June  2015, 35(6): 2625-2657. doi: 10.3934/dcds.2015.35.2625

Control of crack propagation by shape-topological optimization

1. 

Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstr. 11 (03.322), 91058 Erlangen, Germany

2. 

Institut Élie Cartan Nancy, UMR 7502 Université de Lorraine - CNRS, B.P. 70239, 54506 Vandoeuvre-Lès Nancy Cedex, France

3. 

Systems Research Institute, of the Polish Academy of Sciences, ul. Newelska 6, 01-447 Warszawa, Poland

Received  December 2013 Revised  July 2014 Published  December 2014

An elastic body weakened by small cracks is considered in the framework of unilateral variational problems in linearized elasticity. The frictionless contact conditions are prescribed on the crack lips in two spatial dimensions, or on the crack faces in three spatial dimensions. The weak solutions of the equilibrium boundary value problem for the elasticity problem are determined by minimization of the energy functional over the cone of admissible displacements. The associated elastic energy functional evaluated for the weak solutions is considered for the purpose of control of crack propagation. The singularities of the elastic displacement field at the crack front are characterized by the shape derivatives of the elastic energy with respect to the crack shape within the Griffith theory. The first order shape derivative of the elastic energy functional with respect to the crack shape, i.e., evaluated for a deformation field supported in an open neighbourhood of one of crack tips, is called the Griffith functional.
    The control of the crack front in the elastic body is performed by the optimum shape design technique. The Griffith functional is minimized with respect to the shape and the location of small inclusions in the body. The inclusions are located far from the crack. In order to minimize the Griffith functional over an admissible family of inclusions, the second order directional, mixed shape-topological derivatives of the elastic energy functional are evaluated.
    The domain decomposition technique [42] is applied to the shape [56] and topological [54,55] sensitivity analysis of variational inequalities.
    The nonlinear crack model in the framework of linear elasticity is considered in two and three spatial dimensions. The boundary value problem for the elastic displacement field takes the form of a variational inequality over the positive cone in a fractional Sobolev space. The variational inequality leads to a problem of metric projection over a polyhedric convex cone, so the concept of conical differentiability applies to shape and topological sensitivity analysis of variational inequalities under consideration.
Citation: Günter Leugering, Jan Sokołowski, Antoni Żochowski. Control of crack propagation by shape-topological optimization. Discrete and Continuous Dynamical Systems, 2015, 35 (6) : 2625-2657. doi: 10.3934/dcds.2015.35.2625
References:
[1]

A. Ancona, Sur les espaces de Dirichlet: Principes, fonction de Green, J. Math. Pures Appl., 54 (1975), 75-124.

[2]

N. Bubner, J. Sokołowski and J. Sprekels, Optimal boundary control problems for shape memory alloys under state constraints for stress and temperature, Numer. Funct. Anal. Optim., 19 (1998), 489-498. doi: 10.1080/01630569808816840.

[3]

I. I. Argatov and J. Sokołowski, Asymptotics of the energy functional in the Signorini problem under small singular perturbation of the domain, Zh. Vychisl. Mat. Mat. Fiz., 43 (2003), 744-758; translation in Comput. Math. Math. Phys., 43 (2003), 710-724.

[4]

Z. Belhachmi, J.-M. Sac-Epée and J. Sokołowski, Approximation par la méthode des élement finit de la formulation en domaine régulière de problems de fissures, C. R. Acad. Sci. Paris, Ser I, 338 (2004), 499-504. doi: 10.1016/j.crma.2004.01.008.

[5]

Z. Belhachmi, J. M. Sac-Epée and J. Sokołowski, Mixed finite element methods for smooth domain formulation of crack problems, SIAM Journal on Numerical Analysis, 43 (2005), 1295-1320. doi: 10.1137/S0036142903429729.

[6]

A. Beurling and J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. U.S.A., 45 (1959), 208-215. doi: 10.1073/pnas.45.2.208.

[7]

P. Destuynder, Remarques sur le contrôle de la propagation des fissures en régime stationnaire, C. R. Acad. Sci. Paris Sèr. II Méc. Phys. Chim. Sci. Univers Sci. Terre, 308 (1989), 697-701.

