a comparison of mechanical property prediction ...

J. J. Crookston et al., A comparison of mechanical property prediction techniques using conformal tetrahedra and voxel-based finite element meshes for textile composite unit cells

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A COMPARISON OF MECHANICAL PROPERTY PREDICTION TECHNIQUES USING CONFORMAL TETRAHEDRA AND VOXEL-BASED FINITE ELEMENT MESHES FOR TEXTILE COMPOSITE UNIT CELLS J.J. Crookston, W. Ruijter, A.C. Long, N.A. Warrior & I.A. Jones Polymer Composites Research Group, School M3, University of Nottingham, University Park, Nottingham, NG7 2RD, UK. Email: [email protected] Abstract Predictive modelling of the mechanical behaviour of textile composites has been the subject of considerable research activity in recent years. For approaches based on finite element (FE) modelling of the textile structure at the meso-scale, obtaining a numerical model that is within the capabilities of available computing hardware and has an acceptable level of error due to discretisation (both in terms of the geometric description and the solution) continues to be challenging. This paper presents a comparison of the methodology and results of two different approaches to this problem, focussing on the relationship between computational cost and the error due to discretisation. Both approaches are based on the TexGen geometric modelling software package, developed at the University of Nottingham [1]. One approach uses a voxeltype discretisation technique with adaptive mesh refinement (AMR), while the other employs a conformal mesh of tetrahedral finite elements. Both methods are designed to allow automated model generation and processing. It is noted that the predicted stiffness converges more quickly for the conformal meshing technique, but also that the voxel technique has fewer limitations in terms of the reinforcement geometries which can be analysed. 1. Introduction It is well-known that the use of textile reinforcements in engineering composites can result in significant labour savings compared with traditional unidirectional lamination techniques, while retaining high mechanical performance. However, the internal geometry of the textile reinforcement adds another layer of complexity to the problem of mechanical modelling. Various methods to address these complexities have been proposed over a number of years. Analytical methods such as, for example, those suggested in [2-4] offer fast solutions with a low computational effort, but are generally restricted in terms of the geometry which can be analysed, the assumed stress and strain fields, or the prediction of damage. Conversely, numerical models such as those proposed in [5-9] offer greater flexibility in these areas at the expense of much higher computational cost and, typically, greater effort in generation and processing of models. Additionally, various methods have been proposed to simplify the process of numerical modelling, such as the binary model [10] and various approaches employing regular grid discretisation, or voxels [11-14]. 2. Textile modelling In this study, geometric modelling of the reinforcement is undertaken using the TexGen software, which was developed at the University of Nottingham and which has been released as an opensource project [15] such that the engineering community may inspect the algorithms and contribute towards further improvements and development. Development has been undertaken using other open-source software tools to provide the basis of user interaction with the kernel such that TexGen is platform independent. User access to TexGen can be via a graphical user interface (GUI) or from within programming languages either using the Python scripting language or by linking TexGen libraries directly when compiling user programs. The last method is known as an application programming interface (API).

“Finite element modelling of textiles and textile composites”, St-Petersburg, 26-28 September 2007


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The basis of TexGen is the description of yarns using a centreline and superimposed cross section. The shape and size of the cross sections may change locally; this is exploited in the functions for interference correction, which modify the textile according to geometric considerations to avoid interpenetration of yarns. When cross sections change, the fibre volume fraction (Vf) within the yarn is recalculated based on the updated cross section, i.e. the total amount of reinforcement in the textile remains consistent. A recent development implemented within TexGen has been to incorporate a distribution of Vf within the yarn cross section, following the observations of Koissin et al.[16], such that the fibres are closely packed in the centre of the yarn and the Vf reduces towards the sides. These observations are reproduced in Figure 1.

(a) (b) Figure 1 (a) Cross-sectional micrograph of a woven fabric; (b) measured distribution of fibre volume fraction within the yarn. Reproduced from Koissin et al [16]. For the purposes of the present study, two plain weave textile geometries are considered. One is a test case containing large gaps between the yarns is used to investigate numerical aspects of the discretisation procedures since this permits the use of coarse meshes for both techniques. The second model has small gaps and is based on the Vetrotex fabric RT600, a woven E-glass fabric with a surface density of 600gm-2; this model is used for validation against experimental data. The second model is shown in Figure 2. Since this study is primarily numerical, models with a single layer of fabric are employed and the quarter symmetry of the plain weave is exploited. The composite utilises an unsaturated polyester matrix. Material models and mechanical data have been published previously [17]. The assumed fibre distribution is such that the Vf at the yarn edges is approximately 85% of that in the centre. The Vf distribution is plotted in Figure 2b.

(a) (b) Figure 2 TexGen model of Vetrotex RT600 plain woven E-glass reinforcement: (a) geometric model; (b) distribution of fibre volume fraction within the yarns. 3. Approaches to FE modelling In developing mechanical modelling techniques for textile composites, the goal of the current work is to produce automated techniques which are directly compatible with TexGen. In practice, the implementation is done by calling TexGen functions from either the Python scripting interface or the API. In this paper, two methods are presented: a technique to generate a mesh of



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tetrahedral elements conforming to the geometry of the unit cell, and an alternative approach using a regular grid of hexahedral elements employing h-type adaptive mesh refinement. 3.1. FE model generation using conformal meshes One approach for modelling the internal geometry of a repeating or representative volume of fabric reinforced composite consists of generating an FE mesh such that there are element faces which are coincident with the boundaries between the impregnated yarns (which are assumed to be a homogeneous continuum) and the matrix, creating a distinct interface between the two phases. This method was reported in the literature more than ten years ago, e.g. by Chapman and Whitcomb [5]. Due to the complex nature of the matrix volume, it is not usually possible to create a mesh of hexahedral elements for general unit cells without the application of a specialised technique such as that developed by Zako et al.[7]. However, there has been significant progress in recent years on automated tetrahedral mesh generation, resulting in the availability of relatively robust algorithms in engineering software packages. Tetrahedral meshes are employed in the present study. Following the creation of a geometric model of the reinforcement architecture, various challenges exist including mesh generation and assignment of appropriate material properties and principal orientation vectors. The approach investigated here exploits the Python scripting interface of TexGen in conjunction with that of the pre-processing application for the Abaqus FE package (Abaqus CAE). There is an emerging trend towards FE systems which are based around a geometry paradigm, rather than the traditional mesh paradigm. While the mesh is still used to obtain a solution, the role of a modern pre-processor is to manage CAD data and to allow the user to assign material definitions to a geometric region rather than to a set of elements, to define constraints on surfaces rather than on sets of nodes, etc. In such systems the mesh is not operated on directly by the user. Many such systems also offer the facility to build geometric entities, effectively incorporating a basic CAD system. Since the TexGen libraries can be called from within any Python program, and since the creation of CAD-like geometries can be scripted from Python within Abaqus CAE, it is possible effectively to rebuild the TexGen model with native CAD objects within the preprocessing system. This is performed by looping over the textile hierarchy (textiles, yarns, points on the centreline, points around the cross-section), retrieving coordinate data and lofting surfaces over the yarns. Subsequently it is possible to undertake boolean operations to create the intricate matrix volume, to select surfaces for the application of boundary conditions and to define suitable analysis procedures. It should be noted at this stage that there are geometric considerations associated with the reconstruction of the textile geometry. In its internal operation, TexGen uses b-splines to represent yarn centrelines and polylines (series of straight lines) to form the cross sections around the yarns. Although yarn cross sections are usually defined according to a mathematical function, the edges themselves are composed of a series of points joined by straight lines. It is these polylines which are manipulated during interference correction and which are used when calculating the yarn area to determine local Vf. When reconstructing the yarn cross sections in Abaqus CAE, curved splines are generated which pass through the points on the polyline. Figure 3a illustrates schematically that the area of the cross-section bounded by a pair of splines is greater than that of the cross section bounded by the polylines. The effects of this are twofold: firstly, yarns which do not touch in the TexGen model may intersect in the reconstructed form and, when local fibre volume fractions are assigned later in the model generation process, the increased cross-sectional area results in an increase in the total amount of reinforcement in the model.



