An integrated two-step strategy for an optimal design of liquid-cooled channel layout based on the MMC–density approach

Abstract

This paper proposes an integrated two-step strategy for an optimal design of liquid-cooled channel layout based on the moving morphable component (MMC)-density approach. The proposed strategy intends to take the advantage of both the MMC approach for its high flexibility in searching a physically reasonable layout and the density approach for its better capacity of topology description. On the basis of the above-mentioned strategy, an intermediate layout is obtained through MMC approach and further optimized as initial solution of density approach step. Through density approach step, the final layout shows smoother boundary while retaining reasonable feature size. The original contributions of this paper are as follows: (i) An assembled quadratic Bézier curves component is proposed to describe the largely curved channel with limited numbers of optimization variables and computation order. (ii) Benefited from explicit geometric description, adaptive mesh refinement (AMR) is applied in MMC approach step for the first time. The application of AMR, from the numerical point of view, has two key ingredients to be highlighted: (i) the accuracy of solution in fluid–solid boundary region can be ensured with relatively limited computational cost. (ii) The contradiction that the difference step of MMC updating needs to be both as small as possible and integer multiple of the mesh size can be avoided. The performance of our methodology is demonstrated by numerical examples aiming for maximal heat exchange with power dissipation constraint. The main finding reveals that the proposed strategy can offer reasonable channel layout with better thermal performance, compared with conventional density approach. The whole numerical implementation relies on OpenFOAM and PETSc open-source software packages.

Appendix: Sensitivity analysis

In this paper, the adjoint equations and adjoint BCs are derived by continuous adjoint method and subsequently discretized and solved. Using Lagrange’s method, the optimization problem given by Eqs. (17) and (18) can be substituted by an unconstrained optimization problem. The Lagrangian ${\mathcal {L}}({\varvec{u}}, p, T, \gamma )$ can be formulated as (Sato et al. 2018):

$$\begin{aligned} {\mathcal {L}} =&\varPsi +\int _{\varOmega }[\rho ( {\varvec{u}} \cdot \nabla ) {\varvec{u}}-\mu \nabla \cdot \nabla {\varvec{u}}+\nabla p+\alpha {\varvec{u}}] \cdot {\varvec{v}} {\text {d}}\varOmega \\&\quad +\int _{\varOmega }(-\nabla \cdot {\varvec{u}})q {\text {d}}\varOmega +\int _{\varOmega }[\rho c({\varvec{u}}\cdot \nabla T)-\nabla \cdot k \nabla T -Q]t {\text {d}}\varOmega \ \\&\quad +\int _{\varGamma _{\text {in}}}({\varvec{u}}-{\varvec{u}}_{\text {in}}) \cdot {\varvec{v}} _{\text {in}} {\text {d}}\varGamma +\int _{\varGamma _{\text {wall}}} {\varvec{u}}\cdot {\varvec{v}} _{\text {wall}} {\text {d}}\varGamma +\int _{\varGamma _{\text {out}}}(-p{\varvec{I}}+\mu \nabla {\varvec{u}})\cdot {\mathbf {n}} \cdot {\varvec{v}} _{\text {out}} {\text {d}}\varGamma \\&\quad +\int _{S_{\text {in}}}(T-T_{\text {in}})t_{\text {in}} {\text {d}} S + \int _{S_{\text {wall}}}k\nabla T t_{\text {wall}}\cdot {\varvec{n}} {\text {d}} S, \end{aligned}$$

(27)

where ${\varvec{v}}, q, t, {\varvec{v}}_{\text {in}},{\varvec{v}}_{\text {wall}}, {\varvec{v}}_{\text {out}}, t_{\text {in}}, t_{\text {wall}}$ represent the corresponding Lagrangian multipliers. Through partial integration, Eq. (27) can be inferred as

