On Partitions of Unity Property of Nodal Shape Functions: Rigid-Body-Movement Reproduction and Mass Conservation

This paper discusses the Partitions of Unity (PU) property that is one of the most important properties of nodal shape functions used in various numerical methods via discretization, including element-based and/or meshfree methods, such as FEM, S-FEM, S-PIM, EFG, XFEM, etc. The significance of the PU property and the possible consequences of using shape functions that do not possess the PU property in a numerical method are examined in theory. It proves that the PU property is a necessary (not sufficient in general) condition to enable the basic feature of rigid-body-movement production for static problems, and it is both necessary and sufficient condition mass conservation for dynamic problems for solids. This paper offers a fundamental insight into the lack of essential solution properties when formulating a computational method based on discretization techniques with shape functions that do not possess the PU property.

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