Some memories of Tinsley Oden and the extension of the twist-Kirchhoff elements to the buckling analysis of plates

The senior author begins with recollections of his long friendship and professional relationship with Tinsley Oden. Then we discuss some recent developments in the twist-Kirchhoff theory of plates. More specifically, we extend the first two members of the twist-Kirchhoff family of arbitrary quadrilateral elements to the linear buckling analysis of plates. While standard polynomial <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sup>-continuous approximation functions are adopted for the transverse displacements, Raviart-Thomas vector-field approximations are considered for the rotations. Continuity of the normal components of the rotation vector across mesh edges is preserved by using the Piola transformation to map the rotations from the parent domain to the physical domain. The elements possess a unique combination of efficiency and robustness in that minimal quadrature rules are sufficient to guarantee stability without rank deficiency. We numerically study the convergence and accuracy of the elements on square, circular and skew plate buckling problems.