0
Research Papers

Topology Optimization Under Stress Relaxation Effect Using Internal Element Connectivity Parameterization

[+] Author and Article Information
Meisam Takalloozadeh

Engineering Department,
Shiraz University,
Zand Blvd., Shiraz 7134851156, Iran
e-mail: takalloozadeh@shirazu.ac.ir

Gil ho Yoon

Engineering Center 202-2,
Hanyang University,
222 Wangsimni-ro,
Seoul, Seong dong-gu 133-791, Korea
e-mails: ghy@hanyang.ac.kr,
gilho.yoon@gmail.com

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received April 17, 2018; final manuscript received September 20, 2018; published online January 7, 2019. Assoc. Editor: Kyung Choi.

J. Comput. Nonlinear Dynam 14(2), 021006 (Jan 07, 2019) (11 pages) Paper No: CND-18-1171; doi: 10.1115/1.4041578 History: Received April 17, 2018; Revised September 20, 2018

The creep phenomenon has enormous effect on the stress and displacement distribution in the structures. Redistribution of the stress field is one of these effects which is called stress relaxation. The importance of stress relaxation in the design of structures is increasing due to engineering applications especially in high temperature. However, this phenomenon has remained absent from the structural optimization studies. In the present study, the effect of stress relaxation due to high temperature creep is considered in topology optimization (TO). Internal element connectivity parameterization (I-ECP) method is utilized for performing TO. This method is shown to be effective to overcome numerical instabilities in nonlinear problems. Time-dependent adjoint sensitivity formulation is implemented for I-ECP including creep effect. Several benchmark problems are solved, and the optimum layouts obtained by linear and nonlinear methods are compared to show the efficiency of the proposed method and to show the effect of stress relaxation on the optimum layout.

Copyright © 2019 by ASME
Your Session has timed out. Please sign back in to continue.

