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# Three-dimensional reconstruction of geometry of woven reinforcement of a composite material with dual kriging algorithms

Anna Madra, Piotr Breitkopf, François Trochu

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Anna Madra, Piotr Breitkopf, François Trochu. Three-dimensional reconstruction of geometry of woven reinforcement of a composite material with dual kriging algorithms. 12e Colloque national en calcul des structures, CSMA, May 2015, Giens, France. hal-01517299

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# Three-dimensional reconstruction of geometry of woven reinforcement of a composite material with dual kriging algorithms

A. Madra<fnref n="1" />,<fnref n="2" />, P. Breitkopf<fnref n="1" />, F. Trochu<fnref n="2" />

**Résumé** — Composite materials with woven fibrous reinforcements are difficult to model numerically due to high complexity of their internal geometry. Design of components made with composites requires modeling of both fabrication process and mechanical behavior. This creates a need for accurate three-dimensional geometry reconstructions taking into account fiber deformations during draping and closing of the mold. This work provides a solution based on dual kriging of parametric surfaces with nugget effect to create smooth meshes of fiber tow envelopes for further modeling.

**Mots clés** — woven composites, X-ray microtomography, mesh, numerical methods.

## 1 Introduction

Due to their inherent complexity textile reinforced composite materials are difficult to model numerically. This pertains not only to the case of mechanical simulation of final composite parts but also to the deformations of the material during fabrication. The element necessary for realization of any of such simulations is a good representation of the three-dimensional geometry of textile reinforcement [8]. Whereas approximate, theoretical models are now widely available, eg. as described in [2] or [3], they still lack the precise information on the textile deformation during draping and their deformation due to pressure in the mold not to mention limited capacities for reconstruction of 3D textiles. On the other side, techniques like X-ray microtomography is capable of providing exact geometries but on the cost of very large space complexity and noisiness of the model [4].

Here proposed is a method of transformation of the results obtained from X-ray microtomography into three-dimensional surface meshes of fiber tow envelopes. The volumetric data from microtomography is segmented and serves as an input to dual kriging algorithms for surface reconstruction. The obtained geometry is much more succinct and thus strongly reduces the memory needed to process the geometries. It also simplifies the calculation of relative changes in position and shape of particular fiber tows in a textile reinforcement, as may be the case with characterization of textile reinforcement in compression.

## 2 Three-dimensional textile reconstructions

To date textile reinforcements of composite materials have been characterized mostly by performing measurements on raster data of type pixel (for 2D scans and microographies) or voxel (mainly 3D X-ray tomography and microtomography [5]). Most of these approaches limit themselves to definition of the centroid path of a given fiber tow and some orthogonal cross-sections. Already for simple, two-dimensional textiles this represents a problem as the cross-section geometry of a given fiber tow is strongly deformed when in contact with another tow as compared to the state when it is not in contact. Further problems occur when textile is positioned in the mold and/or in contact with another layers of reinforcement. All of the deformations that occur can have an important impact on the final mechanical properties of the part for two reasons : first depending on the configuration of the fiber tows they may be more difficult to impregnate with resin and the lack of impregnation leads directly to the presence of voids which generally are the origin of fractures [6]; secondly, fiber orientation and volume fraction in

<footnote n="1">¹ Laboratoire Roberval, Université de Technologie Compiègne, {anna.madra,piotr.breitkopf}@utc.fr</footnote>

<footnote n="2">² CCHP, Ecole Polytechnique de Montréal, {anna.madra, trochu}@polymtl.ca</footnote>
the material decide on the rigidity and tensile strength of a final composite material.

Yet another problem occurs with three-dimensional textile weaves. These require X-ray microtomography to provide the idea of the general structure of the reinforcement. Still, no fully automatized approaches are currently available to transform microtomography data into vectorized meshes which take into account the error of classification of a given piece of raster data as a part of geometry of a given fiber tow.

## 3 Dual kriging with nugget effect

Kriging is an interpolation and estimation method especially popular in geostatistics for assessment of eg. distribution of ore or population characteristics. Its use for geometrical modeling is not as popular and is usually realized with dual kriging formulation. A given function $U(x) \in \Re$ may be represented as

$$U(x) = a(x) + W(x) \quad (1)$$

where $a(x)$ is the general trend of the function and $W(x)$ represent the fluctuations. The general trend is the averaged behavior of a function described with the variance of the estimation error.

