American Institute of Mathematical Sciences

Advanced
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

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).   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

show all references

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

[1]

Adil Bagirov, Sona Taheri, Soodabeh Asadi. A difference of convex optimization algorithm for piecewise linear regression. Journal of Industrial & 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 & 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 & 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 & 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 & 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 & 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 & 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 & Management Optimization, 2012, 8 (1) : 189-227. doi: 10.3934/jimo.2012.8.189
[9]

Kewei Zhang. On equality of relaxations for linear elastic strains. Communications on Pure & Applied Analysis, 2002, 1 (4) : 565-573. doi: 10.3934/cpaa.2002.1.565
[10]

Charles Fefferman. Interpolation by linear programming I. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 477-492. doi: 10.3934/dcds.2011.30.477
[11]

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 & Heterogeneous Media, 2013, 8 (3) : 783-802. doi: 10.3934/nhm.2013.8.783
[12]

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 & Management Optimization, 2020, 16 (2) : 965-990. doi: 10.3934/jimo.2018188
[13]

Nicoletta Del Buono, Cinzia Elia, Roberto Garrappa, Alessandro Pugliese. Preface: "Structural Dynamical Systems: Computational aspects". Discrete & Continuous Dynamical Systems - B, 2018, 23 (7) : i-i. doi: 10.3934/dcdsb.201807i
[14]

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
[15]

Lorena Bociu, Steven Derochers, Daniel Toundykov. Feedback stabilization of a linear hydro-elastic system. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1107-1132. doi: 10.3934/dcdsb.2018144
[16]

Anulekha Dhara, Aparna Mehra. Conjugate duality for generalized convex optimization problems. Journal of Industrial & Management Optimization, 2007, 3 (3) : 415-427. doi: 10.3934/jimo.2007.3.415
[17]

Igor Griva, Roman A. Polyak. Proximal point nonlinear rescaling method for convex optimization. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 283-299. doi: 10.3934/naco.2011.1.283
[18]

Nobuko Sagara, Masao Fukushima. trust region method for nonsmooth convex optimization. Journal of Industrial & Management Optimization, 2005, 1 (2) : 171-180. doi: 10.3934/jimo.2005.1.171
[19]

Yong Wang, Wanquan Liu, Guanglu Zhou. An efficient algorithm for non-convex sparse optimization. Journal of Industrial & Management Optimization, 2019, 15 (4) : 2009-2021. doi: 10.3934/jimo.2018134
[20]

Zhongliang Deng, Enwen Hu. Error minimization with global optimization for difference of convex functions. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1027-1033. doi: 10.3934/dcdss.2019070

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences