Validity of Lagrange (Bézier) and rational Bézier quads of degree 2

Paul-Louis George (1) , Houman Borouchaki (1, 2)

1 Gamma3 - Automatic mesh generation and advanced methods
2 UTT - Université de Technologie de Troyes

Finite elements of degree two or more are needed to solve various PDE problems. This paper discusses a method to validate such meshes for the case of quadrilateral elements of degree 2. The first section of this paper comes back to Bézier curve and Bézier quadrilateral patches of degree 2. The way in which a Bézier quad patch and a Q2 finite element quad are related is introduced. The two possible quads are discussed, the 9‐node (or complete) quad together with the 8‐node (or Serendipity) quad. A validity condition, the positivity of the Jacobian, is exhibited for these two elements. The discussion continues with a rational Bézier quad patch that can be used as a finite element. Extension to arbitrary degrees is given.

Keywords: Q2 finite element quads; 9-node and 8-node quads; Q2 mesh; Q2 finite element; Béziercurve; Bézier quad; rational Bézier curve; rational Bézier quad; Jacobian; high-orderelement

Domains: Numerical Analysis [cs.NA]

Paul-Louis George, Houman Borouchaki. Validity of Lagrange (Bézier) and rational Bézier quads of degree 2. International Journal for Numerical Methods in Engineering, 2014, 99 (8), pp.611-632. ⟨10.1002/nme.4696⟩. ⟨hal-02286200⟩