We describe the equations of motion of an incompressible elastic body $Ω$ in 3-space acted on by an external pressure force, and the Newton iteration scheme that proves the well-posedness of the resulting initial value problem for its equations of motion on $C^{k,α}$ spaces. We use the first iterate of this Newton scheme as an approximation to the actual vibration motion of the body, and given a (finite) triangulation $K$ of it, produce an algorithm that computes it, employing the direct sum of the space of PL vector fields associated to the oriented edges and faces of the first barycentric subdivision $K'$ of $K$ (the metric duals of the Whitney forms of $K'$ in degree one, and the metric duals of the local Hodge $*$ of the Whitney forms in degree two, respectively) as the discretizing space. These vector fields, which capture the algebraic topology properties of $Ω$, encode them into the solution of the weak version of the linearized equations of motion about a stationary point, the essential component in the finding of the first iterate in the alluded Newton scheme. This allows for the selection of appropriate choices of $K$, relative to the geometry of $Ω$, for which the algorithm produces solutions that accurately describe the vibration of thin plates in a computationally efficient manner. We use these to study the resonance modes of the vibration of these plates, and carry out several relevant simulations, the results of which are all consistent with known vibration patterns of thin plates derived experimentally.
Static bending and free vibration of cross ply laminated composite plates using NURBS based finite element method and unified formulation
The University of New South Wales（新南威尔士大学）；University of Science, HCMC（胡志明市科学大学）；Faculdade de Engenharia da Universidade do Porto（波尔图大学工程学院）；King Abdulaziz University（阿卜杜勒阿齐兹国王大学）；Politecnico di Torino（都灵理工大学）
A new locking-free polygonal plate element for thin and thick plates based on Reissner-Mindlin plate theory and assumed shear strain fields
University of Chile（智利大学）；Indian Institute of Technology-Madras（印度理工学院马德拉斯分校）；University of Luxembourg（卢森堡大学）；Cardiff University（卡迪夫大学）；Duy Tan University（Duy Tan 大学）
