Skip to main content
SpringerLink
Log in

Motivation and Framework

  • Chapter
  • First Online:
Optimal Design through the Sub-Relaxation Method

Part of the book series: SEMA SIMAI Springer Series ((SEMA SIMAI,volume 11))

  • 455 Accesses

Abstract

It is not difficult to motivate, from a practical point of view, the kind of situations we would like to deal with and analyze. We have selected a typical example in heat conduction, but many other examples are as valid as this one. Suppose we have two very different materials at our disposal: the first, with conductivity α 1 = 1, is a good and expensive conductor; the other is a cheap material, almost an insulator with conductivity coefficient α 0 = 0. 001. These two materials are to be used to fill up a given design domain Q, which we assume to be a unit square for simplicity (Fig. 1.1), in given proportions t 1, t 0, with t 1 + t 0 = 1. Typically, t 1 < t 0 given that the first material is much more expensive than the second. We will take, for definiteness, t 1 = 0. 4, t 0 = 0. 6. The thermal device is isolated all over ∂ Q, except for a small sink Γ 0 at the middle of the left side where we normalize temperature to vanish, and there is a uniform source of heat all over Q of size unity. The mixture of the two materials is to be decided so that the dissipated energy is as small as possible.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Alali, B., Milton, G.W.: Effective conductivities of thin-interphase composites. J. Mech. Phys. Solids 61, 2680–2691 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Albin, N., Cherkaev, A., Nesi, V.: Multiphase laminates of extremal effective conductivity in two dimensions. J. Mech. Phys. Solids 55, 1513–1553 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Allaire, G.: Shape Optimization by the Homogenization Method. Applied Mathematical Sciences, vol. 146. Springer, New York (2002)

    Google Scholar 

  4. Allaire, G.: A brief introduction to homogenization and miscellaneous applications. In: Mathematical and numerical approaches for multiscale problem. In: ESAIM Proceedings, vol. 37, pp. 1–49. EDP Sciences, Les Ulis (2012)

    Google Scholar 

  5. Allaire, G., Bonnetier, E., Francfort, G., Jouve, F.: Shape optimization by the homogenization method. Numer. Math. 76, 27–68 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  6. Allaire, G., Dapogny, Ch., Frey, P.: Topology and geometry optimization of elastic structures by exact deformation of simplicial mesh. C. R. Math. Acad. Sci. Paris 349, 999–1003 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  8. Allaire, G., Francfort, G.: Existence of minimizers for non-quasiconvex functionals arising in optimal design. Ann. Inst. H. Poincaré Anal. Non Linéaire 15, 301–339 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Allaire, G., de Gournay, F., Jouve, F., Toader, A.M.: Structural optimization using topological and shape sensitivity via a level set method. Control Cybernet. 34, 59–80 (2005)

    MathSciNet  MATH  Google Scholar 

  10. Allaire, G., Gutiérrez, S.: Optimal design in small amplitude homogenization. M2AN Math. Model. Numer. Anal. 41, 543–574 (2007)

    Google Scholar 

  11. Allaire, G., Jouve, F., Maillot, H.: Topology optimization for minimum stress design with the homogenization method. Struct. Multidiscip. Optim. 28, 87–98 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  12. Allaire, G., Kelly, A.: Optimal design of low-contrast two-phase structures for the wave equation. Math. Models Methods Appl. Sci. 21, 1499–1538 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Allaire, G., Kohn, R.V.: Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions. Quart. Appl. Math. 51, 675–699 (1993)

    MathSciNet  MATH  Google Scholar 

  14. Allaire, G., Kohn, R.V.: Optimal bounds on the effective behavior of a mixture of two well-ordered elastic materials. Quart. Appl. Math. 51, 643–674 (1993)

    MathSciNet  MATH  Google Scholar 

  15. Allaire, G., Münch, A., Periago, F.: Long time behavior of a two-phase optimal design for the heat equation. SIAM J. Control Optim. 48, 5333–5356 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Ambrosio, L., Buttazzo, G.: An optimal design problem with perimeter penalization. Calc. Var. Partial Differ. Equ. 1, 55–69 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  18. Belhachmi, Z., Bucur, D., Buttazzo, G., Sac-Epée, J. M.: Shape optimization problems for eigenvalues of elliptic operators. ZAMM Z. Angew. Math. Mech. 86, 171–184 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bendsoe, M.P.: Optimization of Structural Topology, Shape, and Material. Springer, Berlin (1995)