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Matrix volume fraction in Abaqus CAE representation

0.641 0.640 0.639 0.638 0.637 0.636 0.635 20

40

60

80

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No. points around yarn cross section (a) (b) Figure 3 (a) The difference in cross-sectional dimensions between yarns described using polylines and those using splines, illustrated on ¼ of the cross section; (b) the convergence of total matrix volume within the unit cell as the number of points around the section is increased. This shows that there is a requirement to increase the number of points used when generating the spline in order to constrain it to the desired shape. This has the disadvantage of swelling the geometric dataset which, aside from increasing storage requirements slightly, also serves to slow down the CAE system considerably, particularly for models containing large numbers of yarns. In order to determine the number of points which are required a simple study was performed, reproducing the model with different numbers of points to examine convergence of the volume of the matrix. The results are shown in Figure 3b. Two effects are present within these data: firstly, increasing the number of points within TexGen results in a polyline which is closer to its mathematical definition and hence smoother; secondly, the spline representation in Abaqus exhibits higher fidelity to the polyline both as it is constrained to pass through more points and as the polyline becomes smoother. Although these effects interact, Figure 3b can be used to provide a basis for convergence; in the remainder of this study 60 points were used to describe each cross section. Although the mesh becomes almost transparent to the user in such CAE systems, some considerable care and effort must be taken to ensure that a mesh which is generated is of sufficient quality to obtain reliable solution results. While this may be achieved directly by interactive mesh generation by the user, time savings may be afforded by using a scripted approach which is integrated with the geometry creation. The first step in the mesh generation process is to provide a line mesh used to indicate the edge length in the resulting 3D mesh (a seed). Two approaches are investigated here: the first is to use uniform seed lengths throughout the model, and the second to define seeds according to the smallest gap between each edge and its neighbours. In the latter approach, a factor is applied to the minimum gap which is approximately equivalent to the aspect ratio of the resulting elements at the narrowest point. Figure 4 shows the outer surface of a model where uniform and localised seeding have been used, each technique employing the same minimum seed size.



(a)

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(b)

Figure 4 Meshes produced using (a) uniform edge seeding throughout the model (32k nodes) and (b) local edge seeding according to the distance between adjacent edges (14k nodes). It can be seen from Figure 4 that there is approximately a factor of two between the number of degrees of freedom produced using these two seeding schemes. However, this factor will exhibit a significant dependence not only on the textile geometry, but also on the location of the domain boundaries since the seeding is performed on geometry feature edges. Feature edges are present where surfaces meet with discontinuous curvature, i.e. where yarns intersect the domain boundaries and, in the examples presented in this paper, at the yarn tips. Potential methods to enhance this local seeding technique will be the topic of further study. Since the Vf varies continuously within the yarn due to the changes in local cross section and due to the transverse fibre distribution, material properties must be specified locally. This is possible since a TexGen model can be queried by passing a list of spatial coordinates to a function which returns a corresponding list of fibre volume fractions and orientation vectors. Due to operational considerations, this lookup is presently performed at the element centroids, although it could be performed at the Gauss points with some computational benefit due to the increased sample density. The subject of sampling and stiffness integration is addressed further in the context of voxel models later. It should be noted, however, that because the Vf is not constant within the yarns, the effective Vf in the FE model will be a function of sampling and hence, in this case, of mesh refinement. The convergence of Vf with mesh refinement for the two edge seeding schemes was measured, and the results are presented in Figure 5. It can be seen that both methods converge to almost the same value (within 0.03% absolute Vf ), and that even in the coarsest stages the variation is below 0.15% absolute Vf.



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Fibre volume fraction, Vf

0.221

0.220 True Vf (TexGen)

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0.218 uniform edge seeding local edge seeding 0.217 100

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Number of degrees of freedom

Figure 5 Convergence of fibre volume fraction with conformal mesh refinement for different edge seeding schemes. In addition to verifying the total Vf in the model, it is necessary to evaluate the convergence of the solution to the mechanics problem. In the first instance, this is achieved by monitoring the global stiffness predicted by the model. As stated earlier, simple uniaxial loading was simulated; hence the stiffness was evaluated from the applied displacement and the total reaction force on that face. The convergence behaviour following mesh refinement is shown in Figure 6. As would be expected, these data echo the artefacts of the Vf convergence in addition to the solution itself. Although not as ‘flat’ as the convergence curves for Vf, the variation in Young’s modulus is below 50 MPa from the coarsest model with uniform seeding (3k DOF) and from around 7k DOF with local seeding. Hence it is apparent that, at least for the test case considered here and with the algorithms employed so far, the use of uniform edge seeds actually offers greater numerical efficiency than that of localised seeds. The error in stiffness due to discretisation was observed to be lower for meshes employing uniform seeds even with fewer DOF than those using local seeds.

