Nonlinear shape-manifold learning approach: concepts, tools and applications.

*(English)*Zbl 1390.68548Summary: In this paper, we present the concept of a “shape manifold” designed for reduced order representation of complex “shapes” encountered in mechanical problems, such as design optimization, springback or image correlation. The overall idea is to define the shape space within which evolves the boundary of the structure. The reduced representation is obtained by means of determining the intrinsic dimensionality of the problem, independently of the original design parameters, and by approximating a hyper surface, i.e. a shape manifold, connecting all admissible shapes represented using level set functions. Also, an optimal parameterization may be obtained for arbitrary shapes, where the parameters have to be defined a posteriori. We also developed the predictor-corrector optimization manifold walking algorithms in a reduced shape space that guarantee the admissibility of the solution with no additional constraints. We illustrate the approach on three diverse examples drawn from the field of
computational and applied mechanics.

##### MSC:

68T05 | Learning and adaptive systems in artificial intelligence |

65D17 | Computer-aided design (modeling of curves and surfaces) |

68U07 | Computer science aspects of computer-aided design |

PDF
BibTeX
XML
Cite

\textit{L. Meng} et al., Arch. Comput. Methods Eng. 25, No. 1, 1--21 (2018; Zbl 1390.68548)

Full Text:
DOI

##### References:

[1] | Alfaro I, Yvonnet J, Cueto E, Chinesta F, Doblare M (2006) Meshless methods with application to metal forming. Comput Methods Appl Mech Eng 195(48-49):6661-6675 (Computational Metal Forming) · Zbl 1120.74853 |

[2] | Alkorta, J; Martinez-Esnaola, J; Sevillano, JG, Absence of one-to-one correspondence between elastoplastic properties and sharp-indentation load-penetration data, J Mater Res, 20, 432-437, (2005) |

[3] | Allaire, G; Jouve, F; Toader, AM, Structural optimization using sensitivity analysis and a level-set method, J Comput Phys, 194, 363-393, (2004) · Zbl 1136.74368 |

[4] | Audouze, C; Vuyst, F; Nair, P, Reduced-order modeling of parameterized PDEs using time-space-parameter principal component analysis, Int J Numer Methods Eng, 80, 1025-1057, (2009) · Zbl 1176.76059 |

[5] | Balasubramanian, M; Schwartz, EL, The isomap algorithm and topological stability, Science, 295, 7-7, (2002) |

[6] | Benamara, T; Breitkopf, P; Lepot, I; Sainvitu, C, Adaptive infill sampling criterion for multi-fidelity optimization based on gappy-POD, Struct Multidiscip Optim, (2016) |

[7] | Berkooz, G; Holmes, P; Lumley, JL, The proper orthogonal decomposition in the analysis of turbulent flows, Annu Rev Fluid Mech, 25, 539-575, (1993) |

[8] | Bocciarelli, M; Bolzon, G; Maier, G, Parameter identification in anisotropic elastoplasticity by indentation and imprint mapping, Mech Mater, 37, 855-868, (2005) |

[9] | Bolzon, G; Maier, G; Panico, M, Material model calibration by indentation, imprint mapping and inverse analysis, Int J Solids Struct, 41, 2957-2975, (2004) · Zbl 1119.74408 |

[10] | Breitkopf, P; Kleiber, M, Knowledge engineering enhancement of finite element analysis, Commun Appl Numer Methods, 3, 359-366, (1987) · Zbl 0622.73089 |

[11] | Breitkopf, P; Naceur, H; Rassineux, A; Villon, P, Moving least squares response surface approximation: formulation and metal forming applications, Comput Struct, 83, 1411-1428, (2005) |

[12] | Breitkopf, P; Rassineux, A; Villon, P, An introduction to moving least squares meshfree methods, Revue Europeenne des Elements, 11, 825-867, (2002) · Zbl 1120.74856 |

[13] | Camastra, F, Data dimensionality estimation methods: a survey, Pattern Recognit, 36, 2945-2954, (2003) · Zbl 1059.68100 |

[14] | Castanier, MP; Ottarsson, G; Pierre, C, A reduced order modeling technique for mistuned bladed disks, J Vib Acoust, 119, 439-447, (1997) |

