# Analysis and Evaluation of Fiber Orientation Reconstruction Methods

^{1}

^{2}

^{*}

*Journal of Composites Science*in 2019)

## Abstract

**:**

## 1. Introduction

## 2. Fiber Orientation

## 3. Icosphere

$(\begin{array}{ccccc}-1& & t& & 0\end{array})$ | $(\begin{array}{ccccc}0& & -1& & t\end{array})$ | $(\begin{array}{ccccc}t& & 0& & -1\end{array})$ |

$(\begin{array}{ccccc}1& & t& & 0\end{array})$ | $(\begin{array}{ccccc}0& & 1& & t\end{array})$ | $(\begin{array}{ccccc}t& & 0& & 1\end{array})$ |

$(\begin{array}{ccccc}-1& & -t& & 0\end{array})$ | $(\begin{array}{ccccc}0& & -1& & -t\end{array})$ | $(\begin{array}{ccccc}-t& & 0& & -1\end{array})$ |

$(\begin{array}{ccccc}1& & -t& & 0\end{array})$ | $(\begin{array}{ccccc}0& & 1& & -t\end{array})$ | $(\begin{array}{ccccc}-t& & 0& & 1\end{array})$ |

## 4. Reconstruction Methods

#### 4.1. Spherical Harmonics

#### 4.2. Method of Maximum Entropy

## 5. Evaluation

#### 5.1. Tests Scenarios

#### 5.2. Evaluation Criteria

#### 5.3. Numerical Results for Bingham Distributions

#### 5.4. Numerical Results for Measured Data

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Jeffery, G.B. The Motion of Ellipsoidal Particles Immersed in a Viscous Fluid. Proc. R. Soc. A Math. Phys. Eng. Sci.
**1922**, 102, 161–179. [Google Scholar] [CrossRef] - Jack, D.A. Advanced Analysis of Short Fiber Polymer Composite Material Behavior. Ph.D. Thesis, University of Missouri, Columbia, MO, USA, 2006. [Google Scholar]
- Risken, H. The Fokker-Planck Equation. Methods of Solution and Applications, 2nd ed.; Springer: Berlin, Germany, 1996; ISBN 978-3-540-61530-9. [Google Scholar]
- Einstein, A. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys.
**2005**, 14, 182–193. [Google Scholar] [CrossRef] - Folgar, F.; Tucker, C.L. Orientation Behavior of Fibers in Concentrated Suspensions. J. Reinf. Plast. Compos.
**1984**, 3, 98–119. [Google Scholar] [CrossRef] - Hand, G.L. A theory of anisotropic fluids. J. Fluid Mech.
**1962**, 13, 33–46. [Google Scholar] [CrossRef] - Advani, S.G.; Tucker, C.L. The Use of Tensors to Describe and Predict Fiber Orientation in Short Fiber Composites. J. Rheol.
**1987**, 31, 751–784. [Google Scholar] [CrossRef] - Doi, M. Molecular dynamics and rheological properties of concentrated solutions of rodlike polymers in isotropic and liquid crystalline phases. J. Polym. Sci. Polym. Phys. Ed.
**1981**, 19, 229–243. [Google Scholar] [CrossRef] - Marrucci, G.; Grizzuti, N. Predicted effect of polydispersity on rodlike polymer behaviour in concentrated solutions. J. Non-Newton. Fluid Mech.
**1984**, 14, 103–119. [Google Scholar] [CrossRef] - Agboola, B.O.; Jack, D.A.; Montgomery-Smith, S. Effectiveness of recent fiber-interaction diffusion models for orientation and the part stiffness predictions in injection molded short-fiber reinforced composites. Compos. Part A Appl. Sci. Manuf.
**2012**, 43, 1959–1970. [Google Scholar] [CrossRef] - Montgomery-Smith, S.; He, W.; Jack, D.A.; Smith, D.E. Exact tensor closures for the three-dimensional Jeffery’s equation. J. Fluid Mech.
**2011**, 680, 321–335. [Google Scholar] [CrossRef] - Montgomery-Smith, S.; Jack, D.; Smith, D.E. The Fast Exact Closure for Jeffery’s equation with diffusion. J. Non-Newton. Fluid Mech.