$$\begin{aligned} {\mathcal {L}} =&\varPsi +\int _{\varOmega }\rho ({\varvec{u}}\cdot \nabla ){\varvec{u}}\cdot {\varvec{v}}+\mu \nabla {\varvec{u}} :\nabla {\varvec{v}}-p \nabla \cdot {\varvec{v}} +\alpha {\varvec{u}} \cdot {\varvec{v}}{\text {d}}\varOmega \\&\quad + \int _{\varOmega }q(-\nabla \cdot {\varvec{u}}){\text {d}}\varOmega + \int _{\varOmega }\rho c\nabla Tt\cdot {\varvec{u}}+k \nabla T \cdot \nabla t-Qt {\text {d}} \varOmega \\&\quad + \int _{\varGamma } (- \mu \nabla {\varvec{u}} + p {\varvec{I}})\cdot {\varvec{v}} \cdot {\varvec{n}} {\text {d}} \varGamma - \int _{S} k \nabla Tt \cdot {\varvec{n}} {\text {d}}S\\ {}&\quad +\int _{\varGamma _{\text {in}}}({\varvec{u}}-{\varvec{u}}_{\text {in}}) \cdot {\varvec{v}}_{\text {in}} {\text {d}}\varGamma +\int _{\varGamma _{\text {wall}}} {\varvec{u}}\cdot {\varvec{v}}_{\text {wall}} {\text {d}}\varGamma +\int _{\varGamma _{\text {out}}}(-p{\varvec{I}}+\mu \nabla {\varvec{u}})\cdot {\varvec{n}} \cdot {\varvec{v}}_{\text {out}} {\text {d}}\varGamma \\&\quad +\int _{S_{\text {in}}}(T-T_{\text {in}})t_{\text {in}} {\text {d}} S + \int _{S_{\text {wall}}}k\nabla T t_{\text {wall}}\cdot {\varvec{n}} {\text {d}} S. \end{aligned}$$

(28)

In such case, the Karush–Kuhn–Tucker conditions for PDE constrained optimization problems should be satisfied:

$$\left\{ \begin{array}{l} {\mathcal {L}}_{{\varvec{u}}}=\lim \nolimits _{h \rightarrow 0} \dfrac{{\mathcal {L}}({\varvec{u}} + h {\varvec{w}})-{\mathcal {L}}({\varvec{u}})}{h}=\varPsi _{{\varvec{u}}} + \langle \lambda ,{\varvec{R}},{\varvec{C}} \rangle _{{\varvec{u}}}=0, \\ {\mathcal {L}}_{p}=\lim \nolimits _{h \rightarrow 0} \dfrac{{\mathcal {L}}(p + h r)-{\mathcal {L}}(p)}{h}=\varPsi _{p} + \langle \lambda ,{\varvec{R}},{\varvec{C}} \rangle _{p}=0,\\ {\mathcal {L}}_{T}=\lim \nolimits _{h \rightarrow 0} \dfrac{{\mathcal {L}}(T + h e)-{\mathcal {L}}(T)}{h}=\varPsi _{T} + \langle \lambda ,{\varvec{R}},{\varvec{C}} \rangle _{T}=0, \end{array}\right.$$

(29)

where ${\varvec{w}}, r, e$ are the direction of Gâteaux derivative; R, C represent primal governing equations and primal BCs, respectively. Thus, the derivative versus pseudo density of objective function can be inferred as

$$\begin{aligned} \dfrac{{\text {d}}\varPsi }{{\text {d}}\gamma }&=\varPsi _{{\varvec{u}}}\dfrac{\partial {\varvec{u}}}{\partial \gamma }+\varPsi _{p}\dfrac{\partial p}{\partial \gamma }+\varPsi _{T}\dfrac{\partial T}{\partial \gamma }+\varPsi _{\gamma }\\ {}&\quad +\langle \lambda ,{\varvec{R}},{\varvec{C}}\rangle _{{\varvec{u}}}\dfrac{\partial {\varvec{u}}}{\partial \gamma }+\langle \lambda ,{\varvec{R}},{\varvec{C}} \rangle _{p}\dfrac{\partial p}{\partial \gamma }+\langle \lambda ,{\varvec{R}},{\varvec{C}} \rangle _{T}\dfrac{\partial T}{\partial \gamma }+\langle \lambda ,{\varvec{R}},{\varvec{C}} \rangle _{\gamma }\\ {}&=(\varPsi _{{\varvec{u}}}+\langle \lambda ,{\varvec{R}},{\varvec{C}} \rangle _{{\varvec{u}}})\dfrac{\partial {\varvec{u}}}{\partial \gamma }+(\varPsi _{p}+\langle \lambda ,{\varvec{R}},{\varvec{C}} \rangle _{p})\dfrac{\partial p }{\partial \gamma }+ (\varPsi _{T}+\langle \lambda ,{\varvec{R}},{\varvec{C}} \rangle _{T})\dfrac{\partial T}{\partial \gamma }\\ {}&\quad +\varPsi _{\gamma }+\langle \lambda ,{\varvec{R}},{\varvec{C}} \rangle _{\gamma }\\ {}&=\varPsi _{\gamma }+\langle \lambda ,{\varvec{R}},{\varvec{C}} \rangle _{\gamma }. \end{aligned}$$