    Book  MATH  Google Scholar 

  20. Bendsoe, M.P.: Topology design of structures, materials and mechanisms—status and perspectives. In: System Modelling and Optimization (Cambridge, 1999), pp. 1–17. Kluwer Academic Publishers, Boston, MA (2000)

    Google Scholar 

  21. Bendsoe, M.P.: Recent developments in topology design of materials and mechanisms. In: Canum 2000: Actes du 32e Congrés National d’Analyse Numérique (Port d’Albret). ESAIM Proceedings, vol. 11, pp. 41–60. Société de Mathématiques Appliquées et Industrielles, Paris (2002) (electronic)

    Google Scholar 

  22. Bendsoe, M.P., Díaz, A., Kikuchi, N.: Topology and generalized layout optimization of elastic structures. In: Topology Design of Structures (Sesimbra, 1992). NATO Advanced Science Institutes Series E: Applied Science, vol. 227, pp. 159–205. Kluwer Academic Publishers, Dordrecht (1993)

    Google Scholar 

  23. Bendsoe, M.P., Díaz, A.R., Lipton, R., Taylor, J.E.: Optimal design of material properties and material distribution for multiple loading conditions. Int. J. Numer. Methods Eng. 38, 1149–1170 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  24. Bendsoe, M.P., Guedes, J.M.: Some computational aspects of using extremal material properties in the optimal design of shape, topology and material. Shape design and optimization. Control Cybernet. 23, 327–349 (1994)

    MathSciNet  MATH  Google Scholar 

  25. Bendsoe, M.P., Guedes, J.M., Neves, M.M., Rodrigues, H.C., Sigmund, O.: Aspects of the design of microstructures by computational means. In: Homogenization, 2001 (Naples), GAKUTO International Series Mathematical Sciences and Applications, vol. 18, pp. 99–112. Gakk-tosho, Tokyo (2003)

    MATH  Google Scholar 

  26. Bendsoe, M.P., Hammer, V.B., Lipton, R., Pedersen, P.: Minimum compliance design of laminated plates. In: Homogenization and Applications to Material Sciences (Nice, 1995). GAKUTO International Series Mathematical Sciences and Applications, vol. 9, pp. 45–56. Gakk tosho, Tokyo (1995)

    Google Scholar 

  27. Bendsoe, M.P., Kikuchi, N.: Generating optimal topologies in structural design using a homogenization method. Comput. Methods Appl. Mech. Eng. 71, 197–224 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  28. Bendsoe, M.P., Lund, E., Olhoff, N., Sigmund, O.: Topology optimization—broadening the areas of application. Control Cybernet. 34, 7–35 (2005)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  30. Bendsoe, M.P., Sokolowski, J.: Shape sensitivity analysis of optimal compliance functionals. Mech. Struct. Mach. 23, 35–58 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ben-Tal, A., Bendsoe, M.P.: A new method for optimal truss topology design. SIAM J. Optim. 3, 322–358 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  32. Benveniste, Y., Milton, G.W.: New exact results for the effective electric, elastic, piezoelectric and other properties of composite ellipsoid assemblages. J. Mech. Phys. Solids 51, 1773–1813 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  33. Benveniste, Y., Milton, G.W.: The effective medium and the average field approximations vis-a-vis the Hashin-Shtrikman bounds. I. The self-consistent scheme in matrix-based composites. J. Mech. Phys. Solids 58, 1026–1038 (2010)

    MathSciNet  MATH  Google Scholar 

  34. Benveniste, Y., Milton, G.W.: The effective medium and the average field approximations vis-á-vis the Hashin-Shtrikman bounds. II. The generalized self-consistent scheme in matrix-based composites. J. Mech. Phys. Solids 58, 1039–1056 (2010)

