Investigation of the Effectiveness and Efficiency of the Exact Closure: Comparison With Industrial Closures and Spherical Harmonic Solutions

For stiffness predictions of short fiber reinforced polymer composites, it is essential to understand the orientation during processing. This is often performed through the equation of change of the fiber orientation tensor to simulate the fiber orientation during processing. Unfortunately this approach, while computationally efficient, requires the next higher ordered orientation tensor, thus requiring the use of a closure approximation. Many efforts have been made to develop closures to approximate the fourth-order orientation tensor in terms of the second order orientation tensor. Recently, Montgomery-Smith et al (2010) developed a pair of exact closures, one for systems with dilute suspensions and a second for dense suspensions, where the later works well for a variety of diffusion models. In this paper we compare the fiber orientation results of the Fast Exact Closure (FEC) for dense suspensions to that of the Spherical Harmonic solution, which although considered to be numerically exact does not readily lend itself to implementations in current industrial processing CFD codes. This paper focuses on a series of comparisons of material stiffness predictions between the FEC, current fitted closure models, and the spherical harmonics solution for a thin plate subjected to pure shear. Results for the select flows considered show the similarities between the current class of orthotropic fitted closures and that of the FEC. Although the results are similar between the fitted closures and the FEC, it is important to recognize that the Fast Exact Closure is formed without a fitting process. Consequently, the results are anticipated, in general, to be more robust in implementation.