Collocation in finite element analysis of constrained problems

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Abstract
The concept of three-field displacement finite element formulations of constrained problems is explored in this paper. The approach studied uses collocation within the element domain to enforce the constraints of the problem and resembles mixed formulations because in addition to the approximation of the displacement field, all other auxiliary deformation variables that enter the constraint conditions are interpolated independently. Plate-bending as described by the Reissner–Mindlin theory is used as a model problem for the development of the approach. Two triangular plate elements are formulated based on this methodology, and their performance is investigated using several patch tests. Numerical examples are also considered to study the element behavior under locking conditions (which occur with the classical displacement approach at the thin plate limit). The ability of the elements to overcome locking is established using mathematical arguments and practical examples. The influence of the position of collocation points on the computed results is evaluated through sensitivity studies, with the aim to identify the optimal set. Results are compared with those obtained from the exact solutions and the associated classical displacement model with selective reduced integration of the constrained energy terms. Key words: finite elements, collocation, discrete constraints, mixed methods, thick plates, bending of plates, shear deformations.

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