A displacement‐based finite element formulation for general polyhedra using harmonic shape functions

Abstract

SUMMARYA displacement-based continuous-Galerkin finite element formulation for general polyhedra is presented for applications in nonlinear solid mechanics. The polyhedra can have an arbitrary number of vertices or faces. The faces of the polyhedra can have an arbitrary number of edges and can be nonplanar. The polyhedra can be nonconvex with only the mild restriction of star convexity with respect to the vertex-averaged centroid. Conforming shape functions are constructed using harmonic functions defined on the undeformed configuration, thus requiring the use of a total-Lagrangian finite element formulation in large deformation applications. For nonlinear applications with computationally intensive constitutive models, it is important to minimize the number of element quadrature points. For this reason, an integration scheme is adopted in which the number of quadrature points is equal to the number of vertices. As a first step toward verifying the element behavior in the general nonlinear setting, several linear verification examples are presented using both random Voronoi meshes and distorted hexahedral meshes. For the hexahedral meshes, results for the polyhedral formulation are compared to those of the standard trilinear hexahedral formulation. The element behavior in the nearly incompressible regime is also examined.

Trefftz polygonal finite element for linear elasticity: convergence, accuracy, and properties

“…(b) natural neighbor interpolants [11]; (c) maximum entropy approximant [12]; and (d) harmonic shape functions [8]. As the approximation functions over arbitrary polygonal elements are usually non-polynomial (in particular, rational polynomials) which introduces difficulties in the numerical integration, improving numerical integration over polytopes has gained increasing attention [3,[13][14][15].…”

Serendipity face and edge VEM spaces

“…However it should be clear from the very beginning that VEMs allow much more general geometries. For these more general geometries the comparison should actually be done between VEMs and other methods designed for polytopes, as for instance [11], [15], [19], [21], [23], [26], [27], [28], [31], [33], [34], [37]. The natural comparison, within Finite Elements, of our V f k,k−1,k−1 elements are clearly the BDM spaces as described in (2.20) for triangles (see Figure 1).…”

“…Similarly to other methods for polytopes (see e.g. [4], [15], [26], [34], [35], [36], [37]) they use finite dimensional spaces that, within each element, contain functions that are not polynomials.…”

Divergence free virtual elements for the stokes problem on polygonal meshes

“…We refer to the recent papers and monographs [19,8,17,9,13,15,28,30,31,33,35,34,36,39,40,24,32,21] as a brief representative sample of the increasing list of technologies that make use of polygonal/polyhedral meshes. We mention here in particular the polygonal finite elements, that generalize finite elements to polygons/polyhedrons by making use of generalized non-polynomial shape functions, and the mimetic discretisation schemes, that combine ideas from the finite difference and finite element methods.…”