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The goal of this paper is to give a comprehensive and short review on how tocompute the first and second order topological derivative and potentiallyhigher order topological derivatives for PDE constrained shape functionals. Weemploy the adjoint and averaged adjoint variable within the Lagrangianframework and compare three different adjoint based methods to compute higherorder topological derivatives. To illustrate the methodology proposed in thispaper, we then apply the methods to a linear elasticity model. We compute thefirst and second order topological derivative of the linear elasticity modelfor various shape functionals in dimension $2$ and $3$ using Amstutz' method,the averaged adjoint method and Delfour's method. In contrast to othercontributions regarding this subject, we not only compute the first and secondorder topological derivative, but additionally give some insight on variousmethods and compared their applicability and efficiency with respect to theunderlying problem formulation.

Adjoint based methods to compute higher order topological derivatives with an application to elasticity