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Efficient Geometry Representation Strategies for the Shape Optimization of Profile Extrusion Dies
Jana Sasse, Maximilian Esser, Markus Mügge, Stefan Turek
TL;DR
Profile extrusion die design is challenging because flow-channel geometry must produce uniform outflow despite operating-condition dependence, complex 3D geometry, and melt rheology. The paper investigates adjoint-based optimization with different geometry representations and optimization algorithms. For a complex die, immersed-boundary topology optimization improved flow balancing by 55% using fewer solver calls than other established CFD-based optimization methods.
Problem
Profile-extrusion flow-channel design must produce uniform outflow despite dependence on throughput, temperature, and melt properties, while complex 3D geometry and rheology challenge computational methods.
Source: §1 · p. 1Method
The paper compares geometry representations and sensitivity-based optimization strategies for adjoint optimization of complex profile extrusion dies.
Source: §1 · p. 1, §2.1 · p. 2, §2.2 · p. 3Results
55% improvement in flow balancing was achieved on a complex profile extrusion die using immersed-boundary topology optimization, with fewer solver calls than other established CFD-based optimization methods.
Source: §4 · p. 8Takeaways & Limitations
For complex profile extrusion dies, immersed-boundary topology optimization is the most versatile approach, while aligned-mesh shape optimization can be most reliable when an appropriate aligned mesh is available.
Source: §2.3 · p. 4, §4 · p. 8Lab-trial validation of a manufactured optimized die remains future work, alongside extending the algorithm to fully non-isothermal flows.
Source: §4 · p. 8
Abstract
from arXiv · showhide
The design of profile extrusion dies remains a challenging task due to the complex rheological behavior of polymer melts and the geometric intricacies of flow channels. Traditional manual optimization approaches, which rely heavily on human experience, are inefficient and often employ unvalidated heuristics. To address these challenges, we present a deterministic and explainable framework for automatic die design based on adjoint-based shape optimization. This approach enables the computation of sensitivities that directly indicate beneficial modifications to the flow channel geometry. A major difficulty in such optimization processes lies in generating boundary-conforming meshes that evolve consistently with changing geometries. To overcome this issue, we employ non-boundary conforming geometry representation methods that eliminate the need for an explicit surface representation along physical boundaries. A dedicated reconstruction technique is developed to recover accurate sensitivity information at the virtual interface between fluid and solid regions. The proposed algorithm is demonstrated on 3D geometries with varying complexity, including realistic extrusion die flow channels. Several objective functionals relevant to industrial applications, such as flow balance at the outflow, are considered. The results highlight significant improvements in performance metrics while maintaining numerical robustness. This work showcases the potential of adjoint-based techniques for automated die design in a domain still largely governed by manual trial-and-error procedures, establishing a foundation for data-efficient, sustainable manufacturing workflows using computational rheology.
1. Introduction
Profile extrusion die design remains challenging because the desired uniform outflow depends on operating conditions and melt properties, while complex geometries and rheology complicate computational methods. This paper aims to identify efficient geometry representations and sensitivity-based algorithms for adjoint optimization of complex dies.
1. Introduction
- Uniform outflow velocity depends on throughput, processing temperature, and melt thermo-rheological properties, so die design still relies heavily on manual iterations.The challenge concerns choosing a flow-channel geometry that produces uniform outflow under the operating point.Source: §1 · p. 1
- Adjoint-based sensitivities offer a promising low-computational-cost alternative to established optimization methods, which require significant solver calls.Computational methods also face challenges modeling complex 3D domains and the shear-thinning, viscoelastic, temperature-dependent melt.Source: §1 · p. 1
- The research aims to identify efficient geometry representation and sensitivity-based optimization strategies for complex profile extrusion dies.Source: §1 · p. 1
2. Numerical Modeling
The section compares geometry representations and adjoint optimization pairings for extrusion-die flow channels. It identifies IBM topology optimization as versatile for industrially relevant geometries while retaining aligned-mesh shape optimization as most reliable when suitable meshes are available.
