We compare three nearly optimal quadrilateral finite elements for geometrically exact inextensible-director shell model. Two of them are revisited and one is novel. The assumed natural strain (ANS) element of Ko et al. (Comput Struct 185:1-14, 2017) shows low sensitivity to mesh distortion and excellent convergence behavior for most types of shell problems. The Hu-Washizu element with ANS shear strains of Wagner and Gruttmann (Int J Numer Methods Eng, 64:635-666, 2005) allows for large solution steps and is computationally fast. However, both formulations have undesirable weak spots, which we clearly identify by a comprehensive set of numerical examples. We show that a straightforward combination of both formulations results in a novel element that synergizes the positive features and eliminates the weak spots of its predecessors.

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