Analysis of the fiber laydown quality in spunbond processes with simulation experiments evaluated by blocked neural networks
Abstract
We present a simulation framework for spunbond processes and use a design of experiments to investigate the causeandeffectrelations of process and material parameters onto the fiber laydown on a conveyor belt. The virtual experiments are analyzed by a blocked neural network. This forms the basis for the prediction of the fiber laydown characteristics and enables a quick ranking of the significance of the influencing effects. We conclude our research by an analysis of the nonlinear causeandeffect relations.
spunbond process; machine learning; blocked neural networks; fiber laydown characterization
1 Introduction
Over the last decade, the annual growth rate for global nonwovens production has averaged 7.3 percent. The reason for this is the wide range of applications for nonwovens: They range from hygiene and medicine to construction, home furnishings, clothing, and automobiles. However, the versatility of the fields of applications has the disadvantage that processes should be adapted to the subsequent applications of the nonwovens as far as possible.
Due to their big impact on the nonwovens market – [25] estimates the valuation for the spunbond nonwovens market at USD 11.50 billion in 2018 – spunbond processes experience constant attention in textile research. The review article [20] gives not only a detailed summary of the technical principles and the current market trends of spunbond processes, but also covers the history of the spunbond technology. Apart from that the focus of the latest publications is more on applications of (bicomponent) spunbonded fabrics, e.g., in composites ([6, 17]) or filtration ([12, 21, 28, 31]). More experimental studies that have a big impact on new developments in the spunbond process technology can be found, e.g., in [2, 4, 13]. Hereby, the most important question is how the process conditions influence the nonwoven web quality with respect to mechanical properties like tensile strength or stiffness as well as other properties like cristallinity. We refer to [23] where a broad range of process variables was studied to investigate the relation between process and properties. Current discussions about sustainibility lead to an increasing interest in spunbond processes with polymers from renewable resources like PLA, see, e.g., [29].
According to the current state of art in industry, process and product design is carried out by trial and error on production lines, which is timeconsuming and costintensive. Scaling up experimentalbased optimization results from pilot lines to production plants can be misleading due to the nonlinearity of the influencing factors. In particular the air stream is highly sensitive to varying process conditions and, hence, the turbulent effects of the air within the spunbond processes. In order to support the experimental approach of designing spunbond processes with respect to customerspecific needs, theoretical analyses and simulation methods become a key technology. For example in [16] a physical model is developed describing the fiber properties like fiber diameter, fiber speed, strain rate, stress, temperature and cristallinity dependent of the process conditions. Mathematical models that also describe the laydown of the fibers on the belt can be found in [19] or [30]. A description of the software that implements these models is found here [8].
In this paper, we start by presenting a simulation framework for spunbond processes. With a design of experiments we study the influencing effects of process and material parameters of the fiber laydown on a conveyor belt. Finally, we train a feedforward neural network in order to study a quality criterion of the fiber laydown structure and compare it to a linear in parameter model.
2 Principles of nonwoven production processes and simulation framework of spunbond processes
Spunbond processes follow in principle the process steps extrusion, spinning, drawing, laying down, and bonding. Figure 1 shows a sketch of a typical spunbond process used for the production of nonwovens. On the right hand side of Fig. 1 corresponding physical models describing single process steps are quoted and the currently dominating simulation techniques are listed. Hereby, the enumeration of simulation methods is not complete, but should only be understood as examples.
More precisely, a melted polymer (e.g. polypropylene or polyester) is extruded, filtered, and transported to a socalled spin pack. In the spin pack the melt is forced through hundreds of holes called nozzles or spinnerets. Streams of viscous melt exit the spinnerets and form viscous fibers. The fibers are cooled and stretched by an air flow coming from the side. Then they are driven by compressed air through a channel, the socalled drawing system. After they have left the drawing channel, turbulent air streams entangle them. Finally, they lay down on the conveyor belt and form a random web. Suction underneath the belt prevents the fibers from rebounding. The random web is transported away for further postprocessing steps like mechanical or thermal bonding. We refer to [1] for further details.
