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