Embodiment design (ED) is an early phase of product development. ED problems consist of finding solution principles that satisfy product requirements such as physics behaviors and interactions between components. Constraint satisfaction techniques are useful to solve constraint-based models that are often partial, heterogeneous, and uncertain in ED. This paper proposes new constraint satisfaction techniques to tackle piecewise-defined physics phenomena or skill-based rules and multiple categories of variables arising in design applications. New search heuristics and a global piecewise constraint are introduced in the branch and prune framework. The capabilities of these techniques are illustrated with both academic and real-world problems. Complete models of the latter are presented.