A three‐dimensional C¹ finite element for gradient elasticity

Abstract

SUMMARYIn gradient elasticity strain gradient terms appear in the expression of virtual work, leading to the need for C 1 continuous interpolation in finite element discretizations of the displacement field only. Employing such interpolation is generally avoided in favour of the alternative methods that interpolate other quantities as well as displacement, due to the scarcity of C 1 finite elements and their perceived computational cost. In this context, the lack of three-dimensional C 1 elements is of particular concern. In this paper we present a new C 1 hexahedral element which, to the best of our knowledge, is the first three-dimensional C 1 element ever constructed. It is shown to pass the single element and patch tests, and to give excellent rates of convergence in benchmark boundary value problems of gradient elasticity. It is further shown that C 1 elements are not necessarily more computationally expensive than alternative approaches, and it is argued that they may be more efficient in providing good-quality solutions.

Isogeometric analysis of 2D gradient elasticity

“…For this reason the application of C 1 -continuous finite elements to gradient elasticity is considered only in few works: In [8] the results obtained with two different C 1 -continuous finite elements were compared to the results from a formulation based on the micromorphic theory. A three-dimensional hexahedral element with C 1 -continuity was developed and applied to gradient elasticity in [9]. In [10] the performance of three different C 1 -continuous finite elements and the C 1 Natural Element Method (NEM) was analyzed for nonlinear gradient elasticity.…”

A Novel Highly Accurate Finite-Element Family

“…Among these, we can mention the Galerkin least-squares finite-element method for the solution of the two-dimensional Helmholtz equation [10], the Galerkin residual-free bubbles method [11,12], the smoothed FEM using cubic spline polynomial functions in hexahedral elements [13], and the isogeometric FEM [14]. During the years, very accurate FEM methods suited for structural dynamics and fluid dynamics applications have been developed; among these, it is worth mentioning the Spectral Finite-Element Method (SFEM), based on Lagrange polynomials on the Gauss-Lobatto-Legendre grid [15], the Fourier transform-based and Wavelet transform-based spectral FEM [16,17], the hierarchical -FEM [18][19][20], and 1 -FEM methods based upon isoparametric Hermite elements [21,22]. Since its origin, SFEM have been successfully applied in several fields of the physics and applied sciences: acoustics, fluid dynamics, heat transfer, and structural dynamics.…”

Unified finite element methodology for gradient elasticity

“…The first one comprehends approaches that leave the continuum mechanics equations intact, by using Meshless methods [3][4][5][6][7][8][9][10][11], Penalty methods [12][13][14], Hermitian finite elements [15][16][17][18], next nearest neighbour interaction (instead of the simpler nearest neighbour interaction used in the standard finite element software) [19], etc. The second one includes approaches that transform the governing equations, in order to obtain less demanding continuity requirements; among these is the Ru-Aifantis theorem [20] which splits the original fourth-order p.d.e.…”