Structural parameter optimization of linear elastic systems

Previous Article
Fluctuation and extinction dynamics in host-microparasite systems
CPAA Home
This Issue
Next Article
A particle method and numerical study of a quasilinear partial differential equation

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

Structural parameter optimization of linear elastic systems

David Russell ^1,

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

Received July 2009 Revised May 2010 Published April 2011

Abstract
Related Papers

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 and 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, Springer-Verlag, New York, 2002.
[2]	P. G. Ciarlet, "Mathematical Elasticity; Volume I: Three-Dimensional Elasticity," Studies in Mathematics and Its Applications, 20, Elsevier Inc., 1997.
[3]	M. C. Delfour and J.-P. Zolésio, Evolution equations for shapes and geometries, J. Evol. Equ., 6 (2006), 399-417. 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), 137-164.
[5]	M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries; Analysis, Differential Calculus, and Optimization," Advances in Design and Control, 4, SIAM Publications, Philadelphia, PA, 2001.
[6]	N. Dunford and J. T. Schwartz, "Linear Operators, II Spectral Theory; Self Adjoint Operators in Hilbert Space," Interscience Pub., New York, 1963.
[7]	R. T. Haftka, Z. Gürdal and M. P. Kamat, "Elements of Structural Optimization," Kluwer Academic Publishers, Dordrecht, Boston, London, 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), 671-674. 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, Dunod, Paris, 1968 (French).
[11]	J. E. Marsden and T. J. R. Hughes, "Mathematical Foundations of Elasticity," Dover Publications, Inc., New York, 1994.
[12]	M. Renardy and D. L. Russell, Formability of linear elastic structures with volume - type actuation, Arch. Rat. Mech. & Anal., 149 (1999), 97-122. 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), 181-217. 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), 514-532. 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), 575-588. doi: 10.1007/s10898-006-9116-y.
[16]	P. Villagio, "Qualitative Methods in Elasticity," Noordhoff Intl. Pub., Leyden, 1977 (reprinted by Kluwer). 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), 2441-2468. doi: 10.1002/nme.1798.

show all references

References:

[1]	G. Allaire, "Shape Optimization by the Homogenization Method," Applied Mathematical Sciences, 146, Springer-Verlag, New York, 2002.
[2]	P. G. Ciarlet, "Mathematical Elasticity; Volume I: Three-Dimensional Elasticity," Studies in Mathematics and Its Applications, 20, Elsevier Inc., 1997.
[3]	M. C. Delfour and J.-P. Zolésio, Evolution equations for shapes and geometries, J. Evol. Equ., 6 (2006), 399-417. 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), 137-164.
[5]	M. C. Delfour and J.-P. Zolésio, "Shapes and Geometries; Analysis, Differential Calculus, and Optimization," Advances in Design and Control, 4, SIAM Publications, Philadelphia, PA, 2001.
[6]	N. Dunford and J. T. Schwartz, "Linear Operators, II Spectral Theory; Self Adjoint Operators in Hilbert Space," Interscience Pub., New York, 1963.
[7]	R. T. Haftka, Z. Gürdal and M. P. Kamat, "Elements of Structural Optimization," Kluwer Academic Publishers, Dordrecht, Boston, London, 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), 671-674. 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, Dunod, Paris, 1968 (French).
[11]	J. E. Marsden and T. J. R. Hughes, "Mathematical Foundations of Elasticity," Dover Publications, Inc., New York, 1994.
[12]	M. Renardy and D. L. Russell, Formability of linear elastic structures with volume - type actuation, Arch. Rat. Mech. & Anal., 149 (1999), 97-122. 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), 181-217. 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), 514-532. 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), 575-588. doi: 10.1007/s10898-006-9116-y.
[16]	P. Villagio, "Qualitative Methods in Elasticity," Noordhoff Intl. Pub., Leyden, 1977 (reprinted by Kluwer). 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), 2441-2468. doi: 10.1002/nme.1798.