[8]

G. Fichera, Existence theorems in elasticity, in Festkörpermechanik/Mechanics of Solids, Handbuch der Physik (eds. S. Flügge and C.A. Truesdell) Springer-Verlag, (1984), 347-389. doi: 10.1007/978-3-642-69567-4_3.

[9]

G. Fichera, Boundary value problems of elasticity with unilateral constraints, in Festkörpermechanik/Mechanics of Solids, Handbuch der Physik (eds. S. Flügge and C.A. Truesdell) Springer-Verlag, (1984), 391-424. doi: 10.1007/978-3-642-69567-4_4.

[10]

G. Frémiot, Eulerian semiderivatives of the eigenvalues for Laplacian in domains with cracks, Adv. Math. Sci. Appl., 12 (2002), 115-134.

[11]

G. Frémiot and J. Sokołowski, Hadamard formula in nonsmooth domains and applications, in Partial differential equations on multistructures (Luminy, 1999), Lecture Notes in Pure and Appl. Math., 219 Dekker, (2001), 99-120.

[12]

G. Frémiot and J. Sokołowski, Shape sensitivity analysis of problems with singularities, in Shape optimization and optimal design (Cambridge, 1999), Lecture Notes in Pure and Appl. Math., Dekker, New York 216 (2001), 255-276.

[13]

G. Frémiot, W. Horn, A. Laurain, M. Rao and J. Sokołowski, On the analysis of boundary value problems in nonsmooth domains, Dissertationes Mathematicae, 462 (2009), 149pp. doi: 10.4064/dm462-0-1.

[14]

B. Hanouzet and J.-L. Joly, Méthodes d'ordre dans l'interprétation de certaines inéquations variationnelles et applications, J. Funct. Anal., 34 (1979), 217-249. doi: 10.1016/0022-1236(79)90032-6.

[15]

A. Haraux, How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities, J. Math. Soc. Japan, 29 (1977), 615-631. doi: 10.2969/jmsj/02940615.

[16]

P. Hild, A. Münch and Y. Ousset, On the active control of crack growth in elastic media, Comput. Methods Appl. Mech. Engrg., 198 (2008), 407-419. doi: 10.1016/j.cma.2008.08.010.

[17]

M. Hintermüller and A. Laurain, Optimal shape design subject to variational inequalities, SIAM J. Control Optim., 49 (2011), 1015-1047. doi: 10.1137/080745134.

[18]

M. Hintermüller and V. A. Kovtunenko, From shape variation to topology changes in constrained minimization: a velocity method based concept, in Special issue on Advances in Shape and Topology Optimization: Theory, Numerics and New Applications Areas, (eds. C. Elliott, M. Hintermüller, G. Leugering and J. Sokołowski) Optimization Methods and Software, 26 (2011) , 513-532. doi: 10.1080/10556788.2011.559548.

[19]

D. Hömberg, A. M. Khludnev and J. Sokołowski, Quasistationary problem for a cracked body with electrothermoconductivity, Interfaces Free Bound., 3 (2001), 129-142. doi: 10.4171/IFB/36.

[20]

W. Horn, J. Sokołowski and J. Sprekels, A control problem with state constraints for a phase-field model, Control Cybernet., 25 (1996), 1137-1153.

[21]

A. M. Khludnev and J. Sokołowski, Modelling and Control in Solid Mechanics, Birkhäuser, Basel-Boston-Berlin, 1997, reprinted by Springer in 2012.

[22]

A. M. Khludnev, K. Ohtsuka and J. Sokołowski, On derivative of energy functional for elastic bodies with a crack and unilateral conditions, Quarterly Appl. Math., 60 (2002), 99-109.

[23]

A. M. Khludnev, A. A. Novotny, J. Sokołowski and A. Żochowski, Shape and topology sensitivity analysis for cracks in elastic bodies on boundaries of rigid inclusions, J. Mech. Phys. Solids, 57 (2009), 1718-1732. doi: 10.1016/j.jmps.2009.07.003.

