Shape optimization of a coupled thermal fluid–structure problem in a level set mesh evolution framework

Abstract

Hadamard’s method of shape differentiation is applied to topology optimization of a weakly coupled three physics problem. The coupling is weak because the equations involved are solved consecutively, namely the steady state Navier–Stokes equations for the fluid domain, first, the convection diffusion equation for the whole domain, second, and the linear thermo-elasticity system in the solid domain, third. Shape sensitivities are derived in a fully Lagrangian setting which allows us to obtain shape derivatives of general objective functions. An emphasis is given on the derivation of the adjoint interface condition dual to the one of equality of the normal stresses at the fluid solid interface. The arguments allowing to obtain this surprising condition are specifically detailed on a simplified scalar problem. Numerical test cases are presented using the level set mesh evolution framework of Allaire et al. (Appl Mech Eng 282:22–53, 2014). It is demonstrated how the implementation enables to treat a variety of shape optimization problems.

Appendix A: Proof of Proposition 4

We provide in this appendix a proof of Propositions 4 and  5, or equivalently of (4.14) and (4.15), which is a mere adaptation of the arguments involved in Sect. 3. Using classical arguments based on the implicit function theorem (see e.g. [32]), one proves that under the condition that the linearized version of the state equations (2.1) and (2.3) are well posed (see Remark 10), the mappings ${\varvec{v}}(\varGamma _{{\varvec{\theta }}})\circ (I+{\varvec{\theta }})$, $p(\varGamma _{{\varvec{\theta }}})\circ (I+{\varvec{\theta }})$, $T(\varGamma _{{\varvec{\theta }}})\circ (I+{\varvec{\theta }})$, and ${\varvec{u}}(\varGamma _{{\varvec{\theta }}})\circ (I+{\varvec{\theta }})$ are differentiable with respect to ${\varvec{\theta }}$. Differentiating the variational formulations (4.2)–(4.4), one finds that the Fréchet derivatives ${\dot{{\varvec{v}}}}({\varvec{\theta }}),\dot{p}({\varvec{\theta }}),\dot{T}({\varvec{\theta }})$ and ${\dot{{\varvec{u}}}}({\varvec{\theta }})$ at ${\varvec{\theta }}=0$ solve the following variational problems:

$$\begin{aligned}&\text {Find }({\dot{{\varvec{v}}}}({\varvec{\theta }}),\dot{p}({\varvec{\theta }}))\in V_{{\varvec{v}},p}(\varGamma )\text { such that } \forall ({\varvec{w}}',q')\in V_{{\varvec{v}},p}(\varGamma ),\nonumber \\&\int _{\varOmega _f} [\sigma _f({\dot{{\varvec{v}}}}({\varvec{\theta }}),\dot{p}({\varvec{\theta }})):\nabla {\varvec{w}}'+\rho {\varvec{w}}'\cdot \nabla {\varvec{v}}\cdot {\dot{{\varvec{v}}}}({\varvec{\theta }})+\rho {\varvec{w}}'\cdot \nabla {\dot{{\varvec{v}}}}({\varvec{\theta }})\cdot {\varvec{v}}-q'\mathrm {div}({\dot{{\varvec{v}}}}({\varvec{\theta }}))]\mathrm {d} x\nonumber \\&\quad =\int _{\varOmega _f} [{\varvec{w}}'\cdot \, \mathrm {div}(\varvec{f}_f\otimes {\varvec{\theta }})-(\sigma _f({\varvec{v}},p):\nabla {\varvec{w}}'+\rho {\varvec{w}}'\cdot \nabla {\varvec{v}}\cdot {\varvec{v}})\mathrm {div}({\varvec{\theta }})]\mathrm {d} x\nonumber \\&\qquad +\int _{\varOmega _f} (\sigma _f({\varvec{v}},p):(\nabla {\varvec{w}}'\nabla {\varvec{\theta }})+\sigma _f({\varvec{w}}',q'):(\nabla {\varvec{v}}\nabla {\varvec{\theta }}) +\rho {\varvec{w}}'\cdot \nabla {\varvec{v}}\nabla {\varvec{\theta }}\cdot {\varvec{v}})\mathrm {d} x, \end{aligned}$$

