UNCERTAINTY QUANTIFICATION IN LIQUID COMPOSITE MOULDING PROCESSES

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1 UNCERTAINTY QUANTIFICATION IN LIQUID COMPOSITE MOULDING PROCESSES Bart Verleye 1, Dirk Nuyens 1, Andrew Walbran 3,2, and Ming Gan 2 1 Dept. Computer Science, KU Leuven, Celestijnenlaan 200A, B-3001 Leuven, Belgium 2 Centre for Advanced Composite Materials, University of Auckland, Private Bag 92019, Auckland, New Zealand 3 Lehrstuhl für Carbon Composites, Technische Universität München, Boltzmannstraße 15, Garching bei München, Germany Corresponding author s bart.verleye@cs.kuleuven.be ABSTRACT: Variability of technical textiles plays an important role in Liquid Composite Moulding (LCM) processes and product quality. The numerical simulation of an LCM process depends strongly on the distribution of the volume fraction of the material. To obtain more realistic results, a distribution for the areal mass of the textile can be used as input, i.e., for every point of the mesh, an areal mass value is determined by a random field. The distribution used for the mathematical model has to be selected in agreement with the physical distribution. The random nature of the textile properties, make that different simulations have different results, although the simulations are created with the same macro parameters. This resembles the spread on experimental results and reveals the distribution of the output. To compute the average fill time or required tooling forces and their spreads, one can use the Monte Carlo (MC) method, being repeated random simulations, or the quasi-monte Carlo (QMC) method, consisting of deterministic simulations of the random field. We show the QMC method to converge faster than the MC method and compare the obtained results with the results from a numerical simulation of the same LCM process without the usage of the random field to possibly differ a lot for certain outputs. One such output is the maximum fibre stress which is severely underestimated by the standard method. KEYWORDS: Variability, quasi-monte Carlo, uncertainty quantification INTRODUCTION Recently, variability and uncertainty quantification have been included in the simulation of LCM processes and textile qualification [2,3]. It has been shown that the variability of the permeability of the preform in a mould influences the fill time significantly [7]. Another example is the influence of the variability of textile parameters on the final part quality. To determine the influence of the variability, often experiments or simulations are performed and the average and variance are determined. For the results presented here both the Monte Carlo (MC) method and the quasi-monte Carlo (QMC) method have been used to generate instances of the random field. We show that the QMC method can give more reliable estimates using fewer simulations than the MC method. With these techniques, several parametric studies were performed to determine the influence 265

2 of measured variability on the quality of the textile. These results are compared to the standard numerical simulation without using a random field for the areal mass. In such a simulation the random field values are replaced by the average value. SIMULATION OF THE PROCESS The process is modelled by applying Darcy s law, given by the elliptic PDE (1) over a domain in two physical dimensions. The flow solver uses a so-called 2.5D approach, the geometry is 3D but the flow in the through-thickness dimension is neglected [5]. For a point in, denotes the pressure head of the fluid, is the filtration velocity (or Darcy flux), is the source term and the hydraulic conductivity tensor is a random field dependent on a random event. The conductivity tensor is assumed to follow a log-normal distribution with the following covariance function: (2) where measures the Euclidean distance between the two points. The parameters and are the variance and the correlation length. These two parameters are obtained from physical experiments (described in the next section) which specify the average and standard deviation of the final field. The parameters for a normal random field with covariance from above are then obtained as (3) and the final field is then constructed as (4) (5) where is simulated from the normal random field. One way of solving this PDE is to generate random events in the form of random realizations of the conductivity tensor by Monte Carlo simulations and then calculate the quantities of interest, which can be any linear or non-linear operator on and. It is then possible to undertake uncertainty quantification on these quantities of interest by calculating their mean, variance or any higher moments. The Monte Carlo method is compared to a method using a deterministic low-discrepancy sequence with few random shifts (to determine the accuracy of the simulation), called a randomized quasi-monte Carlo method. This was recently studied in [4] and [6]. The simulated values are generated on a regular 2D grid using the technique of circulant embedding [1]. 266

