Phase field models are often used in situations where sharp interface simulations are difficult, in particular where topological changes can occur. While these are a feature in many applications, we may want to control at least the connectedness of either the phase transition regions or the phases. We present a method which allows us to approximate a relaxation of the perimeter functional under a connectedness constraint in two dimensions and Willmore's curvature energy at connected surfaces in three dimensions. We will give numerical evidence of the effectiveness of the method and describe how it can be implemented in an efficient manner.