This paper presents a novel shape-optimization technique for the design of mixing elements in single-screw extruders. Extruders enable the continuous production of constant-cross-section profiles. Equipped with one or several screw-shaped rotors, the extruder transports solid polymer particles towards the outlet. Due to shear heating, melting is induced and a melt stream is created, which can be further processed. While many variants of multi-screw extruders exist, a significant share of all extrusion machines is made up of single-screw extruders due to their comparatively low operating costs and complexity. While the reduced complexity yields economic benefits, single-screw extruders’ mixing capabilities, i.e., their ability to produce a melt with a homogeneous material and temperature distribution, suffer compared to multi-screw extruders. To compensate for this shortcoming, so-called mixing elements are added to the screw to enhance dynamic mixing by recurring flow reorientations. In view of the largely unintuitive flow characteristics of polymer melts, we present an optimization framework that allows designing these mixing elements numerically based on finite-element simulations of the melt flow. To reduce the computational demand required by shape optimization of a complete mixing section, we only focus on the shape optimization of a single mixing element. This paper presents advances in three aspects of numerical design: (1) A combination of free-form deformation and surface splines is presented, allowing to parameterize the mixing element’s shape by very few variables. (2) The combination of this concept with a linear-elasticity-based mesh update method to deform the computational domain without the need for remeshing is demonstrated. (3) A simple yet robust and sensitive new objective formulation to assess distributive mixing in laminar flows based on a measure for the interfacial area is proposed for the optimization.

本文提出了一种新颖的形状优化技术，用于设计单螺杆挤出机中的混合元件。挤出机能够连续生产等截面型材。配备一个或多个螺旋形转子的挤出机将固体聚合物颗粒输送到出口。由于剪切加热，导致熔化并产生熔体流，可以进一步加工。虽然存在许多多螺杆挤出机的变体，但由于其相对较低的运营成本和复杂性，所有挤出机的很大一部分由单螺杆挤出机组成。虽然降低的复杂性产生了经济效益，但与多螺杆挤出机相比，单螺杆挤出机的混合能力，即它们生产具有均匀材料和温度分布的熔体的能力受到影响。为了弥补这一缺点，在螺杆中添加了所谓的混合元件，以通过重复流动重新定向来增强动态混合。鉴于聚合物熔体在很大程度上不直观的流动特性，我们提出了一个优化框架，该框架允许基于熔体流动的有限元模拟数值设计这些混合元件。为了减少整个混合段形状优化所需的计算量，我们只关注单个混合单元的形状优化。本文介绍了数值设计的三个方面的进展： (1) 提出了自由变形和曲面样条的组合，允许通过很少的变量参数化混合元件的形状。(2) 该概念与基于线性弹性的网格更新方法相结合，无需重新划分网格即可对计算域进行变形。(3) 提出了一种简单而稳健且敏感的新目标公式，用于基于界面面积的测量来评估层流中的分布混合，以进行优化。