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Research Papers: Design Automation

Experimental Validation of Surrogate Models for Predicting the Draping of Physical Interpolating Surfaces

[+] Author and Article Information
Esben Toke Christensen

Department of Mechanical and
Manufacturing Engineering,
Aalborg University,
Fibigerstræde 16,
Aalborg East 9220, Denmark
e-mail: esben@m-tech.aau.dk

Erik Lund

Department of Mechanical and
Manufacturing Engineering,
Aalborg University,
Fibigerstræde 16,
Aalborg East 9220, Denmark
e-mail: el@m-tech.aau.dk

Esben Lindgaard

Department of Mechanical and
Manufacturing Engineering,
Aalborg University,
Fibigerstræde 16,
Aalborg East 9220, Denmark
e-mail: elo@m-tech.aau.dk

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received November 5, 2016; final manuscript received September 5, 2017; published online November 9, 2017. Assoc. Editor: Gary Wang.

J. Mech. Des 140(1), 011401 (Nov 09, 2017) (14 pages) Paper No: MD-16-1742; doi: 10.1115/1.4038073 History: Received November 05, 2016; Revised September 05, 2017

This paper concerns the experimental validation of two surrogate models through a benchmark study involving two different variable shape mould prototype systems. The surrogate models in question are different methods based on kriging and proper orthogonal decomposition (POD), which were developed in previous work. Measurement data used in the benchmark study are obtained using digital image correlation (DIC). For determining the variable shape mould configurations used for the training, and test sets used in the study, sampling is carried out using a novel constrained nested orthogonal-maximin Latin hypercube approach. This sampling method allows for generating a space filling and high-quality sample plan that respects mechanical constraints of the variable shape mould systems. Through the benchmark study, it is found that mechanical freeplay in the modeled system is severely detrimental to the performance of the studied surrogate models. By comparing surrogate model performance for the two variable shape mould systems, and through a numerical study involving simple finite element models, the underlying cause of this effect is explained. It is concluded that for a variable shape mould prototype system with a small degree of mechanical freeplay, the benchmarked surrogate models perform very well.