Young's modulus, E11 (MPa)

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11800 uniform edge seeding local edge seeding 11700 100

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Figure 6 Convergence of Young’s modulus in the warp direction with conformal mesh refinement for different edge seeding schemes.



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3.2. FE model generation using voxel meshes As an alternative approach to the model regeneration and conformal meshing described above, a discretisation technique employing 3D volume elements or voxels has been implemented. In this technique the shape of the mesh is not derived from the internal geometry of the composite repeating unit, but the mesh is constructed entirely from orthogonal elements with material properties defined according to local sampling. The use of adaptive mesh refinement for the analysis of textile composites was first demonstrated by Kim and Swan [18] who also showed the performance of the method as a function of different refinement strategies[19]. The principal advantage of the technique is that it is robust and can analyse any textile model without the need for user intervention in the mesh generation and model preparation phase. In this technique, firstly a coarse grid of finite elements is defined to cover the whole volume. Material properties at the element integration points are determined by querying the Vf and orientation vectors from TexGen and the system is solved according to the boundary conditions which have been applied. A Kelly error estimator [20] is employed to govern the AMR scheme. This is sometimes used in conformal FE analysis to estimate the discretisation error in the solution by calculating the sum of discontinuities in computed strain over element boundaries. Without modification this results in large error estimates around the sudden changes in stiffness present at material boundaries in the multiphase composite. Based on the error estimate, elements are refined locally using an h-type scheme with hanging nodes coupled to the faces of the adjoining elements using constraint equations. Sections through a typical grid are shown in Figure 7. An iterative solver is used, which has the benefit that the initial estimates for each refinement level can be taken from the solution at the previous level, giving a significant advantage in terms of computational cost. A criterion can be defined such that further refinement does not occur once the total estimated discretisation error in the system falls below a certain level, or alternatively (in cases involving more complex textile geometries) such that the numerical problem size remains within the capabilities of the computing hardware.

Figure 7 Local refinement of the voxel grid around yarn boundaries using AMR. Various aspects of this method can be demonstrated to affect the rate of convergence, results of which are plotted in Figure 8. For example, the initial (coarsest) grid may be as coarse as a single element. Together with the geometry of the domain under consideration, the resolution of the initial grid defines both the aspect ratio of the elements and the ratio of the mesh densities in different directions, since subdivision is performed by bisection only. Two initial grid configurations were used on the test geometry, 4x4x1 elements and 1x1x1 elements across the x, y and z dimensions. In this case (since the geometric features are relatively ‘flat’) it can be seen that the 1x1x1 grid, where the elements are flat, exhibits significantly faster convergence than the 4x4x1 grid where the elements have lower aspect ratios. Whether this behaviour has a greater dependence on dimensions of internal features or on those of the domain remains a subject for further investigation. As stated above, Vf and orientation are sampled at each integration point. Consequently it is possible to represent a material distribution which is more complex than the distribution of the solution variables (displacements). Convergence rates of two different integration schemes (23



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and 33) are also presented in Figure 8. In this case no significant difference between the schemes was observed, although Vf can be observed to converge marginally quicker with more sampling points.

Fibre volume fraction, Vf

0.230

0.225

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True Vf (TexGen) initial grid 4x4x1, integration 2x2 initial grid 1x1x1, integration 2x2 initial grid 1x1x1, integration 3x3 conformal mesh

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Young's modulus, E11 (MPa)

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11500 initial grid 4x4x1, integration 2x2 initial grid 1x1x1, integration 2x2 initial grid 1x1x1, integration 3x3 conformal mesh 11000 100

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(b) Figure 8 Convergence of (a) fibre volume fraction and (b) Young’s modulus in the warp direction with voxel mesh refinement for different initial starting grids and stiffness integration schemes. Conformal mesh results using uniform edge seeds are included for reference. It should also be noted that, since the textile geometry in this case is characterised by a high level of internal partitioning, i.e. the ratio of yarn surface area to volume is large, the predicted discretisation error was such that uniform refinement occurred throughout the first five refinement steps until the element size was sufficiently small relative to the feature size. 4. Application to a realistic textile geometry In order to compare results with experimental data, the methods were applied to the geometry with a small gap between the yarns, which was described in Section 2. Table 1 shows the number of degrees of freedom and the stiffness prediction of each of the methods together with experimental measurements made at similar fibre volume fractions. The predicted Young’s modulus falls within the bounds of the experimental data but is close to the bottom of the measured range. It should be noted that the experiments were performed on specimens