[24]

A. M. Khludnev and J. Sokołowski, Smooth domain method for crack problems, Quart. Appl. Math., 62 (2004), 401-422.

[25]

A. M. Khludnev and J. Sokołowski, On solvability of boundary value problems in elastoplasicity, Control and Cybernetics, 27 (1998), 311-330.

[26]

A. M. Khludnev, Optimal control of crack growth in elastic body with inclusions, Europ. J. Mech. A/Solids, 29 (2010), 392-399. doi: 10.1016/j.euromechsol.2009.10.010.

[27]

A. M. Khludnev and V. A. Kovtunenko, Analysis of Cracks in Solids, WIT Press,Southampton-Boston, 2000.

[28]

A. M. Khludnev and G. Leugering, Optimal control of cracks in elastic bodies with thin rigid inclusions, Z. Angew. Math. Mech., 91 (2011), 125-137. doi: 10.1002/zamm.201000058.

[29]

A. M. Khludnev, G. Leugering and M. Specovius-Neugebauer, Optimal control of inclusion and crack shapes in elastic bodies, Journal of Optimization Theory and Applications, 155 (2012), 54-78. doi: 10.1007/s10957-012-0053-2.

[30]

A. M. Khludnev and J. Sokołowski, Griffith formulae for elasticity systems with unilateral conditions in domains with cracks, Eur. J. Mech. A Solids, 19 (2000), 105-119. doi: 10.1016/S0997-7538(00)00138-8.

[31]

A. M. Khludnev and J. Sokołowski, Griffith formula and Rice-Cherepanov's integral for elliptic equations with unilateral conditions in nonsmooth domains, in Optimal control of partial differential equations (Chemnitz, 1998), Internat. Ser. Numer. Math., 133, Birkhäuser, Basel, (1999), 211-219.

[32]

A. M. Khludnev and J. Sokołowski, The Griffith formula and the Rice-Cherepanov integral for crack problems with unilateral conditions in nonsmooth domains, European J. Appl. Math., 10 (1999), 379-394. doi: 10.1017/S0956792599003885.

[33]

A. M. Khludnev, J. Sokołowski and K. Szulc, Shape and topological sensitivity analysis in domains with cracks, Appl. Math., 55 (2010), 433-469. doi: 10.1007/s10492-010-0018-4.

[34]

T. Lewiński and J. Sokołowski, Energy change due to the appearance of cavities in elastic solids, International Journal of Solids and Structures, 40 (2003), 1765-1803.

[35]

G. Leugering, M. Prechtel, P. Steinmann and M. Stingl, A cohesive crack propagation model: Mathematical theory and numerical solution, Commun. Pure Appl. Anal., 12 (2013), 1705-1729. doi: 10.3934/cpaa.2013.12.1705.

[36]

F. Mignot, Contrôle dans les inéquations variationelles elliptiques, J. Functional Analysis, 22 (1976), 130-185. doi: 10.1016/0022-1236(76)90017-3.

[37]

A. Münch and P. Pedregal, Relaxation of an optimal design problem in fracture mechanics: The anti-plane case, ESAIM Control Optim. Calc. Var., 16 (2010), 719-743. doi: 10.1051/cocv/2009019.

[38]

S. A. Nazarov and J. Sokołowski, On asymptotic analysis of spectral problems in elasticity, Lat. Am. J. Solids Struct., 8 (2011), 27-54. doi: 10.1590/S1679-78252011000100003.

[39]

S. A. Nazarov, J. Sokołowski and M. Specovius-Neugebauer, Polarization matrices in anisotropic heterogeneous elasticity, Asymptot. Anal., 68 (2010), 189-221.

[40]

S. A. Nazarov and J. Sokołowski, Shape sensitivity analysis of eigenvalues revisited, Control Cybernet., 37 (2008), 999-1012.

[41]

S. A. Nazarov and J. Sokołowski, Spectral problems in the shape optimisation. Singular boundary perturbations, Asymptot. Anal., 56 (2008), 159-204.

