Multidisciplinary topology optimization design of cold plate for active phased antenna array

Abstract

This paper proposes a multidisciplinary topology optimization design model and method for the cold plate of active phased antenna array. Aiming at the characteristics of multidisciplinary integration of active phased antenna array, a multidisciplinary analysis model considering the thermal effect is established. Based on the multidisciplinary analysis model of antenna array, topology optimization method is used to establish a multidisciplinary optimal design model for the cold plate of antenna array. Since there is no explicit relationship between the electrical performance indicators of the antenna and the topological design variables, the adjoint method is used to derive the adjoint-governing equations that simultaneously consider fluid flow, system heat dissipation, and antenna electrical performance. The corresponding sensitivity information is obtained by solving the adjoint-governing equation. The correctness and effectiveness of the proposed model and method are verified by typical cases.

Appendix A

For the optimization problem listed in Eq. (16), consider the generalized objective function $W$, the Lagrangian function $L$ can be constructed by introduce the adjoint variable ${\varvec{\lambda}} = \left( {\tilde{\user2{u}},\tilde{p},\tilde{T}} \right)$

$$\begin{gathered} L = W + \left\langle {{\varvec{\lambda}},{\varvec{H}}} \right\rangle \hfill \\ \;\;\; = W + \int_{\Omega } {\left( {\rho {\varvec{u}} \cdot \nabla {\varvec{u}} - \mu \nabla \cdot (\nabla {\varvec{u}} + \nabla {\varvec{u}}^{T} ) + \nabla p + \alpha_{f - s} \left( \gamma \right){\varvec{u}}} \right) \cdot \tilde{\user2{u}}d\Omega } \hfill \\ \;\;\;\;\;\;\;\;\; + \int_{\Omega } {\left( { - \rho \nabla \cdot {\varvec{u}}} \right) \cdot \tilde{p}d\Omega } + \int_{\Omega } {\left( {\rho C_{\text{p}} \nabla \cdot {\varvec{u}}T - \nabla \cdot k\left( \gamma \right)\nabla T - Q} \right) \cdot \tilde{T}d\Omega } \hfill \\ \end{gathered},$$

(23)

where $W = \int_{\Omega } {W_{\Omega } d\Omega } + \int_{\Gamma } {W_{\Gamma } d\Gamma }$ is the generalized objective function, $W_{\Omega }$ is the domain integration objective function, $W_{\Gamma }$ is the boundary integration objective function, the adjoint variable ${\varvec{\lambda}} = \left( {\tilde{\user2{u}},\tilde{p},\tilde{T}} \right)$ also known as the Lagrangian multiplier, and the operator $\left\langle \cdot \right\rangle$ represents the domain integration,$\left\langle {{\varvec{\lambda}},{\varvec{H}}} \right\rangle = \int_{\Omega } {\left( {{\varvec{\lambda}} \cdot {\varvec{H}}} \right)d\Omega }$.

For the governing equations of multiphysical fields, theoretically, there are ${\varvec{H}} \equiv 0$. Therefore, the Lagrangian function $L \equiv P_{\text{rad}}$, and the sensitivity of and the objective function to the design variables can also be replaced by the sensitivity of the Lagrangian function to the design variables. It can be obtained from the chain derivation rule

$$\begin{gathered} \frac{dW}{{d\gamma }} = \frac{dL}{{d\gamma }} = \frac{\partial W}{{\partial \gamma }} + \frac{\partial W}{{\partial {\varvec{u}}}}\frac{{\partial {\varvec{u}}}}{\partial \gamma } + \frac{\partial W}{{\partial p}}\frac{\partial p}{{\partial \gamma }} + \frac{\partial W}{{\partial T}}\frac{\partial T}{{\partial \gamma }} \hfill \\ + \frac{{\partial \left\langle {{\varvec{\lambda}},{\varvec{H}}} \right\rangle }}{\partial \gamma } + \frac{{\partial \left\langle {{\varvec{\lambda}},{\varvec{H}}} \right\rangle }}{{\partial {\varvec{u}}}}\frac{{\partial {\varvec{u}}}}{\partial \gamma } + \frac{{\partial \left\langle {{\varvec{\lambda}},{\varvec{H}}} \right\rangle }}{\partial p}\frac{\partial p}{{\partial \gamma }} + \frac{{\partial \left\langle {{\varvec{\lambda}},{\varvec{H}}} \right\rangle }}{\partial T}\frac{\partial T}{{\partial \gamma }} \hfill \\ \end{gathered}.$$