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constructed using two layers of reinforcement, while the boundary conditions employed in the numerical study allowed out of plane deformation as in a single layer. Data presented by the authors as part of a previous parametric study [17] showed a significant difference in the nonlinear stress-strain behaviour (incorporating damage) for similar models analysed using the voxel technique. These data are reproduced in Figure 9. This may be one factor in the slight discrepancy between predicted and measured stiffness; others may include inaccuracies in the geometric idealisation or constituent material data. Table 1 Predicted and experimental data for a realistic geometry. Voxel Conformal Experimental

No. degrees of freedom 366948 476578 -

Vf 0.31867 0.31959 0.318±0.015

E11 (MPa) 16630 16677 17658±1915

Figure 9 Comparison of stress-strain behaviour for a single-layer voxel model (black triangles) and a two-layer voxel model (red crosses) with experimental data. Reproduced from [17]. The distribution of stresses predicted by the two models is compared in Figure 9. It can be seen that values are similar in both cases, and that they increase in regions of higher V f. More detailed comparisons of stress distribution will be the subject of further study.

Figure 10 Stress distributions in the yarns predicted by conformal and voxel models.



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5. Discussion and conclusions Two different modelling methodologies have been used to analyse the same geometric model, which is thought to be representative for 2D woven textile composites. The model was chosen such that it includes the current modelling difficulties posed by very high aspect ratio internal domains as well as Vf variation across and along yarns. The latter feature may affect the relative performance of the two methods since both methods give approximate representations of the material property distribution in the model. It can be seen that the Vf and, consequently, the predicted stiffness converge faster for the meshes obtained through conformal meshing. This means that if a conformal mesh of the ‘true’ geometry is available it is highly likely to give a better prediction. It is the fact that it is frequently not possible to generate a conformal mesh of the ‘true’ geometry which forms the justification for the use of the voxel method. The reason for faster convergence in the case of conformal meshing is thought to be due to the fact that the gap between the yarns is captured accurately even at the coarsest level of mesh refinement. In composites literature the question of whether such a gap should be modelled (i.e. whether it exists physically) and whether it should be continuous or have regions where yarns touch is still being debated and different approaches exist. The conformal method presented here is unlikely to operate on models with regions of touching or intersecting yarns without modification or user intervention; the voxel method does not have this limitation since it does not depend on a consistent geometric formulation, but rather on consistent material sampling. The principal disadvantage of the voxel method is that physical accuracy may be thought to suffer due to representation of the yarn boundaries in an averaged or ‘smeared’ sense. However, since the physical boundary between matrix and impregnated yarn is not one between two distinct homogeneous continua, it may be argued conversely that a representation with a conformal mesh is idealised and unrealistic. Quantification of this issue was given brief consideration by Kim and Swan in [18] but the conclusion was related to highly idealised textiles with constant yarn fibre volume fraction. The conclusions which may be drawn from the comparisons are that:  the modelling strategies are consistent and converge to the same values for Vf and stiffness;  conformal meshes, when available, converge more quickly both in terms of Vf and stiffness;  both methods can operate automatically using a geometric description in TexGen. For the simple textile under consideration both methods are reasonably robust, but for more intricate models conformal mesh generators are known to have limitations in reliability. The extension of this work to compare the capabilities of these two methods for prediction of damage initiation and propagation will be the subject of a future study. Acknowledgments This paper presents the ongoing work of the Polymer Composites Research Group at the University of Nottingham. This work is supported by various grants from UK government bodies including the Engineering and Physical Sciences Research Council (EPSRC) and the Department of Trade and Industry (DTI). The collaborative support of a number of industrial organisations and other academic institutions is also gratefully acknowledged. The authors also wish to acknowledge the assistance of Dr. Martin Sherburn through his support of the TexGen project. References 1. M. Sherburn, A.C. Long, I.A. Jones and C.D. Rudd, ‘TexGen: geometric modelling schema for textile composites’, Proc. 8th Int. Conference on Textile Composites (TEXCOMP 8), Nottingham, UK, Paper T19, Oct 2006.



