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.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20

References

  1. 1.

    Allaire, G.: Conception Optimale de Structures. Mathématiques & Applications (Berlin) [Mathematics and Applications], vol. 58. Springer, Berlin (2007)

    Google Scholar 

  2. 2.

    Allaire, G.: Shape Optimization by the Homogenization Method, vol. 146. Springer Science & Business Media, Berlin (2012)

    Google Scholar 

  3. 3.

    Allaire, G., Pantz, O.: Structural optimization with freefem++. Struct. Multidiscip. Optim. 32(3), 173–181 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Allaire, G., Dapogny, C., Frey, P.: A mesh evolution algorithm based on the level set method for geometry and topology optimization. Struct. Multidiscip. Optim. 48(4), 711–715 (2013)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Allaire, G., Dapogny, C., Frey, P.: Shape optimization with a level set based mesh evolution method. Comput. Methods Appl. Mech. Eng. 282, 22–53 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    Allaire, G., Jouve, F., Toader, A.M.: A level-set method for shape optimization. C. R. Math. 334(12), 1125–1130 (2002)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Allaire, G., Jouve, F., Toader, A.-M.: Structural optimization using sensitivity analysis and a level-set method. J. Comput. Phys. 194(1), 363–393 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Bendsoe, M.P., Sigmund, O.: Topology Optimization. Theory, Methods and Applications. Springer, Berlin (2003)

    Google Scholar 

  9. 9.

    Borrvall, T., Petersson, J.: Topology optimization of fluids in Stokes flow. Int. J. Numer. Methods Fluids 41(1), 77–107 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  10. 10.

    Bourdin, B., Chambolle, A.: Design-dependent loads in topology optimization. ESAIM Control Optim. Calc. Var. 9, 19–48 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Boyer, F.: Trace theorems and spatial continuity properties for the solutions of the transport equation. Differ. Integral Equ. 18(8), 891–934 (2005)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Brezis, H.: Functional Analysis. Sobolev Spaces and Partial Differential Equations. Springer Science & Business Media, Berlin (2010)

    Google Scholar 

  13. 13.

    Bui, C., Dapogny, C., Frey, P.: An accurate anisotropic adaptation method for solving the level set advection equation. Int. J. Numer. Methods Fluids 70(7), 899–922 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    Burger, M.: A framework for the construction of level set methods for shape optimization and reconstruction. Interfaces Free Bound. 5(3), 301–329 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Céa, J.: Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût. ESAIM Math. Model. Numer. Anal. 20(3), 371–402 (1986)

    MATH  Article  Google Scholar 

  16. 16.

    Chukwudozie, C.P.: Shape optimization for drag minimization using the Navier–Stokes equation. Master’s thesis, Louisiana State University (2015)

  17. 17.

    Dapogny, C.: Optimisation de formes, méthode des lignes de niveaux sur maillages non structurés et évolution de maillages. Ph.D. thesis, Université Pierre et Marie Curie-Paris VI (2013)

  18. 18.

    Dapogny, C., Frey, P.: Computation of the signed distance function to a discrete contour on adapted triangulation. Calcolo 49(3), 193–219 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Dapogny, C., Dobrzynski, C., Frey, P.: Three-dimensional adaptive domain remeshing, implicit domain meshing, and applications to free and moving boundary problems. J. Comput. Phys. 262, 358–378 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Dapogny, C., Faure, A., Michailidis, G., Allaire, G., Couvelas, A., Estevez, R.: Geometric constraints for shape and topology optimization in architectural design. Comput. Mech. 59(6), 933–965 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Dapogny, C., Frey, P., Omnès, F., Privat, Y.: Geometrical shape optimization in fluid mechanics using FreeFem++. HAL preprint hal-01481707, March (2017)

  22. 22.

    Dbouk, T.: A review about the engineering design of optimal heat transfer systems using topology optimization. Appl. Therm. Eng. 112, 841–854 (2017)

    Article  Google Scholar 

  23. 23.

    de Gournay, F.: Velocity extension for the level-set method and multiple eigenvalues in shape optimization. SIAM J. Control Optim. 45(1), 343–367 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  24. 24.

    Delfour, M.C., Zolésio, J.-P.: Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization. SIAM, Philadelphia (2011)

    Google Scholar 

  25. 25.