References

Doghri, I. , 2013, Mechanics of Deformable Solids: Linear, Nonlinear, Analytical and Computational Aspects, Springer Science & Business Media, Berlin.
Kawamoto, A. , 2009, “ Stabilization of Geometrically Nonlinear Topology Optimization by the Levenberg–Marquardt Method,” Struct. Multidiscip. Optim., 37(4), pp. 429–433. [CrossRef]
Bendsøe, M. P. , and Kikuchi, N. , 1988, “ Generating Optimal Topologies in Structural Design Using a Homogenization Method,” Comput. Methods Appl. Mech. Eng., 71(2), pp. 197–224. [CrossRef]
Suzuki, K. , and Kikuchi, N. , 1991, “ A Homogenization Method for Shape and Topology Optimization,” Comput. Methods Appl. Mech. Eng., 93(3), pp. 291–318. [CrossRef]
Swan, C. C. , and Kosaka, I. , 1997, “ Voigt–Reuss Topology Optimization for Structures With Linear Elastic Material Behaviours,” Int. J. Numer. Methods Eng., 40(16), pp. 3033–3057. [CrossRef]
Bendsøe, M. P. , 1989, “ Optimal Shape Design as a Material Distribution Problem,” Struct. Optim., 1(4), pp. 193–202. [CrossRef]
Jung, D. , and Gea, H. C. , 2004, “ Topology Optimization of Nonlinear Structures,” Finite Elem. Anal. Des., 40(11), pp. 1417–1427. [CrossRef]
Buhl, T. , Pedersen, C. B. , and Sigmund, O. , 2000, “ Stiffness Design of Geometrically Nonlinear Structures Using Topology Optimization,” Struct. Multidiscip. Optim., 19(2), pp. 93–104. [CrossRef]
Lee, H.-A. , and Park, G.-J. , 2012, “ Topology Optimization for Structures With Nonlinear Behavior Using the Equivalent Static Loads Method,” ASME J. Mech. Des., 134(3), p. 031004. [CrossRef]
Bruns, T. E. , and Tortorelli, D. A. , 2003, “ An Element Removal and Reintroduction Strategy for the Topology Optimization of Structures and Compliant Mechanisms,” Int. J. Numer. Methods Eng., 57(10), pp. 1413–1430. [CrossRef]
Huang, X. , and Xie, Y. , 2008, “ Topology Optimization of Nonlinear Structures Under Displacement Loading,” Eng. Struct., 30(7), pp. 2057–2068. [CrossRef]
Ahmad, Z. , Sultan, T. , Zoppi, M. , Abid, M. , and Park, G. J. , 2017, “ Nonlinear Response Topology Optimization Using Equivalent Static Loads—Case Studies,” Eng. Optim., 49(2), pp. 252–268. [CrossRef]
Nakshatrala, P. B. , Tortorelli, D. A. , and Nakshatrala, K. B. , 2013, “ Nonlinear Structural Design Using Multiscale Topology Optimization—Part I: Static Formulation,” Comput. Methods Appl. Mech. Eng., 261–262, pp. 167–176. [CrossRef]
Xia, L. , and Breitkopf, P. , 2014, “ A Reduced Multiscale Model for Nonlinear Structural Topology Optimization,” Comput. Methods Appl. Mech. Eng., 280, pp. 117–134. [CrossRef]
Xia, L. , and Breitkopf, P. , 2016, “ Recent Advances on Topology Optimization of Multiscale Nonlinear Structures,” Arch. Comput. Methods Eng., 2(2), pp. 227–249. [CrossRef]
Luo, Q. T. , and Tong, L. Y. , 2016, “ An Algorithm for Eradicating the Effects of Void Elements on Structural Topology Optimization for Nonlinear Compliance,” Struct. Multidiscip. Optim., 53(4), pp. 695–714. [CrossRef]
Zhang, X. , Ramos, A. S. , and Paulino, G. H. , 2017, “ Material Nonlinear Topology Optimization Using the Ground Structure Method With a Discrete Filtering Scheme,” Struct. Multidiscip. Optim., 55(6), pp. 2045–2072. [CrossRef]
Osher, S. J. , and Santosa, F. , 2001, “ Level Set Methods for Optimization Problems Involving Geometry and Constraints—I: Frequencies of a Two-Density Inhomogeneous Drum,” J. Comput. Phys., 171(1), pp. 272–288. [CrossRef]
Allaire, G. , Jouve, F. , and Toader, A.-M. , 2002, “ A Level-Set Method for Shape Optimization,” C. R. Math., 334(12), pp. 1125–1130. [CrossRef]
Allaire, G. , Jouve, F. , and Toader, A.-M. , 2004, “ Structural Optimization Using Sensitivity Analysis and a Level-Set Method,” J. Comput. Phys., 194(1), pp. 363–393. [CrossRef]
Ha, S.-H. , and Cho, S. , 2008, “ Level Set Based Topological Shape Optimization of Geometrically Nonlinear Structures Using Unstructured Mesh,” Comput. Struct., 86(13–14), pp. 1447–1455. [CrossRef]
Yoon, G. H. , and Kim, Y. Y. , 2005, “ Element Connectivity Parameterization for Topology Optimization of Geometrically Nonlinear Structures,” Int. J. Solids Struct., 42(7), pp. 1983–2009. [CrossRef]
Yoon, G. H. , and Kim, Y. Y. , 2007, “ Topology Optimization of Material‐Nonlinear Continuum Structures by the Element Connectivity Parameterization,” Int. J. Numer. Methods Eng., 69(10), pp. 2196–2218. [CrossRef]
Yoon, G. H. , and Kim, Y. Y. , 2005, “ The Element Connectivity Parameterization Formulation for the Topology Design Optimization of Multiphysics Systems,” Int. J. Numer. Methods Eng., 64(12), pp. 1649–1677. [CrossRef]
Langelaar, M. , Yoon, G. , Kim, Y. , and Van Keulen, F. , 2011, “ Topology Optimization of Planar Shape Memory Alloy Thermal Actuators Using Element Connectivity Parameterization,” Int. J. Numer. Methods Eng., 88(9), pp. 817–840.
Yoon, G. H. , Kim, Y. Y. , Langelaar, M. , and van Keulen, F. , 2008, “ Theoretical Aspects of the Internal Element Connectivity Parameterization Approach for Topology Optimization,” Int. J. Numer. Methods Eng., 76(6), pp. 775–797. [CrossRef]
Yoon, G. H. , Joung, Y. S. , and Kim, Y. Y. , 2007, “ Optimal Layout Design of Three-Dimensional Geometrically Non-Linear Structures Using the Element Connectivity Parameterization Method,” Int. J. Numer. Methods Eng., 69(6), pp. 1278–1304. [CrossRef]
Yoon, G. H. , 2010, “ Maximizing the Fundamental Eigenfrequency of Geometrically Nonlinear Structures by Topology Optimization Based on Element Connectivity Parameterization,” Comput. Struct., 88(1–2), pp. 120–133. [CrossRef]
van Dijk, N. P. , Yoon, G. H. , van Keulen, F. , and Langelaar, M. , 2010, “ A Level-Set Based Topology Optimization Using the Element Connectivity Parameterization Method,” Struct. Multidiscip. Optim., 42(2), pp. 269–282. [CrossRef]
Moon, S. J. , and Yoon, G. H. , 2013, “ A Newly Developed qp-Relaxation Method for Element Connectivity Parameterization to Achieve Stress-Based Topology Optimization for Geometrically Nonlinear Structures,” Comput. Methods Appl. Mech. Eng., 265, pp. 226–241. [CrossRef]
Richard, B. H. , and Eslami, M. R. , 2008, Thermal Stresses: Advanced Theory and Applications, Springer, Dordrecht, The Netherlands.
Betten, J. , 2008, Creep Mechanics, Springer Science & Business Media, Berlin.
Chen, W. J. , and Liu, S. T. , 2014, “ Topology Optimization of Microstructures of Viscoelastic Damping Materials for a Prescribed Shear Modulus,” Struct. Multidiscip. Optim., 50(2), pp. 287–296. [CrossRef]
Amir, O. , and Sigmund, O. , 2013, “ Reinforcement Layout Design for Concrete Structures Based on Continuum Damage and Truss Topology Optimization,” Struct. Multidiscip. Optim., 47(2), pp. 157–174. [CrossRef]
James, K. A. , and Waisman, H. , 2014, “ Failure Mitigation in Optimal Topology Design Using a Coupled Nonlinear Continuum Damage Model,” Comput. Methods Appl. Mech. Eng., 268, pp. 614–631. [CrossRef]
James, K. A. , and Waisman, H. , 2015, “ Topology Optimization of Viscoelastic Structures Using a Time-Dependent Adjoint Method,” Comput. Methods Appl. Mech. Eng., 285, pp. 166–187. [CrossRef]
Weertman, J. , 1955, “ Theory of Steady‐State Creep Based on Dislocation Climb,” J. Appl. Phys., 26(10), pp. 1213–1217. [CrossRef]
Penny, R. K. , and Marriott, D. L. , 2012, Design for Creep, Chapman & Hall, Boca Raton, FL.
Tegart, W. M. , and Sherby, O. D. , 1958, “ Activation Energies for High Temperature Creep of Polycrystalline Zinc,” Philos. Mag., 3(35), pp. 1287–1296. [CrossRef]
Hayhurst, D. , 1972, “ Creep Rupture Under Multi-Axial States of Stress,” J. Mech. Phys. Solids, 20(6), pp. 381–382. [CrossRef]
Bruggi, M. , 2008, “ On an Alternative Approach to Stress Constraints Relaxation in Topology Optimization,” Struct. Multidiscip. Optim., 36(2), pp. 125–141. [CrossRef]
Sigmund, O. , 2001, “ A 99 Line Topology Optimization Code Written in Matlab,” Struct. Multidiscip. Optim., 21(2), pp. 120–127. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Deformations due to creep effect in a wood structure (Braunschweig, Germany)