3 DETERMINING THE INPUT PARAMETERS We now summarise the physical experimental measurements conducted to determine the variance and correlation length simulation parameters. The material selected for this study is a randomly orientated chopped strand mat (CSM), which exhibits random spatial variations in areal weight (AW) which is defined as the mass per unit area in units of grams per square meter (g/m 2 ). Further details of the selected material are provided in Table 1. Table 1: Properties of the chopped strand mat, as provided by the supplier Property Value Reinforcement type Chopped strand mat Manufacturer Owens Corning Suppplier code M CSM Nominal areal weight 450 g/m 2 +/- 10 % Architecture type Randomly orientated, emulsion bound Spatial AW data for reinforcement layers is obtained using an experimental facility, consisting of a light box, digital camera and enclosure. See [3] for further details, experiments and AW maps figures. AW maps have been generated based on 290x450 mm 2 reinforcement samples, discretised using 7 mm sampling windows. Based on six single layer samples of the CSM reinforcement, average ( ) and standard deviation ( ) AW values of g/m 2 and g/m 2 respectively are obtained based on a Gaussian fit of the data. The variance can be calculated as the square of the standard deviation The mean AW is very close to the nominal value given in Table 1, however the standard deviation corresponds to a 21% variation from the mean, which is larger than that quoted by the supplier. Randomly orientated reinforcements are commonly employed for their homogeneity and isotropy, therefore it is important to quantify the level of variations as they can be used as a measure of quality when comparing reinforcements of similar AW. The correlation length is determined more arbitrarily. The AW maps described above were compared with AW maps with different, generated as described in the previous section. A visual comparison resulted in = as a good estimate. Fig. 1 shows an experimentally obtained AW map (left), and two generated AW maps, one with = (middle) and one with = 0.02 (right). We clearly see that a too high results in too much clustering. Fig. 1: Three areal mass maps: measured (left), for =0.007 (middle) and for =0.02 (right) 267

4 VALIDATION In this section, the QMC method is validated for a realistic RTM application, and the convergence and accuracy of the QMC method compared to those of the MC method. With the technique described in the previous section, the average areal mass, standard deviation and correlation length were determined for the CSM (Table 1,. Table 2: Parameters of the injection simulations Parameter Final height Injection pressure Viscosity Value 3 mm 87 kpa 0.19 Pa.s With these numbers, eight independent simulations using 1024 random areal mass fields were created on a square domain with 25x25 grid points, both with the MC and the QMC method. For the Monte Carlo simulation this is the same as making 8x1024 = 8192 random simulations. For the randomized quasi-monte Carlo method eight independent results are obtained based on 1024 deterministic samplings of the random field. These fields are input for the SimLCM software [5], which is used to simulate an RTM process with the parameters given in Table 2. From these simulations we retain two results: the time needed to fill the mould after the dry compression, and the maximum fibre stress after filling. Fig. 2 presents the results of the simulations with both the MC (left) and QMC (middle) generated input fields for the fill time. In the left and middle panel the running average is plotted for each of the eight independent simulations. One can clearly see that the eight streams of approximations to the mean converge quickly towards each other in case of the QMC simulations. However, in the case of the MC simulation these results differ considerably. This is made clearer by the plot in the right panel which shows the convergence of the relative error on the average by plotting the standard error divided by the estimated value. The results show that the MC and QMC input result in the same average fill time, as expected, however, QMC yields a faster convergence to the result. 268

5 Fig. 2: Results of the Monte Carlo (left) and quasi-monte Carlo (middle) simulation, the running average of the filling time. The convergence for both methods is shown on the right. The fill time computed with a constant areal mass of 450 g/m 2 over the whole domain is 2432s, which is not significantly different from the average fill time of the experiments with a variational areal mass (Table 3). In this case, the inclusion of variability does not give us a better estimation of the average fill time, however, the variance on the fill time is an interesting parameter as it can be used to determine a confidence interval from which a maximum filling time could be set which leads to success in e.g. 98% of the cases. The maximum preform stress at the end of the filling, computed for a constant areal mass of 450 g/m 2 using the standard LCM-processes simulation is 125 kpa. As can be seen in Table 3, this is much less than the average maximum computed with variability using the QMC method, and is certainly much less than the absolute maximum appearing in the simulation. In this case, the use of variability gives us much more information while the result of the standard LCM calculation is almost useless. While only the maximum stress and the fill time have been discussed here, similar results for the minimum stress, variance of stress over the mould, etc. can be obtained using the QMC method. Moreover, for more complicated geometries, also the influence on the robustness of a particular injection scheme can be considered. Table 3: Results of the QMC simulations with an average AW of 450 g/m 2 and both 10% variance (top row) and 20% variance (bottom row) Standard method Fill time (s) 2432 Max. preform stress (Pa) 1.25 E5 Random field simulation by QMC method Average Median Min Max (106) (219) 5.27 E E E E E10 (1.06 E5) 2.03 E E E E E11 (8.40 E5) 269

6 PARAMETRIC STUDY Fig. 3 shows the computed histograms of the fill time for three different cases. The left subfigure is the result of a simulation with an AW of 250 g/m 2 10%. The middle figure is for an AW of 450 g/m 2 10%. The right figure is the histogram of the above discussed simulation, with the areal weight 450 g/m 2 20%. As expected, the average fill time for the lower AW is less than in the case of the higher AW. The histogram for the AW of 250 g/m 2 is not symmetric, as for very low AW, the fill time converges to the one of an empty mould. Note that the bins for the left and right figure were generated automatically, but that for the middle figure the bins of the right figure were used to allow comparison. For the study with higher variance, the average fill time and maximum of the preform stress are also mentioned in Table 3. Also in Table 3 are the medium, the minimum, the maximum and variance of these parameters for both the low and high variance input. For the fill time, we can conclude that the variance on the AW has little influence on the average, however, the spread on the fill time is much larger for a highly variable AW. For the maximum of the preform stress, the variance on the input has a significant influence on both the average and the spread of the maximum. Introducing a variance of 10% results in a four times larger average. Going from a 10% to a 20% variance also doubles the resulting average maximum stress. This shows that variance is an important parameter to consider in simulations that help to choose materials and tooling. Fig. 3: Histograms of the fill time for three different simulations. The middle panel shows the results of the simulations with the parameters provided by the manufacturer (Table 1). For the left panel the AW was changed to 250 g/m 2, for the right panel the variance was changed to 20%. 270

7 CONCLUSIONS The variability of the areal mass of a preform, has a significant influence on LCM processes. To predict robustness and quality of production lines, variability must be included into the simulations. In this paper we presented the use of the quasi-monte Carlo method to introduce the variability, and showed to have faster convergence than with the often used Monte Carlo method in the case of computing the fill time. It was shown that the variability of the areal weight has little influence on the average fill time, however, it does have an influence on the spread of the fill time. The maximal stress in the preform is highly influenced by the variability of the input. In this case, introducing a variability of 10%, results in a four times higher average compared to a standard simulation with constant areal mass. REFERENCES 1. Dietrich CR and Newsam GH (1997), Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix, SIAM Journal on Scientific Computing, 18(4), Endruweit A and Long AC (2006), Influence of stochastic variations in the fibre spacing on the permeability of bi-directional textile fabrics, Composites Part A, 37(5), Gan JM, Bickerton S, Battley M (2012), Quantifying variability within glass fiber reinforcements using an automated optical method, Composites Part A, accepted, 4. Graham IG, Kuo FY, Nuyens D, Scheichl R and Sloan IH (2011), Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications, Journal of Computational Physics, 230(10), Kelly PA and Bickerton SA (2009), Comprehensive filling and tooling force analysis for rigid mold LCM processes, Composites Part A, 40(10). 6. Kuo FY, Schwab C and Sloan IH (2012), Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients, Hausdorff Institute for Mathematics Preprint #2011b08, Bonn, Germany. 7. Sriramula S, Chryssanthopoulos MK (2009), Quantification of uncertainty modelling in stochastic analysis of FRP composites, Composites Part A, 40(11):