Design and 3D simulation of open topping-on structured crochet fabric

2.1 Knitting principle

As shown in Figure 1, the position of each stopper pin is precisely controlled by the stopper mechanism through mechanical or electronic control systems. The lapping movement is achieved through coordinated timing of extension and retraction movements of the stopper pins with positional and trajectory variations of the guide needles, resulting in the formation of weft-insertion structures with specific lengths and configurations. In the warp knitting machine, more versatile loop geometries are enabled by the application of stopper pins through weft yarn motion, whereby aesthetically varied, highly decorative crochet fabrics with superior shape definition are produced.

Figure 1

The Structure of stitch-forming mechanisms: (a) Front view and (b) top view.

The latch needle is composed of a cylindrical or flat-cross-sectioned needle shaft and a movable latch. A needle hook is formed at the apex of the shaft, while the latch is rotatably connected to the shaft through a hinge mechanism, enabling free oscillation for hook opening/closing. Due to the self-acting property of the latch mechanism, yarn feeding during lapping is not directionally restricted, allowing bilateral yarn insertion (both left and right sides). Consequently, more complex knitting trajectories can be accommodated, through which either open or closed loop structures are formed.

Wavy fringe lace is primarily produced through variable strokes of the weft-insertion guide bar, by which fringe loops of varying lengths are formed at the selvage edges. At the weft-turning points, positioning hooks must be employed to prevent yarn displacement, where the positioning hooks and weft-insertion mechanism are required to be precisely coordinated [10].

2.2 Knitting process of open topping-on structures

This fabric is primarily formed through synchronized coordination between knitting needles and stopper mechanisms in the timing sequence. Two sets of stopper mechanisms are required to construct looped structures on both left and right sides, which are then consolidated by pillar stitches. The open topping-on structure process is described as follows:

1. First, both stopper pins are simultaneously extended, and yarn 2 (Figure 1) is wrapped around the stopper pins for one complete cycle by the yarn guide device to form the lower loop structure. Meanwhile, yarn 1 (Figure 1) is normally knitted into the first pillar stitch according to the knitting timing sequence.

2. Subsequently, both stopper pins are simultaneously retracted, causing yarn 2 (Figure 1) to be wrapped around the knitting needles, thereby forming the upper loop structure. Concurrently, yarn 1 (Figure 1) is successively knitted into the second, third, and fourth pillar stitches according to the programmed sequence.

3. The open topping-on structure is consolidated by the ground yarn through pillar stitch formation. By repeating the aforementioned steps, the left open topping-on structure is completed.

3.1 Mathematical model of lapping numerical sequence

In the lapping diagram, a continuous line is used to represent the movement trajectory of the yarn threaded through a single guide needle across multiple courses [13]. The lapping numerical model is generally expressed using a single-needle-bed notation system, where a numerical matrix is defined in equation (1) to represent the lapping sequence.

where i ∈ {1, 2, 3, …, I}, i represents the guide bar number; j ∈ {1, 2, 3,…, m}, j denotes the course number in the repeat pattern; and u ∈ {1, 2, 3, …, n}, u indicates the number of fully threaded yarns. Since two digits represent the lapping movement of one course, the lapping motion of a single yarn is described by two columns of digits. Here P(i,j,u) represents the lapping digit of the uth yarn on the ith guide bar for the j course. For example, i = 1, j = 1, p _i,1,1 and p _i,1,2 denote the two lapping digits of the first yarn in the first course during its lapping motion.

For the types of loops, when p _i,j,2u − p _i,j,2u−1 = 0, the loop type is a Weft-insertion loop; when |p _i,j,2u − p _i,j,2u−1| = 1, the loop type is categorized as either standard knitted loops or upper loop structure of open topping-on structures; and when ∣p _i,j,2u − p _i,j,2u−1∣ = 2, the loop type is identified as the lower loop structure of an open topping-on structure. Open loops are formed when lapping movements in the needle front and needle back directions are identical, or when needle-front lapping occurs with zero lateral movement at the needle back. Conversely, closed loops are produced when opposing lapping directions are executed between the needle front and needle back [14]. It follows that when (p _i,j,2u − p _i,j,2u−1) × (p _i,j−1,2u − p _i,j,2u ) ≥ 0, the loop is an open loop. When (p _i,j,2u − p _i,j,2u−1) × (p _{i,j − 1,2u} − p _i,j,2u ) < 0, the loop is a closed loop.

Regarding the guide bar shogging movement, taking the lapping digits of the first yarn as an example, H(i,j,1) = p _i,j,1 − p _i,j−1,2. Here H(i,j,1) represents the direction and magnitude of the back shog for the ith guide bar at the jth course. If H(i,j,1) > 0, it indicates that the ith guide bar performs a back shog from right to left at the jth course. Conversely, if H(i,j,1) < 0, it means the ith guide bar executes a back shog from left to right at the jth course. When H(i,j,1) = 0, no back shogging occurs. The absolute value ∣H(i,j,1)∣ denotes the distance of the back shog.

