Control of crack propagation by shape-topological optimization. (English) Zbl 1335.35073
Summary: 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 is applied to the shape and topological 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.
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 is applied to the shape and topological 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.
MSC:
| 35J86 | Unilateral problems for linear elliptic equations and variational inequalities with linear elliptic operators |
| 49K40 | Sensitivity, stability, well-posedness |
| 74R10 | Brittle fracture |
| 31C25 | Dirichlet forms |
| 35D30 | Weak solutions to PDEs |
Keywords:
variational inequality; Griffith’s functional; crack propagation; shape derivative; topological derivative; elastic body; unilateral variational problems; linearized elasticity; weak solutionsReferences:
| [1] | A. Ancona, Sur les espaces de Dirichlet: Principes, fonction de Green,, J. Math. Pures Appl., 54, 75 (1975) · Zbl 0286.31008 |
| [2] | N. Bubner, Optimal boundary control problems for shape memory alloys under state constraints for stress and temperature,, Numer. Funct. Anal. Optim., 19, 489 (1998) · Zbl 0907.49012 · doi:10.1080/01630569808816840 |
| [3] | I. I. Argatov, Asymptotics of the energy functional in the Signorini problem under small singular perturbation of the domain,, Zh. Vychisl. Mat. Mat. Fiz., 43, 744 (2003) · Zbl 1074.35007 |
| [4] | Z. Belhachmi, Approximation par la méthode des élement finit de la formulation en domaine régulière de problems de fissures,, C. R. Acad. Sci. Paris, 338, 499 (2004) · Zbl 1116.74424 · doi:10.1016/j.crma.2004.01.008 |
| [5] | Z. Belhachmi, Mixed finite element methods for smooth domain formulation of crack problems,, SIAM Journal on Numerical Analysis, 43, 1295 (2005) · Zbl 1319.74027 · doi:10.1137/S0036142903429729 |
| [6] | A. Beurling, Dirichlet spaces,, Proc. Nat. Acad. Sci. U.S.A., 45, 208 (1959) · Zbl 0089.08201 · 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, 697 (1989) · Zbl 0665.73075 |
| [8] | G. Fichera, Existence theorems in elasticity,, in Festkörpermechanik/Mechanics of Solids, 347 (1984) · 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, 391 (1984) · 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, 115 (2002) · Zbl 1030.49044 |
| [11] | G. Frémiot, Hadamard formula in nonsmooth domains and applications,, in Partial differential equations on multistructures (Luminy, 219, 99 (2001) · Zbl 0993.35016 |
| [12] | G. Frémiot, Shape sensitivity analysis of problems with singularities,, in Shape optimization and optimal design (Cambridge, 216, 255 (2001) · Zbl 0984.49024 |
| [13] | G. Frémiot, On the analysis of boundary value problems in nonsmooth domains,, Dissertationes Mathematicae, 462 (2009) · Zbl 1172.35407 · doi:10.4064/dm462-0-1 |
| [14] | B. Hanouzet, Méthodes d’ordre dans l’interprétation de certaines inéquations variationnelles et applications,, J. Funct. Anal., 34, 217 (1979) · Zbl 0425.49009 · 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, 615 (1977) · Zbl 0387.46022 · doi:10.2969/jmsj/02940615 |
| [16] | P. Hild, On the active control of crack growth in elastic media,, Comput. Methods Appl. Mech. Engrg., 198, 407 (2008) · Zbl 1228.74052 · doi:10.1016/j.cma.2008.08.010 |
| [17] | M. Hintermüller, Optimal shape design subject to variational inequalities,, SIAM J. Control Optim., 49, 1015 (2011) · Zbl 1228.49045 · doi:10.1137/080745134 |
