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Topology optimization of microvascular composites for active-cooling applications using a geometrical reduced-order model

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Abstract

In this study, we develop a design methodology with a basis in gradient-based topology optimization and a geometrical reduced-order thermal/hydraulic model for actively cooled microvascular composite panels. The proposed method is computationally very efficient owing to the suggested simplifications while preserving the required accuracy. The analytical sensitivity for the topology optimization scheme is derived. Several numerical examples are solved to demonstrate the applicability of the proposed method for active-cooling applications. Using topology optimization, the maximum temperature of the composite panel is reduced by up to 59% compared to a benchmark design. The optimization framework is compared to hybrid topology/shape (HyTopS) and shape optimization (SO) methods based on several measures such as maximum and average temperatures, temperature uniformity, network redundancy, and manufacturability. The solution obtained from the proposed TO scheme outperforms the other approaches in terms of the aforementioned measures.

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Notes

  1. In the IGFEM framework, the velocity field is defined as the partial derivative of the nodal coordinates with respect to the design variables.

  2. Note that for this problem, it is not possible to have a uniform design which simultaneously satisfies both the volume fraction and pressure drop constraints.

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Acknowledgements

This work has been supported by Drexel University Career Development and Summer Research Awards. The authors acknowledge the high-performance computing resources (PROTEUS: the Drexel Cluster) and support at the Drexel University. The authors also acknowledge the support from the Villum Foundation through the VILLUM Investigator Project InnoTop.

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Correspondence to Ahmad Raeisi Najafi.

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Responsible Editor: Hyunsun Alicia Kim

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We provided comprehensive implementation details, as a result, we are confident that the design methodology is reproducible. For this reason, we have decided not to publish the code. If the information provided in the paper is not sufficient, the readers are welcome to contact the authors for further explanations.

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Appendices

Appendix 1

In this study, we aim to minimize the maximum temperature of the domain. However, the maximum temperature is not differentiable and as is mentioned in Section 3, we replace it with a differentiable alternative, i.e., p-mean temperature. The p-mean temperature converges to the maximum temperature when p approaches infinity, i.e., \(\lim _{p\rightarrow \infty }\vert \vert T \vert \vert _{p}=T_{max}\). However, using large values for p can be problematic. First of all, since we use the Gauss-Dunavant quadrature (Dunavant 1985) to precisely calculate the integration of p-mean temperature, the larger values of p require more quadrature points which increase the computational cost. Moreover, the large values of p make the p-mean temperature ill-conditioned or less smooth. Small values of p can also be problematic. Choosing a small value for p may not enable ||T||p to capture reliably the trend in Tmax. This issue may lead to local areas of high temperature similar to the stress concentration regions in the structural optimization problems.

In this section, we consider the reference design of CHF1 problem in Section 4.1. The maximum temperature of the panel in this design is 76.1°C. We evaluate the value of p-mean temperature using p equal to 2, 5, 8, 10, 12, 17, and 19. The numbers of quadrature points that we need to consider based on Gauss-Dunavant quadrature (Dunavant 1985) are 3, 7, 16, 33, 61, and 73 for p equal to 2, 5, 8, 10, 12, 17, and 19, respectively. To compute the percent of the difference between the maximum temperature and p-mean temperature, we define a variable Td = (||Tp||− Tmax)/Tmax. Figure 17 compares the values of percent of the difference between the p-mean and maximum temperatures(Td) and the number of quadrature points (NQ) for different values of p. Based on the results, we decided to select p = 10 in this study. Note that we did not observe the problems of smoothness and being ill-conditioned for ||Tp|| with p = 10 in the problems solved in this study.

Fig. 17
figure 17

The percent of difference between the maximum and p-mean temperatures of the domain (Td) and the number of required Gauss-Dunavant quadrature points for different values of p in the p-mean

Appendix 2

As explained in the main text, the major role of Dmin in (10) is preventing numerical issues. But we need to make sure that the value that we choose for Dmin would not adversely impact the objective value. Thus, to find a suitable value for Dmin, we considered the reference design of each problem and we computed the amount of error produced in the value of the objective function when we consider a nonzero value for minimum diameter. We define the error as \(e=(|\theta ^{(@ D_{min}=0)}-\theta ^{(@ D_{min}\neq 0)}|)/\theta ^{(@ D_{min}=0)}\). Obviously, if the minimum diameter is 0, the amount of error will be 0 and as the minimum diameter increases, the amount of error rises. Figure 18 shows the results for the reference designs of UHF1 and CHF1. Based on the presented results, we select 60 μm and 40 μm for the minimum diameter of the problems UHF1 and CHF1, respectively. The amount of error (e) until these two values are less than 0.01%.

Fig. 18
figure 18

The percent of error amount of error produced in the value of objective function when we consider a nonzero value for minimum diameter for the reference designs of UHF1 and CHF1 problems

Appendix 3

In this section, we aim to investigate the effect of removing the microchannels with the flow rates lower than a certain threshold on the objective function for the optimization problem UHF1 solved in Section 4.1. For that problem, we select a mass flow rate threshold of 3 ⋅ 10− 3 gr/s. We reevaluate the p-mean temperature value for all of the designs obtained in each iteration of the optimization problem UHF1 before removing the microchannels. Figure 19 compares the p-mean temperature values obtained from the cases where we remove the low mass flow rate microchannels with the cases where we do not remove any microchannel. The results are very close to each other. Thus, we can conclude that removing the microchannels with the mass flow rates lower than the prescribed threshold has a negligible effect on the p-mean temperature value.

Fig. 19
figure 19

Comparison of the p-mean temperature values obtained from the cases where we removed the low mass flow rate microchannels with the cases where we do not remove any microchannel

Appendix 4

A verification study of the analytic adjoint sensitivity analysis is performed by comparing the computed sensitivity with the central finite difference method for the reference design of the problem UHF1 in Section 4.1. The error between the adjoint and finite difference sensitivity analysis is given by \(\epsilon =\mid \frac {((d\theta /d\alpha )^{Adj}-(d\theta /d\alpha )^{FD})}{(d\theta /d\alpha )^{Adj}}\mid \), where ∣ ∗∣ indicates the absolute value of *, and superscripts of Adj and FD indicate adjoint method and finite difference approach, respectively. The error is plotted in Fig. 20 for a sequence of perturbations from △α = 10− 2 to 10− 12. The amount error is very small (≈ 10− 11) and we can conclude that the sensitivity analysis is correctly derived and implemented in this study.

Fig. 20
figure 20

The relative error, 𝜖, between the adjoint sensitivity analysis in Section 3 and approximated central finite difference sensitivity plotted as function of the magnitude of the perturbation in the design parameter, △α. The relative error has a small value, which indicates that the sensitivity analysis is correctly derived and implemented

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Pejman, R., Sigmund, O. & Najafi, A.R. Topology optimization of microvascular composites for active-cooling applications using a geometrical reduced-order model. Struct Multidisc Optim 64, 563–583 (2021). https://doi.org/10.1007/s00158-021-02951-x

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