Antoni Zochowski

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.

A stress versus compliance constraint in a minimum weight design

Abstract The paper contains a study of the stress-constrained minimum-weight design of the beam in the framework of [1]. The existence of optimal shape is considered. The numerical examples are given and stress-constrained designs are compared to compliance-constrained ones.

Minimum-weight design with displacements constraints in 2-dimensional elasticity☆

Abstract The optimum weight-bounded deflection design of a beam, modelled by means of a set of 2-dimensional elasticity equations is studied. The iterative algorithm based on differentiation with respect to the domain formulae is proposed. The examples of one-sidedly the two-sidedly clamped beams with different loading conditions and solved numerically. The results are compared with those of a straightforward approach.

Optimal Perforation Design in 2-Dimensional Elasticity

ABSTRACT In the paper the design of perforations of two-dimensional bodies is studied. Examples considered consist of an elastic structure and heat radiator. The methods used are extensions of previous work by the author, involving the domain differentiation technique. Variational properties of the commonly used homogenization formulas are derived and compared to the finite element approximation. On this basis, a new set of formulas is obtained, approximating another functional. The paper constitutes a preliminary step in the design of composites.

A variational interpretation of homogenization and its extensions

Abstract In this paper the variational properties of the commonly used homogenization formulae are derived and compared to the finite element approximation. On this basis a new set of formulae is obtained, approximating another functional. The results are applied to heat and elasticity equations and the accuracy of both methods compared. The flexibility of the approach in application to various boundary value problems is demonstrated.

Drag minimization for the obstacle in compressible flow using shape derivatives and finite volumes

In the paper the shape optimization problem for the static, compressible Navier-Stokes equations is analyzed. The drag minimizing of an obstacle immersed in the gas stream is considered. The continuous gradient of the drag is obtained by application of the sensitivity formulas derived in the works of one of the co-authors. The numerical approximation scheme uses mixed Finite Volume - Finite Element formulation. The novelty of our numerical method is a particular choice of the regularizing term for the non-homogeneous Stokes boundary value problem, which may be tuned to obtain the best accuracy. The convergence of the discrete solutions for the model under considerations is proved. The non-linearity of the model is treated by means of the fixed point procedure. The numerical example of an optimal shape is given.

Linearised coupling of elasticity and Navier-Stokes equations

In the paper we consider a coupled problem of the linearly elastic body immersed in the flowing fluid which is modelled by means of incompressible Navier-Stokes equations. The approach is based on transforming the variable domain occupied by the fluid to the fixed one corresponding to the un-deformed elastic inclusion. The main idea consists in performing the linearisation of this transformation, similarly as in linear elasticity one assumes the smallness of strains. We base our approach on the observation that the gradients of the suitably defined transformation are bounded by traces of strains on the surface of the elastic body.

Shape optimization problem for the coupled model of linear elasticity with Navier-Stokes equation.

We consider a coupled model of linear elasticity with Navier-Stokes equations. Two subdomains Ω₁ and Ω₂ are considered. In Ω₁ there is a linear elasticity model. In Ω₂ there is the fluid transport which is modeled by nonlinear Navier-Stokes equations. A propitiate interface conditions should be provided. We want to determine the shape and topological derivatives in Ω₂.

Shape-topological differentiability of energy functionals in domains with cracks

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 the boundary variations of an inclusion.

Application of topological derivative to accelerate genetic algorithm in shape optimization for semilinear elliptic equation

A new variant of the genetic algorithm for finding the location and size of small holes in the domain described by the non-linear boundary value problem is considered. The expansion of the shape functional defined in non-linear domain is provided in order to determine the form of topological derivative. The value of topological derivative is used for computing the probability density applied later to generate location and size of holes by the genetic algorithm.