(A.1)

$$\begin{aligned}&\text {Find }\dot{T}({\varvec{\theta }})\in V_T(\varGamma )\text { such that } \forall S'\in V_T(\varGamma ),\,\nonumber \\&\int _{\varOmega _s} k_s\nabla \dot{T}({\varvec{\theta }})\cdot \nabla S'\mathrm {d} x+\int _{\varOmega _f} (k_f\nabla \dot{T}({\varvec{\theta }})\cdot \nabla S+\rho c_p S'{\dot{{\varvec{v}}}}({\varvec{\theta }})\cdot \nabla T)\mathrm {d} x\nonumber \\&= -\int _{\varOmega _f}\rho c_p S'{\varvec{v}}\cdot \nabla \dot{T}({\varvec{\theta }})\mathrm {d} x\nonumber \\&\quad +\int _{\varOmega _s} [\mathrm {div}(Q_s{\varvec{\theta }})S'+k_s(\nabla {\varvec{\theta }}+\nabla {\varvec{\theta }}^T-\mathrm {div}({\varvec{\theta }})I)\nabla T\cdot \nabla S']\mathrm {d} x\nonumber \\&\quad +\int _{\varOmega _f}[ \mathrm {div}(Q_f{\varvec{\theta }})S'+k_f(\nabla {\varvec{\theta }}+\nabla {\varvec{\theta }}^T-\mathrm {div}({\varvec{\theta }})I)\nabla T\cdot \nabla S']\mathrm {d} x\nonumber \\&\quad +\int _{\varOmega _f} (-\rho c_p S'{\varvec{v}}\cdot \nabla T\mathrm {div}({\varvec{\theta }})+\rho c_p S'{\varvec{v}}\cdot \nabla {\varvec{\theta }}^T\nabla T)\mathrm {d} x, \end{aligned}$$

(A.2)

$$\begin{aligned}&\text {Find }{\dot{{\varvec{u}}}}({\varvec{\theta }})\in V_{{\varvec{u}}}(\varGamma )\text { such that } \forall \varvec{r}'\in V_{\varvec{u}}(\varGamma ),\,\nonumber \\&\int _{\varOmega _s} Ae({\dot{{\varvec{u}}}}({\varvec{\theta }})):\nabla \varvec{r}'\mathrm {d} x= \int _{\varOmega _s}\alpha \dot{T}({\varvec{\theta }})\mathrm {div}(\varvec{r}')\mathrm {d} x\nonumber \\&\quad +\int _{\varOmega _s}[-\mathrm {div}({\varvec{\theta }})\sigma _s({\varvec{u}},T_s):\nabla \varvec{r}'+\mathrm {div}(\varvec{f}_s\otimes {\varvec{\theta }})\cdot \varvec{r}']\mathrm {d} x\nonumber \\&\quad +\int _{\varOmega _s} (\sigma _s({\varvec{u}},T_s):(\nabla \varvec{r}\nabla {\varvec{\theta }})+Ae(\varvec{r}):(\nabla {\varvec{u}}\nabla {\varvec{\theta }}))\mathrm {d} x-\int _{\varGamma }\varvec{r}'\cdot \sigma _f({\dot{{\varvec{v}}}}({\varvec{\theta }}),\dot{p}({\varvec{\theta }}))\cdot {\varvec{n}}\mathrm {d}s ,\nonumber \\ \end{aligned}$$

(A.3)

where $\sigma _f({\dot{{\varvec{v}}}}({\varvec{\theta }}),\dot{p}({\varvec{\theta }}))\cdot {\varvec{n}}$ is an element of the dual space $H^{-1/2}_{00}(\varGamma ,{\mathbb {R}}^d)$ of $H^{1/2}_{00}(\varGamma ,{\mathbb {R}}^d)$ whose action is given by [differentiating (4.5) with respect to ${\varvec{\theta }}$]:

$$\begin{aligned}&\forall \varvec{r}'\in H^{1/2}_{00}(\varGamma ,{\mathbb {R}}^d),\, -\int _{\varGamma }\varvec{r}'\cdot \sigma _f({\dot{{\varvec{v}}}}({\varvec{\theta }}),\dot{p}({\varvec{\theta }}))\cdot {\varvec{n}}\mathrm {d}s\nonumber \\&\quad =\int _{\varOmega _f}( \mathrm {div}(\varvec{f}_f\otimes {\varvec{\theta }})\cdot {\tilde{\varvec{r}}} -(\rho {\tilde{\varvec{r}}}\cdot \nabla {\varvec{v}}\cdot {\varvec{v}}+\sigma _f({\varvec{v}},p):\nabla {\tilde{\varvec{r}}})\mathrm {div}({\varvec{\theta }}))\mathrm {d} x\nonumber \\&\qquad +\int _{\varOmega _f}(\rho {\tilde{\varvec{r}}}\cdot \nabla {\varvec{v}}\nabla {\varvec{\theta }}\cdot {\varvec{v}}+ \sigma _f({\varvec{v}},p):(\nabla {\tilde{\varvec{r}}}\nabla {\varvec{\theta }}) +\sigma _f({\tilde{\varvec{r}}},{\tilde{q}}):(\nabla {\varvec{v}}\nabla {\varvec{\theta }}))\mathrm {d} x\nonumber \\&\qquad -\int _{\varOmega _f}( \sigma _f({\dot{{\varvec{v}}}}({\varvec{\theta }}),\dot{p}({\varvec{\theta }})):\nabla {\tilde{\varvec{r}}}+\rho {\tilde{\varvec{r}}} \cdot \nabla {\varvec{v}}\cdot {\dot{{\varvec{v}}}}({\varvec{\theta }})+\rho {\tilde{\varvec{r}}}\cdot \nabla {\dot{{\varvec{v}}}}({\varvec{\theta }})\cdot {\varvec{v}}-{\tilde{q}} \mathrm {div}({\dot{{\varvec{v}}}}({\varvec{\theta }})))\mathrm {d} x,\nonumber \\ \end{aligned}$$

(A.4)

for any extension $({\tilde{\varvec{r}}},{\tilde{q}})\in V_{{\varvec{v}},p}(\varGamma )$ satisfying ${\tilde{\varvec{r}}}=\varvec{r}'$ on $\varGamma$. Note that the above expression is independent of the extension because of (A.1) with ${\varvec{w}}'={\tilde{\varvec{r}}}$ and $q'={\tilde{q}}$. Then by definition of ${\mathfrak {J}}$:

$$\begin{aligned}&J(\varGamma _{{\varvec{\theta }}},{\varvec{v}}(\varGamma _{{\varvec{\theta }}}),p(\varGamma _{{\varvec{\theta }}}),T(\varGamma _{{\varvec{\theta }}}),{\varvec{u}}(\varGamma _{{\varvec{\theta }}}))\nonumber \\&\quad ={\mathfrak {J}}({\varvec{\theta }},{\varvec{v}}(\varGamma _{{\varvec{\theta }}})\circ (I+{\varvec{\theta }}),p(\varGamma _{{\varvec{\theta }}})\circ (I+{\varvec{\theta }}),T(\varGamma _{{\varvec{\theta }}})\circ (I+{\varvec{\theta }}),{\varvec{u}}(\varGamma _{{\varvec{\theta }}})\circ (I+{\varvec{\theta }})),\qquad \end{aligned}$$

(A.5)

whence the chain rule yields:

$$\begin{aligned}&\frac{\mathrm {d}}{\mathrm {d}{\varvec{\theta }}} [J(\varGamma _{{\varvec{\theta }}},{\varvec{v}}(\varGamma _{{\varvec{\theta }}}),p(\varGamma _{{\varvec{\theta }}}),T(\varGamma _{{\varvec{\theta }}}),{\varvec{u}}(\varGamma _{{\varvec{\theta }}})) ] ({\varvec{\theta }}) \nonumber \\&\quad =\frac{\partial {\mathfrak {J}}}{\partial {\varvec{\theta }}}({\varvec{\theta }}) +\frac{\partial {\mathfrak {J}}}{\partial ({\varvec{v}},p)}({\dot{{\varvec{v}}}}({\varvec{\theta }}),\dot{p}({\varvec{\theta }}))+\frac{\partial {\mathfrak {J}}}{\partial T}( \dot{T}({\varvec{\theta }}))+\frac{\partial {\mathfrak {J}}}{\partial {\varvec{u}}}( {\varvec{u}}({\varvec{\theta }})). \end{aligned}$$