| [18] | M. Hintermüller, 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, 26, 513 (2011) · Zbl 1245.49058 · doi:10.1080/10556788.2011.559548 |
| [19] | D. Hömberg, Quasistationary problem for a cracked body with electrothermoconductivity,, Interfaces Free Bound., 3, 129 (2001) · Zbl 0985.35095 · doi:10.4171/IFB/36 |
| [20] | W. Horn, A control problem with state constraints for a phase-field model,, Control Cybernet., 25, 1137 (1996) · Zbl 1023.49500 |
| [21] | A. M. Khludnev, <em>Modelling and Control in Solid Mechanics</em>,, Birkhäuser (1997) · Zbl 0865.73003 |
| [22] | A. M. Khludnev, On derivative of energy functional for elastic bodies with a crack and unilateral conditions,, Quarterly Appl. Math., 60, 99 (2002) · Zbl 1075.74040 |
| [23] | A. M. Khludnev, Shape and topology sensitivity analysis for cracks in elastic bodies on boundaries of rigid inclusions,, J. Mech. Phys. Solids, 57, 1718 (2009) · Zbl 1425.74042 · doi:10.1016/j.jmps.2009.07.003 |
| [24] | A. M. Khludnev, Smooth domain method for crack problems,, Quart. Appl. Math., 62, 401 (2004) · Zbl 1067.74056 |
| [25] | A. M. Khludnev, On solvability of boundary value problems in elastoplasicity,, Control and Cybernetics, 27, 311 (1998) · Zbl 0970.74026 |
| [26] | A. M. Khludnev, Optimal control of crack growth in elastic body with inclusions,, Europ. J. Mech. A/Solids, 29, 392 (2010) · Zbl 1480.74272 · doi:10.1016/j.euromechsol.2009.10.010 |
| [27] | A. M. Khludnev, <em>Analysis of Cracks in Solids</em>,, WIT Press (2000) · Zbl 1004.74035 |
| [28] | A. M. Khludnev, Optimal control of cracks in elastic bodies with thin rigid inclusions,, Z. Angew. Math. Mech., 91, 125 (2011) · Zbl 1370.74136 · doi:10.1002/zamm.201000058 |
| [29] | A. M. Khludnev, Optimal control of inclusion and crack shapes in elastic bodies,, Journal of Optimization Theory and Applications, 155, 54 (2012) · Zbl 1287.49046 · doi:10.1007/s10957-012-0053-2 |
| [30] | A. M. Khludnev, Griffith formulae for elasticity systems with unilateral conditions in domains with cracks,, Eur. J. Mech. A Solids, 19, 105 (2000) · Zbl 0966.74061 · doi:10.1016/S0997-7538(00)00138-8 |
| [31] | A. M. Khludnev, Griffith formula and Rice-Cherepanov’s integral for elliptic equations with unilateral conditions in nonsmooth domains, in, Optimal control of partial differential equations (Chemnitz, 133, 211 (1999) · Zbl 0935.35057 |
| [32] | A. M. Khludnev, The Griffith formula and the Rice-Cherepanov integral for crack problems with unilateral conditions in nonsmooth domains,, European J. Appl. Math., 10, 379 (1999) · Zbl 0945.74058 · doi:10.1017/S0956792599003885 |
| [33] | A. M. Khludnev, Shape and topological sensitivity analysis in domains with cracks,, Appl. Math., 55, 433 (2010) · Zbl 1224.49014 · doi:10.1007/s10492-010-0018-4 |
| [34] | T. Lewiński, Energy change due to the appearance of cavities in elastic solids,, International Journal of Solids and Structures, 40, 1765 (2003) · Zbl 1035.74009 |
| [35] | G. Leugering, A cohesive crack propagation model: Mathematical theory and numerical solution,, Commun. Pure Appl. Anal., 12, 1705 (2013) · Zbl 1345.74100 · doi:10.3934/cpaa.2013.12.1705 |
| [36] | F. Mignot, Contrôle dans les inéquations variationelles elliptiques,, J. Functional Analysis, 22, 130 (1976) · Zbl 0364.49003 · doi:10.1016/0022-1236(76)90017-3 |
| [37] | A. Münch, Relaxation of an optimal design problem in fracture mechanics: The anti-plane case,, ESAIM Control Optim. Calc. Var., 16, 719 (2010) · Zbl 1221.49074 · doi:10.1051/cocv/2009019 |