In this study we put our focus on the entangeling and laydown phase of fibers within spunbond processes. Hereto, we simulate the air stream starting at the end of the drawing unit to the suction beneath the conveyor belt. Fig. 2 shows the used simulation domain with geometry dimensions. As physical model for the air stream we use the Reynoldsaveraged NavierStokes equations (or RANS equations) with a socalled kturbulence model. The moving conveyor belt is modelled as a porous medium according to Darcy’s law. All simulations of the air stream (CFD simulations) are performed as stationary 2d simulations assuming periodic boundary conditions in cross machine direction (CD). The following considerations therefore hold true only for the center regions of a spunbond process and can not be used to investigate the boundary effects of the fiber laydown themselves. Of course, the study can be extended in the future in order to analyze the boundary effects.
For the fiber dynamics driven by the turbulent air we use a modeling framework based on the theory of Cosserat rods, where the fiber is modeled as a onedimensional object, as the fiber diameter is negligibly small compared to its length. The fiber is modeled by a curve describing its centerline and an orientation of its crosssection, given by a set of directors forming an orthogonal basis. This general framework consists of equations for the fiber’s kinematics and dynamics, and is complemented with a material model and geometry model specific to the spunbond process, cf. [30]. The material model assumes elastic, inextensible behavior of the fiber in the lower part of the production process, whereas the geometry model assumes a circular crosssection of the fiber with a constant radius. Using these assumptions we arrive at a simplified string model for the fiber, given as:
(1) 
Here, the fiber centerline is a function of the material parameter as well as time , is the line density [] (also called titer) of the fiber, is the bending stiffness [], is the tangential contact force [], and includes all external line forces []. Note that the material parameters and are prescribed along the fiber, i.e., as input of the simulation, whereas the centerline and contact force are computed by the simulation, i.e. output.
The line force is the sum of all external forces, including aerodynamic forces, gravity, and contact forces arising from the deposition of the fibers onto the conveyor belt. Modeling these forces is crucial for the correct simulation of the fibers, see [22] for the aerodynamic forces, [10, 27] for the contact forces. For the contact with the conveyor belt, the contact forces take a simple form as {linenomath*}
(2)  
(3) 
where the belt is given as a plane with normal through the point , and the magnitude of the force is computed as a Lagrange multiplier to the nonpenetration constraint (3). Further, this is completed by a friction model.
For the simulation the resulting partial differential equations are discretized in time and space and then integrated using the implicit Euler method. This requires solving a nonlinear system of equations for each time step, which in turn is solved using Newton’s method. More details about the discretization scheme and an industrial application of this simulation method for the dynamics of staple fibers can be found in [9].
3 Design of experiments and corresponding CFD and fiber laydown simulation results
The model of the fiber dynamics and laydown requires a description of the external forces acting on the fibers as well as specification of their material properties. Hereto, we use the geometry with the mesh specified in Fig. 2 as starting point for all following trials. We perform a series of simulations using a design of experiments (DoE) in two steps: first we create a base data set of the effects of the material parameters, which is then augmented by the CFD parameters. Thus, for the first part of the DoE, we vary the material properties while using a fixed CFD data set corresponding to the central values , , and fixed process parameters. Hereby, denotes the inlet speed of the air at the end of the drawing unit, while denotes the pressure of the suction under the conveyor belt (compare Fig. 2). The numerical simulation parameters are also kept constant, i.e., we use a discretization time , discretization length , and total simulation time , producing of fiber length at a spinning speed of . For the material parameters, we use a Latin Hypercube scheme with 100 points in the following ranges:

E modulus in the range to ,

density in the range to , and

line density (titer) in the range to .
The distribution of these material parameters is shown in Fig. 4, where we have used a normalized scale for better visualization.
In a second step, we augment the DoE by an additional simulations where all five parameters are varied, i.e., the two process parameters of the CFD simulation and the three material parameters of the fiber dynamics simulation. Due to the very high computation costs of the CFD simulation, again only the nine discrete values as given in Fig. 4 are used. Also, the three material parameters are no longer continuously sampled but instead correspond to the normalized values , and each. With this augmentation we get a data set of simulation settings which gives a good sampling of both the extreme values as well as the interior of the 5dimensional input parameter range.
