This work discusses simplification algorithms for the generation of a multiresolution family of solid representations from an initial polyhedral solid. We introduce the Discretized Polyhedra Simplification (DPS), a framework for polyhedra simplification using space decomposition models. The DPS is based on a new error measurement and provides a sound scheme for error-bounded, geometry and topology simplification while preserving the validity of the model. A method following this framework, Direct DPS, is presented and discussed. Direct DPS uses an octree for topology simplification and error control, and generates valid solid representations. Our method is also able to generate approximations which do not interpenetrate the original model, either being completely contained in the input solid or bounding it. Unlike most of the current methods, our algorithm can deal and also produces faces with arbitrary complexity. An extension of the Direct method for appearance preservation, called Hybrid DPS, is also discussed.