  We formulate a general shape and topology optimization problem in structural
optimization by using a phase field approach. This problem is considered in
view of well-posedness and we derive optimality conditions. We relate the
diffuse interface problem to a perimeter penalized sharp interface shape
optimization problem in the sense of $\Gamma$-convergence of the reduced
objective functional. Additionally, convergence of the equations of the first
variation can be shown. The limit equations can also be derived directly from
the problem in the sharp interface setting. Numerical computations demonstrate
that the approach can be applied for complex structural optimization problems.
