) and P01 (A0 B0 C0 ).
representation of an af®ne transformation is 0 1 a11 a12 0 u0 v0 1 u v 1 @ a21 a22 0 A a31 a32 1
(1)
This transformation has the property that parallel lines representing the weft and warp alignment of the 2D triangle ABC will remain parallel within the deformed 3D triangle A0 B0 C0 . It has been described elsewhere [4] that a 2D rectangular woven element can be transformed to the types of shapes experienced by woven fabrics with the following sequence of transformations (Fig. 5). First the rectangular element is differentially scaled in the u and v directions by the scaling factors Su and Sv . Secondly, the resultant shape is subjected to a collapsing shear of jv , with respect to the warp direction in which the lengths of the sides are preserved. For a particular af®ne transformation, the af®ne transformation matrix will now be referred to as Mcomp where this matrix is comprised as follows: u0
v0
where M comp
1 u 0
v 1 M comp
Su B @0
0 Sv
10 1 0 CB 0 A@ sin jv
0
0
1
0
Su B @ Sv sin jv 0
0 cos jv
0 0 Sv cos jv 0
1
0
0 C 0A
1 0 C 0A 1 (2)
1
This matrix is of af®ne form as described in Eq. (1). For this model, all the 2D and 3D co-ordinates for a particular deformation of a triangle on the original surface will be known. It is, therefore, necessary to synthesise the net transformation into its constituent scaling factors (Su and Sv ) and shear angle (jv ) in order to accurately determine energy composition. A particular af®ne transformation as described by Fig. 5 (a) will be speci®ed by the mapping of two corners only as the third corner is at the origin for both triangles i.e. u1 ; v1 ! u01 ; v01 ;
u2 ; v2 ! u02 ; v02
(3)
J. McCartney et al. / Journal of Materials Processing Technology 107 (2000) 312±318
Eq. (1) can now be rewritten to include these two mappings and provide a means for solving for Su ; Sv and jv . 10 1 0 1 0 0 0 1 a11 a12 0 0 0 1 @ u0 v0 1 A @ u1 v1 1 A@ a21 a22 0 A 1 1 u02 v02 1 u2 v 2 1 a31 a32 1 or U 0 UA
The shear energy Er, is given as Z Z 0:5Kr j2v du dv 0:5A Kr j2v Er
315
(9)
where Kr is the shear modulus. Hence, given the point mappings described by Eq. (3), the strain constants Ksu and Ksv and the shear modulus Kr , the total strain energy (Es Er ) required to deform the triangle can be calculated.
Hence A U ÿ1 U 0
5. Energy minimisation algorithm
where
0
v1 ÿ v2 v2 1 @ u2 ÿ u1 ÿu2 det U u1 v2 ÿ u2 v1 0
U ÿ1
1 ÿv1 u1 A 0
and det U u1 v2 ÿ u2 v1 This results in A
1 det U 0 v2 u01 ÿ v1 u02 B @ u1 u02 ÿ u2 u01
v2 v01 ÿ v1 v02
0
u1 v02 ÿ u2 v01
0
0
u1 v2 ÿ u2 v1
0
1 C A (4)
By comparing Eqs. (2) and (4), Su a11
v2 u01 ÿ v1 u02 u1 v2 ÿ u2 v1
(5)
jv tanÿ1
u1 u02 ÿ u2 u01 u1 v02 ÿ u2 v01
(6)
Sv
a21 u1 u02 ÿ u2 u01 sin jv u1 v2 ÿ u2 v1 sin jv
(7)
Postle and Norton [5] have discussed the mechanics of fabric deformation and have identi®ed four modes to characterise this. These are fabric strain, fabric bending, in-plane yarn bending or shear and yarn twist. For this model, only the variation associated with strain and shear is considered. It is assumed that the energy associated with bending across triangle edges is a constant. For strain energy Es, Z Z 0:5Ksu Su ÿ 12 du dv Es Z Z 0:5Ksv Sv ÿ 12 du dv 0:5AfKsu Su ÿ 12 Ksv Sv ÿ 12 g
(8)
where A is the area of the 2D triangle and Ksu and Ksv are the strain constants for the weft and warp directions respectively.
