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Numerical shape optimization to decrease failure probability of ceramic structures

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Computing and Visualization in Science

Abstract

Ceramic is a material frequently used in industry because of its favorable properties. Common approaches in shape optimization for ceramic structures aim to minimize the tensile stress acting on the component, as it is the main driver for failure. In contrast to this, we follow a more natural approach by minimizing the component’s probability of failure under a given tensile load. Since the fundamental work of Weibull, the probabilistic description of the strength of ceramics is standard and has been widely applied. Here, for the first time, the resulting failure probabilities are used as objective functions in PDE constrained shape optimization. To minimize the probability of failure, we choose a gradient based method combined with a first discretize then optimize approach. For discretization finite elements are used. Using the Lagrangian formalism, the shape gradient via the adjoint equation is calculated at low computational cost. The implementation is verified by comparison of it with a finite difference method applied to a minimal 2d example. Furthermore, we construct shape flows towards an optimal / improved shape in the case of a simple beam and a bended joint.

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References

  1. Weibull, E.: A statistical theory of the strength of materials. Ingeniörsvetenskapsakedemiens Handlingar 151, 1–45 (1939)

    Google Scholar 

  2. Munz, D., Fett, T.: Ceramics—Mechanical Properties, Failure Behaviour, Materials Selection. Springer, Berlin (1999)

    Google Scholar 

  3. Nemeth, N.N., Manderscheid, J., Gyekenyeshi, J.: Ceramic analysis and reliability evaluation of structures (CARES). Report TP-2916, NASA (1990)

  4. Roudi, S., Riesch-Oppermann, H., Kraft, O.: Advanced probabilistic tools for the uncertainty as- sessment in failure and lifetime prediction of ceramic components. Materialwiss. Werkstofftech. 36, 171–176 (2005)

    Article  Google Scholar 

  5. Lohbauer, U., Petschelt, A., Greil, P.: Lifetime prediction of cad/cam dental ceramics. J. Biomed. Mater. Res. 63(6), 780–785 (1973)

    Article  Google Scholar 

  6. Allaire, G.: Shape Opimization by the Homogenisation Method. Springer, Berlin (2001)

    Google Scholar 

  7. Bucur, D., Buttazzo, G.: Variational Methods in Shape Optimization Problems. Birkhäuser, Basel (2005)

    Book  Google Scholar 

  8. Chenais, D.: On the existence of a solution in a domain identification problem. J. Math. Anal. Appl. 52, 189–289 (1975)

    Article  MathSciNet  Google Scholar 

  9. Haslinger, J., Mäkinen, R.A.E.: Introduction to Shape Optimization. SIAM, New Delhi (2003)

    Book  Google Scholar 

  10. Sokolovski, J., Zolesio, J.P.: Introduction to Shape Optimization—Shape Sensitivity Analysis. Springer, Berlin (1992)

    Book  Google Scholar 

  11. Bolten, M., Gottschalk, H., Schmitz, S.: Minimal failure probability for ceramic design via shape control. J. Optim. Theory Appl. 166(3), 983–1001 (2015)

    Article  MathSciNet  Google Scholar 

  12. Gottschalk, H., Schmitz, S.: Optimal reliability in design for fatigue life. SIAM J. Control Optim. 52(5), 2727–2752 (2015)

    Article  MathSciNet  Google Scholar 

  13. Schmitz, S.: A Local and Probabilistic Model for Low-Cycle Fatigue: New Aspects of Structural Analysis. Hartung-Gorre, Konstanz (2014)

    Google Scholar 

  14. Mohammadi, B., Pironneau, O.: Applied Shape Optimization for Fluids. Oxford university Press, Oxford (2001)

    MATH  Google Scholar 

  15. Frey, C., Nürnberger, D., Kersken, H.: The discrete adjoint of a turbomachinery rans solver. In: Proceedings of ASME-GT2009 (2009)

