ĭiscretizations with triangular elements are much more flexible than tensor-product element structures. Since then, many ways of (approximately or in same cases exactly) enforcing C 1 and G 1 continuity conditions have been developed for NURBS, employing static condensation , mortar methods, ,, penalty methods , as well as Nitsche-type and discontinuous Galerkin methods, ,. G 1 (geometric continuity) conditions allow for step-wise changes of the membrane strain, as opposed to C 1 (parametric continuity) conditions. , developed rotation-free NURBS elements with exactly calculated surface normal and combined them with penalty-based G 1 continuity conditions at patch boundaries, enabling the analysis of a wide range of unstiffened and stiffened plates and shells. The advent of isogeometric analysis spurred numerous developments that take advantage of the high continuity of NURBS basis functions. Explicit shape functions were developed much later . Another problem was the calculation of the shape functions of the quintic triangle, which required the solution of a system of linear equations at every integration point for every element. But it was soon discovered that the derivative degrees of freedom complicate the imposition of boundary conditions and lead to inconsistencies at step-wise thickness and material changes, impairing the rate of convergence and limiting the range of problems that can be analysed effectively. These Hermite elements represent a C 1 (smooth) mid-surface, which enables the calculation of the exact surface normal and of its derivatives at the quadrature points, resulting in a Kirchhoff–Love formulation that is free from transverse-shear locking. During the 60’s of the last century, the first rotation-free Kirchhoff–Love plate elements, such as the Hsieh–Clough–Tocher cubic macro triangle and quintic Hermite triangle (e.g. ), were developed.
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