The energy model described above is able to associate energy values to the geometric distortions that must occur to a 2D pattern when forced to assume a 3D non-developable shape. These energy values are sensitive to the orientation of the 2D pattern in relation to the woven structure of the fabric concerned. For any particular application, there will be an in®nite number of possible 2D patterns Ð all with associated energy distributions that must be applied to the pattern in order to take up the 3D shape. However, if it is required to adopt the minimum energy criterion then this would indicate a unique solution. The authors have developed an algorithm whereby incremental trial movements aligned with the x 0 , ÿx0 , y0 and ÿy0 directions on the 2D pattern are applied to each node. The movement offering the greatest reduction in energy (if there is one) is then registered with that node. The node with the greatest reduction in energy registered then has the movement implemented. The node concerned and all connected nodes have the trial movements re-calculated and the process is repeated. This iterative process continues until no further energy reductions are possible or are considered negligible. In this way, areas of high energy concentrations can be removed by dissipating the energy requirements into surrounding triangles. 6. Numerical example 6.1. Initial energy state of triangles To illustrate the energy model and algorithm performance, the hexahedron example will be considered further. As described above an initial pattern was derived by concentrating all of the distortion in triangles P1 and P6 only. The other triangles do not require any distortion in order to assume their respective 3D shapes. As an example, consider triangle P1. The actual initial comparison between the 2D and the 3D distorted form is depicted in Fig. 4 where each triangle has the same node (A and A0 ) located at the (u; v) origin and the weft direction is aligned with the u axis. With this alignment, the equivalent woven element distortion can be extracted and is displayed in Fig. 5(b).
316
J. McCartney et al. / Journal of Materials Processing Technology 107 (2000) 312±318
From this diagram in can be seen that the geometric distortion is conveyed by the co-ordinate mappings as described by Eq. (3) and in this case is
From Eq. (8), the strain energy can be calculated with the area A, of the ¯attened triangle P01 estimated as A 56:16 mm. Then,
1:0; 1:0 ! 0:99; 0:78;
Es 0:5AfKsu Su ÿ 12 Ksv Sv ÿ 12 g
1:0; 0:0 ! 1:14; 0:00
Therefore, u1 1:0; v1 1:0; u2 1:0; v2 0:0; u01 0:99; v01 0:78; u02 1:14; v02 0:0 From Eq. (5), the scaling in the weft direction Su can be determined. 0 v2 u1 ÿ v1 u02 1:0 1:14 1:141 Su a11 u1 v 2 ÿ u2 v 1 1:0 From Eq. (6), the collapsing shear angle jv , can be found. 0 0 ÿ1 u1 u2 ÿ u2 u1 ÿ1 1:14 ÿ 0:99 jv tan tan ÿ0:78 u1 v02 ÿ u2 v01 ÿ0:193 From Eq. (7), the scaling in the warp direction Sv can be determined. a21 u2 u02 ÿ u2 u01 1:14 ÿ 0:99 Sv sin jv u1 v2 ÿ u2 v1 sin jv ÿsin ÿ0:193 0:795 Hence an examination of the weave element distortion has provided the scaling and shear values that can be applied to the triangle P1 as a whole now. To calculate the energy value for triangle P1, it is now necessary to specify precise values for the strain constants and shear modulus. From material characteristics data, the following typical values were selected. Ksu 10:0 N=mm; Ksv 10:0 N=mm; Kr 1:0 N=mm
0:5 56:16f10:0 0:1412 10:0 ÿ0:2052 g 17:37 N mm From Eq. (9), the shear strain energy can be calculated for the triangle P01 as Er 0:2AKr j2v 0:5 56:161:0 ÿ0:1932 1:05 N mm Hence the total energy involved in making triangle P01 assume the 3D shape is 18.42 Nmm. Since the geometric distortion experienced by triangle P06 is the same and its orientation with respect to the weave structure is symmetrical with that of P01 , then the energy values will initially be identical for these two triangles. 6.2. Application of algorithm Since all of the distortion that is required to make the 2D pattern in Fig. 3 assume the 3D hexahedron shape is concentrated in only two triangles, this represents a clearly sub optimal solution. The algorithm described above was applied to this situation in an attempt to reduce the total energy applied to all six triangles. The progress of this algorithm is shown in Fig. 6 and Fig. 7. These illustrate the reduction in total energy as other triangles start to share the distortion required to assume the 3D shape. As can be seen from the intermediate stages displayed and the ®nal pattern shape, some symmetry exists in the pattern. This is because each triangle has a symmetrical equivalent triangle. Hence triangle P1 is matched with triangle P6, P2 with P5 and
Fig. 6. Algorithm progress in reducing polygon strain energy.
