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Hoppe, Ronald H. W.; Petrova, Svetozara I.

Optimal shape design in biomimetics based on homogenization and adaptivity. (English) Zbl 1106.74390

Math. Comput. Simul. 65, No. 3, 257-272 (2004).
Summary: Optimal shape design of microstructured materials has recently attracted a great deal of attention in materials science. The shape and the topology of the microstructure have a significant impact on the macroscopic properties. The paper is devoted to the shape optimization of new biomorphic microcellular silicon carbide ceramics produced from natural wood by biotemplating. This is a novel technology in the field of biomimetics which features a material synthesis from biologically grown materials into ceramic composites by fast high-temperature processing. We are interested in finding the best material-and-shape combination in order to achieve the optimal prespecified performance of the composite material. The computation of the effective material properties is carried out using the homogenization method. Adaptive mesh-refinement technique based on the computation of recovered stresses is applied in the microstructure to find the homogenized elasticity coefficients. Numerical results show the reliability of the implemented a posteriori error estimators.

Cited in 5 Documents

MSC:

74Q05 Homogenization in equilibrium problems of solid mechanics
90C30 Nonlinear programming
65K10 Numerical optimization and variational techniques
49Q10 Optimization of shapes other than minimal surfaces
74B05 Classical linear elasticity

Keywords:

Shape optimization; Primal–dual method; Homogenization; Elasticity problem; Adaptive refinement
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References:

[1] N. Bakhvalov, G. Panasenko, Averaging Processes in Periodic Media, Nauka, Moscow, 1984.; N. Bakhvalov, G. Panasenko, Averaging Processes in Periodic Media, Nauka, Moscow, 1984. · Zbl 0607.73009
[2] M.P. Bendsøe, Optimization of Structural Topology, Shape, and Material, Springer, Berlin, Germany, 1995.; M.P. Bendsøe, Optimization of Structural Topology, Shape, and Material, Springer, Berlin, Germany, 1995. · Zbl 0822.73001
[3] A. Bensoussan, J.L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Elsevier, Amsterdam, 1978.; A. Bensoussan, J.L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Elsevier, Amsterdam, 1978. · Zbl 0404.35001
[4] A.V. Fiacco, G.P. McCormick, Nonlinear Programming. Sequential Unconstrained Minimization Techniques, Wiley, New York, 1968 (republished by SIAM, Philadelphia, Pennsylvania, 1990).; A.V. Fiacco, G.P. McCormick, Nonlinear Programming. Sequential Unconstrained Minimization Techniques, Wiley, New York, 1968 (republished by SIAM, Philadelphia, Pennsylvania, 1990). · Zbl 0193.18805
[5] V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer, 1994.; V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer, 1994. · Zbl 0801.35001
[6] J. Sokolowski, J.-P. Zolesio, Introduction to shape optimization, Springer Series in Computational Mathematics, vol. 16, Springer, 1992.; J. Sokolowski, J.-P. Zolesio, Introduction to shape optimization, Springer Series in Computational Mathematics, vol. 16, Springer, 1992. · Zbl 0761.73003
[7] Bendsøe, M. P.; Kikuchi, N., Generating optimal topologies in structural design using a homogenization method, Comput. Methods Appl. Mech. Eng., 71, 197-224 (1988) · Zbl 0671.73065
[8] Byrd, R. H.; Hribar, M. E.; Nocedal, J., An interior point algorithm for large scale nonlinear programming, SIAM J. Optim., 9, 4, 877-900 (1999) · Zbl 0957.65057
[9] Byrne, C. E.; Nagle, D. C., Carbonization of wood for advanced materials applications, Carbon, 35, 2, 259-266 (1997)
[10] Carstensen, C.; Funken, S. A., Averaging technique for FE—a posteriori error control in elasticity. Part I. Conforming FEM, Comput. Methods Appl. Mech. Eng., 190, 2483-2498 (2001) · Zbl 0981.74063
[11] El-Bakry, A. S.; Tapia, R. A.; Tsuchiya, T.; Zhang, Y., On the formulation and theory of the Newton interior-point method for nonlinear programming, J. Optim. Theory Appl., 89, 507-541 (1996) · Zbl 0851.90115
[12] K. Eriksson, D. Estep, P. Hansbo, C. Johnson, Introduction to adaptive methods for differential equations, Acta Num. (1995), 105-158.; K. Eriksson, D. Estep, P. Hansbo, C. Johnson, Introduction to adaptive methods for differential equations, Acta Num. (1995), 105-158. · Zbl 0829.65122
[13] Forsgren, A.; Gill, P. E.; Wright, M. H., Interior methods for nonlinear optimization, SIAM Rev., 44, 4, 525-597 (2002) · Zbl 1028.90060
[14] D.M. Gay, M.L. Overton, M.H. Wright, A primal-dual interior method for nonconvex nonlinear programming, in: Y. Yuan (Eds.), Advances in Nonlinear Programming, Kluwer, Dordrecht, 1998, pp. 31-56.; D.M. Gay, M.L. Overton, M.H. Wright, A primal-dual interior method for nonconvex nonlinear programming, in: Y. Yuan (Eds.), Advances in Nonlinear Programming, Kluwer, Dordrecht, 1998, pp. 31-56. · Zbl 0908.90236
[15] Greil, P.; Lifka, T.; Kaindl, A., Biomorphic cellular silicon carbide ceramics from wood. I. Processing and microstructure. II. Mechanical properties, J. Eur. Ceram. Soc., 18, 1961-1973 (1998), and 1975-1983
[16] Hoppe, R. H.W.; Petrova, S. I.; Schulz, V., Primal-dual Newton-type interior-point method for topology optimization, J. Optim. Theory Appl., 114, 3, 545-571 (2002) · Zbl 1026.90108
[17] Kikuchi, N.; Chung, K. Y.; Torigaki, T.; Taylor, J. E., Adaptive finite element methods for shape optimization of linearly elastic structures, Comput. Methods Appl. Mech. Eng., 57, 67-91 (1986) · Zbl 0578.73081
[18] Ota, T.; Takahashi, M.; Hibi, T.; Ozawa, M.; Suzuki, S.; Hikichi, Y.; Suzuki, H., Biomimetic process for producing SiC wood, J. Am. Ceram. Soc., 78, 3409-3411 (1995)
[19] Sieber, H.; Hoffmann, C.; Kaindl, A.; Greil, P., Biomorphic cellular ceramics, Adv. Eng. Mater., 2, 105-109 (2000)
[20] Suzuki, K.; Kikuchi, N., A homogenization method for shape and topology optimization, Comput. Methods Appl. Mech. Eng., 93, 291-318 (1991) · Zbl 0850.73195
[21] Wittum, G., On the convergence of multigrid methods with transforming smoothers. Theory with applications to the Navier-Stokes equations, Num. Math., 57, 15-38 (1989) · Zbl 0698.65061
[22] Zienkiewicz, O. C.; Zhu, J. Z., A simple error estimator and adaptive procedure for practical engineering analysis, Int. J. Num. Methods Eng., 24, 337-357 (1987) · Zbl 0602.73063
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