**2011**, 166, 343–353. [Google Scholar] [CrossRef] - Chaubal, C.V.; Leal, L. A closure approximation for liquid-crystalline polymer models based on parametric density estimation. J. Rheol.
**1998**, 42. [Google Scholar] [CrossRef] - Cintra, J.S.; Tucker, C.L. Orthotropic closure approximations for flow-induced fiber orientation. J. Rheol.
**1995**, 39, 1095–1122. [Google Scholar] [CrossRef] - De Frahan, H.H.; Verleye, V.; Dupret, F.; Crochet, M.J. Numerical prediction of fiber orientation in injection molding. Polym. Eng. Sci.
**1992**, 32, 254–266. [Google Scholar] [CrossRef] - Jack, D.A.; Schache, B.; Smith, D.E. Neural network-based closure for modeling short-fiber suspensions. Polym. Compos.
**2010**, 31, 1125–1141. [Google Scholar] [CrossRef] - Qadir, N.U.; Jack, D.A. Modeling fibre orientation in short fibre suspensions using the neural network-based orthotropic closure. Compos. Part A Appl. Sci. Manuf.
**2009**, 40, 1524–1533. [Google Scholar] [CrossRef] - Feng, J.; Chaubal, C.V.; Leal, L.G. Closure approximations for the Doi theory: Which to use in simulating complex flows of liquid-crystalline polymers? J. Rheol.
**1998**, 42, 1095–1119. [Google Scholar] [CrossRef] - Hinch, E.J.; Leal, L.G. Constitutive equations in suspension mechanics. Part 2. Approximate forms for a suspension of rigid particles affected by Brownian rotations. J. Fluid Mech.
**1976**, 6, 187–208. [Google Scholar] [CrossRef] - Chung, D.H.; Kwon, T.H. Invariant-based optimal fitting closure approximation for the numerical prediction of flow-induced fiber orientation. J. Rheol.
**2002**, 46, 169–194. [Google Scholar] [CrossRef] - Altan, M.C.; Subbiah, S.; Güçeri, S.I.; Pipes, R.B. Numerical prediction of three-dimensional fiber orientation in Hele-Shaw flows. Polym. Eng. Sci.
**1990**, 30, 848–859. [Google Scholar] [CrossRef] - Jack, D.A.; Smith, D.E. An invariant based fitted closure of the sixth-order orientation tensor for modeling short-fiber suspensions. J. Rheol.
**2005**, 49, 1091–1115. [Google Scholar] [CrossRef] - Moldex3D. Fiber Function Overview. Available online: http://support.moldex3d.com/r15/en/sync/sync-for-nx/functionoverview_1.html (accessed on 15 January 2019).
- Autodesk Simulation. Moldflow’s Fiber Orientation Models (Theory). Available online: http://help.autodesk.com/view/MFIWS/2014/ENU/?guid=GUID-6B3A7386-DE57-450E-BF94-B10BD629EC9B (accessed on 15 January 2019).
- Li, T.; Luyé, J.F. Optimization of Fiber Orientation Model Parameters in the Presence of Flow-Fiber Coupling. J. Compos. Sci.
**2018**, 2, 73. [Google Scholar] [CrossRef] - Han, J.H.; Khawaja, A.; Kilic, M.H.; Ingram, K. Identification of fiber orientation prediction error by moldflow on compression molded discontinuous long fiber composites using computed tomography x-ray. In Proceedings of the CAMX 2015–Composites and Advanced Materials Expo, Beijing, China, 27–29 October 2015. [Google Scholar]
- Férec, J.; Heniche, M.; Heuzey, M.C.; Ausias, G.; Carreau, P.J. Numerical solution of the Fokker–Planck equation for fiber suspensions: Application to the Folgar–Tucker–Lipscomb model. J. Non-Newton. Fluid Mech.