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References

Christensen, E. T. , Forrester, A. I. J. , Lund, E. , and Lindgaard, E. , 2017, “ Developing Meta-Models for Fast and Accurate Prediction of the Draping of Physical Surfaces,” ASME J. Comput. Inf. Sci. Eng., submitted.
Munro, C. , and Walczyk, D. F. , 2007, “ Reconfigurable Pin-Type Tooling: A Survey of Prior Art and Reduction to Practice,” ASME J. Manuf. Sci. Eng., 129(3), pp. 551–565. [CrossRef]
Liu, C. , Li, M. , and Fu, W. , 2008, “ Principles and Apparatus of Multi-Point Forming for Sheet Metal,” Int. J. Adv. Manuf. Technol., 35(11–12), pp. 1227–1233. [CrossRef]
Rietbergen, D. , 2008, “ Adjustable Mould for Architectural Freely Curved Glass,” Challenging Glass Conference, Delft, The Netherlands, May 22–23, pp. 523–530.
Walczyk, D. F. , Hosford, J. F. , and Papazian, J. M. , 2003, “ Using Reconfigurable Tooling and Surface Heating for Incremental Forming of Composite Aircraft Parts,” ASME J. Manuf. Sci. Eng., 125(2), pp. 333–343. [CrossRef]
Grünewald, S. , Janssen, B. , Schipper, H. R. , Vollers, K. J. , and Walraven, J. C. , 2012, “ Deliberate Deformation of Concrete After Casting,” Second International Conference on Flexible Formwork (ICFF), Bath, UK, June 27–29, Paper No. 12. https://repository.tudelft.nl/islandora/object/uuid:c46bd896-432d-4430-b1eb-8770dfe4724f?collection=research
Kalogirou, S. A. , 2001, “ Artificial Neural Networks in Renewable Energy Systems Applications: A Review,” Renewable Sustainable Energy Rev., 5(4), pp. 373–401. [CrossRef]
Carr, J. C. , Fright, W. R. , and Beatson, R. K. , 1997, “ Surface Interpolation With Radial Basis Functions for Medical Imaging,” IEEE Trans. Med. Imaging, 16(1), pp. 96–107. [CrossRef] [PubMed]
Klass, V. , Behm, M. , and Lindbergh, G. , 2015, “ Capturing Lithium-Ion Battery Dynamics With Support Vector Machine-Based Battery Model,” J. Power Sources, 298, pp. 92–101. [CrossRef]
Shi, X. , Teixeira, Â. P., Zhang, J. , and Soares, C. G. , 2014, “ Kriging Response Surface Reliability Analysis of a Ship-Stiffened Plate With Initial Imperfections,” Struct. Infrastruct. Eng., 11(11), pp. 1450–1465. [CrossRef]
Hamdaoui, M. , Oujebbour, F.-Z. , Habbal, A. , Breitkopf, P. , and Villon, P. , 2014, “ Kriging Surrogates for Evolutionary Multi-Objective Optimization of CPU Intensive Sheet Metal Forming Applications,” Int. J. Mater. Form., 8(3), pp. 469–480. [CrossRef]
Montgomery, D. C. , 2012, Design and Analysis of Experiments, 8th ed., Wiley, Hoboken, NJ.
MacKay, D. J. C. , 2004, Information Theory, Inference, and Learning Algorithms, Cambridge University Press, Cambridge, UK.
Forrester, A. I. J. , and Keane, A. J. , 2009, “ Recent Advances in Surrogate-Based Optimization,” Prog. Aerosp. Sci., 45(1–3), pp. 50–79. [CrossRef]
Smola, A. , and Schölkopf, B. , 2004, “ A Tutorial on Support Vector Regression,” Stat. Comput., 14(3), pp. 199–222. [CrossRef]
Forrester, A. I. J. , Sóbester, A. , and Keane, A. J. , 2008, Engineering Design Via Surrogate Modelling: A Practical Guide, Wiley, Hoboken, NJ.
Shlens, J. , 2014, “A Tutorial on Principal Component Analysis,” e-print arXiv:1404.1100. https://arxiv.org/abs/1404.1100
Xiao, M. , Breitkopf, P. , Coelho, R. F. , Knopf-Lenoir, C. , Sidorkiewicz, M. , and Villon, P. , 2010, “ Model Reduction by CPOD and Kriging,” Struct. Multidiscip. Optim., 41(4), pp. 555–574. [CrossRef]
Rayas-Sanchez, J. E. , 2004, “ EM-Based Optimization of Microwave Circuits Using Artificial Neural Networks: The State-of-the-Art,” IEEE Trans. Microwave Theory Tech., 52(1), pp. 420–435. [CrossRef]
Joseph, V. R. , and Ying, H. , 2008, “ Orthogonal-Maximin Latin Hypercube Designs,” Stat. Sin., 18(1), pp. 171–186. http://www3.stat.sinica.edu.tw/statistica/oldpdf/A18n17.pdf
Christensen, E. T. , Lund, E. , and Lindgaard, E. , “ Constrained Orthogonal-Maximin Nested Latin Hypercube Sampling,” Technometrics, submitted.
Jones, R. M. , 1999, Mechanics of Composite Materials, Vol. 2, 2nd ed., Taylor & Francis, Abingdon, UK.
Alexandrov, N. M. , Alexandrov, N. M. , Lewis, R. M. , Gumbert, C. R. , Green, L. L. , and Newman, P. A. , 2000, “Optimization With Variable-Fidelity Models Applied to Wing Design,” NASA Langley Research Center, Hampton, VA, ICASE Report No. NASA/CR-1999-209826. http://www.google.co.in/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0ahUKEwjs74XGiObWAhUlxYMKHW8RA7YQFgglMAA&url=http%3A%2F%2Fwww.dtic.mil%2Fcgi-bin%2FGetTRDoc%3FAD%3DADA371451&usg=AOvVaw15l7ZSfdh2cdJo0AnxqriA
Marduel, X. , Tribes, C. , and Trépanier, J.-Y. , 2006, “ Variable-Fidelity Optimization: Efficiency and Robustness,” Optim. Eng., 7(4), pp. 479–500. [CrossRef]
Le Gratiet, L. , 2011, “ Building Kriging Models Using Hierarchical Codes With Different Levels of Accuracy,” 11th Annual Meeting of the European Network for Business and Industrial Statistics, Coimbra, Portugal, Sept. 11–15, pp. 1–8. https://hal-univ-diderot.archives-ouvertes.fr/hal-00654710/document
Fernández-Godino, M. G. , Park, C. , Kim, N.-H. , and Haftka, R. T. , 2016, “ Review of Multi-Fidelity Models,” e-print arXiv:1609.07196 https://arxiv.org/abs/1609.07196.
Rasmussen, C. E. , 2006, Gaussian Processes for Machine Learning, MIT Press, Cambridge, MA.
Forrester, A. I. J. , Sóbester, A. , and Keane, A. J. , 2015, “Optimization Codes,” accessed Oct. 10, 2017, https://optimizationcodes.wordpress.com
Turner, C. J. , and Crawford, R. H. , 2009, “ N-Dimensional Nonuniform Rational B-Splines for Metamodeling,” ASME J. Comput. Inf. Sci. Eng., 9(3), p. 031002. [CrossRef]
Qian, P. Z. G. , 2009, “ Nested Latin Hypercube Designs,” Biometrika, 96(4), pp. 957–970. [CrossRef]
Morris, M. D. , and Mitchell, T. J. , 1995, “ Exploratory Designs for Computational Experiments,” J. Stat. Plann. Inference, 43(3), pp. 381–402. [CrossRef]

