Locking effects in the finite element approximation of plate models

Abstract

We analyze the robustness of various standard finite element schemes for a hierarchy of plate models and obtain asymptotic convergence estimates that are uniform in terms of the thickness <italic>d</italic> . We identify <italic>h</italic> version schemes that show <italic>locking</italic> , i.e., for which the asymptotic convergence rate deteriorates as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d right-arrow 0"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d \to 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and also show that the <italic>p</italic> version is free of locking. In order to isolate locking effects from boundary layer effects (which also arise as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="d right-arrow 0"> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo stretchy="false"> → </mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">d \to 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> ), our analysis is carried out for the periodic case, which is free of boundary layers. We analyze in detail the lowest model of the hierarchy, the well-known Reissner-Mindlin model, and show that the locking and robustness of finite element schemes for higher models of the hierarchy are essentially identical to the Riessner-Mindlin case.