Skip to main content

2021 | OriginalPaper | Buchkapitel

Machine Learning Control Design for Elastic Composite Materials

verfasst von : Sebastián Ossandón, Mauricio Barrientos, Camilo Reyes

Erschienen in: Computational Science – ICCS 2021

Verlag: Springer International Publishing

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

A novel numerical method, based on a machine learning approach, is used to solve an inverse problem involving the Dirichlet eigenfrequencies for the elasticity operator in a bounded domain filled with a composite material. The inhomogeneity of the material under study is characterized by a vector which is designed to control the constituent mixture of homogeneous elastic materials that compose it. Using the finite element method, we create a training set for a forward artificial neural network, solving the forward problem. A forward nonlinear map of the Dirichlet eigenfrequencies as a function of the vector design parameter is then obtained. This forward relationship is inverted and used to obtain a training set for an inverse radial basis neural network, solving the aforementioned inverse problem. A numerical example showing the applicability of this methodology is presented.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Literatur
1.
Zurück zum Zitat Al-Ajlouni, A.-F., Schilling, R.-J., Harris, S.-L.: Identification of nonlinear discrete-time systems using raised-cosine radial basis function networks. Int. J. Syst. Sci. 35(4), 211–221 (2004)MathSciNetCrossRef Al-Ajlouni, A.-F., Schilling, R.-J., Harris, S.-L.: Identification of nonlinear discrete-time systems using raised-cosine radial basis function networks. Int. J. Syst. Sci. 35(4), 211–221 (2004)MathSciNetCrossRef
2.
Zurück zum Zitat Alves, C.-J.S., Antunes, P.-R.S.: The method of fundamental solutions applied to the calculation of eigenfrequencies and eigenmodes of \(2D\) simply connected shapes. CMC-Comput. Mater. Con. 2(4), 251–265 (2005) Alves, C.-J.S., Antunes, P.-R.S.: The method of fundamental solutions applied to the calculation of eigenfrequencies and eigenmodes of \(2D\) simply connected shapes. CMC-Comput. Mater. Con. 2(4), 251–265 (2005)
3.
Zurück zum Zitat Ammari, H., Kang, H., Nakamura, G., Tanuma, K.: Complete asymptotic expansions of solutions of the system of elastostatics in the presence of an inclusion of small diameter and detection of an inclusion. J. Elast. 67, 97–129 (2002)MathSciNetCrossRef Ammari, H., Kang, H., Nakamura, G., Tanuma, K.: Complete asymptotic expansions of solutions of the system of elastostatics in the presence of an inclusion of small diameter and detection of an inclusion. J. Elast. 67, 97–129 (2002)MathSciNetCrossRef
4.
Zurück zum Zitat Andrieux, S., Ben Abda, A., Bui, H.-D.: Sur l’identification de fissures planes via le concept d’écart à la réciprocité en élasticité. C.R. Acad. Sci. Paris, Série II 324, 1431–1438 (1997) Andrieux, S., Ben Abda, A., Bui, H.-D.: Sur l’identification de fissures planes via le concept d’écart à la réciprocité en élasticité. C.R. Acad. Sci. Paris, Série II 324, 1431–1438 (1997)
5.
Zurück zum Zitat Babuška, I., Osborn, J.-E.: Eigenvalue Problems, Handbook of Numerical Analysis: Finite Element Methods (Part 1), vol. 2, Ciarlet, P.G., Lions, J.L. (eds.). North-Holland, Amsterdam (2000) Babuška, I., Osborn, J.-E.: Eigenvalue Problems, Handbook of Numerical Analysis: Finite Element Methods (Part 1), vol. 2, Ciarlet, P.G., Lions, J.L. (eds.). North-Holland, Amsterdam (2000)
6.
Zurück zum Zitat Ballard, P., Constantinescu, A.: On the inversion of subsurface residual stresses from surface stress measurements. J. Mech. Phys. Solids 42, 1767–1788 (1994)MathSciNetCrossRef Ballard, P., Constantinescu, A.: On the inversion of subsurface residual stresses from surface stress measurements. J. Mech. Phys. Solids 42, 1767–1788 (1994)MathSciNetCrossRef
7.
Zurück zum Zitat Baymani, M., Effati, S., Kerayechian, A.: A feed-forward neural network for solving Stokes problem. Acta Appl. Math. 116(1), 55–64 (2011)MathSciNetCrossRef Baymani, M., Effati, S., Kerayechian, A.: A feed-forward neural network for solving Stokes problem. Acta Appl. Math. 116(1), 55–64 (2011)MathSciNetCrossRef
8.
Zurück zum Zitat Baymani, M., Effati, S., Niazmand, H., Kerayechian, A.: Artificial neural network method for solving the Navier-Stokes equations. Neural Comput. Appl. 26, 765–773 (2015)CrossRef Baymani, M., Effati, S., Niazmand, H., Kerayechian, A.: Artificial neural network method for solving the Navier-Stokes equations. Neural Comput. Appl. 26, 765–773 (2015)CrossRef
9.
Zurück zum Zitat Ben Abdallah, J.: Inversion gaussienne appliquée à la correction paramétrique de modèles structuraux. Ph.D. thesis, Ecole Polytechnique, Paris, France (1995) Ben Abdallah, J.: Inversion gaussienne appliquée à la correction paramétrique de modèles structuraux. Ph.D. thesis, Ecole Polytechnique, Paris, France (1995)
12.
Zurück zum Zitat Boffi, D., Gastaldi, L.: Some remarks on finite element approximation of multiple eigenvalues. Appl. Numer. Math. 79, 18–28 (2014)MathSciNetCrossRef Boffi, D., Gastaldi, L.: Some remarks on finite element approximation of multiple eigenvalues. Appl. Numer. Math. 79, 18–28 (2014)MathSciNetCrossRef
14.
Zurück zum Zitat Choi, H.-J.: A numerical solution for the inhomogeneous Dirichlet boundary value problem on a nonconvex polygon. Appl. Math. Comput. 341, 31–45 (2019)MathSciNetMATH Choi, H.-J.: A numerical solution for the inhomogeneous Dirichlet boundary value problem on a nonconvex polygon. Appl. Math. Comput. 341, 31–45 (2019)MathSciNetMATH
15.
Zurück zum Zitat Ciarlet, P.-G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978) Ciarlet, P.-G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)
16.
Zurück zum Zitat Eigel, M., Peterseim, D.: Simulation of composite materials by a nertwork FEM with error control. Comput. Methods Appli. Math. 15(1), 21–37 (2014)CrossRef Eigel, M., Peterseim, D.: Simulation of composite materials by a nertwork FEM with error control. Comput. Methods Appli. Math. 15(1), 21–37 (2014)CrossRef
17.
Zurück zum Zitat Girosi, F., Jones, M., Poggio, T.: Regularization theory and neural networks architectures. J. Neural Comput. 7, 219–269 (1995)CrossRef Girosi, F., Jones, M., Poggio, T.: Regularization theory and neural networks architectures. J. Neural Comput. 7, 219–269 (1995)CrossRef
18.
Zurück zum Zitat Girosi, F., Poggio, T.: A theory of networks for approximation and learning, MIT Artificial Intelligence Laboratory, A.I. Memo No. 1140, C.B.I.P Paper No. 31 (1989) Girosi, F., Poggio, T.: A theory of networks for approximation and learning, MIT Artificial Intelligence Laboratory, A.I. Memo No. 1140, C.B.I.P Paper No. 31 (1989)
19.
Zurück zum Zitat Griffiths, G.-W., Plociniczak, L., Schiesser, W.-E.: Analysis of cornea curvature using radial basis functions - part I: methodology. Comput. Biol. Med. 77, 274–284 (2016)CrossRef Griffiths, G.-W., Plociniczak, L., Schiesser, W.-E.: Analysis of cornea curvature using radial basis functions - part I: methodology. Comput. Biol. Med. 77, 274–284 (2016)CrossRef
