Industrial application of topology optimization for forced convection based on Darcy flow

Abstract

Designing the layout of flow channels for forced convection is a complicated task as a compromise between hydraulic and thermal performance must be found and a wide variety of topology, shape, and size can be manufactured. The authors demonstrate the capability of Altair OptiStruct™ to create a basic layout by topology optimization. The flow analysis is based on the linear potential Darcy model to capture the incompressible steady-state flow. The flow resistance is modeled as porous media, which permeability distinguishes between the solid and fluid domain. The resulting velocity is used in the thermal convection–diffusion equation. In the thermal analysis, an additional discontinuity capturing term is discussed to prevent from numerical over- and undershooting of the temperature field. The optimization utilizes standard techniques like the density method, SIMP interpolation, robust approach, adjoint sensitivity analysis, and dual optimization. As application example, the cooling of automotive battery packs is shown. The cooling is realized by fluid flow through cooling channels manufactured by roll bonding. This manufacturing process allows for complex channel patterns, whereas the production cost is low compared to additive manufacturing or brazed cooling systems especially for high-volume applications. The temperature on the battery modules is optimized to have a uniformly low value, while keeping mechanical losses in the flow low.

Appendix A Sensitivity analysis

1.1 A.1 Nodal pressure

The response $p_i=\varvec {e}_i^T \varvec {p}$ is the flow pressure of node i. The flow adjoint can be calculated by

$$\begin{aligned} \varvec{\lambda }_p = -\varvec {K}_p^{-1} \varvec {e}_i \end{aligned}$$

(A1)

and plugged into the sensitivity equation

$$\begin{aligned} \frac{\mathrm {d} p_i}{\mathrm {d} \tilde{\varvec {x}}} = \varvec{\lambda }_p^T \frac{\partial \varvec {K}_p}{\partial \tilde{\varvec {x}}} \varvec {p}. \end{aligned}$$

(A2)

1.2 A.2 Thermal compliance

The response $T_c=\frac{1}{2} \varvec {t}^T \varvec {f}_t$ is the global thermal compliance. The thermal adjoint is given by

$$\begin{aligned} \varvec{\lambda }_t = \frac{1}{2} \left( \varvec {K}_t + \varvec {C} + \varvec {D} \right) ^{-T} \varvec {f}_t. \end{aligned}$$

(A3)

The flow adjoint can be calculated by

$$\begin{aligned} \varvec{\lambda }_p = -\varvec {K}_p^{-1} \left( \varvec{\lambda }_t^T \frac{\partial \varvec {C} + \varvec {D}}{\partial \varvec {p}} \varvec {t} \right) ^T \end{aligned}$$

(A4)

and plugged into the sensitivity equation

$$\begin{aligned} \begin{aligned} \frac{\mathrm {d} T_c}{\mathrm {d} \tilde{\varvec {x}}}&= \varvec{\lambda }_p^T \frac{\partial \varvec {K}_p}{\partial \tilde{\varvec {x}}} \varvec {p} \\&+ \varvec{\lambda }_t^T \left[ \frac{\partial \varvec {K}_t + \varvec {C} + \varvec {D}}{\partial \tilde{\varvec {x}}} \varvec {t} - \frac{\partial \varvec {f}_t}{\partial \tilde{\varvec {x}}} \right] . \end{aligned} \end{aligned}$$

(A5)

1.3 A.3 Nodal temperature

The response $t_i=\varvec {e}_i^T \varvec {t}$ is the temperature of node i. The thermal adjoint is given by

$$\begin{aligned} \varvec{\lambda }_t = - \left( \varvec {K}_t + \varvec {C} + \varvec {D} \right) ^{-T} \varvec {e}_i. \end{aligned}$$

(A6)

The flow adjoint can be calculated by

$$\begin{aligned} \varvec{\lambda }_p = -\varvec {K}_p^{-1} \left( \varvec{\lambda }_t^T \frac{\partial \varvec {C} + \varvec {D}}{\partial \varvec {p}} \varvec {t} \right) ^T \end{aligned}$$

(A7)

and plugged into the sensitivity equation

$$\begin{aligned} \begin{aligned} \frac{\mathrm {d} t_i}{\mathrm {d} \tilde{\varvec {x}}}&= \varvec{\lambda }_p^T \frac{\partial \varvec {K}_p}{\partial \tilde{\varvec {x}}} \varvec {p} \\&\quad+ \varvec{\lambda }_t^T \left[ \frac{\partial \varvec {K}_t + \varvec {C} + \varvec {D}}{\partial \tilde{\varvec {x}}} \varvec {t} - \frac{\partial \varvec {f}_t}{\partial \tilde{\varvec {x}}} \right] . \end{aligned} \end{aligned}$$

(A8)