[42]

A. A. Novotny and J. Sokołowski, Topological Derivatives in Shape Optimization, Series: Interaction of Mechanics and Mathematics, Springer-Verlag, 2013. doi: 10.1007/978-3-642-35245-4.

[43]

P. Plotnikov and J. Sokołowski, Compressible Navier-Stokes Equations. Theory and Shape Optimization, Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], 73 Birkhäuser/Springer Basel AG, Basel, 2012. doi: 10.1007/978-3-0348-0367-0.

[44]

M. Prechtel, G. Leugering, P. Steinmann and M. Stingl, Towards optimization of crack resistance of composite materials by adjusting of fiber shapes, Engineering Fracture Mechanics, 78 (2011), 944-960. doi: 10.1016/j.engfracmech.2011.01.007.

[45]

M. Prechtel, P. Leiva Ronda, R. Janisch, G. Leugering, A. Hartmaier and P. Steinmann, Cohesive Element Model for Simulation of Crack Growth in Composite Materials, International Conference on crack paths, Vicenza, Italy, 2009.

[46]

M. Prechtel, P. Leiva Ronda, R. Janisch, A. Hartmaier, G. Leugering, P. Steinmann and M. Stingl, Simulation of fracture in heterogeneous elastic materials with cohesive zone models, Int. J. Fract., 168 (2010), 15-29. doi: 10.1007/s10704-010-9552-z.

[47]

G. Leugering, M. Prechtel, P. Steinmann and M. Stingl, A cohesive crack propagation model: Mathematical theory and numerical solution, Commun. Pure Appl. Anal., 12 (2013), 1705-1729. doi: 10.3934/cpaa.2013.12.1705.

[48]

V. V. Saurin, Shape design sensitivity analysis for fracture conditions, Computers and Structures, 76 (2001), 399-405. doi: 10.1016/S0045-7949(99)00154-6.

[49]

S. Seelecke, C. Büskens, I. Müller and J. Sprekels, Real-time optimal control of shape memory alloy actuators in smart structure, in Online Optimization of Large Scale Systems: State of the Art, (eds. M. Grötschel, J. Rambau), Springer Verlag, (2001), 93-104.

[50]

J. Sokołowski and J. Sprekels, Control problems for shape memory alloys with constraints on the shear strain, in Control of partial differential equations (Trento, 1993), Lecture Notes in Pure and Appl. Math., 165, Dekker, New York, (1994), 189-195.

[51]

J. Sokołowski and J. Sprekels, Control problems with state constraints for shape memory alloys, Math. Methods Appl. Sci., 17 (1994), 943-952. doi: 10.1002/mma.1670171204.

[52]

J. Sokołowski and J. Sprekels, Dynamical shape control of nonlinear thin rods, in Optimal control of partial differential equations (Irsee, 1990), Lecture Notes in Control and Inform. Sci., 149, Springer, Berlin, (1991), 202-208. doi: 10.1007/BFb0043225.

[53]

J. Sokołowski and J. Sprekels, Dynamical shape control and the stabilization of nonlinear thin rods, Math. Methods Appl. Sci., 14 (1991), 63-78. doi: 10.1002/mma.1670140104.

[54]

J. Sokołowski and A. Żochowski, Modelling of topological derivatives for contact problems, Numer. Math., 102 (2005), 145-179. doi: 10.1007/s00211-005-0635-0.

[55]

J. Sokołowski and A. Żochowski, Topological derivatives for optimization of plane elasticity contact problems, Engineering Analysis with Boundary Elements, 32 (2008), 900-908.

[56]

J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization. Shape Sensitivity Analysis, Springer Ser. Comput. Math. 16, Springer, Berlin, 1992, reprinted in 2013.

show all references

References:
[1]

A. Ancona, Sur les espaces de Dirichlet: Principes, fonction de Green, J. Math. Pures Appl., 54 (1975), 75-124.

[2]

N. Bubner, J. Sokołowski and J. Sprekels, Optimal boundary control problems for shape memory alloys under state constraints for stress and temperature, Numer. Funct. Anal. Optim., 19 (1998), 489-498. doi: 10.1080/01630569808816840.