Figures

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Fig. 1

Illustration of the principle of the treated variable shape mould systems

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Fig. 2

Schematic illustration of the stiffness characteristic of the nonlinear springs that are used to model mechanical freeplay in the finite element model

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Fig. 3

Flow diagram illustrating the structure of the surrogate models

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Fig. 4

Illustration of the principle in nested Latin hypercube sampling in two-dimensional. L1 samples represent samples obtained in the first layer of five points. L2 samples represent samples of the second layer of ten points. Note that the L1 samples are part of the set of the L2 samples.

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Fig. 5

Illustration of the problem with global sampling using an ordinary Latin hypercube approach in one-dimensional. Configuration 1 represents the worst case configuration, with respect to radius of curvature, for an amplitude of hmax. Setting hmax as the sampling amplitude is, however, too restrictive since configuration 2, with amplitude 2hmax, cannot be reached although it is an acceptable configuration with respect to curvature.

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Fig. 6

An example of a nested Latin hypercube sample design in two dimensions satisfying three constraints. L1 samples illustrate the first level in the design and L2 samples illustrate the second. Constraint boundaries (black line) and the infeasible regions (gray) are shown as well.

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Fig. 7

Histogram showing the distribution of the displacement difference between the actuators and the interpolating membrane

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Fig. 8

Model performance for finite element model 1 (no mechanical freeplay). UM is short for underlying model, SM1 for surrogate model 1 (kriging), and SM2 for surrogate model 2 (kriging and POD). The models are trained on samples of HNLH and are benchmarked on samples of HNLH,test.

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Fig. 9

Model performance for variable shape mould 1 (poor tolerances). UM is short for underlying model, SM1 for surrogate model 1 (kriging), and SM2 for surrogate model 2 (kriging and POD). The models are trained on samples of HNLH and are benchmarked on samples of HNLH,test.

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Fig. 10

Model performances for finite element model 2 (mechanical freeplay included). UM is short for underlying model, SM1 for surrogate model 1 (kriging), and SM2 for surrogate model 2 (kriging and POD). The models are trained on samples of HNLH and are benchmarked on samples of HNLH,test.

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Fig. 11

Behavior of the first POD coefficient while varying the displacements of actuators 13 (center actuator) and 12 (neighbor to 13) over the range from 0 to 8 mm. The remaining actuators are set according to the first geometry in XNLH. The POD coefficient and the actuator displacements are normalized in the plot. The POD behavior is shown with mechanical freeplay present on the left and without on the right.

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Fig. 12

Model performances for the low amplitude data for variable shape mould 2 (good tolerances). UM is short for underlying model, SM1 for surrogate model 1 (kriging), and SM2 for surrogate model 2 (kriging and POD). The models are trained on samples of HNLH and are benchmarked on samples of HNLH,test.

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Fig. 13

Model performances for the high amplitude data for variable shape mould 2 (good tolerances). UM is short for underlying model, SM1 for surrogate model 1 (kriging), and SM2 for surrogate model 2 (kriging and POD). The models are trained on samples of HCNLH and are benchmarked on samples of HCNLH,test.

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