[15] | Chen, X; Ogasawara, N; Zhao, M; Chiba, N, On the uniqueness of measuring elastoplastic properties from indentation: the indistinguishable mystical materials, J Mech Phys Solids, 55, 1618-1660, (2007) · Zbl 1176.74029 |

[16] | Cheng, YT; Cheng, CM, Scaling, dimensional analysis, and indentation measurements, Mater Sci Eng R Rep, 44, 91-149, (2004) |

[17] | Chinesta, F; Ammar, A; Cueto, E, Proper generalized decomposition of multiscale models, Int J Numer Methods Eng, 83, 1114-1132, (2010) · Zbl 1197.76093 |

[18] | Chinesta, F; Ladeveze, P; Cueto, E, A short review on model order reduction based on proper generalized decomposition, Arch Comput Methods Eng, 18, 395-404, (2011) |

[19] | Coelho, RF; Breitkopf, P; Knopf-Lenoir, C, Bi-level model reduction for coupled problems, Struct Multidiscip Optim, 39, 401-418, (2009) · Zbl 1274.74225 |

[20] | Cordier, L; Majd, BA; Favier, J, Calibration of pod reduced order models using Tikhonov regularization, Int J Numer Methods Fluids, 63, 269-296, (2010) · Zbl 1425.76181 |

[21] | Cortes, C; Vapnik, V, Support vector machine, Mach Learn, 20, 273-297, (1995) · Zbl 0831.68098 |

[22] | Couplet, M; Basdevant, C; Sagaut, P, Calibrated reduced-order pod-Galerkin system for fluid flow modeling, J Comput Phys, 207, 192-220, (2005) · Zbl 1177.76283 |

[23] | Cox TF, Cox MA (2000) Multidimensional scaling. CRC Press, Boca Raton · Zbl 1004.91067 |

[24] | D’Acquisto, L; Fratini, L, An optical technique for springback measurement in axisymmetrical deep drawing operations, J Manuf Process, 3, 29-37, (2001) |

[25] | Dulong, JL; Druesne, F; Villon, P, A model reduction approach for real-time part deformation with nonlinear mechanical behavior, Int J Interact Des Manuf, 1, 229-238, (2007) |

[26] | Duvigneau R (2006) Adaptive parameterization using free-form deformation for aerodynamic shape optimization. INRIA Research Report RR-5949 |

[27] | Eggertsen, PA; Mattiasson, K, On the modelling of the bendingunbending behaviour for accurate springback predictions, Int J Mech Sci, 51, 547-563, (2009) |

[28] | Forrester, AIJ; Keane, AJ, Recent advances in surrogate-based optimization, Prog Aerosp Sci, 45, 50-79, (2009) |

[29] | Freund Y, Mason L (1999) The alternating decision tree learning algorithm. In: icml, vol 99, pp 124-133 · Zbl 1286.74107 |

[30] | Furey, TS; Cristianini, N; Duffy, N; Bednarski, DW; Schummer, M; Haussler, D, Support vector machine classification and validation of cancer tissue samples using microarray expression data, Bioinformatics, 16, 906-914, (2000) |

[31] | Gao, T; Zhang, W; Duysinx, P, A bi-value coding parameterization scheme for the discrete optimal orientation design of the composite laminate, Int J Numer Methods Eng, 91, 98-114, (2012) · Zbl 1246.74043 |

[32] | Geuzaine, C; Remacle, JF, Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities, Int J Numer Methods Eng, 79, 1309-1331, (2009) · Zbl 1176.74181 |

[33] | Ghnatios, C; Chinesta, F; Cueto, E; Leygue, A; Breitkopf, P; Villon, P, Methodological approach to efficient modeling and optimization of thermal processes taking place in a die: application to pultrusion, Compos Part A, 42, 1169-1178, (2011) |

[34] | Ghnatios, C; Masson, F; Huerta, A; Cueto, E; Leygue, A; Chinesta, F, Proper generalized decomposition based dynamic data-driven control of thermal processes, Comput Methods Appl Mech Eng, 213, 29-41, (2012) |

[35] | Gibson, RF, A review of recent research on nanoindentation of polymer composites and their constituents, Compos Sci Technol, 105, 51-65, (2014) |

[36] | Goldberg, DE; Holland, JH, Genetic algorithms and machine learning, Mach Learn, 3, 95-99, (1988) |