(30)

The Gâteaux derivative of the Lagrangian with respect to state variables is given as

$$\begin{aligned} \left\{ \begin{aligned} {\mathcal {L}}_{{\varvec{u}}}&=\varPsi _{{\varvec{u}}}+\int _{\varOmega }\rho ({\varvec{w}}\cdot \nabla ){\varvec{u}}\cdot {\varvec{v}}+({\varvec{u}}\cdot \nabla ){\varvec{w}}\cdot {\varvec{v}}+\mu \nabla {\varvec{w}} :\nabla {\varvec{v}}+\alpha {\varvec{w}} \cdot {\varvec{v}}{\text {d}}\varOmega&\\ {}&\quad + \int _{\varOmega }q(-\nabla \cdot {\varvec{w}}){\text {d}}\varOmega +\int _{\varOmega }\rho c\nabla Tt\cdot {\varvec{w}}{\text {d}} \varOmega + \int _{\varGamma } (- \mu \nabla {\varvec{w}})\cdot {\varvec{n}} \cdot {\varvec{v}} {\text {d}} \varGamma&\\ {}&+\int _{\varGamma _{\text {in}}} {\varvec{w}}\cdot {\varvec{v}}_{\text {in}} {\text {d}}\varGamma +\int _{\varGamma _{\text {wall}}} {\varvec{w}}\cdot {\varvec{v}}_{\text {wall}} {\text {d}}\varGamma + \int _{\varGamma _{\text {out}}}\mu \nabla {\varvec{w}}\cdot {\varvec{n}} \cdot {\varvec{v}}_{\text {out}} {\text {d}}\varGamma&\\ {\mathcal {L}}_{p}&=\varPsi _{p}+\int _{\varOmega }- r \nabla \cdot {\varvec{v}}{\text {d}}\varOmega +\int _{\varGamma }r{\varvec{v}}\cdot {\varvec{n}}{\text {d}}\varGamma -\int _{\varGamma _{\text {out}}}r{\varvec{n}}\cdot {\varvec{v}}_{\text {out}}{\text {d}}\varGamma&\\ {\mathcal {L}}_{T}&=\varPsi _{T}+\int _{\varOmega }({\varvec{u}}\cdot \nabla e) t+k\nabla e \nabla t {\text {d}} \varOmega&\\ {}&-\int _{S}k\nabla et\cdot {\varvec{n}} {\text {d}}S+\int _{S_{\text {in}}}et_{\text {in}}{\text {d}}S+\int _{S_{\text {wall}}}k\nabla et_{\text {wall}}\cdot {\varvec{n}}{\text {d}}S,&\end{aligned}\right. \end{aligned}$$

(31)