(A.6)

One then uses the adjoint equations (4.9)–(4.11) with $\varvec{r}'={\dot{{\varvec{u}}}}({\varvec{\theta }}),\,S'=\dot{T}({\varvec{\theta }}),{\varvec{w}}'={\dot{{\varvec{v}}}}({\varvec{\theta }}),q'=\dot{p}({\varvec{\theta }})$ as test functions to obtain:

$$\begin{aligned}&\frac{\partial {\mathfrak {J}}}{\partial {\varvec{u}}}({\dot{{\varvec{u}}}}({\varvec{\theta }}))=\int _{\varOmega _s} Ae(\varvec{r}):\nabla {\dot{{\varvec{u}}}}({\varvec{\theta }})\mathrm {d} x=\int _{\varOmega _s} A e({\dot{{\varvec{u}}}}({\varvec{\theta }})):\nabla \varvec{r}\mathrm {d} x, \end{aligned}$$

(A.7)

$$\begin{aligned}&\frac{\partial {\mathfrak {J}}}{\partial T}( \dot{T}({\varvec{\theta }}))=\int _{\varOmega _s} k_s\nabla S\cdot \nabla \dot{T}({\varvec{\theta }})\mathrm {d} x+\int _{\varOmega _f} (k_f\nabla S\cdot \nabla \dot{T}({\varvec{\theta }})+\rho c_p S{\varvec{v}}\cdot \nabla \dot{T}({\varvec{\theta }}))\mathrm {d} x\nonumber \\&\quad -\int _{\varOmega _s} \alpha \dot{T}({\varvec{\theta }})\mathrm {div}(\varvec{r})\mathrm {d} x, \end{aligned}$$

(A.8)

$$\begin{aligned}&\frac{\partial {\mathfrak {J}}}{\partial ({\varvec{v}},p)}({\dot{{\varvec{v}}}}({\varvec{\theta }}),\dot{p}({\varvec{\theta }}))\nonumber \\&\quad =\int _{\varOmega _f}(\sigma _f({\varvec{w}},q):\nabla {\dot{{\varvec{v}}}}({\varvec{\theta }})+ \rho {\varvec{w}}\cdot \nabla {\varvec{v}}\cdot {\dot{{\varvec{v}}}}({\varvec{\theta }})+\rho {\varvec{w}}\cdot \nabla {\dot{{\varvec{v}}}}({\varvec{\theta }})\cdot {\varvec{v}}\nonumber \\&\qquad - \ \dot{p}({\varvec{\theta }})\mathrm {div}({\varvec{w}}))\mathrm {d} x+\int _{\varOmega _f}\rho c_p S\nabla T\cdot {\dot{{\varvec{v}}}}({\varvec{\theta }})\mathrm {d} x\nonumber \\&\quad =\int _{\varOmega _f} (\sigma _f({\dot{{\varvec{v}}}}({\varvec{\theta }}),\dot{p}({\varvec{\theta }})):\nabla {\varvec{w}}+\rho {\varvec{w}}\cdot \nabla {\dot{{\varvec{v}}}}({\varvec{\theta }})\cdot {\varvec{v}}+\rho {\varvec{w}}\cdot \nabla {\varvec{v}}\cdot {\dot{{\varvec{v}}}}({\varvec{\theta }}) \nonumber \\&\qquad - \ q\mathrm {div}({\dot{{\varvec{v}}}}({\varvec{\theta }})))\mathrm {d} x+\int _{\varOmega _f}\rho c_p S\nabla T\cdot {\dot{{\varvec{v}}}}({\varvec{\theta }})\mathrm {d} x. \end{aligned}$$

(A.9)

