Shape-topological differentiability of energy functionals for unilateral problems in domains with cracks and applications. (English) Zbl 1319.49068
Hoppe, Ronald (ed.), Optimization with PDE constraints. ESF networking program ‘OPTPDE’. Cham: Springer (ISBN 978-3-319-08024-6/hbk; 978-3-319-08025-3/ebook). Lecture Notes in Computational Science and Engineering 101, 243-284 (2014).
Summary: A review of results on first order shape-topological differentiability of energy functionals for a class of variational inequalities of elliptic type is presented.
The velocity method in shape sensitivity analysis for solutions of elliptic unilateral problems is established in the monograph [J. Sokołowski and J.-P. Zolésio, Introduction to shape optimization: shape sensitivity analysis. Springer Series in Computational Mathematics 16. Berlin: Springer (1992; Zbl 0761.73003)]. The shape and material derivatives of solutions to frictionless contact problems in solid mechanics are obtained. In this way, the shape gradients of the associated integral functionals are derived within the framework of nonsmooth analysis. In the case of energy type functionals, classical differentiability results can be obtained, because the shape differentiability of solutions is not required to obtain the shape gradient of the shape functional [loc. cit.]. Therefore, for cracks the strong continuity of solutions with respect to boundary variations is sufficient in order to obtain first order shape differentiability of the associated energy functional. This simple observation which is used in [loc. cit.] for the shape differentiability of multiple eigenvalues is further applied in [A. M. Khludnev and J. Sokołowski, Eur. J. Appl. Math. 10, No. 4, 379–394 (1999; Zbl 0945.74058); Eur. J. Mech., A, Solids 19, No. 1, 105–119 (2000; Zbl 0966.74061)] to derive the first order shape gradient of the energy functional with respect to perturbations of the crack tip. A domain decomposition technique in shape-topology sensitivity analysis for problems with unilateral constraints on the crack faces (lips) is presented for the shape functionals. We introduce the Griffith shape functional as the distributed shape derivative of the elastic energy evaluated in a domain with a crack, with respect to the crack length. We are interested in the dependence of this functional on domain perturbations far from the crack. As a result, the directional shape and topological derivatives of the nonsmooth Griffith shape functional are obtained with respect to boundary variations of an inclusion.
For the entire collection see [Zbl 1297.49002].
The velocity method in shape sensitivity analysis for solutions of elliptic unilateral problems is established in the monograph [J. Sokołowski and J.-P. Zolésio, Introduction to shape optimization: shape sensitivity analysis. Springer Series in Computational Mathematics 16. Berlin: Springer (1992; Zbl 0761.73003)]. The shape and material derivatives of solutions to frictionless contact problems in solid mechanics are obtained. In this way, the shape gradients of the associated integral functionals are derived within the framework of nonsmooth analysis. In the case of energy type functionals, classical differentiability results can be obtained, because the shape differentiability of solutions is not required to obtain the shape gradient of the shape functional [loc. cit.]. Therefore, for cracks the strong continuity of solutions with respect to boundary variations is sufficient in order to obtain first order shape differentiability of the associated energy functional. This simple observation which is used in [loc. cit.] for the shape differentiability of multiple eigenvalues is further applied in [A. M. Khludnev and J. Sokołowski, Eur. J. Appl. Math. 10, No. 4, 379–394 (1999; Zbl 0945.74058); Eur. J. Mech., A, Solids 19, No. 1, 105–119 (2000; Zbl 0966.74061)] to derive the first order shape gradient of the energy functional with respect to perturbations of the crack tip. A domain decomposition technique in shape-topology sensitivity analysis for problems with unilateral constraints on the crack faces (lips) is presented for the shape functionals. We introduce the Griffith shape functional as the distributed shape derivative of the elastic energy evaluated in a domain with a crack, with respect to the crack length. We are interested in the dependence of this functional on domain perturbations far from the crack. As a result, the directional shape and topological derivatives of the nonsmooth Griffith shape functional are obtained with respect to boundary variations of an inclusion.
For the entire collection see [Zbl 1297.49002].
MSC:
| 49Q10 | Optimization of shapes other than minimal surfaces |
| 49Q12 | Sensitivity analysis for optimization problems on manifolds |
| 49J40 | Variational inequalities |
| 49J52 | Nonsmooth analysis |
| 35J86 | Unilateral problems for linear elliptic equations and variational inequalities with linear elliptic operators |
| 35R35 | Free boundary problems for PDEs |
| 74M15 | Contact in solid mechanics |
| 74R99 | Fracture and damage |
