An optimized yarn-level geometric model for Finite Element Analysis of weft-knitted fabrics
Graphical abstract
Introduction
The modeling and simulation of textiles has gained increased interest in recent years. These simulation efforts include the modeling and analysis of both woven (Cirio et al., 2014) and knitted (Kaldor et al., 2008) fabrics. Our research focuses on advancing knitted textiles as a substrate for next-generation smart fabrics. A major thrust of this research is the development of design tools that will automate the specification of optimized knitted structures. A critical component of these tools are modeling and simulation technologies that are capable of accurately predicting the properties of knitted cloth, given the properties of its yarns and the stitch patterns/commands used to knit the yarns into a fabric. To attain these goals we have begun to simulate knitted materials at the yarn-level using Finite Element Analysis (FEA) (Liu et al., 2017, Liu et al., 2018, Liu et al., 2019). FEA simulations require proper geometric models as input, which must meet stringent constraints in order for the simulations to be correctly computed. Namely the models should have a physically plausible shape, but more importantly they should not interpenetrate and only touch at point contacts.
Many of the previous models that represent the shape of yarns in a knitted fabric have been purely geometric in nature. These models have defined parameterized control points that specify tube centerlines in 3D space. While these models may be sufficient for visualization purposes, in that they present some possible geometric configurations of the constituent yarns in a knitted fabric, they are inadequate for physically meaningful FEA simulations. They are deficient because, being only geometric representations, they lack the needed topological form and interconnections that define the correct structure of a knitted fabric. For example, these yarn models may interpenetrate, bend in unusual ways or may not include appropriately defined yarn contacts, making them unsuitable for FEA modeling. In order to produce valid initial geometric conditions for our simulation studies we have implemented a yarn-level model of knitted fabrics that incorporates mechanical properties and spatial constraints with the underlying geometric representation of the yarns; thus producing initial geometric models with the correct topology structure that not only do not interpenetrate, but include yarns that meet at point contacts, and additionally have a feasible, physically-accurate overall shape. Our techniques are able to produce parameterized yarn-level geometric models of weft-knitted fabrics consisting of an arbitrary pattern of knit and purl stitches that provide the proper geometric initial conditions needed for a variety of FEA simulations.
The loop is a fundamental structural element of knitted textiles. A new loop is formed when a yarn is drawn through a previously existing loop, as seen in Fig. 11. When this is repeated across a row, and then subsequently again in other rows, the fabric is formed (Spencer, 1983). When the yarn is drawn through the loop(s) held on a needle from back to front, a knit stitch is created, as shown in Fig. 2(a). A knit stitch can be distinguished by the small “v” shapes visible on the fabric, formed by the vertical yarns of the stitches. When the yarn is drawn through the loop(s) held on a needle from front to back, a purl stitch is created (Fig. 2(b)), which is distinguishable by the rows of small arcs visible on the fabric, formed by the horizontal heads and tails of the stitches. It is important to note that knit and purl stitches are structurally the same, differing only by a 180∘ rotation. The designation of the front and back sides of the fabric determines their labeling.
Producing physically-accurate geometric models of yarns in a knitted fabric is framed as an optimization problem. In this computing context, a single “cost” function is defined that captures the various required features of the final geometric model. The function is specified in such a way that finding the variable values that produces a minimum function evaluation generates the desired geometric result (Witkin et al., 1987). The features incorporated into our model, and therefore the associated cost function, include maintaining yarn rest length, minimizing curvature and creating contact points between crossing yarns. The variables that are modified to minimize the cost function are the spline control points that define the centerlines of the tubes used to represent the yarns.
Given that we intend to simulate rather large swatches (e.g. 100 × 100 stitches) of knitted fabrics consisting of knit and purl stitches, our method takes a cell-based approach to generating yarn-level geometric models. Noting that there are only a finite number of local combinations of knit and purl stitches, we have defined eight unique stitch (and therefore yarn geometry) configurations that can be combined to produce any size fabric composed of an aribitary combination of knit and purl stitches. Defining and enforcing strict boundary conditions, both positions and tangents, allows us to perform efficient optimizations on eight small cells, which are then copied and tiled to produce the overall fabric, rather than performing an expensive and lengthy global optimization over the whole fabric.
Once the optimization problem is formulated for a particular set of fabric parameters the cost function is minimized for the eight cells using a quasi-Newton method, which produces the centerline splines that meet the requirements and constraints of the specified FEA initial conditions in each cell. Given a specific stitch pattern, the cell geometries are merged together to produce a single geometric model of the yarns in the associated fabric. The model is then output in a format that can be read by standard FEA software packages and used for the subsequent analyses. Our model has been utilized as the inputs to numerous FEA simulations of knitted fabrics. We provide several output examples with a variety of stitch patterns and for a range of stitch size parameters, which demonstrate the effectiveness of our approach to produce geometric models that are suitable initial conditions for FEA simulations.
Section snippets
Previous work
Knitted cloth has been modeled in a number of ways in the literature. One of the earliest methods, proposed by Peirce (1947), assumed the trajectory of the yarn path swept a circular cross-section over circular arcs connected by straight lines. From this model he was able to derive the relation between yarn length and number of courses/wales per inch. Leaf and Glaskin (1955) proposed a few modifications to the model to achieve better estimates of these quantities. Munden (1959) assumed that the
Optimized yarn-level geometric model
The centerlines of the yarn paths in our model are defined with a set of Catmull-Rom (C-R) spline control points (Catmull and Rom, 1974). It is assumed that the yarn has a circular cross-section around the centerline spline. The model is built up in units. The smallest component of the model, which we call a cell, consists of two C-R splines, as seen in Fig. 3(a), one which represents the left “leg” of a stitch and the other represents the left half of a stitch's “head”. These two splines can
Results
All results included in the paper were generated on an Apple iMac with a 4.2 GHz Intel Core i7 processor and 32 GB of RAM. The computations were done in MATLAB using the unconstrained optimization function fminunc, with parallel computing enabled which allows for simultaneous processing on all 4 cores of our system's CPU. 45 seconds to 1.5 minutes, depending on the stitch parameters, of compute time is needed to optimize all unit cells, and to generate and export the yarn geometry. The values
Conclusion
In this paper, we presented a yarn-level model for knitted fabrics that can be used for FEA simulations. The model is optimized to avoid inter-penetrations while minimizing curvature and maintaining the length of the yarns. The final locations of the yarns' control points are obtained using a quasi-Newton method. These points are used to create cylindrical surfaces representing the surface of the yarns. This information is then written into an IGES file which can then be read by an FEA program
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We would like to thank Bahareh Shakibajaromi, Eric Markowicz, Dani Liu and Chelsea Amanatides for their contributions to earlier versions of this work. The FE analyses reported here were run on hardware supported by Drexel's University Research Computing Facility. The research was partially supported by National Science Foundation grants CMMI-1344205 and CMMI-1537720.
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