This article addresses the simulation of non-smooth dynamics problems with unilateral contact constraints between rigid and deformable bodies. It proposes a modified CD-Lagrange scheme with a singular mass matrix. This scheme is explicit, and based on a contact condition on velocity. The formulation is designed for a 1D impact problem between deformable and rigid body with unilateral constraint. The singular mass matrix allows to get a more accurate energy balance on the discrete system, especially during non-smooth events. An extension is presented then for the 3D cases. Its implementation is easy, and fully compatible with large deformations or non-linear materials. Indeed it consists only in adding a numerical parameter for each contact node. The energy balance for the singular 3D formulation is improved compared to the consistent one.