  16. Bendsøe, M.P., Sigmund, O.: Topology Optimization. Springer, Berlin (2004)

    Book  Google Scholar 

  17. Conti, S., Held, H., Pach, M., Rumpf, M., Schultz, R.: Shape optimization under uncertainty—a stochastic programming perspective. SIAM J. Optim. 19(4), 1610–1632 (2008)

    Article  MathSciNet  Google Scholar 

  18. Batendorf, S.B., Crosse, J.G.: A statistical theory for the fracture of brittle structures subject to nonuniform polyaxial stress. J. Appl. Mech. 41, 459–465 (1974)

    Article  Google Scholar 

  19. Duran, R.G., Muschietti, M.A.: The korn inrequality for jones domains. Electron. J. Diff. Equ. 127, 1–10 (2004)

    MATH  Google Scholar 

  20. Brückner-Foit, A., Fett, T., Munz, D., Schirmer, K.: Discrimination of multiaxiality criteria with the brasilian disk test. J. Eur. Ceram. Soc. 17, 689–696 (1997)

    Article  Google Scholar 

  21. Gross, D., Seelig, T.: Fracture Mechanics with an Introduction to Micromechanics. Springer, Berlin (2006)

    MATH  Google Scholar 

  22. Watanabe, S.: On discontinuous additive functionals and lévy measures of a markov process. Jpn. J. Math. 34, 53–70 (1964)

    Article  Google Scholar 

  23. Kallenberg, O.: Random Measures. Akademie, Berlin (1983)

    MATH  Google Scholar 

  24. Bäker, M., Harders, H., Rösler, J.: Mechanisches Verhalten der Werkstoffe, 3rd edn. Vieweg+Teubner, Berlin (2008)

    Google Scholar 

  25. Braess, D.: Finite elements. Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  26. Trefethen, L.N., Weideman, J.A.C.: The exponentially convergent trapezoidal rule. SIAM Rev. 56(3), 385–458 (2014)

    Article  MathSciNet  Google Scholar 

  27. Ivanov, S.N., Khazanov, E.N., Taranov, A.V., Mikhailova, I.S., Gropyanov, V.M., Abramovich, A.A.: Grain boundaries and elastic properties of aluminum-oxide and stainless-steel-based cermets. Phys. Solid State 43(4), 665–669 (2001)

    Article  Google Scholar 

  28. Nazarov, S., Plamenevskij, B.A.: Elliptic Boundary Value Problems in Domains with Piecewise Smooth Boundary. De Gruyter, Berlin (1994)

    Book  Google Scholar 

  29. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions i. Commun. Pure Appl. Math. 12(4), 623–727 (1959)

    Article  MathSciNet  Google Scholar 

  30. Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions ii. Commun. Pure Appl. Math. 17(1), 35–92 (1964)

    Article  MathSciNet  Google Scholar 

  31. Ciarlet, P.: Mathematical Elasticity-Volume I: Three-Dimensional Elasticity Studies in mathematics and its applications. North-Holland, Amsterdam (1988)

    MATH  Google Scholar 

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Acknowledgements

Hanno Gottschalk thanks Sebastian Schmitz from Siemens Energy, Gas Turbine Engineering Department, for interesting discussions. This work was supported by BMBF (German Federal Ministry for Education and Research) as part of the project GIVEN (05M18PXA).

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  1. School of Mathematics and Natural Science, Bergische Universität Wuppertal, Wuppertal, Germany

    Matthias Bolten, Hanno Gottschalk, Camilla Hahn & Mohamed Saadi

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  1. Matthias Bolten
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  2. Hanno Gottschalk
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  3. Camilla Hahn
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  4. Mohamed Saadi
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Correspondence to Matthias Bolten.

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Bolten, M., Gottschalk, H., Hahn, C. et al. Numerical shape optimization to decrease failure probability of ceramic structures. Comput. Visual Sci. 21, 1–10 (2019). https://doi.org/10.1007/s00791-019-00315-z

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