[37] | Golub, GH; Reinsch, C, Singular value decomposition and least squares solutions, Numer Math, 14, 403-420, (1970) · Zbl 0181.17602 |

[38] | Gonzalez, D; Cueto, E; Chinesta, F, Computational patient avatars for surgery planning, Ann Biomed Eng, (2015) |

[39] | Hagan MT, Demuth HB, Beale MH, De Jesús O (1996) Neural network design, vol 20. PWS Publishing Company, Boston |

[40] | Hoang, K; Kerfriden, P; Bordas, S, A fast, certified and tuning free two-field reduced basis method for the metamodelling of affinely-parametrised elasticity problems, Comput Methods Appl Mech Eng, 298, 121-158, (2016) |

[41] | Hou, Y; Sapanathan, T; Dumon, A; Culière, P; Rachik, M, A novel artificial dual-phase microstructure generator based on topology optimization, Comput Mater Sci, 123, 188-200, (2016) |

[42] | Hurtado JE (2013) Structural reliability: statistical learning perspectives, vol 17. Springer Science & Business Media, Berlin · Zbl 1086.62116 |

[43] | Ibrahimbegovic, A; Knopf-Lenoir, C; Kucerova, A; Villon, P, Optimal design and optimal control of elastic structures undergoing finite rotations and deformations, Int J Numer Methods Eng, 61, 2428-2460, (2008) · Zbl 1075.74607 |

[44] | Ito, K; Ravindran, S, A reduced-order method for simulation and control of fluid flows, J Comput Phys, 143, 403-425, (1998) · Zbl 0936.76031 |

[45] | Jan S, Zolesio J (1992) Shape sensitivity analysis. Introduction to shape optimization, Springer, Berlin · Zbl 0761.73003 |

[46] | Jolliffe I (2002) Principal component analysis. Wiley Online Library, New York · Zbl 1011.62064 |

[47] | Lassila, T; Rozza, G, Parametric free-form shape design with PDE models and reduced basis method, Comput Methods Appl Mech Eng, 199, 1583-1592, (2010) · Zbl 1231.76245 |

[48] | Quilliec, G; Raghavan, B; Breitkopf, P, A manifold learning-based reduced order model for springback shape characterization and optimization in sheet metal forming, Comput Methods Appl Mech Eng, 285, 621-638, (2015) · Zbl 1423.74780 |

[49] | Legrain, G; Cartraud, P; Perreard, I; Moes, N, An X-FEM and level set computational approach for image-based modelling: application to homogenization, Int J Numer Methods Eng, 86, 915-934, (2011) · Zbl 1235.74297 |

[50] | Li, K; Carden, W; Wagoner, R, Simulation of springback, Int J Mech Sci, 44, 103-122, (2002) · Zbl 0986.74522 |

[51] | Lopez, E; Gonzalez, D; Aguado, J; Abisset-Chavanne, E; Cueto, E; Binetruy, C; Chinesta, F, A manifold learning approach for integrated computational materials engineering, Arch Comput Methods Eng, (2016) · Zbl 1390.74196 |

[52] | Lucia, DJ; Beran, PS; Silva, WA, Reduced-order modeling: new approaches for computational physics, Prog Aerosp Sci, 40, 51-117, (2004) |

[53] | Manzoni, A; Quarteroni, A; Rozza, G, Shape optimization for viscous flows by reduced basis methods and free-form deformation, Int J Numer Methods Fluids, 70, 646-670, (2012) |

[54] | Marteau, J; Bouvier, S; Bigerelle, M, Review on numerical modeling of instrumented indentation tests for elastoplastic material behavior identification, Arch Comput Methods Eng, 22, 577-593, (2015) · Zbl 1348.74004 |

[55] | Meng L, Zhang WH, Zhu JH, Xia L (2014) A biarc-based shape optimization approach to reduce stress concentration effects. Acta Mechanica Sinica 30(3):370-382 · Zbl 1346.74155 |

[56] | Meng, L; Breitkopf, P; Raghavan, B; Mauvoisin, G; Bartier, O; Hernot, X, Identification of material properties using indentation test and shape manifold learning approach, Comput Methods Appl Mech Eng, 297, 239-257, (2015) |

[57] | Meng L, Zhang WH, Zhu JH, Xu Z, Cai SY (2016) Shape optimization of axisymmetric solids with the finite cell method using a fixed grid. Acta Mechanica Sinica 32(3):510-524 · Zbl 1348.74286 |