J. McCartney et al. / Journal of Materials Processing Technology 107 (2000) 312±318
317
Fig. 7. Effect of algorithm progress on 2D pattern and 3D weave.
P3 with P4. Also depicted in Fig. 7 is a plan view of the 3D hexahedron. Since this shape is ®xed, the only variation that is present during the progress of the algorithm is the distribution of the woven structure as it changes between triangles. 7. Conclusion This paper presents an energy model that can be applied to woven materials when they are forced to assume 3D shapes that constitute non-developable surfaces. A simple numerical example is presented which illustrates the application of this energy model and how it can be implemented through a ¯attening algorithm that attempts to minimise the total energy required to apply the material to the 3D shape. Thus the patterns derived are not only sensitive to the 3D shape concerned but also the mechanical properties of the fabric and the orientation of the pattern with respect to the weave.
Acknowledgements The authors would like to acknowledge the support of the Engineering and Physical Sciences Research Council of Great Britain through Grant Ref GR/L35638 and the collaboration of their industrial partners, Marks and Spencer, Coats Viyella Garments and Computer Design Inc. Appendix A. Conversion of 2D pattern Co-ordinates to Weft and Warp Co-ordinates Let the weft and warp axes system (wf ; wp ) be superimposed on the 2D pattern co-ordinate system (x0 , y0 ) with both origins co-incident (Fig. 8). Let the weft and warp increments in each of the directions wf and wp be represented by the vectors wf and wp , respectively. These vectors will have magnitudes given by the wfinc and wpinc increment values. Let wa represent the angle the weft direction makes with the x0 direction. Then these vectors
8. Further work The scope and detail of the deformation model requires further work to incorporate out-of-plane bending or buckling. This mode of deformation will require an examination of many triangles at the same time since it cannot be accurately modelled on an inter-triangle basis. Also Eq. (8) may require some re®ning so that the energy models for extension and compression are handled differently. The work described here is part of a project that is attempting to develop a computer aided design system for garment designers and manufacturers that will provide them with detailed predictions of 2D patterns and simulations of garment appearance. Such CAD systems will require detailed information on fabric characteristics and behaviour in an attempt to remove a large part of the guesswork associated with the production of garment samples.
Fig. 8.
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J. McCartney et al. / Journal of Materials Processing Technology 107 (2000) 312±318
can be de®ned as wfinc cos wa wf wfinc sin wa ÿwpinc sin wa wp wfinc sin wa If a particular point on the (x0 , y0 ) axes has co-ordinates (x01 ; y01 ), then these will be mapped to (wf1 ; wp1 ) on the weft and warp axes as follows: 0 0 x1 x1 w w wf1 wf p1 p y01 y01
References [1] I.D. Faux, M.J. Pratt, Computational Geometry for Design and Manufacture, Ellis Harwood, Chichester, p. 116, 1985. [2] M.M. Lipschutz, Differential Geometry, McGraw Hill, New York, p. 117, 1969. [3] G. Wolberg, Digital Image Warping, IEEE Computer Society Press, pp. 41±94, 1990. [4] D.X. Gong, B.K. Hinds, Energy based drape modelling of fabric, in: The Third Biennial World Conference on Integrated Design and Process Technology, Berlin, 6±9 July, 1998. [5] R. Postle, A.H. Norton, Mechanics of complex fabric deformation and drape, J. Appl. Polym. Sci. Appl. Polym. Symp. 47 (1991) 323±340.
