  We present a cut finite element method for shape optimization in the case of
linear elasticity. The elastic domain is defined by a level-set function, and
the evolution of the domain is obtained by moving the level-set along a
velocity field using a transport equation. The velocity field is the largest
decreasing direction of the shape derivative that satisfies a certain
regularity requirement and the computation of the shape derivative is based on
a volume formulation. Using the cut finite element method no re--meshing is
required when updating the domain and we may also use higher order finite
element approximations. To obtain a stable method, stabilization terms are
added in the vicinity of the cut elements at the boundary, which provides
control of the variation of the solution in the vicinity of the boundary. We
implement and illustrate the performance of the method in the two--dimensional
case, considering both triangular and quadrilateral meshes as well as finite
element spaces of different order.