Grahic Jump Location
Fig. 2

Creep in structural TO problems category

Grahic Jump Location
Fig. 3

An element in I-ECP

Grahic Jump Location
Fig. 4

Typical creep curve

Grahic Jump Location
Fig. 5

Schematic diagram for nonlinear analyzing algorithm

Grahic Jump Location
Fig. 6

Cantilever beam problem

Grahic Jump Location
Fig. 7

von Mises stress field for (a) β=0 (σmax=13.2827 N/m2) and (b) β=10(σmax=10.7875 N/m2)

Grahic Jump Location
Fig. 8

Displacement—Load history at load point (β=0 and β=10)

Grahic Jump Location
Fig. 9

(a) Simple plate with 15 elements and (b) the optimum layout for β=0

Grahic Jump Location
Fig. 10

Oscillating in the optimum layout for β=500

Grahic Jump Location
Fig. 11

The relation between stress and xe using different (a) penalty and (b) sdiag

Grahic Jump Location
Fig. 12

(a) The relation between stress and xe for σ=(1/xe)EeBDuin and (b) the optimum layout for β=500

Grahic Jump Location
Fig. 13

Optimum layouts and stress distributions: (a) optimum layout for β=0, C=31.9620 J, (b) von Mises stress (N/m2),σmax=0.2563 N/m2, (c) optimum layout for β=200, C=38.0078 J, (d) von Mises stress (N/m2), σmax=0.1642 N/m2, (e) optimum layout for β=300, C=41.4702 J, and (f) von Mises stress (N/m2), σmax=0.1593 N/m2

Grahic Jump Location
Fig. 14

(a) Design domain and the boundary condition of beam example, (b) optimum layout for β=0 (C=24.3057 J), and (c) optimum layout for β=50 (C=27.0388 J)

Grahic Jump Location
Fig. 15

Von Mises stress in high stress elements

Grahic Jump Location
Fig. 16

Optimum layouts under β=50 for the maximum displacement umax=24.3  (obtained volume=0.455×V0)

Grahic Jump Location
Fig. 17

(a) Design domain and the boundary condition for cantilever box shape, (b) optimum layout for β=0 (C=1.7041 J) and (b)β=50(C=1.7774 J)

Tables

Errata

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In