A novel efficient mixed formulation for strain‐gradient models

Abstract

Summary Various finite elements based on mixed formulations have been proposed for the solution of boundary value problems involving strain‐gradient models. The relevant literature, however, does not provide details on some important theoretical aspects of these elements. In this work, we first present the existing elements within a novel, single mathematical framework, identifying some theoretical issues common to all of them that affect their robustness and numerical efficiency. We then proceed to develop a new family of mixed elements that addresses these issues while being simpler and computationally cheaper. The behavior of the new elements is further demonstrated through two numerical examples.

Trefftz collocation method for two‐dimensional strain gradient elasticity

“…In Figure 3(A), we show the distribution of the field variables at the loaded side of the region in the case of small and large numbers of the collocation points and trial functions. For the less precise solution, we used the number of collocation points M s = 40 and the number of trial functions N = 24 that correspond to the dimensions of the matrix in the numerical solution K 168 × 24 (see (25)). For the more precise solution, we used M s = 160 and N = 204; this is the case of the matrix dimensions K 648 × 204 .…”

“…12,13 The pioneering work on the development of numerical methods in SGET was undertaken by Oden et al in their paper in 1970, 14 where they adopted the finite element method (FEM). Today, a lot of advanced numerical approaches and methods have been proposed in SGET that can be divided into the mixed FEM with C 0 -continuous interpolation and its variants, [15][16][17][18][19][20][21][22][23][24][25][26][27] FEM with C 1 -continuous interpolation, [28][29][30][31] isogeometric analysis, [32][33][34][35][36][37] singular boundary element methods (BEM) 38,39 and meshless Galerkin-type methods. [40][41][42] Among recent works, we can also mention the penalty three-dimensional elements approach 43 and the nonlocal operator method.…”

Strain gradient differential quadrature finite element for moderately thick micro‐plates

“…Babu and Patel 35 proposed a rectangular plate finite element for a single‐parameter strain gradient Kirchhoff plate model to solve the static bending, free vibration, and buckling problems of rectangular nano‐plates. Papanicolopulos et al 36 devised a new family of hybrid finite element for strain gradient‐related boundary value problems, where the Lagrange multiplier method was used. Sze and Hu 37 developed three 24‐DOF four‐node gradient‐ enhanced quadrilateral elements by extending the discrete Kirchhoff method, the relaxed hybrid‐stress method, and the hybrid‐stress method adopted in macro‐scale thin plate analysis.…”

Rot‐free mixed finite elements for gradient elasticity at finite strains

“…Note the exclusion of ∇u in the second term of (14), since the scalar product of rot and gradient tensor functions vanishes. The second term of (14) can be interpreted as constraint term enforcing H to be rot-free, which is a necessary condition for gradient functions.…”