An energy based model for the ¯attening of woven fabrics J. McCartney*, B.K. Hinds, B.L. Seow, D. Gong School of Mechanical and Manufacturing Engineering, The Queen's University of Belfast, Belfast, Northern Ireland BT9 5AH UK
Abstract Applications such as garment manufacture and composite structure fabrication require a two dimensional (2D) woven material to assume a three dimensional (3D) shape. The speci®cation of the process is usually initiated by de®ning the 3D surface. Hence, the problem arises of determining the best 2D pattern. The problem is made more complex by the anistropic nature of woven fabrics which are often used as raw material. Such materials display a variation in mechanical properties with respect to the woven structure. This paper presents a model for determining the optimum 2D pattern for a speci®ed 3D surface where optimality is determined in terms of minimising the energy distribution required to force the 2D pattern to assume the 3D shape. The 3D surface speci®cation is assumed to consist of a polygonal mesh. The model allows af®ne transformations to be applied to the weave structure which can be unique for each polygon in the mesh. Important considerations in the modelling process include the following: 1. The degree to which the speci®ed 3D surface departs from a developable surface. 2. The energy components used to model the woven structure and their sensitivity to weave direction. Essentially, these stem from tensile strain in each direction of the weave and shear strain. 3. The prediction of weave geometry as it reacts to the energy distribution being applied. The model is demonstrated by applying it to a relatively simple pyramidal 3D shape. Energy values are optimised to produce a pattern that requires the minimum overall energy to be applied to the 2D pattern in order for it to assume the 3D shape. This 2D pattern is sensitive to the orientation of the woven structure with predictions being made of how the woven structure will behave in 3D. # 2000 Elsevier Science B.V. All rights reserved. Keywords: Flattening of woven fabrics; Woven fabric element; Af®ne transformation
1. Introduction In many industries, it is required to manufacture three dimensional (3D) structures or surfaces from two dimensional (2D) raw material. It is common for the design function to originate a speci®cation of the 3D shape and then for a technical department to convert this speci®cation into the requisite 2D pattern. Such situations range from ship hull production to garment manufacture. The problem is complicated by the fact that the 3D shape speci®cation might be of a generalised form for which an exact 2D pattern may not exist. In this context, `exact' means that the 2D pattern can assume the 3D shape by bending only of the raw material. Such 3D surfaces are described as being developable [1]. Situations where a non-developable surface have been speci®ed imply that that some amount of distortion has to be applied to the 2D pattern in order for it to assume the 3D shape. There will exist many 2D patterns for a *
Corresponding author. Tel.: 28902-74127; fax: 28906-61729. E-mail address: [email protected] (J. McCartney).
given 3D surface, each requiring different distortion pro®les in order to deform the 2D pattern in a unique way to obtain the 3D shape. In such circumstances, some criterion must be speci®ed to determine the optimum 2D pattern. Speci®cally, this paper examines the situation where the raw material is of a woven construction. This is the case with the design of many garments and composite structures. The major problem here is that the material displays anistropic behaviour regarding its ability to distort relative to the orientation of the weave. The model described below attempts to obtain a minimum energy solution to ®nding a 2D pattern for a non-developable surface, assuming that the raw material is of woven construction. 2. 3D Surface representation It is assumed that the speci®cation of the 3D surface will be in the form of a polygon list that contains pointers to the vertices used to de®ne each triangular polygon. This is an
0924-0136/00/$ ± see front matter # 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 4 - 0 1 3 6 ( 0 0 ) 0 0 6 9 4 - 4
J. McCartney et al. / Journal of Materials Processing Technology 107 (2000) 312±318
313
Fig. 2. Initial 2D ¯attening of hexahedron.