[3]

I. I. Argatov and J. Sokołowski, Asymptotics of the energy functional in the Signorini problem under small singular perturbation of the domain, Zh. Vychisl. Mat. Mat. Fiz., 43 (2003), 744-758; translation in Comput. Math. Math. Phys., 43 (2003), 710-724.

[4]

Z. Belhachmi, J.-M. Sac-Epée and J. Sokołowski, Approximation par la méthode des élement finit de la formulation en domaine régulière de problems de fissures, C. R. Acad. Sci. Paris, Ser I, 338 (2004), 499-504. doi: 10.1016/j.crma.2004.01.008.

[5]

Z. Belhachmi, J. M. Sac-Epée and J. Sokołowski, Mixed finite element methods for smooth domain formulation of crack problems, SIAM Journal on Numerical Analysis, 43 (2005), 1295-1320. doi: 10.1137/S0036142903429729.

[6]

A. Beurling and J. Deny, Dirichlet spaces, Proc. Nat. Acad. Sci. U.S.A., 45 (1959), 208-215. doi: 10.1073/pnas.45.2.208.

[7]

P. Destuynder, Remarques sur le contrôle de la propagation des fissures en régime stationnaire, C. R. Acad. Sci. Paris Sèr. II Méc. Phys. Chim. Sci. Univers Sci. Terre, 308 (1989), 697-701.

[8]

G. Fichera, Existence theorems in elasticity, in Festkörpermechanik/Mechanics of Solids, Handbuch der Physik (eds. S. Flügge and C.A. Truesdell) Springer-Verlag, (1984), 347-389. doi: 10.1007/978-3-642-69567-4_3.

[9]

G. Fichera, Boundary value problems of elasticity with unilateral constraints, in Festkörpermechanik/Mechanics of Solids, Handbuch der Physik (eds. S. Flügge and C.A. Truesdell) Springer-Verlag, (1984), 391-424. doi: 10.1007/978-3-642-69567-4_4.

[10]

G. Frémiot, Eulerian semiderivatives of the eigenvalues for Laplacian in domains with cracks, Adv. Math. Sci. Appl., 12 (2002), 115-134.

[11]

G. Frémiot and J. Sokołowski, Hadamard formula in nonsmooth domains and applications, in Partial differential equations on multistructures (Luminy, 1999), Lecture Notes in Pure and Appl. Math., 219 Dekker, (2001), 99-120.

[12]

G. Frémiot and J. Sokołowski, Shape sensitivity analysis of problems with singularities, in Shape optimization and optimal design (Cambridge, 1999), Lecture Notes in Pure and Appl. Math., Dekker, New York 216 (2001), 255-276.

[13]

G. Frémiot, W. Horn, A. Laurain, M. Rao and J. Sokołowski, On the analysis of boundary value problems in nonsmooth domains, Dissertationes Mathematicae, 462 (2009), 149pp. doi: 10.4064/dm462-0-1.

[14]

B. Hanouzet and J.-L. Joly, Méthodes d'ordre dans l'interprétation de certaines inéquations variationnelles et applications, J. Funct. Anal., 34 (1979), 217-249. doi: 10.1016/0022-1236(79)90032-6.

[15]

A. Haraux, How to differentiate the projection on a convex set in Hilbert space. Some applications to variational inequalities, J. Math. Soc. Japan, 29 (1977), 615-631. doi: 10.2969/jmsj/02940615.

[16]

P. Hild, A. Münch and Y. Ousset, On the active control of crack growth in elastic media, Comput. Methods Appl. Mech. Engrg., 198 (2008), 407-419. doi: 10.1016/j.cma.2008.08.010.

[17]

M. Hintermüller and A. Laurain, Optimal shape design subject to variational inequalities, SIAM J. Control Optim., 49 (2011), 1015-1047. doi: 10.1137/080745134.

