A nonlinear 4-node shell element with one point quadrature and stabilization based on a Hu–Washizu variational formulation

Abstract In this paper a robust and effective 4–node shell element for the structural analysis of thin structures is described. A Hu–Washizu functional with independent displacements, stress resultants and shell strains is the variational basis of the formulation. With application of three basic assumptions within the FE formulation a stabilized one point integrated element comes out automatically. Besides static computations it can be used for dynamic problems with explicit or implicit time integration without further adaptions. The element possesses the correct rank and fulfills the membrane and bending patch test. The stabilization terms can be integrated analytically and do not require an input of problem dependent control parameters. In contrast to several approaches the results are not path dependent for hyperelastic material behavior. This unphysical behavior follows when the material matrix of the last load increment is used in the stabilization matrix to preserve quadratic convergence in the Newton-Raphson iteration. The element possesses the well-known robustness of Hu–Washizu elements with the possibility of very large load steps in nonlinear applications. Another essential advantage of present development is the application to FE2 computations where the computing times are dominated by the solution of the micro problems at the shell integration points. Depending on the used interpolation for the shell strains the new element is approximately by a factor 4 or 9 faster than the corresponding version with full integration.