0
Research Papers: Design Automation

An Inverse Method for Optimizing Elastic Properties Considering Multiple Loading Conditions and Displacement Criteria

[+] Author and Article Information
Danny Smyl

Department of Mechanical Engineering,
Aalto University,
Espoo 02150, Finland
e-mail: danny.smyl@aalto.fi

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received April 2, 2018; final manuscript received June 29, 2018; published online September 7, 2018. Assoc. Editor: Raymundo Arroyave.

J. Mech. Des 140(11), 111411 (Sep 07, 2018) (8 pages) Paper No: MD-18-1283; doi: 10.1115/1.4040788 History: Received April 02, 2018; Revised June 29, 2018

Significant research effort has been devoted to topology optimization (TO) of two- and three-dimensional structural elements subject to various design and loading criteria. While the field of TO has been tremendously successful over the years, literature focusing on the optimization of spatially varying elastic material properties in structures subject to multiple loading states is scarce. In this article, we contribute to the state of the art in material optimization by proposing a numerical regime for optimizing the distribution of the elastic modulus in structural elements subject to multiple loading conditions and design displacement criteria. Such displacement criteria (target displacement fields prescribed by the designer) may result from factors related to structural codes, occupant comfort, proximity of adjacent structures, etc. In this work, we utilize an inverse problem based framework for optimizing the elastic modulus distribution considering N target displacements and imposed forces. This approach is formulated in a straight-forward manner such that it may be applied in a broad suite of design problems with unique geometries, loading conditions, and displacement criteria. To test the approach, a suite of optimization problems are solved to demonstrate solutions considering N = 2 for different geometries and boundary conditions.

FIGURES IN THIS ARTICLE
<>
Copyright © 2018 by ASME
Your Session has timed out. Please sign back in to continue.