**2008**, 155, 20–29. [Google Scholar] [CrossRef] - Russell, T.; Heller, B.; Jack, D.A.; Smith, D.E. Prediction of the Fiber Orientation State and the Resulting Structural and Thermal Properties of Fiber Reinforced Additive Manufactured Composites Fabricated Using the Big Area Additive Manufacturing Process. J. Compos. Sci.
**2018**, 2, 26. [Google Scholar] [CrossRef] - Friebel, C.; Doghri, I.; Legat, V. General mean-field homogenization schemes for viscoelastic composites containing multiple phases of coated inclusions. Int. J. Solids Struct.
**2006**, 43, 2513–2541. [Google Scholar] [CrossRef][Green Version] - Müller, V.; Böhlke, T. Prediction of effective elastic properties of fiber reinforced composites using fiber orientation tensors. Compos. Sci. Technol.
**2016**, 130, 36–45. [Google Scholar] [CrossRef] - Weber, B.; Kenmeugne, B.; Clement, J.C.; Robert, J.L. Improvements of multiaxial fatigue criteria computation for a strong reduction of calculation duration. Comput. Mater. Sci.
**1999**, 15, 381–399. [Google Scholar] [CrossRef] - Jack, D.A.; Smith, D.E. Elastic Properties of Short-fiber Polymer Composites, Derivation and Demonstration of Analytical Forms for Expectation and Variance from Orientation Tensors. J. Compos. Mater.
**2008**, 42, 277–308. [Google Scholar] [CrossRef] - Huang, T.S.; Kohonen, T.; Schroeder, M.R.; Lotsch, H.K.V.; Wu, N. The Maximum Entropy Method; Springer: Berlin, Germany, 1997; ISBN 978-3-642-64484-9. [Google Scholar]
- PART Engineering GmbH. CONVERSE Documentation V 4.0.6; PART Engineering GmbH: Bergisch Gladbach, Germany, 2019. [Google Scholar]
- Shannon, C.E. A mathematical theory of communication. Bell Syst. Tech. J.
**1948**, 27, 379–423. [Google Scholar] [CrossRef] - Hine, P.J.; Lusti, H.R.; Gusev, A.A. On the possibility of reduced variable predictions for the thermoelastic properties of short fibre composites. Compos. Sci. Technol.
**2004**, 64, 1081–1088. [Google Scholar] [CrossRef] - Müller, V. Micromechanical Modeling of Short-Fiber Reinforced Composites. Ph.D. Thesis, Karlsruher Institut für Technologie (KIT), Karlsruhe, Germany, 2016. [Google Scholar]
- Breitenberger, E. Analogues of the normal distribution on the circle and the sphere*. Biometrika
**1963**, 50, 81–88. [Google Scholar] [CrossRef] - Bingham, C. An Antipodally Symmetric Distribution on the Sphere. Ann. Stat.
**1974**, 2, 1201–1225. [Google Scholar] [CrossRef] - Kraft, D. A software package for sequential quadratic programming. In Forschungsbericht-Deutsche Forschungs-und Versuchsanstalt fur Luft-und Raumfahrt (DFVLR); Royal Society of Chemistry: Cambridge, UK, 1988. [Google Scholar]
- Nocedal, J.; Wright, S.J. Numerical Optimization, 2nd ed.; Springer Science + Business Media LLC: New York, NY, USA, 2006; ISBN 978-0-387-40065-5. [Google Scholar]
- Kullback, S.; Leibler, R.A. On Information and Sufficiency. Ann. Math. Stat.
**1951**, 1951, 79–86. [Google Scholar] [CrossRef] - Rolland, H.; Saintier, N.; Robert, G. Damage mechanisms in short glass fibre reinforced thermoplastic during in situ microtomography tensile tests. Compos. Part B Eng.
**2016**, 90, 365–377. [Google Scholar] [CrossRef][Green Version] - Lenke, K. Kurzfaserverstärktes Polyamid—Charakterisierung der Mikroschädigungsentwicklung unter Zweiachsiger Mechanischer Last. Ph.D. Thesis, Technische Universität Berlin, Berlin, Germany, 2016. [Google Scholar]