[18]

M. Hintermüller and V. A. Kovtunenko, From shape variation to topology changes in constrained minimization: a velocity method based concept, in Special issue on Advances in Shape and Topology Optimization: Theory, Numerics and New Applications Areas, (eds. C. Elliott, M. Hintermüller, G. Leugering and J. Sokołowski) Optimization Methods and Software, 26 (2011) , 513-532. doi: 10.1080/10556788.2011.559548.

[19]

D. Hömberg, A. M. Khludnev and J. Sokołowski, Quasistationary problem for a cracked body with electrothermoconductivity, Interfaces Free Bound., 3 (2001), 129-142. doi: 10.4171/IFB/36.

[20]

W. Horn, J. Sokołowski and J. Sprekels, A control problem with state constraints for a phase-field model, Control Cybernet., 25 (1996), 1137-1153.

[21]

A. M. Khludnev and J. Sokołowski, Modelling and Control in Solid Mechanics, Birkhäuser, Basel-Boston-Berlin, 1997, reprinted by Springer in 2012.

[22]

A. M. Khludnev, K. Ohtsuka and J. Sokołowski, On derivative of energy functional for elastic bodies with a crack and unilateral conditions, Quarterly Appl. Math., 60 (2002), 99-109.

[23]

A. M. Khludnev, A. A. Novotny, J. Sokołowski and A. Żochowski, Shape and topology sensitivity analysis for cracks in elastic bodies on boundaries of rigid inclusions, J. Mech. Phys. Solids, 57 (2009), 1718-1732. doi: 10.1016/j.jmps.2009.07.003.

[24]

A. M. Khludnev and J. Sokołowski, Smooth domain method for crack problems, Quart. Appl. Math., 62 (2004), 401-422.

[25]

A. M. Khludnev and J. Sokołowski, On solvability of boundary value problems in elastoplasicity, Control and Cybernetics, 27 (1998), 311-330.

[26]

A. M. Khludnev, Optimal control of crack growth in elastic body with inclusions, Europ. J. Mech. A/Solids, 29 (2010), 392-399. doi: 10.1016/j.euromechsol.2009.10.010.

[27]

A. M. Khludnev and V. A. Kovtunenko, Analysis of Cracks in Solids, WIT Press,Southampton-Boston, 2000.

[28]

A. M. Khludnev and G. Leugering, Optimal control of cracks in elastic bodies with thin rigid inclusions, Z. Angew. Math. Mech., 91 (2011), 125-137. doi: 10.1002/zamm.201000058.

[29]

A. M. Khludnev, G. Leugering and M. Specovius-Neugebauer, Optimal control of inclusion and crack shapes in elastic bodies, Journal of Optimization Theory and Applications, 155 (2012), 54-78. doi: 10.1007/s10957-012-0053-2.

[30]

A. M. Khludnev and J. Sokołowski, Griffith formulae for elasticity systems with unilateral conditions in domains with cracks, Eur. J. Mech. A Solids, 19 (2000), 105-119. doi: 10.1016/S0997-7538(00)00138-8.

[31]

A. M. Khludnev and J. Sokołowski, Griffith formula and Rice-Cherepanov's integral for elliptic equations with unilateral conditions in nonsmooth domains, in Optimal control of partial differential equations (Chemnitz, 1998), Internat. Ser. Numer. Math., 133, Birkhäuser, Basel, (1999), 211-219.

[32]

A. M. Khludnev and J. Sokołowski, The Griffith formula and the Rice-Cherepanov integral for crack problems with unilateral conditions in nonsmooth domains, European J. Appl. Math., 10 (1999), 379-394. doi: 10.1017/S0956792599003885.

[33]

A. M. Khludnev, J. Sokołowski and K. Szulc, Shape and topological sensitivity analysis in domains with cracks, Appl. Math., 55 (2010), 433-469. doi: 10.1007/s10492-010-0018-4.

[34]

T. Lewiński and J. Sokołowski, Energy change due to the appearance of cavities in elastic solids, International Journal of Solids and Structures, 40 (2003), 1765-1803.

