Finite element analysis of free material optimization problem. (English) Zbl 1099.65112

Summary: Free material optimization solves an important problem of structural engineering, i.e. to find the stiffest structure for given loads and boundary conditions. Its mathematical formulation leads to a saddle-point problem. It can be solved numerically by the finite element method. The convergence of the finite element method can be proved if the spaces involved satisfy suitable approximation assumptions. An example of a finite-element discretization is included.


65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
74G15 Numerical approximation of solutions of equilibrium problems in solid mechanics
74P05 Compliance or weight optimization in solid mechanics
Full Text: DOI EuDML Link


[1] G. Allaire, S. Aubry, and F. Jouve: Eigenfrequency optimization in optimal design. Comput. Methods Appl. Mech. Engrg. 190 (2001), 3565-3579. · Zbl 1004.74063
[2] A. Ben-Tal, M. Ko?vara, A. Nemirovski, and J. Zowe: Free material optimization via semidefinite programming: the multi-load case with contact conditions. SIAM J. Optim. 9 (1999), 813-832. · Zbl 0969.74051
[3] M. P. Bends?e: Optimization of Structural Topology, Shape and Material. Springer-Verlag, Heidelberg-Berlin, 1995.
[4] M. P. Bends?e, A. D?az: Optimization of material properties for Mindlin plate design. Structural Optimization 6 (1993), 268-270.
[5] M. P. Bends?e, A. D?az, R. Lipton, and J. E. Taylor: Optimal design of material properties and material distribution for multiple loading conditions. Internat. J. Numer. Methods Engrg. 38 (1995), 1149-1170. · Zbl 0822.73047
[6] M. P. Bends?e, J. M. Guades, R. B. Haber, P. Pedersen and J. E. Taylor: An analytical model to predict optimal material properties in the context of optimal structural design. J. Appl. Mech. 61 (1994), 930-937. · Zbl 0831.73036
[7] M. P. Bends?e, J. M. Guades, S. Plaxton, and J. E. Taylor: Optimization of structures and material properties for solids composed of softening material. Int. J. Solids Struct. 33 (1995), 1179-1813. · Zbl 0919.73114
[8] M. Brdi?ka: Mechanics of Continuum. N?SAV, Praha, 1959. (In Czech.)
[9] J. Cea: Lectures on Optimization. Springer-Verlag, Berlin-Heidelberg-New York, 1978. · Zbl 0409.90050
[10] P. G. Ciarlet: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam-New York-Oxford, 1978.
[11] I. Ekeland, R. Temam: Convex Analysis and Variational Problems. North-Holland, Amsterdam-Oxford, 1976.
[12] L. C. Evans, R. F. Garpiery: Measure Theory and Fine Properties of Functions. CRC Press, London, 1992.
[13] J. Haslinger: Finite element analysis for unilateral problems with obstacles on the boundary. Apl. Mat. 22 (1977), 180-188. · Zbl 0434.65083
[14] J. Haslinger, P. Neittaanm?ki: Finite Element Approximation for Optimal Shape, Material, and Topology Design. John Wiley & Sons, Chichester, 1996.
[15] M. Ko?vara, J. Zowe: Free Material Optimization. Doc. Math. J. DMV, Extra Volume ICM III (1998), 707-716. · Zbl 0905.73043
[16] J. Ne?as: Les m?thodes directes en th?orie des ?quations elliptiques. Academia, Praha, 1967.
[17] J. Ne?as, I. Hlav??ek: Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction. Elsevier, Amsterdam-Oxford-New York, 1981.
[18] J. Petersson, J. Haslinger: An approximation theory for optimum sheet in unilateral contact. Quart. Appl. Math. 56 (1998), [pp309-325.] · Zbl 0960.74051
[19] U. Ringertz: On finding the optimal distribution of material properties. Structural Optimization 5 (1993), 265-267.
[20] J. Zowe, M. Ko?vara, and M. P. Bends?e: Free material optimization via mathematical programming. Math. Program. Series B 79 (1997), 445-466. · Zbl 0886.90145
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.