Fig. 1. Example hexahedron surface.
approximation process usually since the original design may have been of a continuous form. As a simple example, consider the hexahedron shown in Fig. 1. A list speci®cation of this is given in Tables 1 and 2. 3. 2D Pattern representation Now consider the problem of constructing a one-piece pattern for the hexahedron in Fig. 1. There is no pattern that will take up this shape without some form of distortion. Let the 2D axes system for such patterns be represented by (x0 , y0 ). From classical treatment of surface geometry [2], this shape can be considered as an approximation to an elliptical surface. As such, it will display an angular defect of 41.218 when neighbouring triangles are sequentially laid on the 2D plane (Fig. 2). If a one-piece pattern is required, then some Table 1 3D Surface node list Node
1 2 3 4 5 6 0
distortion has to take place. As an initial attempt, force the two locations (600 ) for node 6 to a mean position (60 ). The resultant pattern will have triangles P02 , P03 , P04 and P05 remaining undistorted but triangles P01 and P06 will have some degree of distortion introduced. The one-piece pattern that results can now be superimposed onto an underlying woven structure. This structure can be speci®ed by the following parameters Weft increment, wfinc . Warp increment, wpinc . Weft angle, wa . In effect, these parameters constitute a weft and warp axes system (wf ; wp ) for the pattern. Assuming that the (wf ; wp ) origin is located at the same position as the (x0 , y0 ) origin, then Appendix A provides a transformation which can be used to provide (wf ; wp ) co-ordinates from (x0 , y0 ) co-ordinates. If the following values are assumed for the warp and weft geometry, wfinc 1:0;
wpinc 1:0;
wa 50
Table 3 can be compiled. This speci®cation only has to contain the new co-ordinates for each node since the polygon composition is the same as Tables 1 and 2.
3D co-ordinates X
Y
Z
5.00 ÿ5.00 ÿ10.00 ÿ5.00 5.00 10.00 0.00
8.67 8.67 0.00 ÿ8.67 ÿ8.67 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 5.00
4. Deformation of a woven fabric element At this stage a 2D pattern has been precisely speci®ed. This is displayed in Fig. 3 where alternate triangles are shaded lighter and darker to highlight the triangular Table 3 Initial pattern node co-ordinates Node
Table 2 3D Surface polygon speci®cation
2D pattern x
Polygon
Node pointers
P1 P2 P3 P4 P5 P6
6, 1, 2, 3, 4, 5,
1, 2, 3, 4, 5, 6,
0 0 0 0 0 0
10 20 30 40 50 60 00
0
3.13 ÿ6.71 ÿ11.18 ÿ6.71 3.13 10.46 0.00
2D weft and warp y
0
10.73 8.94 0.00 ÿ8.94 ÿ10.73 0.00 0.00
wx
wy
9.80 1.58 ÿ7.91 ÿ11.07 ÿ5.38 7.40 0.00
5.38 11.07 7.91 ÿ1.58 ÿ9.80 ÿ7.40 0.00
314
J. McCartney et al. / Journal of Materials Processing Technology 107 (2000) 312±318
Fig. 5. (a) Af®ne transformation; (b) af®ne transformation for woven element in triangle P1.
Fig. 3. Hexahedron 2D pattern on weft and warp axes.
composition. For this pattern to assume the 3D shape of the hexahedron, strain must be applied consistent with the geometric distortions implied above. i.e. that only triangles P01 and P06 are distorted. The approach adopted here is to calculate the energy required to perform such distortions on an individual triangle basis. The process starts by displaying individual triangles on a unifying axes system (u, v) for comparing the shapes of the two forms that exist for each triangle. First the 2D undeformed triangle is positioned so that one of its vertices is coincident with the (u, v) origin with the weft direction aligned with the u direction. Then the 3D deformed triangle is superimposed on the same axes system using the same positioning conditions. Note that the deformed 3D triangle must maintain the same woven structure composition as the 2D in terms of the total number of woven elements and how the weft and warp directions intersect with the edges of the triangle. This process is depicted in Fig. 4 for triangle P1. The 2D triangle (P01 ) is A0 B0 C0 and the deformed triangle on the 3D surface (P1) is ABC. This unifying process enables the speci®cation of the geometric transformation that a representative rectangular woven element must undergo in order for the complete triangle A0 B0 C0 to deform to ABC. The total energy required for the distortion can be calculated by determining the energy to distort the representative weave element and then applying the same transformation to the triangle as a whole. The distortion of the representative woven element can be represented by an af®ne transformation [3]. The general
Fig. 4. Unifying (u,v) axes system for triangles P1 (ABC) and P01 (A0 B0 C0 ).
