## Thermal Topology Optimization Automated Design of Conformal Cooling Channels

With the use of conformal cooling channels, rather than straight ones, cycle time of the die casting process can be shortened, and the quality of the parts can be improved. But how can you find the ideal geometry of the channels?

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Within the overall objective of the Digital Cell vision, Bühler strives for a process cycle time reduction of 40%. In order to reach this ambitious goal, fundamental consideration in the design process of dies related to the thermal management must be done. The use of conformal cooling channels can shorten the cycle time of the die casting process and improve the quality of the part.

Cooling is a very important aspect of die casting, as the cooling phase accounts for a large percentage of the casting cycle. Also, the conditions, under which the cast part solidifies and cools down have a large influence on the quality of the part. That is why it is important to carefully consider the cooling of part and mold when designing a die casting die.

### Conformal Cooling

With the rise of metal additive manufacturing (AM), it has become feasible to 3D print die casting dies (or parts of them), including cooling channels from the beginning. In that way, the channels are no longer constrained to follow straight paths. Using conformal cooling channels, cooling rates can be increased, and the part surface temperature can be distributed more evenly. This improves both cycle time and part quality.

Obviously, the space of possible channel designs is considerably larger with AM than with conventional manufacturing of the die. Hence, designing a cooling system that fully exploits the freedom of design can be very difficult and time consuming. For that reason, different approaches have been taken to develop computational tools to assist with the design task, or even to have the channels designed automatically.

### Density-Based Topology Optimization

Density-based topology optimization was first introduced in structural optimization. The optimization problem is formulated as a material distribution problem: A certain amount of material is available to be distributed in the design domain, such that a predefined objective function, e.g. the compliance of the structure, is optimized. The design domain is divided into finite elements, each of which is characterized by a relative density value between 0 and 1, where 0 stands for void regions and 1 for regions that are filled with material. Densities between 0 and 1 are interpreted as composite materials whose properties are interpolated as functions of the density. In structural optimization, this is primarily done for Young’s modulus or the elasticity tensor. The element densities are the design variables, which can be optimized using a gradient-based algorithm. In the context of a thesis, density-based topology optimization was utilized for the design of cooling channels.

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In the problem discussed here, the geometry of the cast part is given. The design domain is a box that fully encloses the part geometry. The domain is divided into a mesh of brick-shaped elements. For each element, the thermal conductivity is interpolated analogously to Young’s modulus in structural optimization. Here the Solid Isotropic Material with Penalization (SIMP) model is used. SIMP implicitly penalizes intermediate densities with the aim of driving the densities to 1 or 0. For each element interface, the convection coefficient is interpolated from the density difference across the interface. A temperature field is computed via a stationary thermal finite elements analysis.

The design variables are modified iteratively using the Method of Moving Asymptotes (MMA). For each iteration, the objective function, which penalizes deviations of the part surface temperature from an ideal value, is evaluated. Also, the gradients of the objective function and the constraint functions, with respect to the design variables, are computed. This is done using the adjoint method. From all these values, the next design point is computed via MMA.

Now, the question at hand is: How can cooling channels be obtained with this approach? Two possible solutions were investigated: optimization constraints and post-processing.

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### “Channel constraints”

With MMA, it is relatively simple to add multiple constraints to the optimization problem. Consequently, different constraint functions were defined, based on a few requirements expected to be satisfied by proper cooling channels. It was the intent of this approach, that the optimization would directly generate a channel design. However, this did not work. The algorithm did not manage to satisfy the constraint functions on the entire domain. There always remained areas, where the constraints were not fulfilled. The main reason for this seems to be, that the investigated constraints added an individual constraint function for each finite element. Because of this, the number of constraints became too large for proper handling by MMA.

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### Post-processing

Since the constraint functions did not lead to success, the following approach was taken: The optimization algorithm was used to optimize the element densities, followed by a post-processing procedure. The process is illustrated with a simple example in the image gallery. The optimization algorithm yields a design with regions of low density close to thicker segments of the part. At the border of these regions, a cooling surface is generated. Up to this point, the entire procedure is automated. The final cooling channels are now drawn manually on the cooling surface.

A stationary thermal analysis of the design shows that the resulting part surface temperature is close to the desired value on most of the part surface. The presented approach makes some simplifying assumptions, for example that the temperature field in the die can be sufficiently approximated based on a steady-state thermal analysis, where the design-dependent convection boundary conditions are imposed using a simple interpolation scheme. Nevertheless, the resulting designs seem plausible in that the cooling system approaches the thicker regions of the part more closely than thinner ones and simulation results are satisfactory. The method may be used to obtain a first estimate of the optimal cooling channel design and incorporate into a comprehensive design tool, which can contribute in the long run to a more efficient design process with positive impact on the performance of die casting cells.

^{*}Lukas Sägesser is Master Student at ETH Zürich. This work was carried out as part of a Bachelor Thesis, “Density-Based Topology Optimization of Conformal Cooling Channels” at the Engineering Design and Computing Laboratory (EDAC), ETH Zürich in collaboration with Bühler.

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