3.2 Mathematical model of yarn threading

The threading arrangement diagram is utilized to describe the yarn threading status of each guide needle on the guide bars. During process design, the lapping numerical sequence for needle-front lateral movements at each course is composed of two digits, necessitating the horizontal expansion of each yarn’s threading information across two wales. With individual yarns as units, the threading configuration is represented by a block matrix C(i,j) as formulated in equation (2).

where i ∈ {1, 2, 3, …, I}, i represents the guide bar number; j ∈ {1, 2, 3, …, m}, j denotes the course number in the repeat pattern; and u ∈ {1, 2, 3, …, n}, u indicates the number of fully threaded yarns. C(i, j, k) is constructed as the threading sub-matrix for the kth yarn on the ith guide bar, as formulated in equation (3).

where c _i,j,2k–1 and c _i,j,2k represent the threading condition of the kth yarn on the ith guide bar at the jth course of the front needle bed. Within the block sub-matrix C(i, j, k), all elements share the same value (either 0 or 1), determined by the yarn’s threading state.

Taking the threading configuration of the first guide bar (GB1) in Table 3 as an example, when the number of courses is specified as j = 2, the threading matrix is constructed as follows:

The symbolic representation method is as follows: GB1: 3A, 1*, 1A, 2*, 1A, 1*, 3A, 10* or in abbreviated form: GB1: 3-in-1 empty, 1-in-2 empty, 1-in-1 empty, and 3-in-10 empty. The alphabetic characters (A–Z) are designated to represent different yarn types.

4.1 Pillar stitch loop model

Pillar stitch loops are modeled through the following computational process: Key points from physical specimens are measured, and type-value points are programmatically defined. Bézier curve fitting is then applied to generate smooth loop profiles. The fabric structure is consolidated using pillar stitch organization, with the loop model illustrated in Figure 3. Eight control points (S₀–S ₇) are utilized to construct individual loops, where h is defined as the loop height and w as the loop spacing.

Figure 3

Pillar stitch loop model.

4.2 Standard weft-insertion structure model

Since no needle-front lapping is performed by the weft-insertion guide bars (with only needle-back lapping executed), the yarns are clamped between the main bodies and underlaps of the loop-forming stitches [15]. In open topping-on structure fabrics, all weft-insertion structures are consolidated by pillar stitches to form complete constructions, as illustrated in Figure 4. Through physical measurements, a two-point weft-insertion model is established as shown in Figure 5.

Figure 4

Physical diagram of weft-insertion loops.

Figure 5

Weft-insertion loop model: (a) Front view and (b) side view.

4.3 Open topping-on structure model

This structure is recognized as one of the more complex configurations in crochet fabrics. Initially, an idealized loop geometric model is established based on its physical morphology. Through combined analysis with actual fabric measurements, the number and positions of loop type-value points are subsequently determined.

In this study, microscopic measurements are employed to determine loop structural dimensions, with comprehensive high-magnification observation and metrological analysis of the fabric’s surface morphology being conducted to satisfy loop measurement requirements. To enhance measurement accuracy and minimize errors caused by yarn type and diameter variations, dimensional measurements are referenced to yarn centerlines [16]. As shown in Figure 6, through systematic observation of these loop structures’ variation patterns, the 3D model of lower loops in open topping-on structures is constructed with six type-value points, while upper loops are represented by six type-value points.

Figure 6

Physical diagram of open topping-on crochet fabric: (a) Lower loop structure and (b) upper loop structure.

When pillar stitches are fully threaded, the distance between two adjacent pillar stitches is defined as gw, the height of a single pillar stitch loop as gh, and the loop radius as d. During type-value point determination, the coordinate system is first established based on entry and exit point coordinates. The positions of all type-value points are then calculated through the proportional distance relationships and positional references to the pillar stitch organization, ultimately generating the loop geometric model.

The loops are categorized into left loops and right loops. Taking the right loop as an example (as shown in Figure 6), it is divided into the lower loop structure and upper loop structure.

For the lower loop arc structure, the type-value points are designated as P ₀ to P ₅, with their x-coordinates defined as P _0x to P _5x , y-coordinates as P _0y to P _5y , and z-coordinates as P _0z to P _5z . Similarly, for the upper loop structure, the type-value points are assigned as Q ₀ to Q ₅, with their x-coordinates specified as Q _0x to Q _5x , y-coordinates as Q _0y to Q _5y , and z-coordinates as Q _0z to Q _5z . The grid point width and height are established as gw and gh, respectively. The coordinates of all type-value points are systematically presented in Table 1.

Based on the type-value points in Table 1, the loop model shown in Figure 7 is constructed.

Figure 7

The model of open topping-on crochet fabric physical picture: (a) Front view and (b) side view.

4.4 Fringe structure model

As shown in Figure 8, based on the analysis of the fringe structure’s morphology and knitting principles, the edge formation is achieved by increasing the number of type-value points on the weft-insertion loops. An idealized loop model is then constructed.

Figure 8

Physical diagram of fringe loops.