representation of an af®ne transformation is 0 1 a11 a12 0 u0 v0 1 u v 1 @ a21 a22 0 A a31 a32 1
(1)
This transformation has the property that parallel lines representing the weft and warp alignment of the 2D triangle ABC will remain parallel within the deformed 3D triangle A0 B0 C0 . It has been described elsewhere [4] that a 2D rectangular woven element can be transformed to the types of shapes experienced by woven fabrics with the following sequence of transformations (Fig. 5). First the rectangular element is differentially scaled in the u and v directions by the scaling factors Su and Sv . Secondly, the resultant shape is subjected to a collapsing shear of jv , with respect to the warp direction in which the lengths of the sides are preserved. For a particular af®ne transformation, the af®ne transformation matrix will now be referred to as Mcomp where this matrix is comprised as follows: u0
v0
where M comp
1 u 0
v 1 M comp
Su B @0
0 Sv
10 1 0 CB 0 A@ sin jv
0
0
1
0
Su B @ Sv sin jv 0
0 cos jv
0 0 Sv cos jv 0
1
0
0 C 0A
1 0 C 0A 1 (2)
1
This matrix is of af®ne form as described in Eq. (1). For this model, all the 2D and 3D co-ordinates for a particular deformation of a triangle on the original surface will be known. It is, therefore, necessary to synthesise the net transformation into its constituent scaling factors (Su and Sv ) and shear angle (jv ) in order to accurately determine energy composition. A particular af®ne transformation as described by Fig. 5 (a) will be speci®ed by the mapping of two corners only as the third corner is at the origin for both triangles i.e. u1 ; v1 ! u01 ; v01 ;
u2 ; v2 ! u02 ; v02
(3)
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Eq. (1) can now be rewritten to include these two mappings and provide a means for solving for Su ; Sv and jv . 10 1 0 1 0 0 0 1 a11 a12 0 0 0 1 @ u0 v0 1 A @ u1 v1 1 A@ a21 a22 0 A 1 1 u02 v02 1 u2 v 2 1 a31 a32 1 or U 0 UA
The shear energy Er, is given as Z Z 0:5Kr j2v du dv 0:5A Kr j2v Er
315
(9)
where Kr is the shear modulus. Hence, given the point mappings described by Eq. (3), the strain constants Ksu and Ksv and the shear modulus Kr , the total strain energy (Es Er ) required to deform the triangle can be calculated.
Hence A U ÿ1 U 0
5. Energy minimisation algorithm
where
0
v1 ÿ v2 v2 1 @ u2 ÿ u1 ÿu2 det U u1 v2 ÿ u2 v1 0
U ÿ1
1 ÿv1 u1 A 0
and det U u1 v2 ÿ u2 v1 This results in A
1 det U 0 v2 u01 ÿ v1 u02 B @ u1 u02 ÿ u2 u01
v2 v01 ÿ v1 v02
0
u1 v02 ÿ u2 v01
0
0
u1 v2 ÿ u2 v1
0
1 C A (4)
By comparing Eqs. (2) and (4), Su a11
v2 u01 ÿ v1 u02 u1 v2 ÿ u2 v1
(5)
jv tanÿ1
u1 u02 ÿ u2 u01 u1 v02 ÿ u2 v01
(6)
Sv
a21 u1 u02 ÿ u2 u01 sin jv u1 v2 ÿ u2 v1 sin jv
(7)
Postle and Norton [5] have discussed the mechanics of fabric deformation and have identi®ed four modes to characterise this. These are fabric strain, fabric bending, in-plane yarn bending or shear and yarn twist. For this model, only the variation associated with strain and shear is considered. It is assumed that the energy associated with bending across triangle edges is a constant. For strain energy Es, Z Z 0:5Ksu Su ÿ 12 du dv Es Z Z 0:5Ksv Sv ÿ 12 du dv 0:5AfKsu Su ÿ 12 Ksv Sv ÿ 12 g
(8)
where A is the area of the 2D triangle and Ksu and Ksv are the strain constants for the weft and warp directions respectively.