(24)

Because the relationship between the state variable $\left( {{\varvec{u}},p,T,} \right)$ and the design variable $\gamma$ is hidden in the governing equation ${\varvec{H}}$, and it cannot be directly obtained. Therefore, the above formula is sorted out

$$\begin{gathered} \frac{dW}{{d\gamma }} = \frac{dL}{{d\gamma }} = \frac{\partial W}{{\partial \gamma }} + \frac{{\partial \left\langle {{\varvec{\lambda}},{\varvec{H}}} \right\rangle }}{\partial \gamma } + \left( {\frac{\partial W}{{\partial {\varvec{u}}}} + \frac{{\partial \left\langle {{\varvec{\lambda}},{\varvec{H}}} \right\rangle }}{{\partial {\varvec{u}}}}} \right)\frac{{\partial {\varvec{u}}}}{\partial \gamma } \hfill \\ + \left( {\frac{\partial W}{{\partial p}} + \frac{{\partial \left\langle {{\varvec{\lambda}},{\varvec{H}}} \right\rangle }}{\partial p}} \right)\frac{\partial p}{{\partial \gamma }} + \left( {\frac{\partial W}{{\partial T}} + \frac{{\partial \left\langle {{\varvec{\lambda}},{\varvec{H}}} \right\rangle }}{\partial T}} \right)\frac{\partial T}{{\partial \gamma }} \hfill \\ \end{gathered}.$$

(25)

According to KKT condition of constrained optimization problem of partial differential equation (Deng et al. 2018)

$$\left\{ \begin{gathered} \frac{\partial L}{{\partial {\varvec{u}}}} = \mathop {\lim }\limits_{h \to 0} \frac{{L({\varvec{u}} + h\hat{\user2{u}}) - L({\varvec{u}})}}{h} = \frac{\partial W}{{\partial {\varvec{u}}}} + \frac{{\partial \left\langle {{\varvec{\lambda}},{\varvec{H}}} \right\rangle }}{{\partial {\varvec{u}}}} = 0 \hfill \\ \frac{\partial L}{{\partial p}} = \mathop {\lim }\limits_{h \to 0} \frac{{L(p + h\hat{p}) - L(p)}}{h} = \frac{\partial W}{{\partial p}} + \frac{{\partial \left\langle {{\varvec{\lambda}},{\varvec{H}}} \right\rangle }}{\partial p} = 0 \hfill \\ \frac{\partial L}{{\partial T}} = \mathop {\lim }\limits_{h \to 0} \frac{{L(T + h\hat{T}) - L(T)}}{h} = \frac{\partial W}{{\partial T}} + \frac{{\partial \left\langle {{\varvec{\lambda}},{\varvec{H}}} \right\rangle }}{\partial T} = 0 \hfill \\ \end{gathered} \right.,$$

(26)

where$\hat{\user2{u}}$, $\hat{p},$ and $\hat{T}$ are respectively the direction of the state variables ${\varvec{u}}$,$p,$ and $T$.

Substitute Eq. (26) into Eq. (25), the sensitivity expression of the objective function to the design variable can be obtained:

$$\frac{dW}{{d\gamma }} = \frac{dL}{{d\gamma }} = \frac{\partial W}{{\partial \gamma }} + \frac{{\partial \left\langle {{\varvec{\lambda}},{\varvec{H}}} \right\rangle }}{\partial \gamma },$$