[58] | Millan, D; Rosolen, A; Arroyo, M, Nonlinear manifold learning for meshfree finite deformation thin-shell analysis, Int J Numer Methods Eng, 93, 685-713, (2013) · Zbl 1352.74176 |

[59] | Millan, D; Rosolen, A; Arroyo, M, Nonlinear manifold learning for model reduction in finite elastodynamics, Comput Methods Appl Mech Eng, 261-262, 181-131, (2013) · Zbl 1352.74176 |

[60] | Minsky, M, No article title, Steps toward artificial intelligence. Proc IRE, 49, 8-30, (1961) |

[61] | Moes, N; Stolz, C; Bernard, P; Chevaugeon, N, A level set based model for damage growth: the thick level set approach, Int J Numer Methods Eng, 86, 358-380, (2011) · Zbl 1235.74302 |

[62] | Montgomery DC, Peck EA, Vining GG (2006) Introduction to linear regression. Wiley, New Jersey · Zbl 1229.62092 |

[63] | Moon, Y; Kang, S; Cho, J; Kim, T, Effect of tool temperature on the reduction of the springback of aluminum sheets, J Mater Process Technol, 132, 365-368, (2003) |

[64] | Moussa, C; Hernot, X; Bartier, O; Delattre, G; Mauvoisin, G, Identification of the hardening law of materials with spherical indentation using the average representative strain for several penetration depths, Mater Sci Eng A, 606, 409-416, (2014) |

[65] | Murat F, Simon J (1976) Sur le controle par un domaine geometrique. Pre-publication du Laboratoire d’Analyse Numerique, no 76015, Universite de Paris 6 · Zbl 0181.17602 |

[66] | Nayroles, B; Touzot, G; Villon, P, Generalizing the finite element method: diffuse approximation and diffuse elements, Comput Mech, 10, 307-318, (1992) · Zbl 0764.65068 |

[67] | Oshier, S; Sethian, JA, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J Comput Phys, 79, 12-49, (1988) · Zbl 0659.65132 |

[68] | Olof Persson, P; Strang, G, A simple mesh generator in Matlab, SIAM Rev, 46, 329-345, (2004) · Zbl 1061.65134 |

[69] | Peterson, JS, The reduced basis method for incompressible viscous flow calculations, SIAM J Sci Stat Comput, 10, 777-786, (1989) · Zbl 0672.76034 |

[70] | Prud’homme, C; Rovas, DV; Veroy, K; Machiels, L; Maday, Y; Patera, AT; Turinici, G, Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods, J Fluids Eng Trans ASME, 124, 70-80, (2002) |

[71] | Quarteroni, A; Rozza, G; Manzoni, A, Certified reduced basis approximation for parametrized partial differential equations and applications, J Math Ind, 1, 1-49, (2011) · Zbl 1273.65148 |

[72] | Raghavan, B; Breitkopf, P; Tourbier, Y; Villon, P, Towards a space reduction approach for structural shape optimization, Struct Multidiscip Optim, (2013) |

[73] | Raghavan, B; Hamdaoui, M; Xiao, M; Breitkopf, P; Villon, P, A bi-level meta-modeling approach for structural optimization using modified POD bases and diffuse approximation, Comput Struct, 127, 19-28, (2012) |

[74] | Raghavan, B; Quilliec, G; Breitkopf, P; Rassineux, A; Roelandt, JM; Villon, P, Numerical assessment of springback for the deep drawing process by level set interpolation using shape manifolds, Int J Mater Form, 7, 487-501, (2014) |

[75] | Raghavan, B; Xia, L; Breitkopf, P; Rassineux, A; Villon, P, Towards simultaneous reduction of both input and output spaces for interactive simulation-based structural design, Comput Methods Appl Mech Eng, 265, 174-185, (2013) · Zbl 1286.74107 |

[76] | Raghavan, B; Xia, L; Breitkopf, P; Rassineux, A; Villon, P, Towards simultaneous reduction of both input and output spaces for interactive simulation-based structural design, Comput Methods Appl Mech Eng, (2013) · Zbl 1286.74107 |