    Dilgen, C., Dilgen, S., Fuhrman, D., Sigmund, O., Lazarov, B.: Topology optimization of turbulent flows. Comput. Methods Appl. Mech. Eng. 331, 363–393 (2018)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements, vol. 159. Springer Science & Business Media, Berlin (2013)

    Google Scholar 

  27. 27.

    Evans, L.C., Gariepy, R.F.: Measure Theory and Fine Properties of Functions. CRC Press, Boca Raton (2015)

    Google Scholar 

  28. 28.

    Frey, P., George, P.L.: Mesh Generation: Application to Finite Elements. Wiley Online Library, New York (2000)

    Google Scholar 

  29. 29.

    Giacomini, M., Pantz , O., Trabelsi, K.: Volumetric expressions of the shape gradient of the compliance in structural shape optimization. arXiv preprint. arXiv:1701.05762 (2017)

  30. 30.

    Girault, V., Raviart, P.-A.: Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms, vol. 5. Springer Science & Business Media, Berlin (2012)

    Google Scholar 

  31. 31.

    Hecht, F.: New development in freefem++. J. Numer. Math. 20(3–4), 251–266 (2012)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Henrot, A., Pierre, M.: Variation et optimisation de formes: une analyse géométrique, vol. 48. Springer Science & Business Media, Berlin (2006)

    Google Scholar 

  33. 33.

    Hiptmair, R., Paganini, A., Sargheini, S.: Comparison of approximate shape gradients. BIT Numer. Math. 55(2), 459–485 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Jameson, A.: Aerodynamic design via control theory. In: Recent Advances in Computational Fluid Dynamics (Princeton, NJ, 1988), Lecture Notes in Engineering, vol. 43, pp. 377–401. Springer, Berlin (1989)

  35. 35.

    Kimmel, R., Sethian, J.A.: Computing geodesic paths on manifolds. Proc. Natl. Acad. Sci. 95(15), 8431–8435 (1998)

    MathSciNet  MATH  Article  Google Scholar 

  36. 36.

    Lions, J.-L., Magenes, E.: Non-Homogenous Boundary Value Problems and Applications, vol. 1, p. 01. Springer, New York (1972)

    Google Scholar 

  37. 37.

    Marck, G., Privat, Y.: On some shape and topology optimization problems in conductive and convective heat transfers. In: Papadrakakis, M., Karlaftis, M.G., Lagaros, N.D. (eds.) OPTI 2014, An International Conference on Engineering and Applied Sciences Optimization, pp. 1640–1657, June (2014)

  38. 38.

    Marck, G., Nemer, M., Harion, J.-L.: Topology optimization of heat and mass transfer problems: laminar flow. Numer. Heat Transf. Part B Fundam. 63(6), 508–539 (2013)

    MATH  Article  Google Scholar 

  39. 39.

    Mohammadi, B., Pironneau, O.: Applied Shape Optimization for Fluids. Numerical Mathematics and Scientific Computation, 2nd edn. Oxford University Press, Oxford (2010)

    Google Scholar 

  40. 40.

    Murat, F., Simon, J.: Sur le contrôle par un domaine géométrique, rapport la 189, univ. Paris VI (1976)

  41. 41.

    Osher, S.J., Santosa, F.: Level set methods for optimization problems involving geometry and constraints: I. Frequencies of a two-density inhomogeneous drum. J. Comput. Phys. 171(1), 272–288 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  42. 42.

    Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)

    MathSciNet  MATH  Article  Google Scholar 

  43. 43.

    Pantz, O.: Sensibilité de l’équation de la chaleur aux sauts de conductivité. C. R. Math. 341(5), 333–337 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  44. 44.

    Papazoglou, P.: Topology optimization of heat exchangers. Master’s thesis, TU Delft (2015)

  45. 45.

    Pironneau, O.: On optimum profiles in Stokes flow. J. Fluid Mech. 59(1), 117–128 (1973)

    MathSciNet  MATH  Article  Google Scholar 

  46. 46.

    Pironneau, O.: Optimal Shape Design for Elliptic Systems. Springer Science & Business Media, Berlin (2012)

    Google Scholar 

  47. 47.

    Plotnikov, P., Sokołowski, J.: Compressible Navier–Stokes Equations: Theory and Shape Optimization, vol. 73. Springer Science & Business Media, Berlin (2012)

    Google Scholar 

  48. 48.

