Keywords: strain gradient theory, Kirchhoff nanoplates, Finite element method, hierarchical hermite functions, finite cell method, fictitious domain, complex geometries.

Strain gradient plate theories are governed by partial differential equations of higher order than those in classical plate models, requiring higher continuous approximations. In the finite element method, the simplest way to satisfy these continuity requirements is to use Hermite interpolation polynomials. However, the construction of geometry-conforming Hermite elements significantly complicates mesh generation, particularly in complex domains. In this work, the finite cell method is proposed to overcome these difficulties by decoupling the geometry representation from the field approximation. The physical domain is described on a nonconforming integration mesh, while the unknown field variables are approximated on a separate interpolation mesh defined over a simple background domain. To satisfy the continuity requirements inherent to strain gradient theories, a novel hierarchical Hermite polynomial space is introduced. Compared with the classical Hermite elements, the proposed formulation enables p-refinement of the numerical solution, leading to improved refinement flexibility. The methodology is implemented within a Kirchhoff strain gradient plate theory and validated through free vibration problems involving complex geometries. Numerical results demonstrate high convergence of the solution, together with enhanced modeling flexibility enable by the finite cell method.

download the full-text of this paper (PDF, 11 pages, 543 Kb)

go to the previous paper
go to the next paper
return to the table of contents
return to the volume description