Using now Eqs. (A.2) and (A.3) with $\varvec{r}'=\varvec{r},S'=S$ as test functions and (A.4) with $({\tilde{\varvec{r}}},{\tilde{q}})=({\varvec{w}},q)$ as an extension of $\varvec{r}'=\varvec{r}\in H^{1/2}(\varGamma ,{\mathbb {R}}^d)$ to eliminate the bilinear terms, the above three equations rewrite:

$$\begin{aligned}&\frac{\partial {\mathfrak {J}}}{\partial {\varvec{u}}}({\dot{{\varvec{u}}}}({\varvec{\theta }}))=-\int _{\varGamma }\varvec{r}\cdot \sigma _f({\dot{{\varvec{v}}}}({\varvec{\theta }}),\dot{p}({\varvec{\theta }}))\cdot {\varvec{n}}\mathrm {d}s+ \int _{\varOmega _s}\alpha \dot{T}({\varvec{\theta }})\mathrm {div}(\varvec{r})\mathrm {d} x\nonumber \\&\quad +\int _{\varOmega _s} [-\mathrm {div}({\varvec{\theta }})\sigma _s({\varvec{u}},T_s):\nabla \varvec{r}+\mathrm {div}(\varvec{f}_s\otimes {\varvec{\theta }})\cdot \varvec{r}]\mathrm {d} x\nonumber \\&\quad +\int _{\varOmega _s} (\sigma _s({\varvec{u}},T_s):(\nabla \varvec{r}\nabla {\varvec{\theta }})+Ae(\varvec{r}):(\nabla {\varvec{u}}\nabla {\varvec{\theta }}))\mathrm {d} x, \end{aligned}$$

(A.10)

$$\begin{aligned}&\frac{\partial {\mathfrak {J}}}{\partial T}( \dot{T}({\varvec{\theta }}))=-\int _{\varOmega _s}\alpha \dot{T}({\varvec{\theta }})\mathrm {div}(\varvec{r})\mathrm {d} x-\int _{\varOmega _f} \rho c_p S{\dot{{\varvec{v}}}}({\varvec{\theta }})\cdot \nabla T\mathrm {d} x\nonumber \\&\quad +\int _{\varOmega _s} [\mathrm {div}(Q_s{\varvec{\theta }})S+k_s(\nabla {\varvec{\theta }}+\nabla {\varvec{\theta }}^T-\mathrm {div}({\varvec{\theta }})I)\nabla T\cdot \nabla S]\mathrm {d} x\nonumber \\&\quad +\int _{\varOmega _f}[ \mathrm {div}(Q_f{\varvec{\theta }})S+k_f(\nabla {\varvec{\theta }}+\nabla {\varvec{\theta }}^T-\mathrm {div}({\varvec{\theta }})I)\nabla T\cdot \nabla S]\mathrm {d} x\nonumber \\&\quad +\int _{\varOmega _f} (-\rho c_p S{\varvec{v}}\cdot \nabla T\mathrm {div}({\varvec{\theta }})+\rho c_p S{\varvec{v}}\cdot \nabla {\varvec{\theta }}^T\nabla T)\mathrm {d} x, \end{aligned}$$

(A.11)

$$\begin{aligned}&\frac{\partial {\mathfrak {J}}}{\partial ({\varvec{v}},p)}({\dot{{\varvec{v}}}}({\varvec{\theta }}),\dot{p}({\varvec{\theta }}))=\int _{\varGamma }\varvec{r}\cdot \sigma _f({\dot{{\varvec{v}}}}({\varvec{\theta }}),\dot{p}({\varvec{\theta }}))\cdot {\varvec{n}}\mathrm {d}s +\int _{\varOmega _f} \rho c_p S\nabla T\cdot {\dot{{\varvec{v}}}}({\varvec{\theta }})\mathrm {d} x\nonumber \\&\quad +\int _{\varOmega _f}( \mathrm {div}(\varvec{f}_f\otimes {\varvec{\theta }})\cdot {\varvec{w}}-(\rho {\varvec{w}}\cdot \nabla {\varvec{v}}\cdot {\varvec{v}}+\sigma _f({\varvec{v}},p):\nabla {\varvec{w}})\mathrm {div}({\varvec{\theta }}))\mathrm {d} x\nonumber \\&\quad +\int _{\varOmega _f}(\rho {\varvec{w}}\cdot \nabla {\varvec{v}}\nabla {\varvec{\theta }}\cdot {\varvec{v}}+\sigma _f({\varvec{v}},p):(\nabla {\varvec{w}}\nabla {\varvec{\theta }}) +\sigma _f({\varvec{w}}, q):(\nabla {\varvec{v}}\nabla {\varvec{\theta }}))\mathrm {d} x. \end{aligned}$$

