# American Institute of Mathematical Sciences

September  2011, 10(5): 1517-1536. doi: 10.3934/cpaa.2011.10.1517

## Structural parameter optimization of linear elastic systems

 1 418 McBryde Hall, Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123

Received  July 2009 Revised  May 2010 Published  April 2011

The design of elastic structures to optimize strength and economy of materials is a fundamental problem in structural engineering and related areas of applied mathematics. In this article we explore a finite dimensional framework for approximate solution of such design problems based on linear elasticity with a range of elastic coefficients assumed available as design parameters. Solution methods for related optimization problems based on the matrix trace norm are suggested and analyzed, providing existence and uniqueness theorems. Results of computations for sample problems are presented and compared with parallel results in the literature based on other approaches.
Citation: David Russell. Structural parameter optimization of linear elastic systems. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1517-1536. doi: 10.3934/cpaa.2011.10.1517
##### References:
 [1] G. Allaire, "Shape Optimization by the Homogenization Method,", Applied Mathematical Sciences, 146 (2002).   Google Scholar [2] P. G. Ciarlet, "Mathematical Elasticity; Volume I: Three-Dimensional Elasticity,", Studies in Mathematics and Its Applications, 20 (1997).   Google Scholar [3] M. C. Delfour and J.-P. Zolésio, Evolution equations for shapes and geometries,, J. Evol. Equ., 6 (2006), 399.  doi: 10.1007/s00028-006-0257-8.  Google Scholar [4] M. C. Delfour and J.-P. Zolésio, Shape identification via metrics constructed from the oriented distance function,, Control Cybernet, 34 (2005), 137.   Google Scholar [5] M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries; Analysis, Differential Calculus, and Optimization,", Advances in Design and Control, 4 (2001).   Google Scholar [6] N. Dunford and J. T. Schwartz, "Linear Operators, II Spectral Theory; Self Adjoint Operators in Hilbert Space,", Interscience Pub., (1963).   Google Scholar [7] R. T. Haftka, Z. Gürdal and M. P. Kamat, "Elements of Structural Optimization,", Kluwer Academic Publishers, (1990).   Google Scholar [8] J.-Y. Jaffray and J.-Ch. Pomerol, A direct proof of the Kuhn-Tucker necessary optimality theorem for convex and affine inequalities,, SIAM Rev., 31 (1989), 671.  doi: 10.1137/1031131.  Google Scholar [9] C. T. Kelley, "Iterative Methods for Optimization,", SIAM Frontiers in Applied Mathematics, 18 (1999).   Google Scholar [10] J.-L. Lions and E. Magenes, "Probl\emes aux limites nonhomog\enes,", [Nonhomogeneous Boundary Value Problems], I, II (1968).   Google Scholar [11] J. E. Marsden and T. J. R. Hughes, "Mathematical Foundations of Elasticity,", Dover Publications, (1994).   Google Scholar [12] M. Renardy and D. L. Russell, Formability of linear elastic structures with volume - type actuation,, Arch. Rat. Mech. & Anal., 149 (1999), 97.  doi: 10.1007/s002050050169.  Google Scholar [13] J. B. Rosen, The gradient projection method for nonlinear programming; I. Linear constraints,, J. Soc. Indust. Appl. Math., 8 (1960), 181.  doi: 10.1137/0108011.  Google Scholar [14] J. B. Rosen, The gradient projection method for nonlinear programming. II. Nonlinear constraints,, J. Soc. Indust. Appl. Math., 9 (1961), 514.  doi: 10.1137/0109044.  Google Scholar [15] D. L. Russell, The Betti reciprocity principle and the normal boundary component control problem for linear elastic systems,, J. Global Optim., 40 (2008), 575.  doi: 10.1007/s10898-006-9116-y.  Google Scholar [16] P. Villagio, "Qualitative Methods in Elasticity,", Noordhoff Intl. Pub., (1977).  doi: 10.1002/zamm.19790590921.  Google Scholar [17] S. Wang, E. de Sturler and G. H. Paulino, Large-scale topology optimization using preconditioned Krylov 3 subspace methods with recycling,, Internat. J. Numer. Methods Engrg., 69 (2007), 2441.  doi: 10.1002/nme.1798.  Google Scholar

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##### References:
 [1] G. Allaire, "Shape Optimization by the Homogenization Method,", Applied Mathematical Sciences, 146 (2002).   Google Scholar [2] P. G. Ciarlet, "Mathematical Elasticity; Volume I: Three-Dimensional Elasticity,", Studies in Mathematics and Its Applications, 20 (1997).   Google Scholar [3] M. C. Delfour and J.-P. Zolésio, Evolution equations for shapes and geometries,, J. Evol. Equ., 6 (2006), 399.  doi: 10.1007/s00028-006-0257-8.  Google Scholar [4] M. C. Delfour and J.-P. Zolésio, Shape identification via metrics constructed from the oriented distance function,, Control Cybernet, 34 (2005), 137.   Google Scholar [5] M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries; Analysis, Differential Calculus, and Optimization,", Advances in Design and Control, 4 (2001).   Google Scholar [6] N. Dunford and J. T. Schwartz, "Linear Operators, II Spectral Theory; Self Adjoint Operators in Hilbert Space,", Interscience Pub., (1963).   Google Scholar [7] R. T. Haftka, Z. Gürdal and M. P. Kamat, "Elements of Structural Optimization,", Kluwer Academic Publishers, (1990).   Google Scholar [8] J.-Y. Jaffray and J.-Ch. Pomerol, A direct proof of the Kuhn-Tucker necessary optimality theorem for convex and affine inequalities,, SIAM Rev., 31 (1989), 671.  doi: 10.1137/1031131.  Google Scholar [9] C. T. Kelley, "Iterative Methods for Optimization,", SIAM Frontiers in Applied Mathematics, 18 (1999).   Google Scholar [10] J.-L. Lions and E. Magenes, "Probl\emes aux limites nonhomog\enes,", [Nonhomogeneous Boundary Value Problems], I, II (1968).   Google Scholar [11] J. E. Marsden and T. J. R. Hughes, "Mathematical Foundations of Elasticity,", Dover Publications, (1994).   Google Scholar [12] M. Renardy and D. L. Russell, Formability of linear elastic structures with volume - type actuation,, Arch. Rat. Mech. & Anal., 149 (1999), 97.  doi: 10.1007/s002050050169.  Google Scholar [13] J. B. Rosen, The gradient projection method for nonlinear programming; I. Linear constraints,, J. Soc. Indust. Appl. Math., 8 (1960), 181.  doi: 10.1137/0108011.  Google Scholar [14] J. B. Rosen, The gradient projection method for nonlinear programming. II. Nonlinear constraints,, J. Soc. Indust. Appl. Math., 9 (1961), 514.  doi: 10.1137/0109044.  Google Scholar [15] D. L. Russell, The Betti reciprocity principle and the normal boundary component control problem for linear elastic systems,, J. Global Optim., 40 (2008), 575.  doi: 10.1007/s10898-006-9116-y.  Google Scholar [16] P. Villagio, "Qualitative Methods in Elasticity,", Noordhoff Intl. Pub., (1977).  doi: 10.1002/zamm.19790590921.  Google Scholar [17] S. Wang, E. de Sturler and G. H. Paulino, Large-scale topology optimization using preconditioned Krylov 3 subspace methods with recycling,, Internat. J. Numer. Methods Engrg., 69 (2007), 2441.  doi: 10.1002/nme.1798.  Google Scholar
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