[35]

G. Leugering, M. Prechtel, P. Steinmann and M. Stingl, A cohesive crack propagation model: Mathematical theory and numerical solution, Commun. Pure Appl. Anal., 12 (2013), 1705-1729. doi: 10.3934/cpaa.2013.12.1705.

[36]

F. Mignot, Contrôle dans les inéquations variationelles elliptiques, J. Functional Analysis, 22 (1976), 130-185. doi: 10.1016/0022-1236(76)90017-3.

[37]

A. Münch and P. Pedregal, Relaxation of an optimal design problem in fracture mechanics: The anti-plane case, ESAIM Control Optim. Calc. Var., 16 (2010), 719-743. doi: 10.1051/cocv/2009019.

[38]

S. A. Nazarov and J. Sokołowski, On asymptotic analysis of spectral problems in elasticity, Lat. Am. J. Solids Struct., 8 (2011), 27-54. doi: 10.1590/S1679-78252011000100003.

[39]

S. A. Nazarov, J. Sokołowski and M. Specovius-Neugebauer, Polarization matrices in anisotropic heterogeneous elasticity, Asymptot. Anal., 68 (2010), 189-221.

[40]

S. A. Nazarov and J. Sokołowski, Shape sensitivity analysis of eigenvalues revisited, Control Cybernet., 37 (2008), 999-1012.

[41]

S. A. Nazarov and J. Sokołowski, Spectral problems in the shape optimisation. Singular boundary perturbations, Asymptot. Anal., 56 (2008), 159-204.

[42]

A. A. Novotny and J. Sokołowski, Topological Derivatives in Shape Optimization, Series: Interaction of Mechanics and Mathematics, Springer-Verlag, 2013. doi: 10.1007/978-3-642-35245-4.

[43]

P. Plotnikov and J. Sokołowski, Compressible Navier-Stokes Equations. Theory and Shape Optimization, Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], 73 Birkhäuser/Springer Basel AG, Basel, 2012. doi: 10.1007/978-3-0348-0367-0.

[44]

M. Prechtel, G. Leugering, P. Steinmann and M. Stingl, Towards optimization of crack resistance of composite materials by adjusting of fiber shapes, Engineering Fracture Mechanics, 78 (2011), 944-960. doi: 10.1016/j.engfracmech.2011.01.007.

[45]

M. Prechtel, P. Leiva Ronda, R. Janisch, G. Leugering, A. Hartmaier and P. Steinmann, Cohesive Element Model for Simulation of Crack Growth in Composite Materials, International Conference on crack paths, Vicenza, Italy, 2009.

[46]

M. Prechtel, P. Leiva Ronda, R. Janisch, A. Hartmaier, G. Leugering, P. Steinmann and M. Stingl, Simulation of fracture in heterogeneous elastic materials with cohesive zone models, Int. J. Fract., 168 (2010), 15-29. doi: 10.1007/s10704-010-9552-z.

[47]

G. Leugering, M. Prechtel, P. Steinmann and M. Stingl, A cohesive crack propagation model: Mathematical theory and numerical solution, Commun. Pure Appl. Anal., 12 (2013), 1705-1729. doi: 10.3934/cpaa.2013.12.1705.

[48]

V. V. Saurin, Shape design sensitivity analysis for fracture conditions, Computers and Structures, 76 (2001), 399-405. doi: 10.1016/S0045-7949(99)00154-6.

[49]

S. Seelecke, C. Büskens, I. Müller and J. Sprekels, Real-time optimal control of shape memory alloy actuators in smart structure, in Online Optimization of Large Scale Systems: State of the Art, (eds. M. Grötschel, J. Rambau), Springer Verlag, (2001), 93-104.

[50]

J. Sokołowski and J. Sprekels, Control problems for shape memory alloys with constraints on the shear strain, in Control of partial differential equations (Trento, 1993), Lecture Notes in Pure and Appl. Math., 165, Dekker, New York, (1994), 189-195.

[51]

J. Sokołowski and J. Sprekels, Control problems with state constraints for shape memory alloys, Math. Methods Appl. Sci., 17 (1994), 943-952. doi: 10.1002/mma.1670171204.

[52]

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