(A.12)

Formula (4.14) follows by summing up the above three equations. If $H^2$ regularity holds for ${\varvec{v}},{\varvec{u}},T$ and $H^1$ regularity holds for p on their respective domains of definition, then an integration by parts allows to rewrite (4.14) as;

$$\begin{aligned}&\frac{\mathrm {d}}{\mathrm {d}{\varvec{\theta }}} [J(\varGamma _{{\varvec{\theta }}},{\varvec{v}}(\varGamma _{{\varvec{\theta }}}),p(\varGamma _{{\varvec{\theta }}}),T(\varGamma _{{\varvec{\theta }}}),{\varvec{u}}(\varGamma _{{\varvec{\theta }}})) ] ({\varvec{\theta }}) \nonumber \\&\quad =\int _{\varGamma } \varvec{g}_{{\mathfrak {J}}}\cdot {\varvec{\theta }}\mathrm {d}s +\int _{\varGamma } (\varvec{f}_f\cdot {\varvec{w}}-\sigma _f({\varvec{v}},p):\nabla {\varvec{w}}-\rho {\varvec{w}}\cdot \nabla {\varvec{v}}\cdot {\varvec{v}})({\varvec{\theta }}\cdot {\varvec{n}})\mathrm {d}s\nonumber \\&\qquad +\int _{\varGamma }[ {\varvec{n}}\cdot \sigma _f({\varvec{v}},p)\nabla {\varvec{w}}\cdot {\varvec{\theta }}+{\varvec{n}}\cdot \sigma _f({\varvec{w}},q)\nabla {\varvec{v}}\cdot {\varvec{\theta }}+\rho ({\varvec{v}}\cdot {\varvec{n}}){\varvec{w}}\cdot \nabla {\varvec{v}}\cdot {\varvec{\theta }}] \mathrm {d}s\nonumber \\&\qquad +\int _{\varGamma } \left( k_s\nabla T_s\cdot \nabla S_s-k_f\nabla T_f\cdot \nabla S_f+Q_fS-Q_sS_s\right) ({\varvec{\theta }}\cdot {\varvec{n}})\mathrm {d}s\nonumber \\&\qquad +\int _{\varGamma }\left( -k_s(\nabla T_s\cdot {\varvec{\theta }})(\nabla S_s\cdot {\varvec{n}})-k_s(\nabla S_s\cdot {\varvec{\theta }})(\nabla T_s\cdot {\varvec{n}})\right) \mathrm {d}s\nonumber \\&\qquad +\int _{\varGamma }\left( k_f(\nabla T_f\cdot {\varvec{\theta }})(\nabla S_f\cdot {\varvec{n}})+k_f(\nabla S_f\cdot {\varvec{\theta }})(\nabla T_f\cdot {\varvec{n}})\right) \mathrm {d}s \nonumber \\&\qquad +\int _{\varGamma }\left[ (\sigma _s({\varvec{u}},T_s):\nabla \varvec{r}-\varvec{f}_s\cdot \varvec{r})({\varvec{\theta }}\cdot {\varvec{n}})\right. \nonumber \\&\qquad \left. - \ {\varvec{n}}\cdot \sigma _s({\varvec{u}},T_s)\nabla \varvec{r}\cdot {\varvec{\theta }}-{\varvec{n}}\cdot Ae(\varvec{r})\nabla {\varvec{u}}\cdot {\varvec{\theta }}\right] \mathrm {d}s+\int _{\varGamma }\varvec{\varLambda }\cdot {\varvec{\theta }}\mathrm {d} x, \end{aligned}$$

(A.13)

where $\varvec{\varLambda }$ is a $L^1(\varOmega ,{\mathbb {R}}^d)$ function obtained from Green’s identity. The Hadamard structure theorem implies that (A.13) vanishes on compactly supported fields ${\varvec{\theta }}$ or on fields ${\varvec{\theta }}$ tangent to $\varGamma$. This implies that in fact, $\varvec{\varLambda }=0$, and (4.15) follows by removing terms depending on the tangential component of ${\varvec{\theta }}$ on $\varGamma$.