Shape optimization of flow field improving hydrodynamic stability

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• 作者：Takashi Nakazawa ; Hideyuki Azegami
• 关键词：Shape optimization ; Fluid dynamics ; Hydrodynamic stability ; 65K10 ; 65N25
• 刊名：Japan Journal of Industrial and Applied Mathematics
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：33
• 期：1
• 页码：167-181
• 全文大小：2,165 KB
• 参考文献：1.Azegami, H.: A solution to domain optimization problems (in Japanese). Trans. Jpn. Soc. Mech. Eng., Ser. A 60(574), 1479–1486 (1994)
  2.Azegami, H.: Regularized solution to shape optimization problem (in Japanese). Trans. Jpn. Soc. Ind. Appl. Math. 23(2), 83–138 (2014)
  3.Azegami, H., Fukumoto, S., Aoyama, T.: Shape optimization of continua using NURBS as basis functions. Struct. Multidiscip. Opt. 47(2), 247–258 (2013). doi:10.​1007/​s00158-012-0822-4 CrossRef MathSciNet MATH
  4.Azegami, H., Kaizu, S., Takeuchi, K.: Regular solution to topology optimization problems of continua. JSIAM Lett. 3, 1–4 (2011)CrossRef MathSciNet MATH
  5.Azegami, H., Takeuchi, K.: A smoothing method for shape optimization: Traction method using the Robin condition. Int. J. Comput. Methods 3(1), 21–33 (2006)CrossRef MathSciNet MATH
  6.Azegami, H., Wu, Z.Q.: Domain optimization analysis in linear elastic problems: Approach using traction method. JSME Int. J. Ser. A 39(2), 272–278 (1996)
  7.Belson, A.B., Semeraro, O., Rowley, C.W., Henningson, S.D.: Feedback control of instabilities in the two-dimensional Blasius boundary layer: the role of sensors and actuators. Phys. Fluids 25(054106) (2013). doi:10.​1063/​1.​4804390
  8.Camarri, S., Iollo, A.: Feedback control of the vortex-shedding instability based on sensitivity analysis. Phys. Fluids 22(094102) (2010). doi:10.​1063/​1.​3481148
  9.Chenais, D.: On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52, 189–219 (1975)CrossRef MathSciNet MATH
  10.Davis, T.: Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30, 196–199 (2004)CrossRef MATH
  11.Fani, A., Camarri, S., Salvetti, V.M.: Stability analysis and control of the flow in a symmetric channel with a sudden expansion. Phys. Fluids 24(084102) (2012). doi:10.​1063/​1.​4745190
  12.Glowinski, R., Pironneau, O.: On the numerical computation of minimum-drag profile in laminar flow. J. Fluid. Mech. 72(2), 385–389 (1975)CrossRef MATH
  13.Hadamard, J.: Mémoire des savants etragers. Oeuvres de J. Hadamard, chap. Mémoire sur le probléme d’analyse relatif á l’équilibre des plaques élastiques encastrées, Mémoire des savants etragers, Oeuvres de J. Hadamard, pp. 515–629. CNRS, Paris (1968)
  14.Haslinger, J., Mäkinen, R.A.E.: Introduction to Shape Optimization: Theory, Approximation, and Computation. SIAM, Philadelphia (2003)CrossRef
  15.Haslinger, J., Málek, J., Stebel, J.: Shape optimization in problems governed by generalised Navier–Stokes equations: existence analysis. Control Cybern. 34(1), 283–303 (2005)MATH
  16.Huan, J., Modi, V.: Optimum design of minimum drag bodies in incompressible laminar flow using a control theory approach. Inv. Probl. Eng. 1(1), 1–25 (1994)CrossRef
  17.Huan, J., Modi, V.: Design of minimum drag bodies in incompressible laminar flow. Inv. Probl. Eng. 3(1), 233–260 (1996)CrossRef
  18.Imam, M.H.: Three-dimensional shape optimization. Int. J. Num. Meth. Eng. 18, 661–673 (1982)CrossRef MATH
  19.Kaizu, S.: The Gateau derivative of cost functions in the optimal shape problems and the existence of the shape derivatives of solutions of the Stokes problems. JSIAM Lett. 1, 17–20 (2009)CrossRef MathSciNet MATH
  20.Katamine, E., Azegami, H., Tsubata, T., Itoh, S.: Solution to shape optimization problems of viscous flow fields. Int. J. Comput. Fluid Dyn. 19(1), 45–51 (2005)CrossRef MathSciNet MATH
  21.Katamine, E., Nagatomo, Y., Azegami, H.: Shape optimization of 3d viscous flow fields. Inv. Probl. Sci. Eng. 17(1), 105–114 (2009)CrossRef MATH
  22.Marquet, O., Sipp, D., Jacquin, L.: Sensitivity analysis and passive control of cylinder flow. J. Fluid. Mech. 615, 221–252 (2008)CrossRef MathSciNet MATH
  23.Mizushima, J., Shiotani, Y.: Structural instability of the bifurcation diagram for two-dimensional flow in a channel with a sudden expansion. Comput. Methods Appl. Mech. Eng. 420, 131–145 (2000)MathSciNet MATH
  24.Mohammadi, B., Pironneau, O.: Applied shape optimization for fluids. Oxford University Press, Oxford (2001)MATH
  25.Pironneau, O.: On optimum profiles in Stokes flow. J. Fluid. Mech. 59(1), 117–128 (1973)CrossRef MathSciNet MATH
  26.Pironneau, O.: On optimum design in fluid mechanics. J. Fluid. Mech. 64(1), 97–110 (1974)CrossRef MathSciNet MATH
  27.Pironneau, O.: Optimal Shape Design for Elliptic Systems. Springer-Verlag, New York (1984)CrossRef MATH
  28.Sano, M., Sakai, H.: Numerical determination of minimum drag profile in Stokes flow (in case of two-dimensional finite region and constant cross-sectional area). Trans. Jpn. Soc. Aeronaut. Space Sci. 26(71), 1–9 (1983)
  29.Shapira, M., Degani, D., Weihs, D.: Stability and existence of multiple solutions for viscous flow in suddenly enlarged channels. Comput. Fluids 18, 239–258 (1990)CrossRef MATH
  30.Sipp, D., Marquet, O., Meliga, P., Barbagallo, A.: Dynamics and control of global instabilities in open flows: a linearized approach. Appl. Mech. Rev. 63(030801), 1–26 (2010). doi:10.​1115/​1.​4001478
  31.Sokolowski, J., Zolésio, J.P.: Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer-Verlag, New York (1992)CrossRef MATH
  
