Jan Sokolowski

On the Topological Derivative in Shape Optimization

In this paper the topological derivative for an arbitrary shape functional is defined. Examples are provided for elliptic equations and the elasticity system in the plane. The topological derivative can be used for solving shape optimization problems in structural mechanics.

Asymptotic analysis of shape functionals

Abstract A family of boundary value problems is considered in domains Ω(e)=Ω⧹ ω e ⊂ R n , n⩾3, with cavities ωe depending on a small parameter e∈(0,e0]. An approximation U (e,x) , x∈Ω(e) , of the solution u(e,x), x∈Ω(e) , to the boundary value problem is obtained by an application of the methods of matched and compound asymptotic expansions. The asymptotic expansion is constructed with precise a priori estimates for solutions and remainders in Holder spaces, i.e., pointwise estimates are established as well. The asymptotic solution U (e,x) is used in order to derive the first term of the asymptotic expansion with respect to e for the shape functional J (Ξ(e))= J e (u)≅ J e ( U ) . In particular, we obtain the topological derivative T (x) of the shape functional J (Ξ) at a point x∈Ω . Volume and surface functionals are considered in the paper.

Topological Derivatives in Shape Optimization

The topological derivative is defined as the first term (correction) of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of singular domain perturbations, such as holes, inclusions, defects, source-terms and cracks. Over the last decade, topological asymptotic analysis has become a broad, rich and fascinating research area from both theoretical and numerical standpoints. It has applications in many different fields such as shape and topology optimization, inverse problems, imaging processing and mechanical modeling including synthesis and/or optimal design of microstructures, fracture mechanics sensitivity analysis and damage evolution modeling. Since there is no monograph on the subject at present, the authors provide here the first account of the theory which combines classical sensitivity analysis in shape optimization with asymptotic analysis by means of compound asymptotic expansions for elliptic boundary value problems. This book is intended for researchers and graduate students in applied mathematics and computational mechanics interested in any aspect of topological asymptotic analysis. In particular, it can be adopted as a textbook in advanced courses on the subject and shall be useful for readers interested on the mathematical aspects of topological asymptotic analysis as well as on applications of topological derivatives in computation mechanics.

Optimality Conditions for Simultaneous Topology and Shape Optimization

New optimality conditions are derived for a class of shape optimization problems. The conditions are established on the boundary by an application of the boundary variations technique and in the interior of an optimal domain by exploiting the topological derivative method. An example is provided for which the classical second order sufficient optimality conditions are verified for an optimal simply connected domain. However, the value of the cost can be improved by the topology variations, and therefore, the optimal solution can be substantially changed by applying the topology optimization.

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.

Sensitivity analysis of control constrained optimal control problems for distributed parameter systems

The differential stability of solutions to control constrained quadratic optimal control problems for distributed parameter systems is investigated in this paper. The form of the right-derivatives of optimal controls for such problems with respect to the real parameter is derived. The differential sensitivity of optimal controls with respect to the perturbations of the coefficients of the state equation as well as to the deformations of the domain of integration is considered. The right-derivative of an optimal control with respect to the parameter is obtained in the form of an optimal control to the auxiliary control constrained optimal control problem.

A Level Set Method in Shape and Topology Optimization for Variational Inequalities

A Level Set Method in Shape and Topology Optimization for Variational Inequalities The level set method is used for shape optimization of the energy functional for the Signorini problem. The boundary variations technique is used in order to derive the shape gradients of the energy functional. The conical differentiability of solutions with respect to the boundary variations is exploited. The topology modifications during the optimization process are identified by means of an asymptotic analysis. The topological derivatives of the energy shape functional are employed for the topology variations in the form of small holes. The derivation of topological derivatives is performed within the framework proposed in (Sokołowski and Żochowski, 2003). Numerical results confirm that the method is efficient and gives better results compared with the classical shape optimization techniques.

Inhomogeneous Boundary Value Problems for Compressible Navier–Stokes Equations: Well-Posedness and Sensitivity Analysis

In this paper compressible, stationary Navier–Stokes equations are considered. A framework for analysis of such equations is established. In particular, the well-posedness for inhomogeneous boundary value problems of elliptic-hyperbolic type is shown. Analysis is performed for small perturbations of the so-called approximate solutions that take form (1.12). The approximate solutions are determined from Stokes problem (1.11). The small perturbations are given by solutions to (1.13). The uniqueness of solutions for problem (1.13) is proved, and, in addition, the differentiability of solutions with respect to the coefficients of differential operators is shown. The results on the well-posedness of nonlinear problems are interesting on their own and are used to obtain the shape differentiability of the drag functional for incompressible Navier–Stokes equations. The shape gradient of the drag functional is derived in the classical and useful for computations form; an appropriate adjoint state is introduced to ...

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.

Differential stability of solutions to constrained optimization problems

In this paper the differential stability of solutions to constrained optimization problems is investigated. The form of right-derivatives of optimal solutions to such problems, with respect to a real parameter, is derived. The right-derivative of the optimal control with respect to parameter for an optimal control problem for parabolic equation is obtained in the form of the optimal solution to an auxiliary optimal control problem. A method for determination of the second right-derivative of the optimal solutions to constrained optimization problems is proposed. Several examples are provided.