    MathSciNet  MATH  Google Scholar 

  35. Briane, M., Casado-Díaz, J., Murat, F.: The div-curl lemma “trente ans aprés”: an extension and an application to the G-convergence of unbounded monotone operators. J. Math. Pures Appl. 91, 476–494 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  36. Briane, M., Milton, G.W., Treibergs, A.: Which electric fields are realizable in conducting materials? ESAIM Math. Model. Numer. Anal. 48, 307–323 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  37. Bucur, D., Buttazzo, G.: Variational Methods in Shape Optimization Problems. Progress in Nonlinear Differential Equations and Their Applications, vol. 65. Birkhäuser Boston, Boston, MA (2005)

    Google Scholar 

  38. Bucur, D., Buttazzo, G., Henrot, A.: Minimization of λ 2(Ω) with a perimeter constraint. Indiana Univ. Math. J. 58, 2709–2728 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  39. Buttazzo, G.: On the existence of minimizing domains for some shape optimization problems. In: Actes du 29éme Congrés d’Analyse Numérique: CANum’97 (Larnas, 1997). ESAIM Proceedings, vol. 3, pp. 51–64. Société de Mathématiques Appliquées et Industrielles, Paris (1998) (electronic)

    Google Scholar 

  40. Buttazzo, G., Dal Maso, G.: Shape optimization for Dirichlet problems: relaxed solutions and optimality conditions. Bull. Am. Math. Soc. (N.S.) 23, 531–535 (1990)

    Google Scholar 

  41. Buttazzo, G., Dal Maso, G.: Shape optimization for Dirichlet problems: relaxed formulation and optimality conditions. Appl. Math. Optim. 23, 17–49 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  42. Buttazzo, G., Dal Maso, G.: An existence result for a class of shape optimization problems. Arch. Rational Mech. Anal. 122, 183–195 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  43. Buttazzo, G., Maestre, F.: Optimal shape for elliptic problems with random perturbations. Discrete Cont. Dyn. Syst. 31, 1115–1128 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  44. Buttazzo, G., Santambrogio, F., Varchon, N.: Asymptotics of an optimal compliance-location problem. ESAIM Control Optim. Calc. Var. 12, 752–769 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  45. Casado-Díaz, J., Couce-Calvo, J., Luna-Laynez, M., Martín-Gómez, J.D.: Optimal design problems for a non-linear cost in the gradient: numerical results. Appl. Anal. 87, 1461–1487 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  46. Casado-Díaz, J., Castro, C., Luna-Laynez, M., Zuazua, E.: Numerical approximation of a one-dimensional elliptic optimal design problem. Multiscale Model. Simul. 9, 1181–1216 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  47. Cherkaev, A.V.: Relaxation of problems of optimal structural design. Int. J. Solids Struct. 31, 2251–2280 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  48. Cherkaev, A.: Variational Methods for Structural Optimization. Applied Mathematical Sciences, vol. 140. Springer, New York (2000)

    Google Scholar 

  49. Cherkaev, A., Krog, L.A., Kucuk, I.: Stable optimal design of two-dimensional elastic structures. Recent advances in structural modelling and optimization. Control Cybernet. 27, 265–282 (1998)

    MathSciNet  MATH  Google Scholar 

  50. Cherkaev, A.V., Lurie, K.A., Milton, G.W.: Invariant properties of the stress in plane elasticity and equivalence classes of composites. Proc. R. Soc. Lond. Ser. A 438, 519–529 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  51. Cherkaev, A., Palais, R.: Optimal design of three-dimensional axisymmetric elastic structures. Structural Dynamic Systems Computational Techniques and Optimization, 237–267, Gordon and Breach International Series in Engineering, Technology and Applied Sciences, vol. 9. Gordon and Breach, Amsterdam (1999)

    Google Scholar 

  52. Christensen, P.W., Klarbring, A.: An introduction to structural optimization. In: Solid Mechanics and Its Applications, vo.l. 153. Springer, New York (2009)

