Virtual clustering analysis for long fiber reinforced composites

Abstract

Virtual clustering analysis (VCA) is an effective data-driven numerical homogenization method. In its original version (Tang et al. in Comput Mech 62(6):1443–1460, 2018), a homogeneous reference material is selected, and the governing system of heterogeneous material is reformulated into a Lippmann–Schwinger equation. For effective treatment of representative volume element (RVE) where long fibers penetrate the matrix, we propose to introduce an inhomogeneous reference material in this work. Since analytical expressions are not available in general, the interaction tensor and reference strain field are computed numerically. We validate the consistency with the original version of VCA in the first example, and demonstrate the effectiveness of the proposed strategy by several numerical examples. While better resolving the boundary traction, the proposed virtual clustering analysis for long fiber reinforced composites (VCA-L) algorithm correctly predicts homogenized properties in numerical tests.

Appendices

Appendices

SOM algorithm

In SOM, we have a data set $\varvec{X}$, with each data $\varvec{x} \in \mathbb {R}^{m}$, and designate k clusters. We prescribe a norm $\Vert \cdot \Vert$ for the m-dimensional vector space, and a distance $d_{i,j}$ for the integer indices $i,j=1,\ldots ,k$. In this paper, we take $\Vert \cdot \Vert$ as Euclidean norm, and $d_{i,j}=|i-j|$. An iterative algorithm with N iterations is as follows.

• 1. Determine a spatial arrangement of the k clusters. In all our tests, we use $1\times k$ for this arrangement.

• 2. Initialize the weight vector $\varvec{w}_{j}$ for each cluster, $j=1,\ldots ,k$. It is one possible way to pick k data from the input data randomly and use them as the weight vectors of the k clusters.

• 3. Draw a sample x from the input data with a certain probability, and find the winning cluster i(x) at time-step n ruled by

  $$\begin{aligned} i(\varvec{x})=\arg \underset{1\le j \le k}{\min }\Vert \varvec{x}(n) - \varvec{w}_{j}\Vert . \end{aligned}$$

• 4. Adjust the weight vectors of all affected clusters

  $$\begin{aligned} \varvec{w}_{j}(n+1) = \varvec{w}_{j}(n) + \eta (n)h_{j,i(\varvec{x})}(n)\left( \varvec{x}(n)-\varvec{w}_{j}(n)\right) , \end{aligned}$$

  where $\eta (n)$ is a learning rate parameter, and $h_{j,i(\varvec{x})}$ is the neighborhood function centered around the winning cluster i(x). We choose a Gaussian function

  $$\begin{aligned} h_{j,i(\varvec{x})}(n) = \exp \left( - \frac{d^{2}_{j,i(\varvec{x})}}{2k^{2}\exp (-6 n(\log {k})/N)} \right) , \end{aligned}$$

  and a decreasing learning function

  $$\begin{aligned} \eta (n) = 0.1 \exp (-\frac{3n}{N}). \end{aligned}$$

• 5. Go back Step 3 until it reaches N.

Relationship with the finite element method

As described in Sect. 2.6, the RVE is usually discretized by regular pixels/voxels. Given that the finite element method is adopted as the direct numerical simulation method, two procedures are conducted to give a suitable transfer and comparison of data between these two types of discretizations.

The first procedures is to generate a FEM mesh compatible with the regular pixels/voxels. In this work, CPE4 element is adopted in the computation of finite element method. Furthermore, the element is square, which can be converted by the pixel.

The second procedure is the element average technique. Each pixel/voxel only contains one sampling point, corresponding to the Gauss sampling points in the finite element method. To transfer the data from the Gauss sampling points to the single sampling point, element average technique is used to compress the data. That is, for a pixel/voxel, a quantity in the single sampling point corresponds to averaging this quantity in its corresponding element.