    Richter, T.: Fluid–Structure Interactions: Models, Analysis and Finite Elements. Lecture Notes in Computational Science and Engineering, vol. 118. Springer, Cham (2017)

    Google Scholar 

  49. 49.

    Sethian, J.A.: A fast marching level set method for monotonically advancing fronts. Proc. Natl. Acad. Sci. 93(4), 1591–1595 (1996)

    MathSciNet  MATH  Article  Google Scholar 

  50. 50.

    Sethian, J.A., Wiegmann, A.: Structural boundary design via level set and immersed interface methods. J. Comput. Phys. 163(2), 489–528 (2000)

    MathSciNet  MATH  Article  Google Scholar 

  51. 51.

    Sokolowski, J., Zolésio, J.-P.: Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer Series in Computational Mathematics, vol. 16. Springer, Berlin (1992)

    Google Scholar 

  52. 52.

    Strain, J.: Semi-Lagrangian methods for level set equations. J. Comput. Phys. 151(2), 498–533 (1999)

    MathSciNet  MATH  Article  Google Scholar 

  53. 53.

    Takezawa, A., Nishiwaki, S., Kitamura, M.: Shape and topology optimization based on the phase field method and sensitivity analysis. J. Comput. Phys. 229(7), 2697–2718 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  54. 54.

    Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces, vol. 3. Springer Science & Business Media, Berlin (2007)

    Google Scholar 

  55. 55.

    Temam, R.: Navier Stokes Equations: Theory and Numerical Analysis, vol. 45. North-Holland Publishing Company, Amsterdam (1977)

    Google Scholar 

  56. 56.

    Wang, M.Y., Wang, X., Guo, D.: A level set method for structural topology optimization. Comput. Methods Appl. Mech. Eng. 192(1), 227–246 (2003)

    MathSciNet  MATH  Article  Google Scholar 

  57. 57.

    Welker, K.: Suitable spaces for shape optimization. arXiv preprint. arXiv:1702.07579 (2017)

  58. 58.

    Xia, Q., Wang, M.Y.: Topology optimization of thermoelastic structures using level set method. Comput. Mech. 42(6), 837–857 (2008)

    MATH  Article  Google Scholar 

  59. 59.

    Yoon, G.H.: Topological layout design of electro-fluid-thermal-compliant actuator. Comput. Methods Appl. Mech. Eng. 209–212, 28–44 (2012)

    Article  Google Scholar 

  60. 60.

    Yoon, G.H.: Stress-based topology optimization method for steady-state fluid-structure interaction problems. Comput Methods Appl. Mech. Eng. 278, 499–523 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  61. 61.

    Zhao, H.: A fast sweeping method for eikonal equations. Math. Comput. 74(250), 603–627 (2005)

    MathSciNet  MATH  Article  Google Scholar 

  62. 62.

    Zhao, X., Zhou, M., Sigmund, O., Andreasen, C.: A “poor man’s approach” to topology optimization of cooling channels based on a darcy flow model. Int. J. Heat Mass Transf. 116, 1108–1123 (2018)

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the Association Nationale de la Recherche et de la Technologie (ANRT) [Grant number CIFRE 2017/0024]. G. A. is a member of the DEFI project at INRIA Saclay Ile-de-France. The work of G. A. is partially supported by the SOFIA project, funded by BPI (Banque Publique d’Investissement). The work of C. D. is partially supported by the IRS-CAOS Grant from Université Grenoble-Alpes.

Author information

Affiliations

Authors

Corresponding author

Correspondence to G. Allaire.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix A: Proof of Proposition 4

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 \(\varGamma \).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Feppon, F., Allaire, G., Bordeu, F. et al. Shape optimization of a coupled thermal fluid–structure problem in a level set mesh evolution framework. SeMA 76, 413–458 (2019). https://doi.org/10.1007/s40324-018-00185-4

Download citation

Keywords

  • Topology and shape optimization
  • Adjoint methods
  • Fluid structure interaction
  • Convective heat transfer
  • Adaptive remeshing

Mathematics Subject Classification

  • 49Q10 Optimization of shapes other than minimal surfaces
  • 74P15 Topological methods
  • 74P20 Geometrical methods