References

Zegard, T. , and Paulino, G. H. , 2016, “Bridging Topology Optimization and Additive Manufacturing,” Struct. Multidiscip. Optim., 53(1), pp. 175–192. [CrossRef]
Deaton, J. D. , and Grandhi, R. V. , 2014, “A Survey of Structural and Multidisciplinary Continuum Topology Optimization: Post 2000,” Struct. Multidiscip. Optim., 49(1), pp. 1–38. [CrossRef]
Bendsoe, M. P. , and Sigmund, O. , 2013, Topology Optimization: Theory, Methods, and Applications, Springer Science & Business Media, Berlin.
Sigmund, O. , 2001, “A 99 Line Topology Optimization Code Written in MATLAB,” Struct. Multidiscip. Optim., 21(2), pp. 120–127. [CrossRef]
Zhou, M. , and Rozvany, G. , 1991, “The Coc Algorithm, Part Ii: Topological, Geometrical and Generalized Shape Optimization,” Comput. Methods Appl. Mech. Eng., 89(1–3), pp. 309–336. [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]
Maute, K. , Tkachuk, A. , Wu, J. , Qi, H. J. , Ding, Z. , and Dunn, M. L. , 2015, “Level Set Topology Optimization of Printed Active Composites,” ASME J. Mech. Des., 137(11), p. 111402. [CrossRef]
van Dijk, N. P. , Maute, K. , Langelaar, M. , and Van Keulen, F. , 2013, “Level-Set Methods for Structural Topology Optimization: A Review,” Struct. Multidiscip. Optim., 48(3), pp. 437–472. [CrossRef]
Yamada, T. , Izui, K. , Nishiwaki, S. , and Takezawa, A. , 2010, “A Topology Optimization Method Based on the Level Set Method Incorporating a Fictitious Interface Energy,” Comput. Methods Appl. Mech. Eng., 199(45–48), pp. 2876–2891. [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]
Wang, M. Y. , Wang, X. , and Guo, D. , 2003, “A Level Set Method for Structural Topology Optimization,” Comput. Methods Appl. Mech. Eng., 192(1–2), pp. 227–246. [CrossRef]
Blank, L. , Garcke, H. , Sarbu, L. , Srisupattarawanit, T. , Styles, V. , and Voigt, A. , 2012, “Phase-Field Approaches to Structural Topology Optimization,” Constrained Optimization and Optimal Control for Partial Differential Equations, Springer, Basel, Switzerland, pp. 245–256.
Dedè, L. , Borden, M. J. , and Hughes, T. J. , 2012, “Isogeometric Analysis for Topology Optimization With a Phase Field Model,” Arch. Comput. Methods Eng., 19(3), pp. 427–465. [CrossRef]
Takezawa, A. , Nishiwaki, S. , and Kitamura, M. , 2010, “Shape and Topology Optimization Based on the Phase Field Method and Sensitivity Analysis,” J. Comput. Phys., 229(7), pp. 2697–2718. [CrossRef]
Czarnecki, S. , and Lewiński, T. , 2017, “On Material Design by the Optimal Choice of Young's Modulus Distribution,” Int. J. Solids Struct., 110, pp. 315–331. [CrossRef]
Ben-Tal, A. , Kocvara, M. , Nemirovski, A. , and Zowe, J. , 1999, “Free Material Design via Semidefinite Programming: The Multiload Case With Contact Conditions,” SIAM J. Optim., 9(4), pp. 813–832. [CrossRef]
Zuo, W. , and Saitou, K. , 2017, “Multi-Material Topology Optimization Using Ordered Simp Interpolation,” Struct. Multidiscip. Optim., 55(2), pp. 477–491. [CrossRef]
Sivapuram, R. , Dunning, P. D. , and Kim, H. A. , 2016, “Simultaneous Material and Structural Optimization by Multiscale Topology Optimization,” Struct. Multidiscip. Optim., 54(5), pp. 1267–1281. [CrossRef]
Tavakoli, R. , 2014, “Multimaterial Topology Optimization by Volume Constrained Allen–Cahn System and Regularized Projected Steepest Descent Method,” Comput. Methods Appl. Mech. Eng., 276, pp. 534–565. [CrossRef]
Guo, X. , Zhang, W. , and Zhong, W. , 2014, “Doing Topology Optimization Explicitly and Geometrically-A New Moving Morphable Components Based Framework,” ASME J. Appl. Mech., 81(8), p. 081009. [CrossRef]
Ramani, A. , 2011, “Multi-Material Topology Optimization With Strength Constraints,” Struct. Multidiscip. Optim., 43(5), pp. 597–615. [CrossRef]
Cramer, A. D. , Challis, V. J. , and Roberts, A. P. , 2016, “Microstructure Interpolation for Macroscopic Design,” Struct. Multidiscip. Optim., 53(3), pp. 489–500. [CrossRef]
Bendsoe, M. P. , Guedes, J. , Haber, R. B. , Pedersen, P. , and Taylor, J. , 1994, “An Analytical Model to Predict Optimal Material Properties in the Context of Optimal Structural Design,” ASME J. Appl. Mech., 61(4), pp. 930–937. [CrossRef]
Ringertz, U. , 1993, “On Finding the Optimal Distribution of Material Properties,” Struct. Optim., 5(4), pp. 265–267. [CrossRef]
Maar, B. , and Schulz, V. , 2000, “Interior Point Multigrid Methods for Topology Optimization,” Struct. Multidiscip. Optim., 19(3), pp. 214–224. [CrossRef]
Kočvara, M. , Zibulevsky, M. , and Zowe, J. , 1998, “Mechanical Design Problems With Unilateral Contact,” ESAIM: Math. Modell. Numer. Anal., 32(3), pp. 255–281. [CrossRef]
Zowe, J. , Kočvara, M. , and Bendsøe, M. P. , 1997, “Free Material Optimization Via Mathematical Programming,” Math. Program., 79(1–3), pp. 445–466.
