Structural parameter optimization of linear elastic systems

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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

Abstract
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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.

Keywords: linear elastic systems, Structural optimization, convex programming..

Mathematics Subject Classification: Primary: 74C05, 74K10, 74K20; Secondary: 90C2.

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).
[2]	P. G. Ciarlet, "Mathematical Elasticity; Volume I: Three-Dimensional Elasticity,", Studies in Mathematics and Its Applications, 20 (1997).
[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.
[4]	M. C. Delfour and J.-P. Zolésio, Shape identification via metrics constructed from the oriented distance function,, Control Cybernet, 34 (2005), 137.
[5]	M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries; Analysis, Differential Calculus, and Optimization,", Advances in Design and Control, 4 (2001).
[6]	N. Dunford and J. T. Schwartz, "Linear Operators, II Spectral Theory; Self Adjoint Operators in Hilbert Space,", Interscience Pub., (1963).
[7]	R. T. Haftka, Z. Gürdal and M. P. Kamat, "Elements of Structural Optimization,", Kluwer Academic Publishers, (1990).
[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.
[9]	C. T. Kelley, "Iterative Methods for Optimization,", SIAM Frontiers in Applied Mathematics, 18 (1999).
[10]	J.-L. Lions and E. Magenes, "Probl\`emes aux limites nonhomog\`enes,", [Nonhomogeneous Boundary Value Problems], I, II (1968).
[11]	J. E. Marsden and T. J. R. Hughes, "Mathematical Foundations of Elasticity,", Dover Publications, (1994).
[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.
[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.
[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.
[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.
[16]	P. Villagio, "Qualitative Methods in Elasticity,", Noordhoff Intl. Pub., (1977). doi: 10.1002/zamm.19790590921.
[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.

show all references

References:

[1]	G. Allaire, "Shape Optimization by the Homogenization Method,", Applied Mathematical Sciences, 146 (2002).
[2]	P. G. Ciarlet, "Mathematical Elasticity; Volume I: Three-Dimensional Elasticity,", Studies in Mathematics and Its Applications, 20 (1997).
[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.
[4]	M. C. Delfour and J.-P. Zolésio, Shape identification via metrics constructed from the oriented distance function,, Control Cybernet, 34 (2005), 137.
[5]	M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries; Analysis, Differential Calculus, and Optimization,", Advances in Design and Control, 4 (2001).
[6]	N. Dunford and J. T. Schwartz, "Linear Operators, II Spectral Theory; Self Adjoint Operators in Hilbert Space,", Interscience Pub., (1963).
[7]	R. T. Haftka, Z. Gürdal and M. P. Kamat, "Elements of Structural Optimization,", Kluwer Academic Publishers, (1990).
[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.
[9]	C. T. Kelley, "Iterative Methods for Optimization,", SIAM Frontiers in Applied Mathematics, 18 (1999).
[10]	J.-L. Lions and E. Magenes, "Probl\`emes aux limites nonhomog\`enes,", [Nonhomogeneous Boundary Value Problems], I, II (1968).
[11]	J. E. Marsden and T. J. R. Hughes, "Mathematical Foundations of Elasticity,", Dover Publications, (1994).
[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.
[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.
[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.
[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.
[16]	P. Villagio, "Qualitative Methods in Elasticity,", Noordhoff Intl. Pub., (1977). doi: 10.1002/zamm.19790590921.
[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.

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