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The standard topological derivative methodology developed by the authors in many papers requires knowledge of point-wise values of solutions to partial differential equations or variational equalities. However, the contact or unilateral problems are studied in the energy space setting, wsznumer2005here point-wise values are not defined. The authors proposed the approach based on domain decomposition...
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...
Formulae for the first order expansions of the Steklov-Poincaré operators in the case of the Laplace operator and of the elasticity boundary value problems in singularly perturbed domains in ℝ3 are presented. Such expansions are required for the evaluation of topological derivatives of the energy shape functionals.
The non stationary, compressible Navier-Stokes equations are considered in a bounded hold-all domain. The nonhomegeneous boundary conditions are prescribed on the boundary of hold-all domain. The existence of the so-called weak normalized solutions for the model is established in [2]. In this talk we consider the associated shape optimization problems for the model. In the stationary case the drag...
In the paper the first order sensitivity analysis is performed for a class of optimal control problems for parabolic-hyperbolic equations. A singular perturbation of geometrical domain of integration is introduced in the form of a circular hole. The Steklov-Poincaré operator on a circle is defined in order to reduce the problem to regular perturbations in the truncated domain. The optimality system...
We consider an elastic body with a rigid inclusion and a crack located at the boundary of the inclusion. It is assumed that non-penetration conditions are imposed at the crack faces which do not allow the opposite crack faces to penetrate each other. The differentiability of the elastic energy with respect to the crack length, for the crack located at the boundary of rigid inclusion, is established.
Shape optimization problem for semilinear elliptic equation is considered. There is an optimal solution which is computed by the Levelset method. To this end the shape derivative of the functional is evaluated. In order to predict the topology changes the topological derivative is employed. Numerical results confirm that the proposed framework for numerical solution of shape optimization problems...
In the series of papers the mathematical theory of shape optimization for compressible Navier-Stokes inhomogeneous boundary value problems is developed. The key part of the theory include the new results on the existence and shape differentiability of the weak solutions to compressible Navier-Stokes equations. In particular, our results lead to the rigorous mathematical framework for the drag minimization...
In the paper the first order sensitivity analysis is performed for a class of optimal control problems for infinite order hyperbolic equations. A singular perturbation of geometrical domain of integration is introduced in the form of a circular hole. The Steklov-Poincaré operator on a circle is defined in order to reduce the problem to regular perturbations in the truncated domain. The optimality...
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