Considering the relative length–width relationship of fringe weft-insertion loops, as exemplified by the lapping sequence “12-12, 18-18”, the fringe structure is characterized where the fringe length is defined as L, and the maximum width is calculated as (L + 12)/12. The type-value points for the fringe structure are systematically presented in Table 2.

Based on the type-value points in Table 2, the loop model shown in Figure 9 is constructed.

Figure 9

Fringe structure loop model: (a) Left-side weft insertion and (b) right-side weft insertion.

5.1 Process design

During the production of open topping-on fabric, GB1 is used to create the pillar stitch foundation, GB2 forms the looped structures, GB3 through GB5 are assigned for weft-insertion patterns, while GB6 is dedicated to fringe formation. The process is executed on a DECORTRONIC-1000-EL machine with E10 gauge (10 needles per inch), producing fabric with 5.0 wales/cm course density and 5.0 courses/cm wale density. Complete technical parameters are provided in Table 3.

5.2 Matrix transformation

After being taken off the machine, the open topping-on fabric is observed to undergo torsional deformations in varying directions and angles due to yarn relaxation. Based on empirical testing, the standard configuration is established as shown in Figure 10, where fringe terminals are documented to rotate around the x-axes, y-axes, and z-axes by defined angular displacements ( − 10 ° < θ x < 10 ° , − 15 ° < θ y < 15 ° , − 20 ° < θ z < 20 ° ) . A Random. next() function is implemented to simulate stochastic fringe twisting behavior.

Figure 10

Physical diagram of fringe terminals in standard configuration.

In simulating the torsional deformation of fabric edges, a matrix transformation algorithm based on rigid body transformation and affine transformation is employed. As shown in equation (5), first, the global coordinate system and the initial coordinates of key points on the fabric are defined, with the midpoint of the line connecting the entry and exit points of the yarn loop selected as the rotation center. Subsequently, all control points A(x,y,z)(N ₁ – N ₆) of the loop are uniformly translated to the local coordinate system through the transformation T ( − O ) . Within the local coordinate system, independent random rotations about the X, Y, and Z axes are applied using the rotation matrices R ( X ) , R ( Y ) , and R ( Z ) , where the rotation angle for each axis is generated within a predetermined range. Finally, the transformed control points are precisely restored to their original spatial positions via the inverse translation T(O), yielding the final coordinates A′(x,y,z) of the twisted loop.

where A₁ represents the matrix of control points translated to the local coordinate system, as shown in equation (6); R denotes the rotation matrix, as shown in equation (10); T ( O ) corresponds to the transformation matrix for restoring the points to their original coordinate system, as shown in equation (14).

The matrix A of type value points is as shown in equation (7).

The translation matrix T ( − O ) is as shown in equation (8).

The translated matrix A ₁ is as shown in equation (9).

Multiply the rotation matrices around different axes to obtain the combined rotation matrix R. The calculation formula of the combined rotation matrix is as shown in equation (10).

The rotation matrix R ( X ) for rotation about the x-axis by angle θₓ is as shown in equation (11).

The rotation matrix R ( Y ) for rotation about the y-axis by angle θ _y is as shown in equation (12).

The rotation matrix R ( Z ) for rotation about the z-axis by angle θ _z is as shown in equation (13).

The translation matrix T ( O ) is as shown in equation (14).

The type-valued point matrix A ′ in the global coordinate system is as shown in equation (15).

During the computational process, the XYZ coordinate origin is positioned at the lower-left corner of the grid. The grid height is defined by j units, while the displacement along the X-axis is calculated as R 1 = − 0.125 × gw + L × gw , the displacement along the Y-axis is calculated as R 2 = gh × j , the displacement along the Z-axis is calculated as R 3 = 22.6 × r .

This specific transformation method is chosen because it can accurately simulate the complex deformations of fabric in its natural state. In particular, when simulating the fringe structure at the fabric edge, the introduction of random rotations more realistically reflects the torsional deformation of fringes under natural conditions, thereby enhancing the realism and natural appearance of the simulation.

5.3 3D fabric simulation

Based on the aforementioned fabric structural design and loop modeling methodology, appropriate process configurations and yarn materials are selected. Through computational analysis using matrix models, the 3D structural simulation of open topping-on crochet fabric is systematically performed, with its digital model being constructed. The simulation is implemented utilizing JavaScript and C# programming languages. The workflow for the design and 3D simulation of open topping-on crochet fabric is illustrated in Figure 11.

Figure 11

Fabric design and 3D simulation flowchart.

As shown in Figure 12, both the physical sample and simulation model of the warp-knitted open-loop fabric are presented.

Figure 12

Open topping-on crochet fabric: (a) Physical photograph, (b) front of the simulation image, (c) back of the simulation drawing, and (d) side view of the simulation diagram.

Based on the above coil modeling, by modifying the process parameters, structure, yarn threading, raw materials, etc., of the fabric, diversified design and simulation of the fabric can be achieved, as shown in Figure 13.

Figure 13

Fabric design simulation drawing: (a) Front of the simulation image, (b) back of the simulation drawing, and (c) side view of the simulation diagram.