From the fiber dynamics simulation, we want to judge the quality of the resulting nonwoven. As the simulation produces endless filaments, the part of the fiber that is still in air is discarded and only the laydown is considered, i.e. the part of the fiber that has already been deposited onto the conveyor belt, see Fig. 7. Of this laydown a "backtracked" version is computed where the transport of the fiber along the belt is subtracted. Thus, we reconstruct a distribution of the fiber below the spinneret. Since the simulation is initialized with a short fiber being spun into a long filament, the fiber initially can have a different behavior while the free end is in air compared to the real process where parts of the fiber are already laid down. We also discard the first part of the fiber laydown to remove this outlier effect, see Fig. 7.
We extract three statistical parameters , and from this laydown which represent the overall stochastic structure of the laydown. Assuming the fiber laydown corresponds to a twodimensional normal distribution, we compute the standard deviations and of the throwing ranges in MD and CD direction. Additionally, we compute a parameter that corresponds to the stochasticity of the fiber deposition, where small values of correspond to a deterministic deposition and to a completely stochastic process. The reader is referred to [19] for details of the computation of .
Using these three statistical properties of the simulated representative fibers, we can parametrize a stochastic surrogate model based on a Wiener process, which in a further step allows us to compute a full virtual nonwoven sample consisting of thousands of fibers. This sample can then be analyzed with regard to homogeneity, base weight distribution, etc., giving a measure of quality. However, in the following, we directly use the parameters extracted from the fiber simulation as a proxy for predicting quality, i.e., we consider these as the output values of our simulations which we want to predict given a set of input values.
4 Study of the influencing parameters by blocked neural networks
In this section, we analyze the influence of the process parameters spinning speed and pressure of the suction as well as of the material parameters E modulus, density, and line density on the fiber laydown characterized by , , and . For simplicity, we denote the process/material parameters as input variables of the spunbond system, while the fiber laydown characteristics are denoted as output variables. A first, rough look at the resulting simulation data shows that the effect of the input variables is of nonlinear nature. Hence, we use a generalized regression model for the following analysis.
A good choice for generalized regressions models are feedforward neural networks. The main advantage of neural networks to other regression models is their universal approximation framework realized by their special architecture. Since many neural networks are constructed by single neurons, we briefly give a short introduction of the mathematical concept of neurons or socalled perceptrons.
A single neuron consists of inputs , with corresponding weights . Sometimes a bias corresponding to a permanent input of s added. A socalled activation function is attached to the definition of a neuron. Since we consider single neurons, we regard only one output variable . Together these parts form the neuron by implementing the following two rules: First, the weighted sum of all input variables is computed, i.e., (including the bias), then the activation function is applied, i.e., {linenomath*}
Graphically, neurons can be presented by one of the following two typical network diagrams.
In [5] or [14] it is shown that fully connected neural networks are able to approximate arbitrary continuous functions with arbitrary accuracy. Furthermore, in [15] it is proven that neural networks with appropriate smooth activation functions are able to approximate the derivatives of the regression functions – useful, e.g., for optimization. Since our main focus of this paper is to gain more insight into the causeandeffectrelations from process/material parameters to the fiber laydown characteristics, we use a blocked neural network approach.
A blocked neural network has one (socalled hidden) layer with blocks of neurons. All neurons in each block have – besides the bias – only one input parameter. We denote the input parameters by , . The number of neurons in each block does not need to be equal. We denote the number of neurons in the block by . The weight coming from the input to the th neuron in block is denoted by . Similarly, the weight of the bias to the th neuron in the th block is abbreviated by . The weights for the summation of the neurons in the hidden layer are denoted by , and , respectively. With we denote the weight for the bias neuron. Then the output of this blocked neural network can be computed for the activation function as
(4) 
Hereby, summarizes all weights, i.e., the parameters of the blocked neural network, while abbreviates the input parameters and the output parameter. The neuron activation function is chosen to be of sigmoidal type, i.e., .
The weights have to be determined by the given input and output data. This is done by minimizing the means squared error over a part of the given data, the socalled training set. The remaining data sets are used to validate the regression model. More precisely, the performance of the neural network is measured by the prediction mean squared error, which is estimated by cross validation (see [3] for more details).
Additionally, we like to analyze the sensitivity of the causeandeffectrelations. Hereto, we compute the first partial derivative of the regression function with respect to each input parameter. A large value of the partial derivative indicates a large influence of the corresponding input parameter, i.e. small changes in the input will lead to large changes in the output. Furthermore, the sign of the partial derivative is important. A positive partial derivative indicates that an increase in the input leads to an increase in the output, while a negative sign means that an increase in the input leads to a decrease in the output. Now the advantage of the blocked neural network approach becomes clear. Computing the partial derivatives with respect to the input parameters can easily be done in such a neural network. Due to [26] we have: {linenomath*}
where is the number of neurons in the th block as defined as in equation (4).
