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optimal design; concentrated forces and moments; continuous load; cost functional; $H^2$-norm of the deflection curve; $L^2$-norm of the normal stress; primary and dual formulations; elastic beam; elastic foundation; existence; convergence

References:

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[2] D. Begis R. Glowinski: **Application de la méthode des éléments finis à l'approximation d'un problème de domaine optimal. Méthodes de résolution des problèmes approchés**. Applied Mathematics & Optimization, 2 (1975), 130-169. DOI 10.1007/BF01447854 | MR 0443372

[3] R. Courant D. Hilbert: **Methoden der matematischen Physik I**. Springer-Verlag 1968, 3. Auflage. MR 0344038

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[6] I. Hlaváček I. Bock J. Lovíšek: **Optimal control of a variational inequality with applications to structural analysis. Optimal design of a beam with unilateral supports**. Applied Mathematics & Optimization, 1984, 111-143. DOI 10.1007/BF01442173 | MR 0743922

[7] J. Chleboun: **Optimal Design of an Elastic Beam on an Elastic Basis**. Thesis (Czech). MFF UK Praha, 1984.

[8] J. Nečas I. Hlaváček: **Mathematical Theory of Elastic and Elasto-Plastic Bodies: An Introduction**. Elsevier, Amsterdam, 1981. MR 0600655

[9] S. Timoshenko: **Strength of Materials, Part II**. D. Van Nostrand Company, Inc. New York 1945. (Czech translation, Technicko-vědecké nakladatelství, Praha 1951.)