2.1. Geometry Representation Strategies
- FBM and IBM both simplify mesh generation, but FBM masks the filtered fluid domain while IBM blocks flow locally using cell pseudo-porosity.Compared with body-fitted meshing, these methods make mesh generation independent of channel complexity, at the cost of interface accuracy; IBM may be more robust to interface artifacts but requires interface reconstruction for later processing.Source: §2.1 · pp. 1–2
2.2. Adjoint-based Optimization
- Adjoint-based optimization minimizes an objective under PDE constraints and provides design sensitivities from the Lagrange functional’s derivative with respect to the design variable.After solving the primal and adjoint problems, the directional derivative can be calculated in two solver calls.Source: §2.2 · p. 2
- Shape optimization moves design-boundary nodes by local normal displacement, whereas topology optimization updates the IBM variable using cell-volume sensitivities.Topology updates may require additional geometric constraints to ensure engineering-viable results.Source: §2.2 · p. 3
2.3. Discussion
- Shape optimization with IBM and topology optimization with FBM are infeasible pairings within the described framework because their components are inherently incompatible.Source: §2.3 · p. 4
- FBM shape optimization can produce wave-patterned sensitivity artifacts when the interface is not co-planar with the background mesh, and smoothing risks removing real results.Topology optimization with an aligned mesh can work with appropriate interface resolution, but is usually infeasible for complex geometries.Source: §2.3 · p. 4, Figure 2.2 · p. 5
- IBM topology optimization is the most versatile choice for industrially relevant cases, while aligned-mesh shape optimization remains most reliable when a suitable mesh is available.The discussion also favors IBM topology optimization for complex geometries because aligned-mesh results depend strongly on the computational mesh.Source: §2.3 · p. 4
3. Application to Profile Extrusion
The application tests an IBM-based optimization on a realistic profile-extrusion die, targeting more balanced outlet flow. The optimization reduces the cost functional and outlet-velocity variation, improving flow balance by 55%.
3.1. Setup and Numerical Model
- The demonstrator is a real die for a 45 mm extruder, with a fixed 15 mm parallel zone and PE-HD melt modeled using Carreau-WLF.The operating point used a throughput of 10 kg/h and processing temperature of 220 °C; the objective was to maximize outlet flow balance.Source: §3.1 · p. 5
- An IBM representation embedded the die in a Cartesian mesh, and the topology update was restricted to interface cells to act as a shape update.Twenty optimization loops were run; the resulting channel was reconstructed with SurfaceNets and exported as a surface mesh.Source: §3.1 · p. 5
- The numerical model used about 48 million degrees of freedom per optimization loop, with the first loop taking 20 minutes on two compute servers.Later steps reused the previous step's solutions to reduce computational cost.Source: §3.1 · p. 6
3.2. Results
- The cost functional fell from 439.435 to 207.800 and outlet velocity standard deviation from 2507.75 to 1128.65, improving flow balance by 55%.The optimization had not fully converged, but the most significant shape changes were complete.Source: §3.2 · p. 6
- Positive sensitivities identify regions to restrict near the upper T-junction, while negative sensitivities identify lower channels to expand where narrow widths caused incomplete filling.These changes address excessive upper-junction velocities and incomplete filling in the lower channel.Source: §3.2 · p. 6
4. Conclusion and Outlook
For complex profile extrusion dies, topology optimization using an immersed boundary method is the most versatile approach described. The algorithm improved flow balancing by 55% using fewer solver calls than established CFD-based optimization methods; further development and lab validation remain future work.
4. Conclusion and Outlook
- Adjoint-based optimization can succeed through different pathways, but methods suited to CFD analysis are not necessarily suited to shape or topology optimization.Source: §4 · pp. 6, 8
- 55% improvement in flow balancing was achieved on a complex profile extrusion die using fewer solver calls than established CFD-based optimization methods.The study identifies topology optimization using an immersed boundary method as the most versatile approach for complex profile extrusion dies.Source: §4 · p. 8
- Future work includes improving geometry representation, extending the algorithm to fully non-isothermal flows, increasing optimization robustness, and manufacturing an optimized die for lab-trial validation.Source: §4 · p. 8