20.
Zurück zum Zitat Hassell, M.-E., Sayas, F.-J.: A fully discrete BEM-FEM scheme for transient acoustic waves. Comput. Methods Appl. Mech. Eng. 309, 106–130 (2016)MathSciNetCrossRef Hassell, M.-E., Sayas, F.-J.: A fully discrete BEM-FEM scheme for transient acoustic waves. Comput. Methods Appl. Mech. Eng. 309, 106–130 (2016)MathSciNetCrossRef
21.
Zurück zum Zitat Leonard, K.-R., Malyarenko, E.-V., Hinders, M.-K.: Ultrasonic Lamb wave tomography. Inverse Prob. 18, 1795–1808 (2002)MathSciNetCrossRef Leonard, K.-R., Malyarenko, E.-V., Hinders, M.-K.: Ultrasonic Lamb wave tomography. Inverse Prob. 18, 1795–1808 (2002)MathSciNetCrossRef
23.
Zurück zum Zitat Mercier, B., Osborn, J., Rappaz, J., Raviart, P.-A.: Eigenvalue approximation by mixed and hybrid methods. Math. Comp. 36, 427–453 (1981)MathSciNetCrossRef Mercier, B., Osborn, J., Rappaz, J., Raviart, P.-A.: Eigenvalue approximation by mixed and hybrid methods. Math. Comp. 36, 427–453 (1981)MathSciNetCrossRef
24.
Zurück zum Zitat Oden, J.-T., Reddy, J.-N.: An Introduction to the Mathematical Theory of Finite Elements. Wiley, New York (1976) Oden, J.-T., Reddy, J.-N.: An Introduction to the Mathematical Theory of Finite Elements. Wiley, New York (1976)
25.
Zurück zum Zitat Ossandón, S., Reyes, C.: On the neural network calculation of the Lamé coefficients through eigenvalues of the elasticity operator. C. R. Mecanique 344, 113–118 (2016)CrossRef Ossandón, S., Reyes, C.: On the neural network calculation of the Lamé coefficients through eigenvalues of the elasticity operator. C. R. Mecanique 344, 113–118 (2016)CrossRef
26.
Zurück zum Zitat Ossandón, S., Reyes, C., Reyes, C.-M.: Neural network solution for an inverse problem associated with the Dirichlet eigenvalues of the anisotropic Laplace operator. Comput. Math. Appl. 72, 1153–1163 (2016)MathSciNetCrossRef Ossandón, S., Reyes, C., Reyes, C.-M.: Neural network solution for an inverse problem associated with the Dirichlet eigenvalues of the anisotropic Laplace operator. Comput. Math. Appl. 72, 1153–1163 (2016)MathSciNetCrossRef
27.
Zurück zum Zitat Ossandón, S., Reyes, C., Cumsille, P., Reyes, C.-M.: Neural network approach for the calculation of potential coefficients in quantum mechanics. Comput. Phys. Commun. 214, 31–38 (2017)MathSciNetCrossRef Ossandón, S., Reyes, C., Cumsille, P., Reyes, C.-M.: Neural network approach for the calculation of potential coefficients in quantum mechanics. Comput. Phys. Commun. 214, 31–38 (2017)MathSciNetCrossRef
28.
Zurück zum Zitat Ossandón, S., Barrientos, M., Reyes, C.: Neural network solution to an inverse problem associated with the eigenvalues of the Stokes operator. C. R. Mecanique 346, 39–47 (2018)CrossRef Ossandón, S., Barrientos, M., Reyes, C.: Neural network solution to an inverse problem associated with the eigenvalues of the Stokes operator. C. R. Mecanique 346, 39–47 (2018)CrossRef
29.
Zurück zum Zitat Rodrigues, D.-E.S., Belinha, J., Pires, F.-M.A., Dinis, L.-M.J.S., Natal Jorge, R.-M.: Homogenization technique for heterogeneous composite materials using meshless methods. Eng. Anal. Boundary Elem. 92, 73–89 (2018)MathSciNetCrossRef Rodrigues, D.-E.S., Belinha, J., Pires, F.-M.A., Dinis, L.-M.J.S., Natal Jorge, R.-M.: Homogenization technique for heterogeneous composite materials using meshless methods. Eng. Anal. Boundary Elem. 92, 73–89 (2018)MathSciNetCrossRef
30.
