Investigation of reinforcement learning for shape optimization of 2D profile extrusion die geometries

Clemens Fricke; Daniel Wolff; Marco Kemmerling; Stefanie Elgeti

doi:10.3934/acse.2023001

Article Contents

2023, Volume 1: 1-35. Doi: 10.3934/acse.2023001 open access

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Investigation of reinforcement learning for shape optimization of 2D profile extrusion die geometries

1.
Institute for Lightweight Design and Structural Biomechanics, TU Wien, Austria
2.
Chair for Computational Analysis of Technical Systems, RWTH Aachen University, Germany
3.
Information Management in Mechanical Engineering, RWTH Aachen University, Germany

^*Corresponding author: Daniel Wolff
^*Corresponding author: Daniel Wolff

Received: December 2022

Revised: January 2023

Early access: February 2023

Published: March 2023

Abstract Full Text(HTML) Figure(21) / Table(6) Related Papers Cited by

Abstract

Profile extrusion is a continuous production process for manufacturing plastic profiles from molten polymer. Especially interesting is the design of the die, through which the melt is pressed to attain the desired shape. However, due to an inhomogeneous velocity distribution at the die exit or residual stresses inside the extrudate, the final shape of the manufactured part often deviates from the desired one. To avoid these deviations, the shape of the die can be computationally optimized, which has already been investigated in the literature using classical optimization approaches [7,32,47].

A new approach in the field of shape optimization is the utilization of RL as a learning-based optimization algorithm. RL is based on trial-and-error interactions of an agent with an environment. For each action, the agent is rewarded and informed about the subsequent state of the environment. While not necessarily superior to classical, e.g., gradient-based or evolutionary, optimization algorithms for one single problem, RL techniques are expected to perform especially well when similar optimization tasks are repeated since the agent learns a more general strategy for generating optimal shapes instead of concentrating on just one single problem.

In this work, we investigate this approach by applying it to two 2D test cases. The flow-channel geometry can be modified by the RL agent using so-called FFD [34], a method where the computational mesh is embedded into a transformation spline, which is then manipulated based on the control-point positions. In particular, we investigate the impact of utilizing different agents on the training progress and the potential of wall time saving by utilizing multiple environments during training.

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Mathematics Subject Classification: Primary: 49Q10, 76D55; Secondary: 35Q30.

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Figure 1. Interaction loop between an agent and an environment during training. In each interaction / training step $ t $, the agent selects an action $ a_t $ according to a policy $ \pi $ based on observations of the current environment's state $ s_t $ and a numerical reward signal $ r_t $. This changes the state of the environment and generates the new observation and the new reward for the next step

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Figure 2. Limited taxonomy of RL agents. Blue items represent categories of agents. Turquoise items are agents trained with the off-policy method. Magenta items are agents trained with the onpolicy method

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Figure 3. Visualization of the RL interaction between an agent and the environment in our $ \texttt{releso} $ framework. The environment has been customized to our shape optimization problem. It comprises the base mesh and the deformation spline used for the geometry parameterization, a FFD module deforming the mesh, the solver computing the governing PDE problem, and a component for postprocessing the simulation results to determine the reward and an observation of the CFD environment for the agent. The arrows correspond to the information flows between the different components: Green arrows represent meshes, yellow arrows spline parameterizations, and red arrow stands for the simulation results. Based on the provided reward and observation, the agent chooses an action to modify the deformation spline of the FFD

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Figure 4. Visualization of the key idea of FFD: An initial geometry (A) is transformed by modifying the control points (dark-blue dots) of a transformation spline (highlighted in light-blue) to obtain a deformed geometry (B)

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Figure 5. Geometry and boundaries of the T-shaped geometry

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Figure 6. (A) shows the deformation spline used for the parameterization of the T-shaped geometry. To make the FFD more generic, the actual geometry is scaled to the parametric space of the transformation spline before applying geometric modifications. Additionally, the possible movement of the control points is illustrated by the orange arrows in (B)

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Figure 7. Comparison of different algorithms trained to optimize the T-shaped geometry following a direct strategy with respect to the episode reward over the trained steps. Each run was repeated twice as indicated using the same color