• 作者单位：Takashi Nakazawa (1)
  Hideyuki Azegami (2)
  
  1. Graduate School of Information Science, Tohoku University, Aoba-ku, Aramaki Aza, Aoba 6-3, Sendai, Miyagi, 980-8579, Japan
  2. Graduate School of Information Science, Nagoya University, Chikusa-ku Hurou-chou, Nagiya, Aichi, 464-8601, Japan
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Applications of Mathematics
  Computational Mathematics and Numerical Analysis
  
• 出版者：Springer Japan
• ISSN：1868-937X

文摘

This paper presents a solution of a shape optimization problem of a flow field for delaying transition from a laminar flow to a turbulent flow. Mapping from an initial domain to a new domain is chosen as the design variable. Main problems are defined by the stationary Navier–Stokes problem and an eigenvalue problem assuming a linear disturbance on the solution of the stationary Navier–Stokes problem. The maximum value of the real part of the eigenvalue is used as an objective cost function. The shape derivative of the cost function is defined as the Fréchet derivative of the cost function with respect to arbitrary variation of the design variable, which denotes the domain variation, and is evaluated using the Lagrange multiplier method. To obtain a numerical solution, we use an iterative algorithm based on the \(H^{1}\) gradient method using the finite element method. To confirm the validity of the solution, a numerical example for two-dimensional Poiseuille flow with a sudden expansion is presented. Results reveal that a critical Reynolds number increases by the iteration of reshaping. Keywords Shape optimization Fluid dynamics Hydrodynamic stability