Zurück zum Zitat Schilling, R.-J., Carroll Jr., J.-J., Al-Ajlouni, A.-F.: Approximation of nonlinear systems with radial basis function neural networks. IEEE Trans. Neural Netw. 12(1), 1–15 (2001)CrossRef Schilling, R.-J., Carroll Jr., J.-J., Al-Ajlouni, A.-F.: Approximation of nonlinear systems with radial basis function neural networks. IEEE Trans. Neural Netw. 12(1), 1–15 (2001)CrossRef
31.
Zurück zum Zitat Sun, J., Zhou, A.: Finite Element Methods for Eigenvalue Problems. CRC Press Taylor & Francis Group, Boca Raton (2016)CrossRef Sun, J., Zhou, A.: Finite Element Methods for Eigenvalue Problems. CRC Press Taylor & Francis Group, Boca Raton (2016)CrossRef
32.
Zurück zum Zitat Xu, W., Xu, B., Guo, F.: Elastic properties of particle-reinforced composites containing nonspherical particles of high packing density and interphase: DEM-FEM simulation and micromechanical theory. Comput. Methods Appl. Mech. Eng. 326, 122–143 (2017)MathSciNetCrossRef Xu, W., Xu, B., Guo, F.: Elastic properties of particle-reinforced composites containing nonspherical particles of high packing density and interphase: DEM-FEM simulation and micromechanical theory. Comput. Methods Appl. Mech. Eng. 326, 122–143 (2017)MathSciNetCrossRef
33.
Zurück zum Zitat Yan, X.: Finite element modeling of consolidation of composite laminates. Acta Mechanica Sinica 22(1), 62–67 (2006)CrossRef Yan, X.: Finite element modeling of consolidation of composite laminates. Acta Mechanica Sinica 22(1), 62–67 (2006)CrossRef
34.
Zurück zum Zitat Zadler, B.-J.: Properties of elastic materials using contacting and non-contacting acoustic spectroscopy. Ph.D. thesis, Colorado School of Mines, Golden, Colorado, USA (2005) Zadler, B.-J.: Properties of elastic materials using contacting and non-contacting acoustic spectroscopy. Ph.D. thesis, Colorado School of Mines, Golden, Colorado, USA (2005)
35.
Zurück zum Zitat Zadler, B.-J., Scales, J.-A.: Monitoring crack-induced changes in elasticity with resonant spectroscopy. J. Appl. Phys. 104(2), 023536 (2008)CrossRef Zadler, B.-J., Scales, J.-A.: Monitoring crack-induced changes in elasticity with resonant spectroscopy. J. Appl. Phys. 104(2), 023536 (2008)CrossRef
36.
Zurück zum Zitat Zhou, L., Ren, S., Liu, C., Ma, Z.: A valid inhomogeneous cell-based smoothed finite element model for the transient characteristics of functionally graded magneto-electro-elastic structures. Compos. Struct. 208, 298–313 (2019)CrossRef Zhou, L., Ren, S., Liu, C., Ma, Z.: A valid inhomogeneous cell-based smoothed finite element model for the transient characteristics of functionally graded magneto-electro-elastic structures. Compos. Struct. 208, 298–313 (2019)CrossRef
37.
Zurück zum Zitat Zienkiewicz, O.-C.: Origins, milestones and directions of the finite element method a personal view. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis: Techniques of Scientific Computing (Part 2), vol. 5. North-Holland, Amsterdam (1997) Zienkiewicz, O.-C.: Origins, milestones and directions of the finite element method a personal view. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis: Techniques of Scientific Computing (Part 2), vol. 5. North-Holland, Amsterdam (1997)
38.
Zurück zum Zitat Zienkiewicz, O.-C.: The Finite Element Method, 5th edn. McGraw-Hill, New York (2000)MATH Zienkiewicz, O.-C.: The Finite Element Method, 5th edn. McGraw-Hill, New York (2000)MATH
Metadaten
Titel
Machine Learning Control Design for Elastic Composite Materials
verfasst von
Sebastián Ossandón
Mauricio Barrientos
Camilo Reyes
Copyright-Jahr
2021
DOI
https://doi.org/10.1007/978-3-030-77964-1_34