Antoni Żochowski

TOPOLOGICAL DERIVATIVES OF SHAPE FUNCTIONALS FOR ELASTICITY SYSTEMS

The form of topological derivative (TD) in three-dimensional (3D) elasticity is derived for an arbitrary shape functional. TD is used for numerical solution of shape optimization problems in structural mechanics in the framework of the so-called bubble method, as well as for numerical solution of shape inverse problems. * Communicated by K.K. Choi.

Modelling of topological derivatives for contact problems

The problem of topology optimization is considered for free boundary problems of thin obstacle types. The formulae for the first term of asymptotics for energy functionals are derived. The precision of obtained terms is verified numerically. The topological differentiability of solutions to variational inequalities is established. In particular, the so-called outer asymptotic expansion for solutions of contact problems in elasticity with respect to singular perturbation of geometrical domain depending on small parameter are derived by an application of nonsmooth analysis. Such results lead to the topological derivatives of shape functionals for contact problems. The topological derivatives are used in numerical methods of simultaneous shape and topology optimization.

On Numerical Solution of Shape Inverse Problems

The new method is proposed for the numerical solution of a class of shape inverse problems. The size and the location of a small opening in the domain of integration of an elliptic equation is identified on the basis of an observation. The observation includes the finite number of shape functionals. The approximation of the shape functionals by using the so-called topological derivatives is used to perform the learning process of an artificial neural network. The results of computations for 2D examples show, that the method allows to determine an approximation of the global solution to the inverse problem, sufficiently closed to the exact solution. The proposed method can be extended to the problems with an opening of general shape and to the identification problems of small inclusions. However, the mathematical theory of the proposed approach still requires futher research. In particular, the proof of global convergence of the method is an open problem.

Topological Derivatives for Contact Problems

Numerical methods of evaluation of topological derivatives are proposed for contact problems in two dimensional elasticity. Problems of topology optimisation are investigated for free boundary problems of boundary obstacle types. The formulae for the first term of asymptotics for energy functionals are derived. The precision of obtained terms is verified numerically. The topological differentiability of solutions to variational inequalities is established. In particular, the so-called outer asymptotic expansion for solutions of contact problems with respect to singular perturbation of geometrical domain depending on small parameter are obtained by an application of nonsmooth analysis. The topological derivatives can be used in numerical methods of simultaneous shape and topology optimisation, in particular, in the level set type methods.

Numerical experiments in drag minimization for compressible Navier-Stokes flows in bounded domains

Abstract In the paper we present the preliminary numerical results of drag minimization using shape optimization for weakly compressible flow. The formula for shape derivative contains all terms. The computations are performed in a bounded, regularly shaped domain.

Shape-Topological Differentiability of Energy Functionals for Unilateral Problems in Domains with Cracks and Applications

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 (Sokolowski and Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer, Berlin/Heidelberg/New York, 1992). 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 the 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 (Sokolowski and Zolesio, Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer, Berlin/Heidelberg/New York, 1992). 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 Sokolowski and Zolesio (Introduction to Shape Optimization: Shape Sensitivity Analysis, Springer, Berlin/Heidelberg/New York, 1992) for the shape differentiability of multiple eigenvalues is further applied in Khludnev and Sokolowski (Eur. J. Appl. Math. 10:379–394, 1999; Eur. J. Mech. A Solids 19:105–120, 2000) 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.

Passive Control of Singularities by Topological Optimization: The Second-Order Mixed Shape Derivatives of Energy Functionals for Variational Inequalities

A class of nonsmooth shape optimization problems for variational inequalities is considered. The variational inequalities model elliptic boundary value problems with the Signorini type unilateral boundary conditions. The shape functionals are given by the first order shape derivatives of the elastic energy. In such a way the singularities of weak solutions to elliptic boundary value problems can be characterized. An example in solid mechanics is given by the Griffith’s functional, which is defined in plane elasticity to measure SIF, the so-called stress intensity factor, at the crack tips. Thus, topological optimization can be used for passive control of singularities of weak solutions to variational inequalities. The Hadamard directional differentiability of metric the projection onto the positive cone in fractional Sobolev spaces is employed to the topological sensitivity analysis of weak solutions of nonlinear elliptic boundary value problems. The first order shape derivatives of energy functionals in the direction of specific velocity fields depend on the solutions to variational inequalities in a subdomain. A domain decomposition technique is used in order to separate the unilateral boundary conditions and the energy asymptotic analysis. The topological derivatives of nonsmooth integral shape functionals for variational inequalities are derived. Singular geometrical domain perturbations in an elastic body \(\Omega\) are approximated by regular perturbations of bilinear forms in variational inequality, without any loss of precision for the purposes of the second-order shape-topological sensitivity analysis. The second-order shape-topological directional derivatives are obtained for the Laplacian and for linear elasticity in two and three spatial dimensions. In the proposed method of sensitivity analysis, the singular geometrical perturbations \(\epsilon \rightarrow \omega _{\epsilon }\subset \Omega\) centred at \(\hat{x} \in \Omega\) are replaced by regular perturbations of bilinear forms supported on the manifold \(\Gamma _{R} =\{ \vert x -\hat{ x}\vert = R\}\) in an elastic body, with R > e > 0. The obtained expressions for topological derivatives are easy to compute and therefore useful in numerical methods of topological optimization for contact problems.

Shape and Topology Sensitivity Analysis for Elastic Bodies with Rigid Inclusions and Cracks

In the keynote lecture we describe how the asymptotic analysis in singularly perturbed domains can be employed to determine effectively the influence of nucleation of small voids on some shape functionals. To this end the classical shape sensitivity analysis is combined with asymptotic expansions in order to determine the singular limits of shape derivatives which are called topological derivatives of shape functionals. The topological derivatives are determined for elastic bodies weakened by cracks on boundaries of rigid inclusions. On the crack faces the nonpenetration conditions are prescribed, such conditions are non linear and assure that the displacements of the crack lips or surfaces cannot penetrate each other. Small voids are located on the finite distance from the crack, so there is no interaction between the crack and the voids.

Topological Derivatives in Plane Elasticity

We present a method for construction of the topological derivatives in plane elasticity. It is assumed that a hole is created in the subdomain of the elastic body which is filled out with isotropic material. The asymptotic analysis of elliptic boundary value problems in singularly perturbed geometrical domains is used in order to derive the asymptotics of the shape functionals depending on the solutions to the boundary value problems. Our method allows for the asymptotic expansions of arbitrary order, since the explicit solutions to the boundary value problems are obtained by the method of elastic potentials. Some numerical results are presented to show the applicability of the proposed method in numerical analysis of elliptic problems.

An approach to infinite domains, singularities and superelements in FEM computations

In the paper the representation method for treating the infinite parts of computational domains is derived. It is based on a discrete formulation of the problems and uses the formal series technique. The method applies to wedge-like domains and static problems in general. For finite domains, it makes possible a uniform treatment of problems with corner singularities and improvement of accuracy in ordinary FEM computations.