    Google Scholar 

  53. Clark, K.E., Milton, G.W.: Modelling the effective conductivity function of an arbitrary two-dimensional polycrystal using sequential laminates. Proc. R. Soc. Edinb. Sect. A 124, 757–783 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  54. de Gournay, F., Allaire, G., Jouve, F.: Shape and topology optimization of the robust compliance via the level set method. ESAIM Control Optim. Calc. Var. 14, 43–70 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  55. Diaz, A.R., Bénard, A.: Designing materials with prescribed elastic properties using polygonal cells. Int. J. Numer. Methods Eng. 57, 301–314 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  56. Diaz, A.R., Sigmund, O.: A topology optimization method for design of negative permeability metamaterials. Struct. Multidiscip. Optim. 41, 163–177 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  57. Donoso, A.: Optimal design modeled by Poisson’s equation in the presence of gradients in the objective. PhD doctoral dissertation, University of Castilla-La Mancha, Ciudad Real (2004)

    Google Scholar 

  58. Duysinx, P., Bendsoe, M.P.: Topology optimization of continuum structures with local stress constraints. Int. J. Numer. Methods Eng. 43, 1453–1478 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  59. Fakharzadeh J.A., Rubio, J.E.: Global solution of optimal shape design problems. Z. Anal. Anwendungen 18, 143–155 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  60. Fakharzadeh J.A., Rubio, J.E.: Shape-measure method for solving elliptic optimal shape problems (fixed control case). Bull. Iranian Math. Soc. 27, 41–64 (2001)

    MathSciNet  MATH  Google Scholar 

  61. Fernandes, P., Guedes, J.M., Rodrigues, H.: Topology optimization of three-dimensional linear elastic structures with a constraint on “perimeter”. Comput. Struct. 73, 583–594 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  62. Francfort, G.A., Milton, G.W.: Sets of conductivity and elasticity tensors stable under lamination. Commun. Pure Appl. Math. 47, 257–279 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  63. Francfort, G., Tartar, L.: Comportement effectif d’un mélange de matériaux élastiques isotropes ayant le même module de cisaillement (French) (Effective behavior of a mixture of isotropic materials with identical shear moduli). C. R. Acad. Sci. Paris Sér. I Math. 312, 301–307 (1991)

    MathSciNet  MATH  Google Scholar 

  64. Fujii, D., Chen, B.C., Kikuchi, N.: Composite material design of two-dimensional structures using the homogenization design method. Int. J. Numer. Methods Eng. 50, 2031–2051 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  65. Gersborg-Hansen, A., Bendsoe, M.P., Sigmund, O.: Topology optimization of heat conduction problems using the finite volume method. Struct. Multidiscip. Optim. 31, 251–259 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  66. Gibiansky, L.V., Cherkaev, A.V.: Design of composite plates of extremal rigidity. In: Topics in the Mathematical Modelling of Composite Materials. Progress in Nonlinear Differential Equations and Their Applications, vol. 31, pp. 95–137. Birkhäuser Boston, Boston, MA (1997)

    Google Scholar 

  67. Gibiansky, L.V., Cherkaev, A.: Microstructures of composites of extremal rigidity and exact bounds on the associated energy density. In: Topics in the Mathematical Modelling of Composite Materials. Progress in Nonlinear Differential Equations and Their Applications, vol. 31, pp. 273–317. Birkhäuser Boston, Boston (1997)

    Google Scholar 

  68. Gibiansky, L.V., Sigmund, O.: Multiphase composites with extremal bulk modulus. J. Mech. Phys. Solids 48, 461–498 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  69. Grabovsky, Y.: The G-closure of two well-ordered, anisotropic conductors. Proc. R. Soc. Edinb. Sect. A 123, 423–432 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  70. Grabovsky, Y.: Bounds and extremal microstructures for two-component composites: a unified treatment based on the translation method. Proc. R. Soc. Lond. Ser. A 452, 919–944 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  71. Grabovsky, Y.: Optimal design problems for two-phase conducting composites with weakly discontinuous objective functionals. Adv. Appl. Math. 27, 683–704 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  72. Grabovsky, Y.: homogenization in an optimal design problem with quadratic weakly discontinuous objective functional. Int. J. Differ. Equ. Appl. 3, 183–194 (2001)