which can be reformulated as

$$\begin{aligned} \left\{ \begin{aligned} {\mathcal {L}}_{{\varvec{u}}}&=\varPsi _{{\varvec{u}}}+\int _{\varOmega }[\rho \nabla {\varvec{u}}\cdot {\varvec{v}}-\rho ({\varvec{u}}\cdot \nabla ){\varvec{v}}-\mu \nabla \cdot \nabla {\varvec{v}}+\alpha {\varvec{v}}+\rho ct\nabla T]\cdot {\varvec{w}}{\text {d}}\varOmega +\int _{\varOmega }\nabla q\cdot {\varvec{w}}{\text {d}}\varOmega&\\ {}&+\int _{\varGamma }\rho ({\varvec{u}}\cdot {\varvec{n}}){\varvec{v}}\cdot {\varvec{w}}+(-q{\varvec{I}}+\mu \nabla {\varvec{v}})\cdot {\varvec{n}}\cdot {\varvec{w}}{\text {d}}\varGamma +\int _{\varGamma }(-\mu \nabla {\varvec{w}})\cdot {\varvec{n}}\cdot {\varvec{v}}{\text {d}}\varGamma&\\ {}&+\int _{\varGamma _{\text {in}}}{\varvec{w}}\cdot {\varvec{v}}_{\text {in}}{\text {d}}\varGamma +\int _{\varGamma _{\text {wall}}}{\varvec{w}}\cdot {\varvec{v}}_{\text {wall}}{\text {d}}\varGamma +\int _{\varGamma _{\text {out}}}\mu \nabla {\varvec{w}}\cdot {\varvec{n}} \cdot {\varvec{v}}_{\text {out}} {\text {d}}\varGamma&\\ {\mathcal {L}}_{p}&=\varPsi _{p}+\int _{\varOmega }-r\nabla \cdot {\varvec{v}}{\text {d}}\varOmega +\int _{\varGamma }r{\varvec{v}}\cdot {\varvec{n}}{\text {d}}\varGamma -\int _{\varGamma _{\text {out}}}r{\varvec{n}}\cdot {\varvec{v}}_{\text {out}}{\text {d}}\varGamma&\\ {\mathcal {L}}_{T}&=\varPsi _{T}-\int _{\varOmega }(\rho c{\varvec{u}}\cdot \nabla t+k\nabla ^{2}t)e{\text {d}}\varOmega&\\ {}&+\int _{S}(\rho c{\varvec{u}}t+k\nabla t)\cdot {\varvec{n}}e{\text {d}}S-\int _{S}k\nabla et\cdot {\varvec{n}}{\text {d}}S&\\ {}&+\int _{S_{\text {in}}}et_{\text {in}}{\text {d}}S+\int _{S_{\text {wall}}}k\nabla et_{\text {wall}}\cdot {\varvec{n}}{\text {d}}S.&\end{aligned}\right. \end{aligned}$$

(32)

Due to the arbitrary of Gáteaux derivative direction, all integral items but Gáteaux derivative parts ought to equal zero. Then, the adjoint equations, the adjoint BCs can be derived by merging Gateaux derivative parts, respectively. Now consider a generic integral-type objective function of the following form:

$$\begin{aligned} \varPsi&=\int \limits _{\varOmega }A({\varvec{u}}, \nabla {\varvec{u}}, p, T, \nabla T, \gamma ){\text {d}}\varOmega \\ {}&\quad +\int \limits _{\varGamma }B_{\text {f}}({\varvec{u}}, \nabla {\varvec{u}}, p, T, \nabla T, \gamma ){\text {d}}\varGamma +\int \limits _{S}B_{t}({\varvec{u}}, \nabla {\varvec{u}}, p, T, \nabla T, \gamma ){\text {d}}S, \end{aligned}$$

(33)

where A, $B_{\text {f}}$, $B_{\text {t}}$ are objective function defined in the domain, the fluid and temperature boundary, respectively. The derivatives versus state variables of the objective function are

$$\begin{aligned} \left\{ \begin{aligned} \varPsi _{{\varvec{u}}}&=\int \limits _{\varOmega }\left( \dfrac{\partial A}{\partial {\varvec{u}}}-\nabla \cdot \dfrac{\partial A}{\partial \nabla {\varvec{u}}}\right) \cdot {\varvec{w}}{\text {d}}\varOmega +\int \limits _{\varGamma _{\text {out}}}(\dfrac{\partial A}{\partial {\varvec{u}}\nabla }\cdot {\varvec{n}}+\dfrac{\partial B_{f}}{\partial {\varvec{u}}})\cdot {\varvec{w}}{\text {d}} \varGamma&\\ \varPsi _{p}&=\int \limits _{\varOmega }\dfrac{\partial A}{\partial p}\cdot r{\text {d}}\varOmega +\int \limits _{\varGamma _{\text {out}}}\dfrac{\partial B_{f}}{\partial p}\cdot r{\text {d}}\varGamma&\\ \varPsi _{T}&=\int \limits _{\varOmega }\left( \dfrac{\partial A}{\partial T}-\nabla \cdot \dfrac{\partial A}{\partial \nabla T}\right) \cdot e{\text {d}}\varOmega +\int \limits _{S_{\text {wall}}}(\dfrac{\partial A}{\partial T}\cdot {\varvec{n}}+\dfrac{\partial B_{t}}{\partial T})\cdot e{\text {d}} S.&\end{aligned}\right. \end{aligned}$$