| [38] | S. A. Nazarov, On asymptotic analysis of spectral problems in elasticity,, Lat. Am. J. Solids Struct., 8, 27 (2011) · doi:10.1590/S1679-78252011000100003 |
| [39] | S. A. Nazarov, Polarization matrices in anisotropic heterogeneous elasticity,, Asymptot. Anal., 68, 189 (2010) · Zbl 1213.35386 |
| [40] | S. A. Nazarov, Shape sensitivity analysis of eigenvalues revisited,, Control Cybernet., 37, 999 (2008) · Zbl 1188.49044 |
| [41] | S. A. Nazarov, Spectral problems in the shape optimisation. Singular boundary perturbations,, Asymptot. Anal., 56, 159 (2008) · Zbl 1157.35006 |
| [42] | A. A. Novotny, <em>Topological Derivatives in Shape Optimization</em>,, Series: Interaction of Mechanics and Mathematics (2013) · Zbl 1276.35002 · doi:10.1007/978-3-642-35245-4 |
| [43] | P. Plotnikov, <em>Compressible Navier-Stokes Equations. Theory and Shape Optimization</em>,, Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], 73 (2012) · Zbl 1260.35002 · doi:10.1007/978-3-0348-0367-0 |
| [44] | M. Prechtel, Towards optimization of crack resistance of composite materials by adjusting of fiber shapes,, Engineering Fracture Mechanics, 78, 944 (2011) · doi:10.1016/j.engfracmech.2011.01.007 |
| [45] | M. Prechtel, <em>Cohesive Element Model for Simulation of Crack Growth in Composite Materials,</em>, International Conference on crack paths (2009) |
| [46] | M. Prechtel, Simulation of fracture in heterogeneous elastic materials with cohesive zone models,, Int. J. Fract., 168, 15 (2010) · Zbl 1283.74073 · doi:10.1007/s10704-010-9552-z |
| [47] | G. Leugering, A cohesive crack propagation model: Mathematical theory and numerical solution,, Commun. Pure Appl. Anal., 12, 1705 (2013) · Zbl 1345.74100 · doi:10.3934/cpaa.2013.12.1705 |
| [48] | V. V. Saurin, Shape design sensitivity analysis for fracture conditions,, Computers and Structures, 76, 399 (2001) · doi:10.1016/S0045-7949(99)00154-6 |
| [49] | S. Seelecke, Real-time optimal control of shape memory alloy actuators in smart structure, in, Online Optimization of Large Scale Systems: State of the Art, 93 (2001) · Zbl 1002.74070 |
| [50] | J. Sokołowski, Control problems for shape memory alloys with constraints on the shear strain, in, Control of partial differential equations (Trento, 165, 189 (1994) · Zbl 0925.93419 |
| [51] | J. Sokołowski, Control problems with state constraints for shape memory alloys,, Math. Methods Appl. Sci., 17, 943 (1994) · Zbl 0809.35140 · doi:10.1002/mma.1670171204 |
| [52] | J. Sokołowski, Dynamical shape control of nonlinear thin rods, in, Optimal control of partial differential equations (Irsee, 149, 202 (1991) · Zbl 0784.49017 · doi:10.1007/BFb0043225 |
| [53] | J. Sokołowski, Dynamical shape control and the stabilization of nonlinear thin rods,, Math. Methods Appl. Sci., 14, 63 (1991) · Zbl 0725.93073 · doi:10.1002/mma.1670140104 |
| [54] | J. Sokołowski, Modelling of topological derivatives for contact problems,, Numer. Math., 102, 145 (2005) · Zbl 1077.74039 · doi:10.1007/s00211-005-0635-0 |
| [55] | J. Sokołowski, Topological derivatives for optimization of plane elasticity contact problems,, Engineering Analysis with Boundary Elements, 32, 900 (2008) · Zbl 1244.74108 |
| [56] | J. Sokołowski, <em>Introduction to Shape Optimization. Shape Sensitivity Analysis</em>,, Springer Ser. Comput. Math. 16, 16 (1992) · Zbl 0761.73003 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