    Google Scholar 

  73. Grabovsky, Y., Kohn, R.V.: Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. I. The confocal ellipse construction. J. Mech. Phys. Solids 43, 933–947 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  74. Grabovsky, Y., Kohn, R.V.: Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. II. The Vigdergauz microstructure. J. Mech. Phys. Solids 43, 949–972 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  75. Grabovsky, Y., Milton, G.W.: Exact relations for composites: towards a complete solution. In: Proceedings of the International Congress of Mathematicians, vol. III (Berlin, 1998). Documenta Mathematica, extra vol. III, pp. 623–632 (1998)

    Google Scholar 

  76. Grabovsky, Y., Milton, G.W., Sage, D.S.: Exact relations for effective tensors of composites: necessary conditions and sufficient conditions. Commun. Pure Appl. Math. 53, 300–353 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  77. Guedes, J.M.: Effective properties for nonlinear composite materials: computational aspects. In: Topology Design of Structures (Sesimbra, 1992). NATO Advanced Science Institutes Series E: Applied Science, vol. 227, pp. 375–394. Kluwer Academic Publishers, Dordrecht (1993)

    Google Scholar 

  78. Haber, R.B., Bendsoe, M.P., Jog, C.S.: Perimeter constrained topology optimization of continuum structures. In: IUTAM Symposium on Optimization of Mechanical Systems (Stuttgart, 1995). Solid Mechanics and Its Applications, vol. 43, pp. 113–120. Kluwer Academic Publishers, Dordrecht (1996)

    Google Scholar 

  79. Holmberg, E., Torstenfelt, B., Klarbring, A.: Stress constrained topology optimization. Struct. Multidiscip. Optim. 48, 33–47 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  80. Kawohl, B., Pironneau, O., Tartar, L., Zolésio, J.P.: Optimal shape design. Lectures given at the Joint C.I.M.-C.I.M.E. Summer School held in Tróia, June 1–6, 1998. Edited by A. Cellina and A. Ornelas. Lecture Notes in Mathematics, 1740. Fondazione C.I.M.E.. Springer, Berlin; Centro Internazionale Matematico Estivo (C.I.M.E.), Florence (2000)

    Google Scholar 

  81. Kikuchi, N.: Shape and topology optimization of elastic structures using the homogenization method. In: Nonlinear Mathematical Problems in Industry, I (Iwaki, 1992). GAKUTO International Series Mathematical Sciences and Applications, vol. 1, pp. 129–148. Gakktosho, Tokyo (1993)

    Google Scholar 

  82. Kikuchi, N., Suzuki, K.: Structural optimization of a linearly elastic structure using the homogenization method. In: Composite Media and Homogenization Theory (Trieste, 1990). Progress in Nonlinear Differential Equations and Their Applications, vol. 5, pp. 183–203. Birkhäuser Boston, Boston, MA (1991)

    Google Scholar 

  83. Kohn, R.V., Lipton, R.: Optimal bounds for the effective energy of a mixture of isotropic, incompressible, elastic materials. Arch. Ration. Mech. Anal. 102, 331–350 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  84. Kohn, R.V., Milton, G.W.: On bounding the effective conductivity of anisotropic composites. In: Homogenization and Effective Moduli of Materials and Media (Minneapolis, MN, 1984–1985). IMA Volumes in Mathematics and its Applications, vol. 1, pp. 97–125. Springer, New York (1986)

    Google Scholar 

  85. Lazarov, B.S., Schevenels, M., Sigmund, O.: Topology optimization considering material and geometric uncertainties using stochastic collocation methods. Struct. Multidiscip. Optim. 46, 597–612 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  86. Lipton, R.: On the effective elasticity of a two-dimensional homogenised incompressible elastic composite. Proc. R. Soc. Edinb. Sect. A 110, 45–61 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  87. Lipton, R.: Optimal bounds on effective elastic tensors for orthotropic composites. Proc. R. Soc. Lond. Ser. A 444, 399–10 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  88. Lipton, R.: Optimal fiber configurations for maximum torsional rigidity. Arch. Ration. Mech. Anal. 144, 79–106 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  89. Lipton, R.: Relaxation through homogenization for optimal design problems with gradient constraints. J. Optim. Theory Appl. 114, 27–53 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  90. Lipton, R.: Stress constrained G closure and relaxation of structural design problems. Quart. Appl. Math. 62, 295–321 (2004)