[1]	Adil Bagirov, Sona Taheri, Soodabeh Asadi. A difference of convex optimization algorithm for piecewise linear regression. Journal of Industrial and Management Optimization, 2019, 15 (2) : 909-932. doi: 10.3934/jimo.2018077
[2]	Radu Ioan Boţ, Anca Grad, Gert Wanka. Sequential characterization of solutions in convex composite programming and applications to vector optimization. Journal of Industrial and Management Optimization, 2008, 4 (4) : 767-782. doi: 10.3934/jimo.2008.4.767
[3]	David L. Russell. Trace properties of certain damped linear elastic systems. Evolution Equations and Control Theory, 2013, 2 (4) : 711-721. doi: 10.3934/eect.2013.2.711
[4]	Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial and Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
[5]	Andrea Braides, Margherita Solci, Enrico Vitali. A derivation of linear elastic energies from pair-interaction atomistic systems. Networks and Heterogeneous Media, 2007, 2 (3) : 551-567. doi: 10.3934/nhm.2007.2.551
[6]	David L. Russell. Coefficient identification and fault detection in linear elastic systems; one dimensional problems. Mathematical Control and Related Fields, 2011, 1 (3) : 391-411. doi: 10.3934/mcrf.2011.1.391
[7]	Yuan Shen, Wenxing Zhang, Bingsheng He. Relaxed augmented Lagrangian-based proximal point algorithms for convex optimization with linear constraints. Journal of Industrial and Management Optimization, 2014, 10 (3) : 743-759. doi: 10.3934/jimo.2014.10.743
[8]	Murat Adivar, Shu-Cherng Fang. Convex optimization on mixed domains. Journal of Industrial and Management Optimization, 2012, 8 (1) : 189-227. doi: 10.3934/jimo.2012.8.189
[9]	Charles Fefferman. Interpolation by linear programming I. Discrete and Continuous Dynamical Systems, 2011, 30 (2) : 477-492. doi: 10.3934/dcds.2011.30.477
[10]	Kewei Zhang. On equality of relaxations for linear elastic strains. Communications on Pure and Applied Analysis, 2002, 1 (4) : 565-573. doi: 10.3934/cpaa.2002.1.565
[11]	Haibo Jin, Long Hai, Xiaoliang Tang. An optimal maintenance strategy for multi-state systems based on a system linear integral equation and dynamic programming. Journal of Industrial and Management Optimization, 2020, 16 (2) : 965-990. doi: 10.3934/jimo.2018188
[12]	Edward S. Canepa, Alexandre M. Bayen, Christian G. Claudel. Spoofing cyber attack detection in probe-based traffic monitoring systems using mixed integer linear programming. Networks and Heterogeneous Media, 2013, 8 (3) : 783-802. doi: 10.3934/nhm.2013.8.783
[13]	Nicoletta Del Buono, Cinzia Elia, Roberto Garrappa, Alessandro Pugliese. Preface: "Structural Dynamical Systems: Computational aspects". Discrete and Continuous Dynamical Systems - B, 2018, 23 (7) : i-i. doi: 10.3934/dcdsb.201807i
[14]	Jonathan Meddaugh. Shadowing as a structural property of the space of dynamical systems. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2439-2451. doi: 10.3934/dcds.2021197
[15]	Fasma Diele, Marina Popolizio, Alessandro Pugliese, Giuseppe Vacca, Ivonne Sgura. Preface special issue on structural dynamical systems. Journal of Computational Dynamics, 2022, 9 (2) : ⅰ-ⅱ. doi: 10.3934/jcd.2022013
[16]	Jean Creignou, Hervé Diet. Linear programming bounds for unitary codes. Advances in Mathematics of Communications, 2010, 4 (3) : 323-344. doi: 10.3934/amc.2010.4.323
[17]	Lorena Bociu, Steven Derochers, Daniel Toundykov. Feedback stabilization of a linear hydro-elastic system. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1107-1132. doi: 10.3934/dcdsb.2018144
[18]	Yi Xu, Wenyu Sun. A filter successive linear programming method for nonlinear semidefinite programming problems. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 193-206. doi: 10.3934/naco.2012.2.193
[19]	Anulekha Dhara, Aparna Mehra. Conjugate duality for generalized convex optimization problems. Journal of Industrial and Management Optimization, 2007, 3 (3) : 415-427. doi: 10.3934/jimo.2007.3.415
[20]	Zhongliang Deng, Enwen Hu. Error minimization with global optimization for difference of convex functions. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1027-1033. doi: 10.3934/dcdss.2019070

2021 Impact Factor: 1.273

Tools

Metrics

Other articles
by authors

on AIMS
David Russell
on Google Scholar
David Russell

[Back to Top]