Construct ‘FE-Meshfree’ Quad4 using mean value coordinates

• A hybrid FE-Meshfree quadrilateral element, named as Quad4-MVC, is proposed.
• Quad4-MVC uses mean value coordinates (MVC) to construct nodal approximations.
• Quad4-MVC does not rely on any arbitrary parameters, which makes Quad4-MVC more stable.
• Compared to Quad4, the present Quad4-MVC stably presents better accuracy.

The present work uses mean value coordinates to construct the shape functions of a hybrid ‘FE-Meshfree’ quadrilateral element, which is named as Quad4-MVC. This Quad4-MVC can be regarded as the development of the ‘FE-Meshfree’ quadrilateral element with radial-polynomial point interpolation (Quad4-RPIM). Similar to Quad4-RPIM, Quad4-MVC has Kronecker delta property on the boundaries of computational domain, so essential boundary conditions can be enforced as conveniently as in the finite element method (FEM). The novelty of the present work is to construct nodal approximations using mean value coordinates, instead of radial basis functions which are used in Quad4-RPIM. Compared to the radial basis functions, mean value coordinates does not utilize any uncertain parameters, which enhances stability of numerical results. Numerical tests in this paper show that the performance of Quad4-RPIM becomes even worse than four-node iso-parametric element (Quad4) when the parameters of radial basis functions are not chosen properly. However, the performance of Quad4-MVC is stably better than Quad4.