    MathSciNet  MATH  Google Scholar 

  91. Lipton, R., Stuebner, M.: Optimization of composite structures subject to local stress constraints. Comput. Methods Appl. Mech. Eng. 196, 66–75 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  92. Lipton, R., Stuebner, M.: Inverse homogenization and design of microstructure for pointwise stress control. Quart. J. Mech. Appl. Math. 59, 139–161 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  93. Lipton, R., Stuebner, M.: Optimal design of composite structures for strength and stiffness: an inverse homogenization approach. Struct. Multidiscip. Optim. 33, 351–362 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  94. Lurie, K. A.: G-closures of material sets in space-time and perspectives of dynamic control in the coefficients of linear hyperbolic equations. Recent advances in structural modelling and optimization. Control Cybernet. 27, 283–294 (1998)

    MathSciNet  MATH  Google Scholar 

  95. Lurie, K. A.: A stable spatio-temporal G-closure and Gm-closure of a set of isotropic dielectrics with respect to one-dimensional wave propagation. Wave Motion 40, 95–110 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  96. Lurie, K. A.: An Introduction to the Mathematical Theory of Dynamic Materials. Advances in Mechanics and Mathematics, vol. 15. Springer, New York (2007)

    Google Scholar 

  97. Lurie, K.A.: On homogenization of activated laminates in 1D-space and time. ZAMM Z. Angew. Math. Mech. 89, 333–340 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  98. Lurie, K.A., Cherkaev, A.V.: Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion. Proc. R. Soc. Edinb. Sect. A 99, 71–87 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  99. Lurie, K.A., Cherkaev, A.V.: Optimization of properties of multicomponent isotropic composites. J. Optim. Theory Appl. 46, 571–580 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  100. Lurie, K.A., Cherkaev, A.V.: Effective characteristics of composite materials and the optimal design of structural elements. (Russian) Adv. Mech. 9, 3–81 (1986)

    Google Scholar 

  101. Lurie, K.A., Cherkaev, A.V.: Effective characteristics of composite materials and the optimal design of structural elements. In: Topics in the Mathematical Modelling of Composite Materials. Progress in Nonlinear Differential Equations and Their Applications, vol. 31, pp. 175–258. Birkhäuser Boston, Boston, MA (1997)

    Google Scholar 

  102. Lurie, K.A., Cherkaev, A.V., Fedorov, A.V.: On the existence of solutions to some problems of optimal design for bars and plates. J. Optim. Theory Appl. 42, 247–281 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  103. Milton, G.W.: Composite materials with Poisson’s ratios close to −1. J. Mech. Phys. Solids 40, 1105–1137 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  104. Milton, G.W.: A link between sets of tensors stable under lamination and quasiconvexity. Commun. Pure Appl. Math. 47, 959–1003 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  105. Milton, G.W.: The Theory of Composites. Cambridge Monographs on Applied and Computational Mathematics, vol. 6. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  106. Milton, G.W.: On optimizing the properties of hierarchical laminates using Pontryagin’s maximum principle. Multiscale Model. Simul. 3, 658–679 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  107. Milton, G.W., Nesi, V.: Optimal G-closure bounds via stability under lamination. Arch. Ration. Mech. Anal. 150, 191–207 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  108. Murat, F., Tartar, L.: Calculus of variations and homogenization. In: Topics in the Mathematical Modelling of Composite Materials. Progress in Nonlinear Differential Equations and Their Applications, vol. 31, pp. 139–173. Birkhäuser Boston, Boston, MA (1997)

    Google Scholar 

  109. Murat, F., Tartar, L.: On the control of coefficients in partial differential equations. In: Topics in the Mathematical Modelling of Composite Materials. Progress in Nonlinear Differential Equations and Their Applications, vol. 31, pp. 1–8. Birkhäuser Boston, Boston, MA (1997)

    Google Scholar 

  110. Nishiwaki, S., Frecker, M.I., Min, S., Kikuchi, N.: Topology optimization of compliant mechanisms using the homogenization method. Int. J. Numer. Methods Eng. 42, 535–559 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  111. Novotny, A.A., Sokolowski, J.: Topological Derivatives in Shape Optimization. Interaction of Mechanics and Mathematics. Springer, Heidelberg (2013)