[77] | Raghavan, B; Xiao, M; Breitkopf, P; Villon, P, Implicit constraint handling for shape optimization using pod-morphing, Eur J Comput Mech, 21, 325-336, (2012) · Zbl 1348.74284 |

[78] | Rokach L, Maimon O (2014) Data mining with decision trees: theory and applications. World Scientific, Singapore · Zbl 1154.68098 |

[79] | Rozza, G; Huynh, D; Patera, A, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations, Arch Comput Methods Eng, 15, 229-275, (2008) · Zbl 1304.65251 |

[80] | Russell S, Norvig P, Intelligence A (1995) A modern approach. Artificial intelligence. Prentice-Hall, Egnlewood Cliffs, pp 25-27 |

[81] | Sahan, RA; Gunes, H; Liakopoulos, A, A modeling approach to transitional channel flow, Comput Fluids, 27, 121-136, (1998) · Zbl 0909.76079 |

[82] | Samuel, AL, Some studies in machine learning using the game of checkers, IBM J Res Dev, 3, 210-229, (1959) |

[83] | Saul LK, Roweis ST (2000) An introduction to locally linear embedding. Unpublished. http://www.cs.toronto.edu/ roweis/lle/publications.html |

[84] | Saul LK, Weinberger KQ, Ham JH, Sha F, Lee DD (2006) Spectral methods for dimensionality reduction. Semisupervised Learn, MIT Press, Cambridge, pp 293-308 · Zbl 1176.76059 |

[85] | Schulz, V, A Riemannian view on shape optimization, Found Comput Math, 14, 483-501, (2012) · Zbl 1296.49037 |

[86] | Sebastiani, F, Machine learning in automated text categorization, ACM Comput Surv (CSUR), 34, 1-47, (2002) |

[87] | Teimouri, R; Baseri, H; Rahmani, B; Bakhshi-Jooybari, M, Modeling and optimization of spring-back in bending process using multiple regression analysis and neural computation, Int J Mater Form, (2012) |

[88] | Toal, DJJ; Keane, AJ, Efficient multipoint aerodynamic design optimization via cokriging, J Aircr, 48, 1685-1695, (2011) |

[89] | Tong, S; Koller, D, Support vector machine active learning with applications to text classification, J Mach Learn Res, 2, 45-66, (2002) · Zbl 1009.68131 |

[90] | Tu, J; Choi, KK; Park, YH, A new study on reliability-based design optimization, J Mech Des, 121, 557-564, (1999) |

[91] | Veiz A, Egerland M (2007) Cad-parametric optimization with optiSLang-ANSYS workbench. In: 4th Weimar optimization and stochastic days · Zbl 1136.74368 |

[92] | Wang, S; Lim, K; Khoo, B; Wang, M, An extended level set method for shape and topology optimization, J Comput Phys, 221, 395-421, (2007) · Zbl 1110.65058 |

[93] | Willcox, K; Peraire, J, Balanced model reduction via the proper orthogonal decomposition, AIAA J, 40, 2323-2330, (2002) |

[94] | Xia, L; Raghavan, B; Breitkopf, P; Zhang, W, Numerical material representation using proper orthogonal decomposition and diffuse approximation, Appl Math Comput, 224, 450-462, (2013) · Zbl 1334.74099 |

[95] | Xiao, M; Breitkopf, P; Coelho, RF; Knopf-Lenoir, C; Sidorkiewicz, M; Villon, P, Model reduction by CPOD and Kriging, Struct Multidiscip Optim, 41, 555-574, (2009) · Zbl 1274.90365 |

[96] | Xie, X; Mirmehdi, M, Radial basis function based level set interpolation and evolution for deformable modelling, Image Vis Comput, 29, 167-177, (2011) |

[97] | Yagawa, G; Okuda, H, Neural networks in computational mechanics, Arch Comput Methods Eng, 3, 435-512, (1996) |

[98] | Zhang, P; Breitkopf, P; Knopf-Lenoir, C; Zhang, W, Diffuse response surface model based on moving Latin hypercube patterns for reliability-based design optimization of ultrahigh strength steel NC milling parameters, Struct Multidiscip Optim, 44, 613-628, (2011) |

[99] | Zhang, WH; Beckers, P; Fleury, C, Unified parametric design approach to structural shape optimization, Int J Numer Methods Eng, 38, 2283-2292, (1995) · Zbl 0854.73040 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.