(34)

After substituting Eq. (34) into Eq. (32), the adjoint equations can be derived:

$$\begin{aligned} \left\{ \begin{aligned} \dfrac{\partial A}{\partial T}-\nabla \cdot \dfrac{\partial A}{\partial \nabla T}&=\rho c{\varvec{u}}\cdot \nabla t+k\nabla ^{2}t&\\ \dfrac{\partial A}{\partial p}&=\nabla \cdot {\varvec{v}}&\\ -\dfrac{\partial A}{\partial {\varvec{u}}}+\nabla \cdot \dfrac{\partial A}{\partial \nabla {\varvec{u}}}&=\rho \nabla {\varvec{u}}\cdot {\varvec{v}}-\rho ({\varvec{u}}\cdot \nabla ){\varvec{v}}-{\mu \nabla \cdot \nabla {\varvec{v}}}+\alpha _{f}{\varvec{v}}+\rho c t\nabla T+\nabla q,&\end{aligned}\right. \end{aligned}$$

(35)

with adjoint BCs:

$$\begin{aligned} \left\{ \begin{aligned} t&=0&\text { on }S_{\text {in}}\\ -\dfrac{\partial A}{\partial \nabla T}\cdot {\varvec{n}}-\dfrac{\partial B_{t}}{\partial T}&=(\rho ct{\varvec{u}}+k\nabla t)\cdot {\varvec{n}}&\text { on } S _{\text {wall}}\\ -\dfrac{\partial B_{\text {f}}}{\partial p}&={\varvec{v}}\cdot {\varvec{n}}&\text { on }\varGamma _{in\cup wall}\\ -\dfrac{\partial A}{\partial {\varvec{u}}\nabla }\cdot {\varvec{n}}-\dfrac{\partial B_{\text {f}}}{\partial {\varvec{u}}}&=\rho ({\varvec{u}}\cdot {\varvec{n}}){\varvec{v}}+(-q{\mathbf {I}}+\mu \nabla {\varvec{v}})\cdot {\varvec{n}}&\text { on } \varGamma _{\text {out}}.\\ \end{aligned}\right. \end{aligned}$$

(36)

Using the adjoint variables, the stationary conditions meet the following conditions for the Lagrangian multipliers:

$$\begin{aligned} \left\{ \begin{aligned} {\varvec{v}}&={\varvec{u}}_{a}&\\q&=p_{a}&\\t&=T_{a}&\\ {\varvec{v}}_{\text {in}}&=-\rho ({\varvec{u}}\cdot {\varvec{n}}){\varvec{v}}\cdot {\varvec{w}}+(-q{\mathbf {I}}+\mu \nabla {\varvec{v}})\cdot {\varvec{n}}\cdot {\varvec{w}}&\\ {\varvec{v}}_{\text {wall}}&=-\rho ({\varvec{u}}\cdot {\varvec{n}}){\varvec{v}}\cdot {\varvec{w}}+(-q{\mathbf {I}}+\mu \nabla {\varvec{v}})\cdot {\varvec{n}}\cdot {\varvec{w}}&\\ {\varvec{v}}_{\text {out}}&={\varvec{v}}&\\ t_{\text {in}}&=-(\rho ct{\varvec{u}}+k\nabla t)\cdot {\varvec{n}}&\\t_{\text {wall}}&=t.&\end{aligned}\right. \end{aligned}$$

(37)

In this paper, the objective function is set as $\int \limits _{\varOmega }T{\text {d}}\varOmega /\left| \varOmega \right|$, that is $A=T$; $B_{\text {f}}=0$, $B_{t}=0$. Substituting above equation into Eqs. (35) and (36), Eqs. (22) and (23) can be derived eventually.