Shape and topological sensitivity analysis in domains with cracks

Abstract

Abstract. Framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. Equilibrium problem for 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 interesting on its own. In particular, the existence of the shape and topological derivatives of the energy functional is obtained. It is shown, in fact, that the level set type method [4] can be applied to shape and topology opimization of the related variational inequalities for elasticity problems in domains with cracks, with the nonpenetration condition prescribed on the crack faces. The results presented in the paper can be used for numerical solution of shape optimization and inverse problems in structural mechanics.Key words. Crack with non-penetration, shape sensitivity, derivative of energy functional, topological derivative AMS subject classifications. Primary 35J85, 74K20 Secondary 35J25, 74M151. Introduction. Shape optimization requires few mathematical results, in the framework of modelling and numerical solution, for any specific class of problems governed by partial differential equations of mathematical physics. Usually, we need to show the well posedness of the specific problem, and also we can propose a numerical method for the effective solution procedure. Hence, in order to solve a shape optimization problem we are obliged to have the results on• the existence and continuous dependence with respect to the shape of solutions to the model, which may result in the existence of optimal shapes, • the differentiability of solutions with respect to the boundary variations, which imply the existence of shape gradients and leads to some necessary conditions for optimality, of the first order and possibly of the second order which leads to the Newton method of shape optimization, • and in addition, perform the asymptotic analysis of the related boundary value problem in singularly perturbed geometrical domains and derive the form of the topological derivative for the shape functional of interest, which allows for the topology changes in the process of numerical optimization, if necessary, • and finally, we may device a numerical method and show its efficiency in numerical examples, and its convergence form the mathematical point of view.

On the analysis of boundary value problems in nonsmooth domains

“…We present two results which are proved in [76]. The smooth domain method is applied to the two-dimensional elasticity and the Kirchhoff plate model.…”

“…It is possible to prove existence of a solution to the problem (5.10)-(5.12) and check that (5.10)-(5.12) is formally equivalent to (5.1)-(5.5) (see [65], [76]). For (5.10)-(5.12) existence can be proved independently of (5.1)-(5.5).…”

“…We should remark that the smooth domain method proposed in this paper can be applied to elastic linear crack problems as well as to many other linear and non-linear elliptic boundary value problems. For elastic bodies with non-linear cracks the smooth domain method was developed in [76].…”

“…A new approach to crack theory for linear elastic bodies with inequality type boundary conditions prescribed on the crack faces was proposed in [76]. The results of this method are summarized in this chapter.…”

Topological sensitivity analysis in heterogeneous anisotropic elasticity problem. Theoretical and computational aspects

“…This derivative measures the sensitivity of the shape functional with respect to the infinitesimal singular domain perturbation and it was rigorously introduced in [3]. Since then, this concept has proven extremely useful in the treatment of a wide range of problems; see, for instance, [4,5,6,7,8,9,10,11]. Concerning the theoretical development of the topological asymptotic analysis, besides the monograph [1], the reader is referred to [12,13,14,15].…”