Jarre, F. , Kočvara, M. , and Zowe, J. , 1996, Interior Point Methods for Mechanical Design Problems, Universität Erlangen-Nürnberg. Institut für Angewandte Mathematik, Nuremberg, Germany.
Allaire, G. , Jouve, F. , and Maillot, H. , 2004, “Topology Optimization for Minimum Stress Design With the Homogenization Method,” Struct. Multidiscip. Optim., 28(2–3), pp. 87–98.
Pereira, J. , Fancello, E. , and Barcellos, C. , 2004, “Topology Optimization of Continuum Structures With Material Failure Constraints,” Struct. Multidiscip. Optim., 26(1–2), pp. 50–66. [CrossRef]
Kočvara, M. , and Stingl, M. , 2007, “Free Material Optimization for Stress Constraints,” Struct. Multidiscip. Optim., 33(4–5), pp. 323–335. [CrossRef]
Kočvara, M. , and Stingl, M. , 2007, “On the Solution of Large-Scale Sdp Problems by the Modified Barrier Method Using Iterative Solvers,” Math. Program., 109(2–3), pp. 413–444. [CrossRef]
Stingl, M. , 2006, On the Solution of Nonlinear Semidefinite Programs by Augmented Lagrangian Methods, Shaker Aachen, Nuremberg, Germany.
Deng, S. , and Suresh, K. , 2017, “Stress Constrained Thermo-Elastic Topology Optimization With Varying Temperature Fields via Augmented Topological Sensitivity Based Level-Set,” Struct. Multidiscip. Optim., 56(6), pp. 1413–1427. [CrossRef]
Haslinger, J. , Kočvara, M. , Leugering, G. , and Stingl, M. , 2010, “Multidisciplinary Free Material Optimization,” SIAM J. Appl. Math., 70(7), pp. 2709–2728. [CrossRef]
Picelli, R. , Townsend, S. , Brampton, C. , Norato, J. , and Kim, H. , 2018, “Stress-Based Shape and Topology Optimization With the Level Set Method,” Comput. Methods Appl. Mech. Eng., 329, pp. 1–23. [CrossRef]
Zhang, S. , Gain, A. L. , and Norato, J. A. , 2017, “Stress-Based Topology Optimization With Discrete Geometric Components,” Comput. Methods Appl. Mech. Eng., 325, pp. 1–21. [CrossRef]
Le, C. , Norato, J. , Bruns, T. , Ha, C. , and Tortorelli, D. , 2010, “Stress-Based Topology Optimization for Continua,” Struct. Multidiscip. Optim., 41(4), pp. 605–620. [CrossRef]
Czarnecki, S. , and Wawruch, P. , 2015, “The Emergence of Auxetic Material as a Result of Optimal Isotropic Design,” Phys. Status Solidi (b), 252(7), pp. 1620–1630. [CrossRef]
Czubacki, R. , and Lewiński, T. , 2015, “Topology Optimization of Spatial Continuum Structures Made of Non-Homogeneous Material of Cubic Symmetry,” J. Mech. Mater. Struct., 10(4), pp. 519–535. [CrossRef]
Brańka, A. , Heyes, D. , and Wojciechowski, K. , 2009, “Auxeticity of Cubic Materials,” Phys. Status Solidi (b), 246(9), pp. 2063–2071. [CrossRef]
Brańka, A. , and Wojciechowski, K. , 2008, “Auxeticity of Cubic Materials: The Role of Repulsive Core Interaction,” J. Non-Cryst. Solids, 354(35–39), pp. 4143–4145. [CrossRef]
Coelho, P. G. , Cardoso, J. B. , Fernandes, P. R. , and Rodrigues, H. C. , 2011, “Parallel Computing Techniques Applied to the Simultaneous Design of Structure and Material,” Adv. Eng. Software, 42(5), pp. 219–227. [CrossRef]
Liu, L. , Yan, J. , and Cheng, G. , 2008, “Optimum Structure With Homogeneous Optimum Truss-like Material,” Comput. Struct., 86(13–14), pp. 1417–1425. [CrossRef]
Cheng, G. , Liu, L. , and Yan, J. , 2006, “Optimum Structure With Homogeneous Optimum Truss-like Material,” III European Conference on Computational Mechanics, Lisbon, Portugal, June 5–8, pp. 481–481.
Gosselin, C. , Duballet, R. , Roux, P. , Gaudillière, N. , Dirrenberger, J. , and Morel, P. , 2016, “Large-Scale 3D Printing of Ultra-High Performance Concrete—A New Processing Route for Architects and Builders,” Mater. Des., 100, pp. 102–109. [CrossRef]
Feng, P. , Meng, X. , Chen, J.-F. , and Ye, L. , 2015, “Mechanical Properties of Structures 3D Printed With Cementitious Powders,” Constr. Build. Mater., 93, pp. 486–497. [CrossRef]
Radman, A. , Huang, X. , and Xie, Y. , 2013, “Topology Optimization of Functionally Graded Cellular Materials,” J. Mater. Sci., 48(4), pp. 1503–1510. [CrossRef]
Liu, W. , and DuPont, J. , 2003, “Fabrication of Functionally Graded Tic/Ti Composites by Laser Engineered Net Shaping,” Scr. Mater., 48(9), pp. 1337–1342. [CrossRef]
Wang, X. , Xu, S. , Zhou, S. , Xu, W. , Leary, M. , Choong, P. , Qian, M. , Brandt, M. , and Xie, Y. M. , 2016, “Topological Design and Additive Manufacturing of Porous Metals for Bone Scaffolds and Orthopaedic Implants: A Review,” Biomaterials, 83, pp. 127–141. [CrossRef] [PubMed]
Hollister, S. J. , 2005, “Porous Scaffold Design for Tissue Engineering,” Nat. Mater., 4(7), p. 518. [CrossRef] [PubMed]
Evgrafov, A. , 2005, “The Limits of Porous Materials in the Topology Optimization of Stokes Flows,” Appl. Math. Optim., 52(3), pp. 263–277. [CrossRef]
Yaji, K. , Yamada, T. , Kubo, S. , Izui, K. , and Nishiwaki, S. , 2015, “A Topology Optimization Method for a Coupled Thermal–Fluid Problem Using Level Set Boundary Expressions,” Int. J. Heat Mass Transfer, 81, pp. 878–888. [CrossRef]