(27)

where

$$\begin{gathered} \left\langle {{\varvec{\lambda}},{\varvec{H}}} \right\rangle = \int_{\Omega } {\left( {\rho {\varvec{u}} \cdot \nabla {\varvec{u}} - \mu \nabla \cdot (\nabla {\varvec{u}} + \nabla {\varvec{u}}^{T} ) + \nabla p + \alpha_{f - s} \left( \gamma \right){\varvec{u}}} \right) \cdot \tilde{\user2{u}}d\Omega } \hfill \\ \;\;\;\; + \int_{\Omega } {\left( { - \rho \nabla \cdot {\varvec{u}}} \right) \cdot \tilde{p}d\Omega } + \int_{\Omega } {\left( {\rho C_{\text{p}} \nabla \cdot {\varvec{u}}T - \nabla \cdot k\left( \gamma \right)\nabla T - Q} \right) \cdot \tilde{T}d\Omega } \hfill \\ \end{gathered}.$$

(28)

Here, the problem of solving the sensitivity of the objective function to the design variable is transformed into the problem of solving the adjoint variable. First, the distribution integral of Eq. (28) is carried out and the Gaussian formula is used for sorting

$$\begin{gathered} \left\langle {{\varvec{\lambda}},{\varvec{H}}} \right\rangle = \int_{\Omega } {\left[ {\rho {\varvec{u}} \cdot \nabla {\varvec{u}} \cdot \tilde{\user2{u}} - \mu (\nabla {\varvec{u}} + \nabla {\varvec{u}}^{T} ):\nabla \tilde{\user2{u}} - p\nabla \cdot \tilde{\user2{u}} + \alpha_{f - s} \left( \gamma \right){\varvec{u}} \cdot \tilde{\user2{u}}} \right]d\Omega } \hfill \\ \;\;\;\;\;\;\;\; + \int_{\Omega } {\left( { - \rho \nabla \cdot {\varvec{u}}} \right) \cdot \tilde{p}d\Omega } + \int_{\Omega } {\left[ {\rho C_{\text{p}} ({\varvec{u}} \cdot \nabla T)\tilde{T} + k\left( \gamma \right)\nabla T \cdot \nabla \tilde{T} - Q\tilde{T}} \right]d\Omega } \hfill \\ \;\;\;\;\;\;\;\; + \int_{\Gamma } {\left[ {p{\mathbf{I}} - \mu (\nabla {\varvec{u}} + \nabla {\varvec{u}}^{T} )} \right] \cdot {\varvec{n}}d\Gamma } - \int_{\Gamma } {k\left( \gamma \right)\tilde{T}\nabla T \cdot {\varvec{n}}d\Gamma } \hfill \\ \end{gathered}.$$

(29)

Substitute Eq. (29) into Eq. (26) and integrate by parts again, it can obtain

$$\left\{ \begin{gathered} \frac{\partial W}{{\partial {\varvec{u}}}} + \int_{\Omega } {\left[ {\nabla \tilde{p} - \mu \nabla \cdot (\nabla \tilde{\user2{u}} + \nabla \tilde{\user2{u}}^{T} ) + \rho \nabla {\varvec{u}} \cdot \tilde{\user2{u}} - \rho {\varvec{u}} \cdot \nabla \tilde{\user2{u}} + \alpha_{f - s} \left( \gamma \right)\tilde{\user2{u}}} \right.} \hfill \\ \left. { + \rho C_{\text{p}} \tilde{T}\nabla T} \right] \cdot \hat{\user2{u}}d\Omega + \int_{{\Gamma_{N} }} {\left\{ {\rho ({\varvec{u}} \cdot {\varvec{n}})\tilde{\user2{u}} + \left[ { - \tilde{p}{\mathbf{I}} + \mu (\nabla \tilde{\user2{u}} + \nabla \tilde{\user2{u}}^{T} )} \right] \cdot {\varvec{n}}} \right\} \cdot \hat{\user2{u}}d\Gamma } = 0 \hfill \\ \frac{\partial W}{{\partial p}} + \int_{\Omega } {\left( { - \hat{p}\nabla \cdot \tilde{\user2{u}}} \right)d\Omega } + \int_{{\Gamma_{D} }} {\hat{p}\tilde{\user2{u}} \cdot {\varvec{n}}d\Gamma } = 0 \hfill \\ \frac{\partial W}{{\partial T}} + \int_{\Omega } { - \left[ {\rho C_{\text{p}} {\varvec{u}} \cdot \nabla \tilde{T} + k\left( \gamma \right)\nabla^{2} \tilde{T}} \right]\hat{T}d\Omega } + \int_{{\Gamma_{N} }} {\left( {\rho C_{\text{p}} \tilde{T}{\varvec{u}} + k\left( \gamma \right)\nabla \tilde{T}} \right) \cdot {\varvec{n}}\hat{T}d\Gamma } = 0 \hfill \\ \end{gathered} \right.$$

