On some aspects of the CNEM implementation in 3D in order to simulate high speed machining or shearing

This paper deals with the implementation in 3D of the constrained natural element method (CNEM) in order to simulate material forming involving large strains. The CNEM is a member of the large family of mesh-free methods, but is at the same time very close to the finite element method. The CNEM’s shape function is built using the constrained Voronoï diagram (the dual of the constrained Delaunay tessellation) associated with a domain defined by a set of nodes and a description of its border. The use of the CNEM involves three main steps. First, the constrained Voronoï diagram is built. Thus, for each node, a Voronoï cell is geometrically defined, with respect of the boundary of the domain. Then, the Sibson-type CNEM shape functions are computed. Finally, the discretization of a generic variational formulation is defined by invoking an “stabilized conforming nodal integration”. In this work, we focus especially on the two last points. In order to compute the Sibson shape function, five algorithms are presented, analyzed and compared, two of them are developed. For the integration task, a discretization strategy is proposed to handle domains with strong non-convexities. These approaches are validated on some 3D benchmarks in elasticity under the hypothesis of small transformations. The obtained results are compared with analytical solutions and with finite elements results. Finally, the 3D CNEM is applied for addressing two forming processes: high speed shearing and machining.