Garcke, H. , Hecht, C. , Hinze, M. , and Kahle, C. , 2015, “Numerical Approximation of Phase Field Based Shape and Topology Optimization for Fluids,” SIAM J. Sci. Comput., 37(4), pp. A1846–A1871. [CrossRef]
Guest, J. K. , and Prévost, J. H. , 2006, “Topology Optimization of Creeping Fluid Flows Using a Darcy–Stokes Finite Element,” Int. J. Numer. Methods Eng., 66(3), pp. 461–484. [CrossRef]
Borrvall, T. , and Petersson, J. , 2003, “Topology Optimization of Fluids in Stokes Flow,” Int. J. Numer. Methods Fluids, 41(1), pp. 77–107. [CrossRef]
Pizzolato, A. , Sharma, A. , Maute, K. , Sciacovelli, A. , and Verda, V. , 2017, “Topology Optimization for Heat Transfer Enhancement in Latent Heat Thermal Energy Storage,” Int. J. Heat Mass Transfer, 113, pp. 875–888. [CrossRef]
Jing, G. , Isakari, H. , Matsumoto, T. , Yamada, T. , and Takahashi, T. , 2015, “Level Set-Based Topology Optimization for 2D Heat Conduction Problems Using Bem With Objective Function Defined on Design-Dependent Boundary With Heat Transfer Boundary Condition,” Eng. Anal. Boundary Elem., 61, pp. 61–70. [CrossRef]
Gersborg-Hansen, A. , Bendsøe, M. P. , and Sigmund, O. , 2006, “Topology Optimization of Heat Conduction Problems Using the Finite Volume Method,” Struct. Multidiscip. Optim., 31(4), pp. 251–259. [CrossRef]
Deng, S. , and Suresh, K. , 2017, “Topology Optimization Under Thermo-Elastic Buckling,” Struct. Multidiscip. Optim., 55(5), pp. 1759–1772. [CrossRef]
Deng, S. , and Suresh, K. , 2016, “Multi-Constrained 3D Topology Optimization Via Augmented Topological Level-Set,” Comput. Struct., 170, pp. 1–12. [CrossRef]
Deng, S. , and Suresh, K. , 2015, “Multi-Constrained Topology Optimization Via the Topological Sensitivity,” Struct. Multidiscip. Optim., 51(5), pp. 987–1001. [CrossRef]
Sokół, T. , and Rozvany, G. , 2013, “On the Adaptive Ground Structure Approach for Multi-Load Truss Topology Optimization,” Tenth World Congress on Structural and Multidisciplinary Optimization, Orlando, FL, May 20–24, pp. 19–24.
Alvarez, F. , and Carrasco, M. , 2005, “Minimization of the Expected Compliance as an Alternative Approach to Multiload Truss Optimization,” Struct. Multidiscip. Optim., 29(6), pp. 470–476. [CrossRef]
Guedes, J. M. , Rodrigues, H. C. , and Bendsøe, M. P. , 2003, “A Material Optimization Model to Approximate Energy Bounds for Cellular Materials Under Multiload Conditions,” Struct. Multidiscip. Optim., 25(5–6), pp. 446–452. [CrossRef]
Oberai, A. A. , Gokhale, N. H. , Doyley, M. M. , and Bamber, J. C. , 2004, “Evaluation of the Adjoint Equation Based Algorithm for Elasticity Imaging,” Phys. Med. Biol., 49(13), p. 2955. [CrossRef] [PubMed]
Tikhonov, A. N. , and Glasko, V. B. , 1965, “Use of the Regularization Method in Non-Linear Problems,” USSR Comput. Math. Math. Phys., 5(3), pp. 93–107. [CrossRef]
Vauhkonen, M. , Vadasz, D. , Karjalainen, P. A. , Somersalo, E. , and Kaipio, J. P. , 1998, “Tikhonov Regularization and Prior Information in Electrical Impedance Tomography,” IEEE Trans. Med. Imaging, 17(2), pp. 285–293. [CrossRef] [PubMed]
Mueller, J. L. , and Siltanen, S. , 2012, Linear and Nonlinear Inverse Problems With Practical Applications, Society of Indian Automobile Manufactures, Auckland, NZ.
Kaipio, J. , and Somersalo, E. , 2006, Statistical and Computational Inverse Problems, Vol. 160, Springer Science & Business Media, New York.
Surana, K. S. , and Reddy, J. , 2016, The Finite Element Method for Boundary Value Problems: Mathematics and Computations, CRC Press, Boca Raton, FL.
Kim, S.-J. , Koh, K. , Lustig, M. , Boyd, S. , and Gorinevsky, D. , 2007, “An Interior-Point Method for Large-Scale l1-Regularized Least Squares,” IEEE J. Sel. Topics Signal Process., 1(4), pp. 606–617. [CrossRef]
Vauhkonen, M. , 1997, “Electrical Impedance Tomography and Prior Information,” PhD Thesis, University of Kuopio, Kuopio, Finland.
An, H.-B. , Wen, J. , and Feng, T. , 2011, “On Finite Difference Approximation of a Matrix-Vector Product in the Jacobian-Free Newton-Krylov Method,” J. Comput. Appl. Math., 236(6), pp. 1399–1409. [CrossRef]
Smyl, D. , Bossuyt, S. , and Liu, D. , Submitted, 2018, “Stacked Elasticity Imaging Approach for Visualizing Defects in the Presence of Background Inhomogeneity,” J. Eng. Mech., (accepted).
Liu, D. , Kolehmainen, V. , Siltanen, S. , Laukkanen, A. , and Seppänen, A. , 2015, “Estimation of Conductivity Changes in a Region of Interest With Electrical Impedance Tomography,” Inverse Probl. Imaging, 9(1), pp. 211–229. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Example 1: Schematic detailing the external forces and structural elements deformed by the design displacement targets (a) Δ1, a cubic polynomial translating the x grid coordinates and (b) Δ2, a pure axial translation along the y-axis

