Open Access
Issue
ESAIM: M2AN
Volume 58, Number 3, May-June 2024
Page(s) 1137 - 1151
DOI https://doi.org/10.1051/m2an/2024036
Published online 26 June 2024
  1. J. Aboudi, Mechanics of Composite Materials: Studies in Applied Mechanics. Elsevier, Amsterdam (1991). [Google Scholar]
  2. G. Allaire, Explicit lamination parameters for three-dimensional shape optimization. Control Cybern. 23 (1994) 309–326. [Google Scholar]
  3. G. Allaire, Shape optimization by the homogenization method, in Applied Mathematical Sciences. Springer (2001). [Google Scholar]
  4. G. Allaire and R.V. Kohn, Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions. Q. Appl. Math. 51 (1993) 675–699. [CrossRef] [Google Scholar]
  5. G. Allaire and R.V. Kohn, Optimal design for minimum weight and compliance in plane stress using extremal microstructures. Eur. J. Mech. A/Solids 12 (1993) 839–878. [Google Scholar]
  6. G. Allaire, E. Bonnetier, G. Francfort and F. Jouve, Shape optimization by the homogenization method. Numer. Math. 76 (1997) 27–68. [Google Scholar]
  7. N. Antonić and N. Balenović, Optimal design for plates and relaxation. Math. Commun. 4 (1999) 111–119. [MathSciNet] [Google Scholar]
  8. N. Antonić and N. Balenović, Homogenisation and optimal design for plates. Z. Math. Mech. 80 (2000) 757–758. [Google Scholar]
  9. M.P. Bendsøe, Optimization of Structural Topology, Shape, and Material. Vol. 414. Springer (1995). [CrossRef] [Google Scholar]
  10. S.C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods. Vol. 3. Springer (2008). [Google Scholar]
  11. K. Burazin, Unique solutions of multiple-state optimal design problems on an annulus. J. Optim. Theory App. 177 (2018) 329–344. [CrossRef] [Google Scholar]
  12. K. Burazin and I. Crnjac, Application of explicit energy bounds in optimization of 3D elastic structures. Optim. Eng. (2023). DOI: 10.1007/s11081-023-09840-w. [Google Scholar]
  13. K. Burazin and J. Jankov, Explicit bounds on the energy and moduli of a composite elastic plate, under review. [Google Scholar]
  14. K. Burazin and J. Jankov, On the effective properties of composite elastic plate. J. Math. Anal. App. 495 (2021) 124696. [CrossRef] [Google Scholar]
  15. K. Burazin and M. Vrdoljak, Exact solutions in optimal design problems for stationary diffusion equation. Acta Appl. Math. 161 (2019) 71–88. [CrossRef] [MathSciNet] [Google Scholar]
  16. K. Burazin, J. Jankov and M. Vrdoljak, Homogenization of elastic plate equation. Math. Modell. Anal. 23 (2018) 190–204. [CrossRef] [Google Scholar]
  17. K. Burazin, I. Crnjac and M. Vrdoljak, Optimality criteria method in 2D linearized elasticity problems. Appl. Numer. Math. 160 (2021) 192–204. [CrossRef] [MathSciNet] [Google Scholar]
  18. K. Burazin, I. Crnjac and M. Vrdoljak, Explicit Hashin-Shtrikman bounds in 3D linearized elasticity, under review. [Google Scholar]
  19. J. Casado-Díaz, C. Castro, M. Luna-Laynez and E. Zuazua, Numerical approximation of a one-dimensional elliptic optimal design problem. Multiscale Model. Simul. 9 (2011) 1181–1216. [CrossRef] [MathSciNet] [Google Scholar]
  20. R. Christensen, Mechanics of Composite Materials. John Wiley, New York (1979). [Google Scholar]
  21. G. Dzierżanowski, Stress energy minimization as a tool in the material layout design of shallow shells. Int. J. Solids Struct. 49 (2012) 1343–1354. [CrossRef] [Google Scholar]