For one dimension kriging is a minimization problem of

$$\frac{1}{2} \sum_{i=1}^{N} \sum_{j=1}^{N} \lambda_i(x) \cdot \lambda_j(x) \cdot K(|x_i - x_j|) - \sum_{i=1}^{N} \lambda_i(x) \cdot K(|x_i - x|) \quad (2)$$

with linear unbiased criteria

$$\begin{cases} \sum_{i=1}^{N} \lambda_i - 1 = 0 \\ \sum_{i=1}^{N} \lambda_i \cdot x_i - x = 0 \end{cases}$$

where $\lambda_i \in \Re^N$ are coefficients in the minimization problem of the expression (1) under unbiased conditions and $K$ is the generalized covariance. The solution of the (2) results with the one-dimensional kriging system which could be further transformed into dual kriging system (for a function with linear trend)

$$\begin{bmatrix} K(|x_i - x_j|) & \begin{bmatrix} 1 & x_1 \\ \vdots & \vdots \\ 1 & x_N \end{bmatrix} & \begin{bmatrix} b_1 \\ \vdots \\ b_N \\ a_0 \\ a_1 \end{bmatrix} \\ 1 & \dots & 1 & 0 & 0 \\ x_1 & \dots & x_N & 0 & 0 \end{bmatrix} = \begin{bmatrix} u_1 \\ \vdots \\ u_N \\ 0 \\ 0 \end{bmatrix} \quad (3)$$

where $x_i$ are the coordinates of the known values $u_i$ of the original function $U(x)$. The values $a_i$ and $b_i$ are the coefficients in the final dual kriging equation of an interpolated function $u(x)$

$$u(x) = a_0 + a_1 \cdot x + \sum_{i=1}^{N} b_i \cdot K(|x - x_i|) \quad (4)$$

In case when the uncertainty, i.e., variance $\sigma^2$ of a value in a given point is known, it can be used to accordingly diminish the impact of this point when solving the kriging system. In such case it is sufficient to add the variance $\sigma^2$ (the so-called nugget effect) to the $K(|x_i - x_i|)$ terms on the diagonal of the matrix in (3).

### 3.1 Parametric kriging of surfaces

The surfaces of revolution may be described by kriging with parameters the geometry of cross-sections and then applying another kriging system along the centroid path to provide the full description of the surface. The resulting dual kriging equation of the surface is (for constant trend function)

$$x(s, t_k) = a_0(t_k) + \sum_{i=1}^{M} b_i(t) K_1(|s - s_i|) \quad (5)$$
where *s* and *t* are parameters for two different kriging directions. The idea is to obtain kriging equations of cross-section parametrized by *s* for each parameter *t*<sub>*k*</sub> and then to perform another kriging along the path described by *t*.

# 4 Coupling of X-ray microtomography with dual kriging

The kriging apparatus described above once calculated provides a method with constant compu-
tational complexity for reconstruction of a given geometry. The critical part of its application is the
good preparation of source data, i.e., image processing (mainly segmentation with learning algorithms
and shape detection) of X-ray microtomography results. This has been realized with software described
in [7] by combining existing and in-house developed tools. The geometry has been then created with
software from [1] and meshed with built-in algorithms. An example of previously obtained, noisy and
detailed mesh of the fiber tow as compared to the one obtained with kriging is shown in Figure 1.

<figure><img src="image_2.png" /><figcaption>FIGURE 1 – Comparison of untreated mesh of fibers tows with smooth geometry envelopes obtained with dual kriging.</figcaption></figure>

Références

[1] Blender Foundation. *Blender - a 3D modelling and rendering package*, 2014.

[2] L. P. Brown, X. Zeng, A. Long, and I. A. Jones. *Recent developments in the realistic geometric modelling of textile structures using TexGen*, International Conference on Digital Technologies for the Textile Industries, 2013.

[3] S. Lomov, D. Ivanov, I. Verpoest, M. Zako, T. Kurashiki, H. Nakai, and S. Hirosawa. *Meso-FE modelling of textile composites: Road map, data flow and algorithms*, Composites Science and Technology, Elsevier, 1870-1891, 2007.

[4] A. Madra, N. El Hajj and M. Benzeggagh. X-ray microtomography applications for quantitative and qualitative analysis of porosity in woven glass fiber reinforced thermoplastic, Composites Science and Technology, Elsevier, 50-58, 2014.

[5] J. Pazmino, V. Carvelli, and S. V. Lomov. *Micro-CT analysis of the internal deformed geometry of a non-crimp 3D orthogonal weave E-glass composite reinforcement*, Composites Part B: Engineering, Elsevier, 2013.

[6] E. Ruiz, V. Achim, S. Soukane, F. Trochu, and J. Breard. *Optimization of injection flow rate to minimize micro/macro-voids formation in resin transfer molded composites*, Composites Science and Technology, Elsevier, 475-486, 2006.

[7] J. Schindelin, I. Arganda-Carreras, E. Frise, V. Kaynig, M. Longair, T. Pietzsch, S. Preibisch, C. Rueden, and S. Saalfeld. *Fiji: an open-source platform for biological-image analysis*, Nature Methods, Macmillan Publishers Limited, 676-682, 2012.

[8] I. Verpoest and S. Lomov. *Virtual textile composites software: Integration with micro-mechanical, permeability and structural analysis*, Composites Science and Technology, Elsevier, 2563-2574, 2005.