Fiber orientation is an important descriptor of the microstructure for short fiber polymer composite materials where accurate and efficient prediction of the orientation state is crucial when evaluating the bulk thermo-mechanical response of the material. Recent macroscopic fiber orientation models have employed the moment-tensor form in representing the fiber orientation state which all require a closure approximation for the higher order orientation tensors. In addition, various models have been developed to account for rotary diffusion due to fiber-fiber and fiber-matrix interactions which can now more accurately simulate the experimentally observed slow fiber kinematics in polymer composite processing. Traditionally explicit numerical IVP-ODE transient solvers like the 4th order Runge-Kutta method have been used to predict the steady-state fiber orientation state. Here we propose a computationally efficient method based on the Newton-Raphson iterative technique for determining steady state orientation tensor values by evaluating the exact derivatives of the moment-tensor evolution equation with respect to the independent components of the orientation tensor. We consider various existing macroscopic fiber orientation models and several closure ap-proximations to ensure the robustness and reliability of the method. The performance and stability of the approach for obtaining physical solutions in various homogeneous flow fields is demonstrated through several examples. Validation of the obtained exact derivatives of the orientation tensor is performed by benchmarking with results of finite difference techniques

翻译：纤维取向是短纤维聚合物复合材料微观结构的重要描述参数，在评估材料整体热力学响应时，准确高效地预测取向状态至关重要。近期宏观纤维取向模型采用矩张量形式表征纤维取向状态，该形式均需对高阶取向张量进行闭合近似。此外，为考虑纤维-纤维及纤维-基体相互作用引起的旋转扩散效应，多种模型已被开发，现能更精确地模拟聚合物复合材料加工过程中实验观测到的缓慢纤维运动学。传统上常采用显式数值初值问题-常微分方程瞬态求解器（如四阶龙格-库塔法）预测稳态纤维取向状态。本文提出一种基于牛顿-拉夫森迭代技术的计算高效方法，通过求解矩张量演化方程对取向张量独立分量的精确导数，以确定稳态取向张量值。我们综合考虑现有多种宏观纤维取向模型及若干闭合近似方案，以确保方法的鲁棒性与可靠性。通过多个算例验证了该方法在不同均匀流场中获得物理解的性能与稳定性。取向张量精确导数的验证通过有限差分技术的结果进行基准测试实现。