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Figure 8. Comparison of different algorithms trained to optimize the T-shaped geometry following an incremental strategy with respect to the episode reward over the trained steps. Each run was repeated twice as indicated using the same color

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Figure 9. Examples of optimal geometries obtained by a PPO agent following an incremental optimization strategy on the T-shaped geometry. One can see that the trained agent has learned a valid strategy which involves modifying the control points that strongly influence the cross-sectional area of the two outflows

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Figure 10. Episode reward over training steps of a PPO agent following an incremental strategy for optimizing the T-shaped geometry when interacting with different numbers of environments

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Figure 11. Episode reward over wall time of a PPO agent following an incremental strategy for optimizing the T-shaped geometry when interacting with different numbers of environments

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Figure 12. Geometry and boundaries of the converging channel geometry

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Figure 13. Deformation spline used for the parameterization of the channel geometry. Additionally, the possible movement of the control points is illustrated by the orange arrows

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Figure 14. Outflow boundary of the converging channel geometry. The outflow is divided into three patches $ \Gamma_{\text{out}, i} $, for which the flow-homogeneity criterion is evaluated

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Figure 15. Comparison of different agents trained to optimize the converging channel geometry following a direct strategy with respect to the episode reward over the trained steps. Each run was repeated twice as indicated using the same color

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Figure 16. Comparison of different agents trained to optimize the converging channel geometry following an incremental strategy with respect to the episode reward over the trained steps. Each run was repeated twice as indicated using the same color

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Figure 17. Examples of optimal geometries obtained by a PPO agent following an incremental optimization strategy for the converging channel geometry. Each of the shown geometries has been generated from a random initial geometry generated by a random perturbation of the control points. Qualitatively the agent has learned to contract the channel as smoothly as possible to assert an optimal flow homogeneity

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Figure 18. Comparison of different agents trained to optimize the converging channel geometry following a direct strategy with respect to the steps per episode over the trained steps. Each run was repeated twice as indicated using the same color

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Figure 19. Comparison of different agents trained to optimize the converging channel geometry following an incremental strategy with respect to the steps per episode over the trained steps. Each run was repeated twice as indicated using the same color

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Figure 20. Episode reward over training steps of a PPO agent following an incremental strategy for optimizing the converging channel geometry when interacting with different numbers of environments

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Figure 21. Episode reward over wall time of a PPO agent following an incremental strategy for optimizing the converging channel geometry when interacting with different numbers of environments

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Table 1. Compatibility of the agents implemented in $ \texttt{stable-baselines3} $ with regard to the direct and incremental shape optimization method

Agent	Incremental	Direct
PPO	$\checkmark$	$\checkmark$
DQN	$\checkmark$	-
SAC	-	$\checkmark$
A2C	$\checkmark$	$\checkmark$
DDPG	-	$\checkmark$

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Table 2. Material properties of the shear-thinning material law for all test cases

Property	Symbol	Value	Unit
zero-shear viscosity	$ A $	10935	kg m⁻¹ s⁻¹
reciprocal transition rate	$ B $	0.433	s⁻¹
slope of viscosity curve in pseudoplastic region	$ C $	0.699	-

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Table 3. Wall-clock times of the agents trained with the direct optimization method to optimize the T-shaped geometry

Agent	Max. training time
PPO	26.1 h
A2C	46.5 h
SAC	26.0 h
DDPG	33.5 h

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Table 4. Wall-clock times of the agents trained with the incremental optimization method for the T-shaped geometry use case.

Agent	Max. training time
PPO	49.9 h
A2C	45.1 h
DQN	45.0 h

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Table 5. Wall-clock times of the agents trained with the direct optimization method for the converging channel use case

Agent	Training time
PPO	64.2 h
A2C	44.0 h
SAC	70.0 h
DDPG	45.0 h

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Table 6. Wall-clock times of the agents trained with the incremental optimization method for the converging channel use case

Agent	Training time
PPO	42.2 h
A2C	43.1 h
DQN	44.7 h

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