Shape and topological sensitivity analysis in domains with cracks

Alexander Khludnev; Jan Sokołowski; Katarzyna Szulc

Applications of Mathematics (2010)

  • Volume: 55, Issue: 6, page 433-469
  • ISSN: 0862-7940

Abstract

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The framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. The equilibrium problem for the elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural mechanics. In the paper, general results on the shape and topology sensitivity analysis of this problem are provided. The results are of interest of their own. In particular, the existence of the shape and topological derivatives of the energy functional is obtained. The results presented in the paper can be used for numerical solution of shape optimization and inverse problems in structural mechanics.

How to cite

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Khludnev, Alexander, Sokołowski, Jan, and Szulc, Katarzyna. "Shape and topological sensitivity analysis in domains with cracks." Applications of Mathematics 55.6 (2010): 433-469. <http://eudml.org/doc/116485>.

@article{Khludnev2010,
abstract = {The framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. The equilibrium problem for the elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural mechanics. In the paper, general results on the shape and topology sensitivity analysis of this problem are provided. The results are of interest of their own. In particular, the existence of the shape and topological derivatives of the energy functional is obtained. The results presented in the paper can be used for numerical solution of shape optimization and inverse problems in structural mechanics.},
author = {Khludnev, Alexander, Sokołowski, Jan, Szulc, Katarzyna},
journal = {Applications of Mathematics},
keywords = {crack with non-penetration; shape sensitivity; derivative of energy functional; topological derivative; crack with non-penetration; shape sensitivity; derivative of energy functional; topological derivative},
language = {eng},
number = {6},
pages = {433-469},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Shape and topological sensitivity analysis in domains with cracks},
url = {http://eudml.org/doc/116485},
volume = {55},
year = {2010},
}

TY - JOUR
AU - Khludnev, Alexander
AU - Sokołowski, Jan
AU - Szulc, Katarzyna
TI - Shape and topological sensitivity analysis in domains with cracks
JO - Applications of Mathematics
PY - 2010
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 55
IS - 6
SP - 433
EP - 469
AB - The framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. The equilibrium problem for the elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural mechanics. In the paper, general results on the shape and topology sensitivity analysis of this problem are provided. The results are of interest of their own. In particular, the existence of the shape and topological derivatives of the energy functional is obtained. The results presented in the paper can be used for numerical solution of shape optimization and inverse problems in structural mechanics.
LA - eng
KW - crack with non-penetration; shape sensitivity; derivative of energy functional; topological derivative; crack with non-penetration; shape sensitivity; derivative of energy functional; topological derivative
UR - http://eudml.org/doc/116485
ER -