    Book  MATH  Google Scholar 

  112. Periago, F.: Optimal design of the time-dependent support of bang-bang type controls for the approximate controllability of the heat equation. J. Optim. Theory Appl. 161, 951–968 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  113. Rodrigues, H.C.: Shape optimal design of elastic bodies using a mixed variational formulation. Comput. Methods Appl. Mech. Eng. 69, 29–44 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  114. Sigmund, O.: Materials with prescribed constitutive parameters: an inverse homogenization problem. Int. J. Solids Struct. 31, 2313–2329 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  115. Sigmund, O.: On the usefulness of non-gradient approaches in topology optimization. Struct. Multidiscip. Optim. 43, 589–596 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  116. Sigmund, O., Maute, K.: Topology optimization approaches. Struct. Multidiscip. Optim. 48, 1031–1055 (2013)

    Article  MathSciNet  Google Scholar 

  117. Sokolowski, J., Zochowski, A.: Shape and topology optimization of distributed parameter systems. Control Cybernet. 42, 217–226 (2013)

    MathSciNet  MATH  Google Scholar 

  118. Sokolowski, J., Zolésio, J. P.: Introduction to shape optimization. In: Shape Sensitivity Analysis. Springer Series in Computational Mathematics, vol. 16. Springer, Berlin (1992)

    Google Scholar 

  119. Sverak, V.: On optimal shape design. J. Math. Pures Appl. 72, 537–551 (1993)

    MathSciNet  MATH  Google Scholar 

  120. Tartar, L.: Estimations fines des coefficients homogénéisés (French) (Fine estimates of homogenized coefficients). In: Ennio De Giorgi colloquium (Paris, 1983). Research Notes in Mathematics, vol. 125, pp. 168–187. Pitman, Boston, MA (1985)

    Google Scholar 

  121. Tartar, L.: Remarks on optimal design problems. Calculus of variations. In: Homogenization and Continuum Mechanics (Marseille, 1993). Series on Advances in Mathematics for Applied Sciences, vol. 18, pp. 279–296. World Scientific Publishers, River Edge, NJ (1994)

    Google Scholar 

  122. Tartar, L.: Remarks on the homogenization method in optimal design methods. In: Homogenization and Applications to Material Sciences (Nice, 1995). GAKUTO International Series Mathematical Sciences and Applications, vol. 9, pp. 393–412. Gakktosho, Tokyo (1995)

    Google Scholar 

  123. Tartar, L.: Estimations of homogenized coefficients. In: Topics in the Mathematical Modelling of Composite Materials. Progress in Nonlinear Differential Equations and Their Applications, vol. 31, pp. 9–20. Birkhäuser Boston, Boston, MA (1997)

    Google Scholar 

  124. Tartar, L.: An introduction to the homogenization method in optimal design. In: Optimal Shape Design (Tróia, 1998). Lecture Notes in Mathematics, vol. 1740, pp. 47–156. Springer, Berlin (2000)

    Google Scholar 

  125. Tartar, L.: The general theory of homogenization. A personalized introduction. Lecture Notes of the Unione Matematica Italiana, vol. 7. Springer/UMI, Berlin/Bologna (2009)

    Google Scholar 

  126. Wang, F., Sigmund, O., Jensen, J.S.: Design of materials with prescribed nonlinear properties. J. Mech. Phys. Solids 69, 156–174 (2014)

    Article  MathSciNet  Google Scholar 

  127. Zowe, J., Kocvara, M., Bendsoe, M.P.: Free Material Optimization via Mathematical Programming. Lectures on Mathematical Programming (ismp97) (Lausanne, 1997). Mathematical Programming, vol. 79, pp. 445–466 (1997)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universidad de Castilla-La Mancha, Ciudad Real, Spain

    Pablo Pedregal

Authors
  1. Pablo Pedregal
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Pedregal, P. (2016). Motivation and Framework. In: Optimal Design through the Sub-Relaxation Method. SEMA SIMAI Springer Series, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-41159-0_1

Download citation

Publish with us

Policies and ethics

Search

Navigation