(30)

Expanding the generalized objective function and reorganizing the Eq. (28), the adjoint control equation and adjoint boundary conditions for solving the adjoint variables can be obtained.

The adjoint Navier–Stokes equations:

$$\left\{ \begin{gathered} \nabla \cdot \tilde{\user2{u}} = \frac{{\partial W_{\Omega } }}{\partial p}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;in\;\Omega \hfill \\ \nabla \tilde{p} - \mu \nabla \cdot (\nabla \tilde{\user2{u}} + \nabla \tilde{\user2{u}}^{T} ) + \rho \nabla {\varvec{u}} \cdot \tilde{\user2{u}} - \rho {\varvec{u}} \cdot \nabla \tilde{\user2{u}} + \alpha_{f - s} \left( \gamma \right)\tilde{\user2{u}} + \rho C_{\text{p}} \tilde{T}\nabla T \hfill \\ = - \frac{{\partial W_{\Omega } }}{{\partial {\varvec{u}}}} + \nabla \cdot \frac{{\partial W_{\Omega } }}{{\partial \nabla {\varvec{u}}}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;in\;\Omega \hfill \\ \tilde{\user2{u}} \cdot {\varvec{n}} = - \frac{{\partial W_{\Gamma } }}{\partial p}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;on\;\Gamma_{D} \hfill \\ \rho ({\varvec{u}} \cdot {\varvec{n}})\tilde{\user2{u}} + \left[ { - \tilde{p}{\mathbf{I}} + \mu (\nabla \tilde{\user2{u}} + \nabla \tilde{\user2{u}}^{T} )} \right] \cdot {\varvec{n}} = - \frac{{\partial W_{\Omega } }}{{\partial \nabla {\varvec{u}}^{T} }} \cdot {\varvec{n}} - \frac{{\partial W_{\Gamma } }}{{\partial {\varvec{u}}}}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;on\;\Gamma_{F} \hfill \\ \end{gathered} \right.$$

(31)

The adjoint heat transfer equations:

$$\left\{ \begin{gathered} \rho C_{\text{p}} {\varvec{u}} \cdot \nabla \tilde{T} + k\left( \gamma \right)\nabla^{2} \tilde{T} = \frac{{\partial W_{\Omega } }}{\partial T} - \nabla \cdot \frac{{\partial W_{\Omega } }}{\partial \nabla T}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;in\;\Omega \hfill \\ \tilde{T} = 0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;on\;\Gamma_{D} \hfill \\ \left( {\rho C_{\text{p}} \tilde{T}{\varvec{u}} + k\left( \gamma \right)\nabla \tilde{T}} \right) \cdot {\varvec{n}} = - \frac{{\partial W_{\Omega } }}{\partial \nabla T} \cdot {\varvec{n}} - \frac{{\partial W_{\Gamma } }}{\partial T}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;on\;\Gamma_{D} \hfill \\ \end{gathered} \right.$$

(32)

By replacing the corresponding term in the generalized objective function with the constraint function in Eq. (16), the adjoint-governing equation and adjoint boundary conditions of the constraint function can be obtained. For the optimization problem listed in Eq. (16), the right-hand side terms corresponding to the adjoint equations and adjoint boundary conditions [Eqs. (31), (32)] are listed in Table 4.

Table 4 The right-hand terms of the adjoint equations and adjoint boundary conditions of the optimization problem listed in Eq. (16)