References

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  1. Belhachmi, Z., Sac-Epée, J. M., Sokołowski, J., 10.1137/S0036142903429729, SIAM J. Numer. Anal. 43 (2005), 1295-1320. (2005) MR2177806DOI10.1137/S0036142903429729
  2. Francfort, G. A., Marigo, J.-J., 10.1016/S0022-5096(98)00034-9, J. Mech. Phys. Solids 46 (1998), 1319-1342. (1998) Zbl0966.74060MR1633984DOI10.1016/S0022-5096(98)00034-9
  3. Fremiot, G., Sokołowski, J., Shape sensitivity analysis of problems with singularities, Shape Optimization and Optimal Design. Proc. IFIP coference, Cambridge 1999. Lect. Notes Pure Appl. Math. 216 J. Cagnol et al. Marcel Dekker New York (2001). (2001) Zbl0984.49024MR1812355
  4. Fulmański, P., Lauraine, A., Scheid, J.-F., Sokołowski, J., 10.2478/v10006-007-0034-z, Int. J. Appl. Math. Comput. Sci., 17 (2007), 413-430. (2007) MR2356899DOI10.2478/v10006-007-0034-z
  5. Garreau, S., Guillaume, Ph., Masmoudi, M., 10.1137/S0363012900369538, SIAM J. Control Optimization 39 (2001), 1756-1778. (2001) Zbl0990.49028MR1825864DOI10.1137/S0363012900369538
  6. Hlaváček, I., Novotný, A. A., Sokołowski, J., .Zochowski, A., 10.1007/s10957-008-9490-3, J. Optim. Theory Appl. 141 (2009), 569-595. (2009) MR2507486DOI10.1007/s10957-008-9490-3
  7. Hoffmann, K.-H., Khludnev, A. M., Fictitious domain method for the Signorini problem in linear elasticity, Adv. Math. Sci. Appl. 14 (2004), 465-481. (2004) MR2111825
  8. Khludnev, A. M., 10.1007/s10808-005-0129-y, J. Appl. Mech. and Tech. Phys. 46 (2005), 717-729 Prikl. Mekh. Tekh. Fiz. 46 (2005), 123-137 Russian. (2005) Zbl1125.74365MR2168113DOI10.1007/s10808-005-0129-y
  9. Khludnev, A. M., Kovtunenko, V. A., Analysis of Cracks in Solids, WIT Press Southampton-Boston (2000). (2000) 
  10. Khludnev, A. M., Kovtunenko, V. A., Tani, A., 10.2969/jmsj/06041219, J. Math. Soc. Japan 60 (2008), 1219-1253. (2008) Zbl1153.49040MR2467876DOI10.2969/jmsj/06041219
  11. Khludnev, A. M., Ohtsuka, K., Sokołowski, J., 10.1090/qam/1878261, Q. Appl. Math. 60 (2002), 99-109. (2002) MR1878261DOI10.1090/qam/1878261
  12. Khludnev, A. M., Sokołowski, J., Modelling and Control in Solid Mechanics. International Series of Numerical Mathematics, Birkhäuser Basel (1997). (1997) MR1433133
  13. Khludnev, A. M., Sokołowski, J., 10.1016/S0997-7538(00)00138-8, Eur. J. Mech. A, Solids 19 (2000), 105-119. (2000) MR1748780DOI10.1016/S0997-7538(00)00138-8
  14. Khludnev, A. M., Sokołowski, J., On differentiation of energy functionals in the crack theory with possible contact between crack faces, J. Appl. Math. Mech. 64 (2000), 464-475. (2000) 
  15. Khludnev, A. M., Sokołowski, J., 10.1090/qam/2086037, Q. Appl. Math. 62 (2004), 401-422. (2004) MR2086037DOI10.1090/qam/2086037
  16. Knees, D., Zanini, C., Mielke, A., 10.1016/j.physd.2009.02.008, Physica D 239 (2010), 1470-1484. (2010) Zbl1201.49013MR2658341DOI10.1016/j.physd.2009.02.008
  17. Kovtunenko, V. A., 10.1016/S0021-8928(03)00021-2, J. Appl. Math. Mech. 67 (2003), 109-123. (2003) MR1997626DOI10.1016/S0021-8928(03)00021-2
  18. Kovtunenko, V. A., 10.1002/mma.449, Math. Methods Appl. Sci. 27 (2004), 163-179. (2004) MR2029877DOI10.1002/mma.449
  19. Laurain, A., Structure of shape derivatives in non-smooth domains and applications, Adv. Math. Sci. Appl. 15 (2005), 199-226. (2005) MR2148282
  20. Lazarev, N. P., Differentiation of energy functional in the equilibrium problem for a body with a crack and Signorini boundary conditions, J. Appl. Industr. Math. 5 (2002), 139-147. (2002) MR1960797
  21. Lewiński, T., Sokołowski, J., 10.1016/S0020-7683(02)00641-8, Int. J. Solids Struct. 40 (2003), 1765-1803. (2003) Zbl1035.74009DOI10.1016/S0020-7683(02)00641-8
  22. Maz'ja, W. G., Nazarov, S. A., Plamenevskii, B. A., Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Vol. I, II, Birkhäuser Basel (2000). (2000) MR1779977
  23. Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity, P. Noordhoff Groningen (1952). (1952) 
  24. Nazarov, S. A., Sokołowski, J., 10.1016/S0021-7824(03)00004-7, J. Math. Pures Appl. Sér. 9 Journal de Mathématiques pures et appliquées 82 (2003), 125-196. (2003) Zbl1031.35020MR1976204DOI10.1016/S0021-7824(03)00004-7
  25. Rudoy, E. M., 10.1023/B:JAMT.0000046033.10086.86, J. Appl. Mech. Techn. Phys. 45 (2004), 843-852. (2004) Zbl1087.74008MR2113099DOI10.1023/B:JAMT.0000046033.10086.86
  26. Rudoy, E. M., 10.1134/S1990478907010103, J. Appl. Ind. Math. 1 (2007), 95-104. (2007) MR2221674DOI10.1134/S1990478907010103
  27. Rudoy, E. M., 10.3103/S0025654407060118, Mech. Solids 42 (2007), 935-946. (2007) MR2466473DOI10.3103/S0025654407060118
  28. Sokołowski, J., Zolesio, J-P., Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer Series in Computational Mathematics, Vol. 16, Springer Berlin (1992). (1992) MR1215733
  29. Sokołowski, J., .Zochowski, A., 10.1137/S0363012997323230, SIAM J. Control Optim. 37 (1999), 1251-1272. (1999) MR1691940DOI10.1137/S0363012997323230
  30. Sokołowski, J., .Zochowski, A., 10.1137/S0363012901384430, SIAM J. Control Optim. 42 (2003), 1198-1221. (2003) MR2044792DOI10.1137/S0363012901384430
  31. Sokołowski, J., .Zochowski, A., 10.1007/s00211-005-0635-0, Numer. Math. 102 (2005), 145-179. (2005) MR2206676DOI10.1007/s00211-005-0635-0
  32. Sokołowski, J., .Zochowski, A., Asymptotic analysis and topological derivatives for shape and topology optimization of elasticity problems in two spatial dimensions, Prépublication IECN 16, 2007. 

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