398
9 Topologic Optimization of Vibrations of Size-Dependent Beams
35. Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Rat. Mech.
Anal. 11, 415–448 (1962)
36. Rajabi, F., Ramezani, S.: A nonlinear microbeam model based on strain gradient elasticity
theory with surface energy. Arch. Appl. Mech. 82, 363–376 (2012)
37. Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414
(1962)
38. Xie, Y., Steven, G.: A simple evolutionary procedure for structural optimization. Comput.
Struct. 49, 885–896 (1993)
39. Eringen, A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972)
40. Aifantis, E.C.: Strain gradient interpretation of size effects. Int. J. Fract. 95, 299–314 (1999)
41. Gurtin, M.E., Weissmuller, J., Larche, F.: The general theory of curved deformable inter-faces
in solids at equilibrium. Philos. Mag. A 78, 1093–1109 (1998)
42. Arbind, A., Reddy, J.N.: Nonlinear analysis of functionally graded microstructure-dependent
beams. Compos. Struct. 98, 272–281 (2013)
43. Arbind, A., Reddy, J.N., Srinivasa, A.R.: Modified couple stress-based third-order theory for
nonlinear analysis of functionally graded beams Lat. Am. J. Sol. Struct. 11, 459–487 (2014)
44. Bendsoe, M., Kikuchi, N.: Generating optimal topologies in optimal design using a homogenization method. Comput. Meth. Appl. Mech. Eng. 71, 197–224 (1988)
45. Bensoussan, A., Lions, J., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures.
North-Holland, Amsterdam (1978)
46. Sanchez-Palencia, E.: Non-Homogeneous Media and Vibration Theory. Springer, Berlin
(1980)
47. Bendsoe, M.: Optimal shape design as a material distribution problem. Struct. Optimiz. 1,
193–202 (1989)
48. Allaire, G., Jouve, F., Toader, A.: Structural optimization using sensitivity analysis and a
level-set method. J. Comput. Phys. 194, 363–393 (2004)
49. Belytschko, T., Xiao, S., Parimi, C.: Topology optimization with implicit functions and regularization. Int. J. Num. Meth. Eng. 57(8), 1177–1196 (2003)
50. Sethian, J., Wiegmann, A.: Structural boundary design via level set and immersed interface
methods. J. Comput. Phys. 163(2), 489–528 (2000)
51. Wang, M., Wang, X., Guo, D.: A level set method for structural topology optimization.
Comput. Meth. Appl. Eng. 192, 227–246 (2003)
52. Zhou, M., Rozvany, G.: Dcoc: an optimality criteria method for large systems, part i: theory.
Struct. Optimiz. 5, 12–29 (1993)
53. Svanberg, K., Werme, M.: Topology optimization by neighbour-hood search method based
on efficient sensitivity calculations. Int. J. Numer. Meth. Eng. 67, 1670–1699 (2006)
54. Svanberg, K.: The method of moving asymptotes - a new method for structural optimization.
Int. J. Num. Meth. Eng. 24, 359–373 (1987)
55. Gibiansky, L., Sigmund, O.: Multiphase elastic composites with extremal bulk modulus. J.
Mech. Phys. Sol. 48, 461–498 (2000)
56. Larsen, U., Sigmund, O., Bouwstra, S.: Design and fabrication of compliant micro mechanisms and structures with negative Poisson’s ratio. J. Microelectromech. Sys. 6(2), 99–106
(1997)
57. Sigmund, O.: Materials with prescribed constitutive parameters: an inverse homogenization
problem. Int. J. Sol. Struct. 31, 2313–2329 (1994)
58. Sigmund, O.: Tailoring materials with prescribed elastic properties. Mech. Mater. 20, 351–368
(1999)
59. Sigmund, O.: A new class of extremal composites. J. Mech. Phys. Sol. 48, 397–428 (2000)
60. Sigmund, O., Torquato, S.: Design of materials with extreme thermal expansion using a threephase topology optimization method. J. Mech. Phys. Sol. 45, 1037–1067 (1997)
61. Jonsmann, J., Sigmund, O., Bouwstra, S.: Compliant thermal microactuators. Sens. Actuat.
76, 463–469 (1999)
62. Li, Q., Steven, G., Xie, Y., Qurein, O.: Shape and topology design for heat conduction by
evolutionary structural optimization. Int. J. Heat Mass Trans. 42, 3361–3371 (1999)