Constitutive relation and coordinate transformation for the transversely isotropic fiber material

Through the fiber is transversely isotropic in its three-dimension local coordinate, we only use its material property in two-dimension local coordinate. Based on this requirement, the constitutive relation in its two-dimension local coordinate is given in the matrix form as follows:

$$\begin{aligned}{}[\varvec{\sigma }^{l}] = [\varvec{C}^{l}] [\varvec{\varepsilon }^{l}], \end{aligned}$$

(53)

where

$$\begin{aligned}{}[\varvec{\sigma }^{l}] = \{ \sigma _{11}^{l} \quad \sigma _{22}^{l} \quad \sigma _{12}^{l} \}^{T}, \end{aligned}$$

(54)

$$\begin{aligned}{}[\varvec{\varepsilon }^{l}] = \{ \varepsilon _{11}^{l} \quad \varepsilon _{22}^{l} \quad \gamma _{12}^{l} \}^{T}, \end{aligned}$$

(55)

$$\begin{aligned}{}[\varvec{C}^{l}] = \begin{bmatrix} \dfrac{E_{1}\left( 1-\nu _{23}\right) }{\Delta } &{} \dfrac{E_{2}\nu _{12}}{\Delta } &{} 0 \\ *** &{} \dfrac{E_{2}\left( E_{1}-E_{2}\nu _{12}^{2}\right) }{\Delta E_{1} \left( 1+\nu _{23}\right) }&{} 0\\ *** &{} *** &{} G_{12}\\ \end{bmatrix}. \end{aligned}$$

(56)

in which the superscript l means that the relevant quantities are evaluated in the local coordinate system; the subscripts 1, 2 and 3 are corresponding to the longitudinal, in-plane transverse and out-of plane transverse directions; $***$ denotes an entry determined by symmetry; and

$$\begin{aligned} \Delta = 1 - \nu _{23} - \frac{2 E_{2} \nu _{12}^{2}}{E_{1}}. \end{aligned}$$

(57)

Then these constitutive quantities (especially the local stiffness $\varvec{C}^{l}$) need transformation from the local coordinate system to the global coordinate system. The coordinate transformation of the stiffness can be expressed as follows:

$$\begin{aligned}{}[\varvec{C}^{g}] = [\varvec{T}_{c}]^{T} [\varvec{C}^{l}] [\varvec{T}_{c}]. \end{aligned}$$

(58)

Here the superscript g denotes that the corresponding quantities are evaluated in the global coordinate system, and $[\varvec{T}_{c}]$ is the transformation matrix

$$\begin{aligned}{}[\varvec{T}_{c}] = \begin{bmatrix} cos^2\theta _{11} &{} cos^{2}\theta _{21} &{} cos\theta _{11}cos\theta _{21} \\ cos^2\theta _{12} &{} cos^2\theta _{22} &{} cos\theta _{12}cos\theta _{22} \\ 2cos\theta _{11}cos\theta _{12} &{} 2cos\theta _{21}cos\theta _{22} &{} cos\theta _{11}cos\theta _{22}+cos\theta _{21}cos\theta _{12} \\ \end{bmatrix},\nonumber \\ \end{aligned}$$

(59)

where $\theta _{ij}$ ($i,j=1,2$) is the angle between the jth axis of the local coordinate system and the ith axis of the global coordinate system. Further, the constitutive relation in the global coordinate system is

$$\begin{aligned}{}[\varvec{\sigma }^{g}] = [\varvec{C}^{g}][\varvec{\varepsilon }^{g}], \end{aligned}$$

(60)

where

$$\begin{aligned}{}[\varvec{\sigma }^{g}] = \{ \sigma _{11}^{g} \quad \sigma _{22}^{g} \quad \sigma _{12}^{g} \}^{T}, \end{aligned}$$

(61)

$$\begin{aligned}{}[\varvec{\varepsilon }^{g}] =\{ \varepsilon _{11}^{g} \quad \varepsilon _{22}^{g} \quad \gamma _{12}^{g} \}^{T}. \end{aligned}$$

(62)