  22. G.A. Francfort and F. Murat, Homogenization and optimal bounds in linear elasticity. Arch. Ration. Mech. Anal. 94 (1986) 307–334. [CrossRef] [Google Scholar]
  23. L.V. Gibiansky and A.V. Cherkaev, Design of composite plates of extremal rigidity. Ioffe Physicotechnical Institute preprint (1984). [Google Scholar]
  24. L.V. Gibiansky and A.V. Cherkaev, Microstructures of composites of extremal rigidity and exact bounds of the associated energy density. Ioffe Physicotechnical Institute preprint (1987). [Google Scholar]
  25. L.V. Gibiansky and A.V. Cherkaev, Design of composite plates of extremal rigidity, in Topics in the Mathematical Modelling of Composite Materials. Modern Birkh¨auser Classics. Birkh¨auser (2018) 95–137. [CrossRef] [Google Scholar]
  26. Z. Hashin and Z. Shtrikman, A variational approach to the theory of the elastic behaviour of multiphase materials. J. Mech. Phys. Solids 11 (1963) 127–140. [CrossRef] [MathSciNet] [Google Scholar]
  27. F. Hecht, New development in FreeFem++. J. Numer. Math. 20 (2012) 251–266. [Google Scholar]
  28. V.V. Jikov, S.M. Kozlov and O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals. Springer Science & Business Media (2012). [Google Scholar]
  29. R. Kohn and G. Strang, Optimal design and relaxation of variational problems I–III. Commun. Pure Appl. Math. 39 (1986) 113–137, 139–182, 353–377. [CrossRef] [Google Scholar]
  30. K.A. Lurie and A.V. Cherkaev, Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion. Proc. R. Soc. Edinburgh Sect. A: Math. 99 (1984) 71–87. [CrossRef] [Google Scholar]
  31. K.A. Lurie and A.V. Cherkaev, Effective characteristics of composite materials and the optimal design of structural elements. Uspekhi Mekhaniki 9 (1986) 3–81. [Google Scholar]
  32. K.A. Lurie, A.V. Cherkaev and A.V. Fedorov, Regularization of optimal design problems for bars and plates I–II. J. Optim. Theory App. 37 (1982) 499–522. [CrossRef] [Google Scholar]
  33. G. Milton, The Theory of Composites. Cambridge University Press (2001). [Google Scholar]
  34. G. Milton, M. Briane and D. Harutyunyan, On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials. Math. Mech. Complex Syst. 5 (2017) 41–94. [CrossRef] [MathSciNet] [Google Scholar]
  35. L.S.D. Morley, The triangular equilibrium element in the solution of plate bending problems. Aeron. Q. 19 (1968) 149–169. [CrossRef] [Google Scholar]
  36. F. Murat and L. Tartar, H-convergence, in séminaire d’analyse fonctionnelle et numérique de l’université d’alger. Lecture Notes (1978). [Google Scholar]
  37. F. Murat and L. Tartar, Calcul des variations et homogénéisation, in: Les méthodes de l’homogenisation théorie et applications en physique, bréausans-nappe. 57 (1985) 319–369. [Google Scholar]
  38. A.A. Novotny, R.A. Feijóo, C. Padra and E. Taroco, Topological derivative for linear elastic plate bending problems. Control Cybern. 34 (2005) 339–361. [Google Scholar]
  39. G.I. Rozvany, Structural Design and Optimality Criteria. Kluwer Academic Publisers (1989). [Google Scholar]
  40. L. Tartar, Remarks on homogenization, in Homogenization and Effective Moduli of Materials and Media. Springer (1986) 228–246. [CrossRef] [Google Scholar]
  41. L. Tartar, The General Theory of Homogenization: A Personalized Introduction. Vol. 7. Springer Science & Business Media (2009). [Google Scholar]
  42. M. Vrdoljak, Classical optimal design in two-phase conductivity problems. SIAM J. Control Optim. 54 (2016) 2020–2035. [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you