The energy model described above is able to associate energy values to the geometric distortions that must occur to a 2D pattern when forced to assume a 3D non-developable shape. These energy values are sensitive to the orientation of the 2D pattern in relation to the woven structure of the fabric concerned. For any particular application, there will be an in®nite number of possible 2D patterns Ð all with associated energy distributions that must be applied to the pattern in order to take up the 3D shape. However, if it is required to adopt the minimum energy criterion then this would indicate a unique solution. The authors have developed an algorithm whereby incremental trial movements aligned with the x 0 , ÿx0 , y0 and ÿy0 directions on the 2D pattern are applied to each node. The movement offering the greatest reduction in energy (if there is one) is then registered with that node. The node with the greatest reduction in energy registered then has the movement implemented. The node concerned and all connected nodes have the trial movements re-calculated and the process is repeated. This iterative process continues until no further energy reductions are possible or are considered negligible. In this way, areas of high energy concentrations can be removed by dissipating the energy requirements into surrounding triangles. 6. Numerical example 6.1. Initial energy state of triangles To illustrate the energy model and algorithm performance, the hexahedron example will be considered further. As described above an initial pattern was derived by concentrating all of the distortion in triangles P1 and P6 only. The other triangles do not require any distortion in order to assume their respective 3D shapes. As an example, consider triangle P1. The actual initial comparison between the 2D and the 3D distorted form is depicted in Fig. 4 where each triangle has the same node (A and A0 ) located at the (u; v) origin and the weft direction is aligned with the u axis. With this alignment, the equivalent woven element distortion can be extracted and is displayed in Fig. 5(b).
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From this diagram in can be seen that the geometric distortion is conveyed by the co-ordinate mappings as described by Eq. (3) and in this case is
From Eq. (8), the strain energy can be calculated with the area A, of the ¯attened triangle P01 estimated as A 56:16 mm. Then,
1:0; 1:0 ! 0:99; 0:78;
Es 0:5AfKsu Su ÿ 12 Ksv Sv ÿ 12 g
1:0; 0:0 ! 1:14; 0:00
Therefore, u1 1:0; v1 1:0; u2 1:0; v2 0:0; u01 0:99; v01 0:78; u02 1:14; v02 0:0 From Eq. (5), the scaling in the weft direction Su can be determined. 0 v2 u1 ÿ v1 u02 1:0 1:14 1:141 Su a11 u1 v 2 ÿ u2 v 1 1:0 From Eq. (6), the collapsing shear angle jv , can be found. 0 0 ÿ1 u1 u2 ÿ u2 u1 ÿ1 1:14 ÿ 0:99 jv tan tan ÿ0:78 u1 v02 ÿ u2 v01 ÿ0:193 From Eq. (7), the scaling in the warp direction Sv can be determined. a21 u2 u02 ÿ u2 u01 1:14 ÿ 0:99 Sv sin jv u1 v2 ÿ u2 v1 sin jv ÿsin ÿ0:193 0:795 Hence an examination of the weave element distortion has provided the scaling and shear values that can be applied to the triangle P1 as a whole now. To calculate the energy value for triangle P1, it is now necessary to specify precise values for the strain constants and shear modulus. From material characteristics data, the following typical values were selected. Ksu 10:0 N=mm; Ksv 10:0 N=mm; Kr 1:0 N=mm
0:5 56:16f10:0 0:1412 10:0 ÿ0:2052 g 17:37 N mm From Eq. (9), the shear strain energy can be calculated for the triangle P01 as Er 0:2AKr j2v 0:5 56:161:0 ÿ0:1932 1:05 N mm Hence the total energy involved in making triangle P01 assume the 3D shape is 18.42 Nmm. Since the geometric distortion experienced by triangle P06 is the same and its orientation with respect to the weave structure is symmetrical with that of P01 , then the energy values will initially be identical for these two triangles. 6.2. Application of algorithm Since all of the distortion that is required to make the 2D pattern in Fig. 3 assume the 3D hexahedron shape is concentrated in only two triangles, this represents a clearly sub optimal solution. The algorithm described above was applied to this situation in an attempt to reduce the total energy applied to all six triangles. The progress of this algorithm is shown in Fig. 6 and Fig. 7. These illustrate the reduction in total energy as other triangles start to share the distortion required to assume the 3D shape. As can be seen from the intermediate stages displayed and the ®nal pattern shape, some symmetry exists in the pattern. This is because each triangle has a symmetrical equivalent triangle. Hence triangle P1 is matched with triangle P6, P2 with P5 and
Fig. 6. Algorithm progress in reducing polygon strain energy.