Grahic Jump Location
Fig. 2

Optimized distribution of EΨ for Example 1, (a) EΨ plotted atop U1 and (b) EΨ plotted atop U2. The right-hand side color bar indicates the magnitude of inhomogeneous EΨ in each image.

Grahic Jump Location
Fig. 3

Example 2: Schematic detailing the external forces and structural elements deformed by the design displacement targets (a) Δ1, a quadratic polynomial translating the x grid coordinates and (b) Δ2 a cubic polynomial translating the x grid coordinates

Grahic Jump Location
Fig. 4

Optimized distribution of EΨ for Example 2, (a) EΨ plotted atop U1 and (b) EΨ plotted atop U2. The right-hand side color bar indicates the magnitude of inhomogeneous EΨ in each image.

Grahic Jump Location
Fig. 5

Example 3: Schematic detailing the external forces and structural elements deformed by the design displacement targets Δ1 and Δ2 for (a) a coarse triangular discretization and (b) a refined triangular discretization

Grahic Jump Location
Fig. 6

Optimized distributions of EΨ for Example 3 for (a) the coarse mesh with Ec > 0, (b) the coarse mesh with Ec > 50, (c) the refined mesh with Ec > 0, and (d) the refined mesh with Ec > 50

Grahic Jump Location
Fig. 7

Minimization curves for the cost function Ψ as a function of the iteration number k considering the lower constraint Ec and the discretization

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