For our analysis, we choose a blocked neural network with two neurons in each of the block. The weights of each block are trained on the training data set by the LevenbergMarquardt Method [11]. We split the data set randomly in training and test data sets with a distribution, respectively. The results are presented in the following figures for each output parameter , , and . First, we present the results for the input process parameters (see Fig. 10). Then we visualize the effect results for the material parameters (see Fig. 11).
We conclude, that the effect of the process parameters is small. The material parameters clearly have more influence, in particular the line density has factor 10 more effect on the fiber laydown characteristics than E modulus and density. Furthermore, we observe the material input parameters have opposite effects in machine and cross machine direction. For example, an increasing density leads to an increasing , i.e. an increasing standard deviation of the throwing range in cross machine direction of the fiber laydown mass, while at the same time the standard deviation in machine direction decreases. This behavior is also true for the causeandeffectrelation of the line density. An increase of the line density leads to an increase of , but a decrease of . Surprisingly, the sign of the causeandeffectrelation for E modulus vs. changes. In lower regimes of the fiber’s E modulus we observe the same effects, i.e. an increase of the E modulus leads to increased values of and . But for higher values of the E modulus the throwing range in machine directions switches to decreased values of for increased values of .
For completeness, the performance of the training of the blocked neural network is visualized in Fig. 12. Note, that the confidence intervals have been estimated by the delta method described in [24].
Comparing the five partial derivatives with each other is not so easy. A scalar quantity summarizing the causeandeffectrelations would be desirable. A popular measure to quantify the sensitivity is the socalled average elasticity (AE). The average elasticity quantifies the percentage change of the output parameter with respect to a one percent change of the input parameters. In practice, the average elasticity is computed for given samples of input and output data as follows.
Let us assume that we have a total number of samples of input/output data and a blocked neural network as a nonlinear regression model approximating this data. Then the average elasticity for the th input parameter is defined (see [32]) as {linenomath*}
Table 1 shows the average elasticity values for each of the five input parameters with respect to the three different output parameters.
air speed  

pressure suction  
E modulus  
density  
line density 
From the average elasticity measures in Tab. 1 we conclude that the line density (titer) has the largest influence on the fiber laydown. Hereby, the impact on the fiber laydown in machine direction () is more than doubled compared ton the influence in cross machine direction (). As expected, changes of the air speed influence the fiber laydown as well. In this case, the impact on the cross machine direction is approximatly three times bigger than the impact in machine direction. On the first glance, the E modulus seems to be of minor impact, since the average elasticity measure is between and , but a look at the effect plots shows an interesting effect that can not be detected by the summarized scalar quantity. As explained above, there is a change of the sign of the partial derivative in the causeandeffectrelation, so in this case the summarized quantity of the AE is not applicable. Again, the pressure of the suction is negligible.
5 Summary and Outlook
In this paper we present the mathematical and physical framework to simulate spunbond processes. Furthermore, we present a fiber laydown criterion that characterizes the laydown on the conveyor belt. We set up a design of experiments DoE for two process parameters of an academic spunbond process and three material parameters of the filaments. The simulation results in this DoE are analyzed by a blocked neural network. The BNN is not only used to predict the fiber laydown characteristics for the five input parameters, but also to analyze the causeandeffectrelations with relevance plots. Additonally, the average elasticity measure leads to a quick ranking of the influencing effects with respect to their statistical significance.
The proposed simulation framework can be applied to optimize spunbond process with respect to homogeneity of the fiber mass distribution on a conveyor belt. However, the considered framework addresses the laydown of one single fiber and its characterization in machine and cross machine direction. For future work the superposition of multiple fibers forming a 3D microstructure and its height distribution should be investigated. Therefore, a threedimensional fiber laydown model based on stochastic differential equations can be used as described in [18] or a construction of 3D nonwovens using a greedy approximation of the distribution of fiber directions as derived in [7].
Acknowledgement
This work was developed in the Fraunhofer Cluster of Excellence “Cognitive Internet Technologies”. The authors would like to thank Walter Arne for performing the CFD simulations.
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