Asymptotic analysis and topological derivatives for shape and topology optimization of elasticity problems in two spactial dimensions

Topological derivatives for elasticity problems are used in shape and topology optimization in structural mechanics. We propose an approach to the asymptotic analysis of singular perturbations of geometrical domains. This approach can be used in order to determine the exact solutions of elasticity boundary value problems in domains with small holes, and determine the explicit asymptotic expansions of solutions with respect to small parameter which describes the radius of internal hole. The elastic potentials of Muskhelishvili gives us an explicite solution in the ring $C(\rho,R)=\{\rho < |x| < R \}$ in the form of complex valued series. The series depends on the small parameter, the radius $\rho$ of the ring, and we are interested in the behavior of the series for the passage $\rho\to 0$. Such analysis leads to the expansion of the elastic energy in the form $$\mathcal{E}(\rho,R)=\mathcal{E}(0,R)+\rho^2\mathcal{E}^1(R)+\rho^4\mathcal{E}^2(R)+\dots\ ,$$ where $\mathcal{E}^1(R)$ is used to determine the first order topological derivatives of shape functionals, and $\mathcal{E}^2(R)$ can be used to determine the second order topological derivatives of shape functionals. In the paper the first order term $\mathcal{E}^1(R)$ is given, however the method is general and can be used to determine the subsequent terms of the energy expansion and the topological derivatives of higher order

On Topological Derivative in Shape Optimization

In the paper the topological derivative for 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

The Topological Derivative Method and Artificial Neural Networks for 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 a 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

Shape and topology sensitivity analysis for cracks in elastic bodies on boundaries of rigid inclusions

We consider a 3D 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. We analyze the variational formulation of the problem and provide shape and topology sensitivity analysis of the solution

Energy change in elastic solids due to a spherical or circular cavity, considering uncertain in put data

In the paper we consider topological derivative of shape functionals for elasticity, which is used to derive the worst and also the maximum range scenarios for behavior of elastic body in case of uncertain material parameters and loading. It turns out that both problems are connected, because the criteria describing this behavior have form of functionals depending on topological derivative of elastic energy. Therefore in the first part we describe the methodology of computing the topological derivative with some new additional conditions for shape functionals depending on stress. For the sake of fulness of presentation the explicit formulas for stress distribution around cavities are provided

Modelling of topological derivatives for contact problems

The problem of topology optimisation 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 {\it 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 {\it topological derivatives} of shape functionals for contact problems. The topological derivatives are used in numerical methods of simultaneous shape and topology optimisation

Topology optimization for unilateral 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 {\it 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

Shape and topology optimization of distributed parameter systems

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