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317
Fig. 7. Effect of algorithm progress on 2D pattern and 3D weave.
P3 with P4. Also depicted in Fig. 7 is a plan view of the 3D hexahedron. Since this shape is ®xed, the only variation that is present during the progress of the algorithm is the distribution of the woven structure as it changes between triangles. 7. Conclusion This paper presents an energy model that can be applied to woven materials when they are forced to assume 3D shapes that constitute non-developable surfaces. A simple numerical example is presented which illustrates the application of this energy model and how it can be implemented through a ¯attening algorithm that attempts to minimise the total energy required to apply the material to the 3D shape. Thus the patterns derived are not only sensitive to the 3D shape concerned but also the mechanical properties of the fabric and the orientation of the pattern with respect to the weave.
Acknowledgements The authors would like to acknowledge the support of the Engineering and Physical Sciences Research Council of Great Britain through Grant Ref GR/L35638 and the collaboration of their industrial partners, Marks and Spencer, Coats Viyella Garments and Computer Design Inc. Appendix A. Conversion of 2D pattern Co-ordinates to Weft and Warp Co-ordinates Let the weft and warp axes system (wf ; wp ) be superimposed on the 2D pattern co-ordinate system (x0 , y0 ) with both origins co-incident (Fig. 8). Let the weft and warp increments in each of the directions wf and wp be represented by the vectors wf and wp , respectively. These vectors will have magnitudes given by the wfinc and wpinc increment values. Let wa represent the angle the weft direction makes with the x0 direction. Then these vectors
8. Further work The scope and detail of the deformation model requires further work to incorporate out-of-plane bending or buckling. This mode of deformation will require an examination of many triangles at the same time since it cannot be accurately modelled on an inter-triangle basis. Also Eq. (8) may require some re®ning so that the energy models for extension and compression are handled differently. The work described here is part of a project that is attempting to develop a computer aided design system for garment designers and manufacturers that will provide them with detailed predictions of 2D patterns and simulations of garment appearance. Such CAD systems will require detailed information on fabric characteristics and behaviour in an attempt to remove a large part of the guesswork associated with the production of garment samples.
Fig. 8.
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can be de®ned as wfinc cos wa wf wfinc sin wa ÿwpinc sin wa wp wfinc sin wa If a particular point on the (x0 , y0 ) axes has co-ordinates (x01 ; y01 ), then these will be mapped to (wf1 ; wp1 ) on the weft and warp axes as follows: 0 0 x1 x1 w w wf1 wf p1 p y01 y01
References [1] I.D. Faux, M.J. Pratt, Computational Geometry for Design and Manufacture, Ellis Harwood, Chichester, p. 116, 1985. [2] M.M. Lipschutz, Differential Geometry, McGraw Hill, New York, p. 117, 1969. [3] G. Wolberg, Digital Image Warping, IEEE Computer Society Press, pp. 41±94, 1990. [4] D.X. Gong, B.K. Hinds, Energy based drape modelling of fabric, in: The Third Biennial World Conference on Integrated Design and Process Technology, Berlin, 6±9 July, 1998. [5] R. Postle, A.H. Norton, Mechanics of complex fabric deformation and drape, J. Appl. Polym. Sci. Appl. Polym. Symp. 47 (1991) 323±340.