On Material Optimization in Continuum Dynamics

  • Konstantin A. Lurie
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 15)


This chapter deals with material optimization in dynamics—the class of problems that necessarily require the involvement of DM to meet the challenges put forth by the temporally changing environment. These problems can be regular and irregular in the sense of Chapter  2. For any particular case, there is no a priori knowledge about the regularity or irregularity of an optimal material layout, so the irregular case should be characterized by a special scenario dictated by the physics accompanying collisions of characteristics. An example of such scenario for the problem of optimal delivery of masses is discussed in detail; it illustrates the way of relaxation of an optimal problem totally different from the homogenization procedure traditionally applicable in statics. Particularly, this result indicates that homogenization in material dynamics is of limited significance compared to its role in statics.


  1. 1.
    Adams, R.: Direct solution of an optimal layout problem for isotropic heat conductions with a volume fraction constraint. J. Optim. Theory Appl. 85(3), 545–561 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Brenier, Y., Grenier, E.: Sticky particles and scalar conservation laws. SIAM J. Numer. Anal. 35(6), 2317–2328 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Castro, C., Palasios, F., Zuazua, E.: An alternative descent method for the optimal control of the inviscid Burgers equation in the presence of shocks. Math. Models Methods Appl. Sci. 18, 369–416 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Chen, G.- Q., Liu, H.: Formation of δ-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids. SIAM J. Math. Anal. 34(4), 925–938 (2003)Google Scholar
  5. 5.
    Cherkaev, A.: In: Variational Methods for Structural Optimization. Applied Mathematical Sciences, vol. 140, p. 545. Springer, New York (2000)Google Scholar
  6. 6.
    Cherkaev, A.: Bounds for effective properties of multimaterial two-dimensional conducting composites and fields in optimal composites. Mech. Mater. 41, 411–433 (2009)CrossRefGoogle Scholar
  7. 7.
    Cherkaev, A.: Optimal three-material wheel assemblage of conducting and elastic composites. Int. J. Eng. Sci. 59, 27–39 (2012)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Cherkaev, A., Dzieranowski, G.: Three-phase plane composites of minimal elastic stress energy: high-porosity structures. Int. J. Solids Struct. 50, 25–26, 4145–4160 (2013)CrossRefGoogle Scholar
  9. 9.
    Cherkaev, A., Zhang, Y.: Optimal anisotropic three-phase conducting composites: plane problem. Int. J. Solids Struct. 48, 20, 2800–2813 (2011)CrossRefGoogle Scholar
  10. 10.
    Chertock, A., Kurganov, A., Rykov, Yu.: A new sticky particle method for pressureless gas dynamics. SIAM J. Numer. Anal. 45(6), 2408–2441 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Dunaevskaya, O.: Topics in the coefficient control of linear hyperbolic equations. MS Thesis, 1–39, WPI (1997)Google Scholar
  12. 12.
    Gibianski, L.V., Lurie, K.A., Cherkaev, A.V.: Optimal focusing of heat flux by non-homogeneous heat conducting medium (a “thermolense” problem). Zh. Tekh. Fiz. (J. Tech. Phys.) 58(1), 67–74 (1998)Google Scholar
  13. 13.
    Goodman, J., Kohn, R.V., Reyna, L.: Numerical study of a relaxed variational problem from optimal design. Comput. Methods Appl. Mech. Eng. 57, 107–127 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lurie, K.A.: The Mayer-Bolza problem for multiple integrals and optimization of the behavior of systems with distributed parameters. Prikladnaya Matematika i Mekhanika (PMM) = Appl. Math. Mech. 27(2), 842–853 (1963)Google Scholar
  15. 15.
    Lurie, K.A.: On the optimal distribution of the resistivity tensor of the working medium in the channel of a MHD generator. Prikladnaya Matematika i Mekhanika (PMM) = Appl. Math. Mech. 34(2), 270–291 (1970)Google Scholar
  16. 16.
    Lurie, K.A.: The extension of optimization problems containing controls in the coefficients. Proc. Roy. Soc. Edinb. 114A, 81–97 (1990)CrossRefzbMATHGoogle Scholar
  17. 17.
    Lurie, K.A.: Direct relaxation of optimal layout problems for plates. J. Optim Theory Appl. 80(1), 93–116 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lurie, K.A.: On material optimization in continuum dynamics. J. Optim. Theory Appl. 167(1), 147–160 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lurie, K.A., Cherkaev, A.V.: Method of deriving exact bounds for effective constants of composites (in Russian). In: Zvolinskii, N.V., Grinfield, M.A., Engelbrecht, Yu. (eds.) Problems of Nonlinear Continuum Mechanics. Valgus, Tallinn (1985)Google Scholar
  20. 20.
    Lurie, K.A., Cherkaev, A.V.: Effective characteristics of composite materials and the optimal design of structural elements (in Russian). Uspekhi Mekhaniki (Advances in Mechanics) 9(2), 3–81 (1986). English translation: In: Cherkaev, A., Kohn, R. (eds.) Topics in the Mathematical Modeling of Composite Materials. Progress in Nonlinear Differential Equations and their Applications, vol. 31, pp. 175–241. Birkh\(\ddot{a}\) user, Basel (1997)Google Scholar
  21. 21.
    Maestre, F., Münch, A., Pedregal, P.: Optimal design under the one-dimensional wave equation. Interfaces Free Bound. 10, 87–116 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Pedregal, P.: Vector variational problems and applications to optimal design. ESAIM Control Optim. Calc. Var. 15, 357–381 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Strang, G.: Research Report, Centre for Mathematical Analysis, Australian National University, pp. 1–21 (1984)Google Scholar
  24. 24.
    Weinan, E., Rykov, Yu.G., Sinai, Ya.G.: Generalized variational principles, global weak solutions and behaviour with random initial data for systems of conservation laws arising in adhesion particle dynamics. Commun. Math. Phys. 177, 349–380 (1996)Google Scholar
  25. 25.
    Zel’dovich, Ya.B., Myshkis, A.D.: Elements of Mathematical Physics (in Russian). Nauka, Moscow (1973)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Konstantin A. Lurie
    • 1
  1. 1.Department of Mathematical SciencesWorcester Polytechnic InstituteWorcesterUSA

Personalised recommendations