9 Topologic Optimization of Vibrations of Size-Dependent Beams
35. Mindlin, R.D., Tiersten, H.F.: Effects of couple-stresses in linear elasticity. Arch. Rat. Mech.
Anal. 11, 415–448 (1962)
36. Rajabi, F., Ramezani, S.: A nonlinear microbeam model based on strain gradient elasticity
theory with surface energy. Arch. Appl. Mech. 82, 363–376 (2012)
37. Toupin, R.A.: Elastic materials with couple-stresses. Arch. Ration. Mech. Anal. 11, 385–414
(1962)
38. Xie, Y., Steven, G.: A simple evolutionary procedure for structural optimization. Comput.
Struct. 49, 885–896 (1993)
39. Eringen, A.C.: Nonlocal polar elastic continua. Int. J. Eng. Sci. 10, 1–16 (1972)
40. Aifantis, E.C.: Strain gradient interpretation of size effects. Int. J. Fract. 95, 299–314 (1999)
41. Gurtin, M.E., Weissmuller, J., Larche, F.: The general theory of curved deformable inter-faces
in solids at equilibrium. Philos. Mag. A 78, 1093–1109 (1998)
42. Arbind, A., Reddy, J.N.: Nonlinear analysis of functionally graded microstructure-dependent
beams. Compos. Struct. 98, 272–281 (2013)
43. Arbind, A., Reddy, J.N., Srinivasa, A.R.: Modified couple stress-based third-order theory for
nonlinear analysis of functionally graded beams Lat. Am. J. Sol. Struct. 11, 459–487 (2014)
44. Bendsoe, M., Kikuchi, N.: Generating optimal topologies in optimal design using a homogenization method. Comput. Meth. Appl. Mech. Eng. 71, 197–224 (1988)
45. Bensoussan, A., Lions, J., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures.
North-Holland, Amsterdam (1978)
46. Sanchez-Palencia, E.: Non-Homogeneous Media and Vibration Theory. Springer, Berlin
(1980)
47. Bendsoe, M.: Optimal shape design as a material distribution problem. Struct. Optimiz. 1,
193–202 (1989)
48. Allaire, G., Jouve, F., Toader, A.: Structural optimization using sensitivity analysis and a
level-set method. J. Comput. Phys. 194, 363–393 (2004)
49. Belytschko, T., Xiao, S., Parimi, C.: Topology optimization with implicit functions and regularization. Int. J. Num. Meth. Eng. 57(8), 1177–1196 (2003)
50. Sethian, J., Wiegmann, A.: Structural boundary design via level set and immersed interface
methods. J. Comput. Phys. 163(2), 489–528 (2000)
51. Wang, M., Wang, X., Guo, D.: A level set method for structural topology optimization.
Comput. Meth. Appl. Eng. 192, 227–246 (2003)
52. Zhou, M., Rozvany, G.: Dcoc: an optimality criteria method for large systems, part i: theory.
Struct. Optimiz. 5, 12–29 (1993)
53. Svanberg, K., Werme, M.: Topology optimization by neighbour-hood search method based
on efficient sensitivity calculations. Int. J. Numer. Meth. Eng. 67, 1670–1699 (2006)
54. Svanberg, K.: The method of moving asymptotes - a new method for structural optimization.
Int. J. Num. Meth. Eng. 24, 359–373 (1987)
55. Gibiansky, L., Sigmund, O.: Multiphase elastic composites with extremal bulk modulus. J.
Mech. Phys. Sol. 48, 461–498 (2000)
56. Larsen, U., Sigmund, O., Bouwstra, S.: Design and fabrication of compliant micro mechanisms and structures with negative Poisson’s ratio. J. Microelectromech. Sys. 6(2), 99–106
(1997)
57. Sigmund, O.: Materials with prescribed constitutive parameters: an inverse homogenization
problem. Int. J. Sol. Struct. 31, 2313–2329 (1994)
58. Sigmund, O.: Tailoring materials with prescribed elastic properties. Mech. Mater. 20, 351–368
(1999)
59. Sigmund, O.: A new class of extremal composites. J. Mech. Phys. Sol. 48, 397–428 (2000)
60. Sigmund, O., Torquato, S.: Design of materials with extreme thermal expansion using a threephase topology optimization method. J. Mech. Phys. Sol. 45, 1037–1067 (1997)
61. Jonsmann, J., Sigmund, O., Bouwstra, S.: Compliant thermal microactuators. Sens. Actuat.
76, 463–469 (1999)
62. Li, Q., Steven, G., Xie, Y., Qurein, O.: Shape and topology design for heat conduction by
evolutionary structural optimization. Int. J. Heat Mass Trans. 42, 3361–3371 (1999)