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2. T. Ishikawa and T.W. Chou, ‘Stiffness and strength behaviour of woven fabric composites’, Journal of Materials Science, 17, 1982, pp. 3211-3220. 3. N.K. Naik and P.S. Shembekar, ‘Elastic behaviour of woven fabric composites: I – Lamina analysis’, Journal of Composite Materials, 26, 1992, pp. 2196-2225. 4. J. Hofstee and F. van Keulen, ‘Elastic stiffness analysis of a thermo-formed plain-weave fabric composite. Part II: analytical models’, Composites Science and Technology, 60, 2000, pp.1249-1261. 5. C. Chapman and J.D. Whitcomb, ‘Effect of assumed tow architecture on predicted moduli and stresses in plain weave composites’, Journal of Composite Materials, 29, 1995, pp. 2134-2159. 6. V. Carvelli and C. Poggi, ‘A homogenisation procedure for the numerical analysis of woven fabric composites’, Composites: Part A, 32, 2001, pp. 1425-1432. 7. M. Zako, Y. Uetsujib, and T. Kurashikia, ‘Finite element analysis of damaged woven fabric composite materials’, Composites Science and Technology, 63, 2003, pp. 507–516. 8. G. Nicoletto and E. Riva, ‘Failure mechanisms in twill-weave laminates: FEM predictions vs. experiments’, Composites Part A, 35, 2004, pp. 787-795. 9. D.S. Ivanov, S.V. Lomov, I. Verpoest and A.A. Tashkinov, ‘Local elastic properties of a shaped textile composite: homogenisation algorithm’, Proc. 6th Int. ESAFORM Conference on Material Forming (ESAFORM-6), Salerno, Italy, 2003. 10.B.N. Cox, W.C. Carter and N.A. Fleck, ‘A binary model of textile composites – I: Formulation’, Acta Metallurgica et Materiala, 42, 1994, pp. 3463-3479. 11.P. Vandeurzen, J. Ivens and I. Verpoest, ‘Micro-stress analysis of woven fabric composites by multilevel decomposition’, Journal of Composites Materials, 32, 1998, pp. 623-651. 12.Prodromou, ‘Mechanical modelling of textile composites utilising a cell method’, Ph.D. Thesis, Katholieke Universiteit Leuven, 2004. 13.H. J. Kim and C.C. Swan, ‘Voxel-based meshing and unit-cell analysis of textile composites’, International Journal for Numerical Methods in Engineering, 56, 2003, pp. 977-1006. 14.A.E. Bogdanovich, ‘Multi-scale modeling, stress and failure analyses of 3-D woven composites’ Journal of Materials Science, 41, 2006, pp. 6547-6590. 15.M. Sherburn, TexGen open source project, online at http://texgen.sourceforge.net/ 16.V. Koissin, D.S. Ivanov, S.V. Lomov and I. Verpoest, ‘Fibre distribution inside yarns of textile composite: geometrical and FE modelling’ Proc. 8th Int. Conference on Textile Composites (TEXCOMP 8), Nottingham, UK, Paper T02, Oct 2006. 17.W. Ruijter, J. Crookston and A. Long, ‘Effects of variable fibre density on mechanical properties of a plain weave glass reinforced polyester’, Proc. 16th Int. Conference on Composite Materials (ICCM-16), Kyoto, Japan, 2007. 18.H.J. Kim and C.C. Swan, ‘Voxel-based meshing and unit-cell analysis of textile composites’, International Journal for Numerical Methods in Engineering, 56, 2003, pp. 977-1006. 19.H.J. Kim and C.C. Swan, ‘Algorithms for automated meshing and unit cell analysis of periodic composites with hierarchical tri-quadratic tetrahedral elements’, International Journal for Numerical Methods in Engineering, 58, 2003, pp. 1683-1711. 20.D.W. Kelly, J.P. Gago, O.C. Zienkiewicz and I. Babuska, ‘A posteriori error analysis and adaptive processes in the finite element method: Part I Error analysis’, International Journal for Numerical Methods in Engineering, 19, 1983, pp. 1593-1619.


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