**Figure 2.**Two different orientation density functions (ODFs) (left: unidirectional with two maximas, right: planar isotropy) with the same fiber orientation tensor to illustrate the ambiguity of the reconstruction.

**Figure 3.**Evolution of icosphere with refinement level 0, 1, and 5. The color represents the relative area of the triangles.

**Figure 4.**Top: Initial ODF and data of first test case. Bottom: Reconstruction of test case with the different reconstruction methods.

**Figure 5.**Top: Initial ODF and data of second test case. Bottom: Reconstruction of test case with the different reconstruction methods.

**Figure 6.**Top: Initial ODF and data of third test case. Bottom: Reconstruction of test case with the different reconstruction methods.

**Figure 7.**Top: Initial ODF and data of fourth test case. Bottom: Reconstruction of test case with the different reconstruction methods.

**Figure 8.**Initial ODF and data of fifth test case. Bottom: Reconstruction of test case with the different reconstruction methods.

**Figure 9.**Color plot of the Kullback–Leibner distance in dependency of covariance (SH2, SH4, SH6, and SH8).

**Figure 10.**Color plot of the Kullback–Leibner distance in dependency of covariance (SH4HY, SH6HY, SH6HYHY, and ME).

**Figure 12.**Measured and reconstructed ODFs from [7].

**Figure 13.**Measured and reconstructed ODFs from [43].

**Figure 14.**Measured and reconstructed ODFs from [44].

Regime | Volume Fraction: ${\mathit{v}}_{\mathit{F}}$ |
---|---|

Dilute | ${v}_{f}<\frac{1}{{a}_{r}^{2}}$ |

Semi-Dilute | $\frac{1}{{a}_{r}^{2}}<{v}_{F}<\frac{1}{{a}_{r}}$ |

Concentrated | $\frac{1}{{a}_{r}}>{v}_{F}$ |

Closure Name | Reference |
---|---|

Linear | [6,7] |

Quadratic | [7,8,9] |

Hybrid | [7,10] |

Exact | [11] |

Fast Exact (FEC) | [11,12] |

Bingham | [13] |

Orthotropic | [10,14] |

Natural | [15] |

Neural Network (NNET) | [16] |

Neural Network Orthotropic (NNORT) | [17] |

Quad R | [18] |

Hinch and Leal W1 (isotropic) | [19] |

Hinch and Leal S2 (strong flow) | [19] |

Hinch and Leal HL1 | [18] |

Hinch and Leal HL2 | [18] |

Hinch and Leal HL1Q | [18] |

Invariant Based optimal fitting (IOBF) | [20] |

Method | Abbreviation | $\widehat{\mathit{\psi}}$ | ${\mathit{a}}_{\mathbf{2}}$ | ${\mathit{a}}_{\mathbf{4}}$ | ${\mathit{a}}_{\mathbf{6}}$ | ${\mathit{a}}_{\mathbf{8}}$ |
---|---|---|---|---|---|---|

Method of Maximum Entropy | ME | $f\left({a}_{2}\right)$ | $f\left(\psi \right)$ | |||

Spherical Harmonics (2nd order) | SH2 | $f\left({a}_{2}\right)$ | $f\left(\psi \right)$ | |||

Spherical Harmonics (4th order) | SH4 | $f\left({a}_{2},\text{}{a}_{4}\right)$ | $f\left(\psi \right)$ | $f\left(\psi \right)$ | ||

Spherical Harmonics (6th order) | SH6 | $f\left({a}_{2},\text{}{a}_{4},\text{}{a}_{6}\right)$ | $f\left(\psi \right)$ | $f\left(\psi \right)$ | $f\left(\psi \right)$ | |

Spherical Harmonics (8th order) | SH8 | $f\left({a}_{2},\text{}{a}_{4},{a}_{6},{a}_{8}\right)$ | $f\left(\psi \right)$ | $f\left(\psi \right)$ | $f\left(\psi \right)$ | $f\left(\psi \right)$ |

Spherical Harmonics (4th order, with hybrid closure) | SH4HY | $f\left({a}_{2},\text{}{\widehat{a}}_{4}\right)$ | $f\left(\psi \right)$ | ${\widehat{a}}_{4}=f\left({a}_{2}\right)$ Hybrid Closure | ||

Spherical Harmonics (6th order, with hybrid closure) | SH6HY | $f\left({a}_{2},\text{}{a}_{4},\text{}{\widehat{a}}_{6}\right)$ | $f\left(\psi \right)$ | $f\left(\psi \right)$ | ${\widehat{a}}_{6}=f\left({a}_{2},{\widehat{a}}_{4}\right)$ Hybrid Closure | |

Spherical Harmonics (6th order, with hybrid closure for 4th and 6th order tensor) | SH6HYHY | $f\left({a}_{2},\text{}{\widehat{a}}_{4},\text{}{\widehat{a}}_{6}\right)$ | $f\left(\psi \right)$ | ${\widehat{a}}_{4}=f\left({a}_{2}\right)$ Hybrid Closure | ${\widehat{a}}_{6}=f\left({a}_{2},{\widehat{a}}_{4}\right)$ Hybrid Closure |

**Table 5.**Comparison of ODFs (with ${\mathsf{\psi}}_{0}$) for the illustration of evaluation criteria.

${\mathit{\psi}}_{\mathbf{0}}$ (Reference) | ${\mathit{\psi}}_{\mathbf{1}}$ | ${\mathit{\psi}}_{\mathbf{2}}$ | ${\mathit{\psi}}_{\mathbf{3}}$ | ${\mathit{\psi}}_{\mathbf{4}}$ |
---|---|---|---|---|

$D=0.0$ | $D=0.0$ | $D=0.346$ | $D=2.767$ | $D=4.461$ |

$E=0.0$ | $E=0.159$ | $E=0.079$ | $E=0.081$ | $E=0.112$ |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Breuer, K.; Stommel, M.; Korte, W. Analysis and Evaluation of Fiber Orientation Reconstruction Methods. *J. Compos. Sci.* **2019**, *3*, 67.
https://doi.org/10.3390/jcs3030067

**AMA Style**

Breuer K, Stommel M, Korte W. Analysis and Evaluation of Fiber Orientation Reconstruction Methods. *Journal of Composites Science*. 2019; 3(3):67.
https://doi.org/10.3390/jcs3030067

**Chicago/Turabian Style**

Breuer, Kevin, Markus Stommel, and Wolfgang Korte. 2019. "Analysis and Evaluation of Fiber Orientation Reconstruction Methods" *Journal of Composites Science* 3, no. 3: 67.
https://doi.org/10.3390/jcs3030067