A three‐dimensional finite element method with arbitrary polyhedral elements

Abstract

SUMMARYThe 'variable-element-topology finite element method' (VETFEM) is a finite-element-like Galerkin approximation method in which the elements may take arbitrary polyhedral form. A complete development of the VETFEM is given here for both two and three dimensions. A kinematic enhancement of the displacement-based formulation is also given, which effectively treats the case of near-incompressibility. Convergence of the method is discussed and then illustrated by way of a 2D problem in elastostatics. Also, the VETFEM's performance is compared to that of the conventional FEM with eight-node hex elements in a 3D finite-deformation elastic-plastic problem. The main attraction of the new method is its freedom from the strict rules of construction of conventional finite element meshes, making automatic mesh generation on complex domains a significantly simpler matter.

On the flexibility of agglomeration based physical space discontinuous Galerkin discretizations

“…Other approaches to deal with non-standard meshes exist and deserve to be mentioned. A standard (continuous) Galerkin discretization on arbitrary polyhedral elements has been proposed by Rashid and Selimotic [18] in the context of elastostatics and elastic-plastic problems of solid mechanics. In [18], the definition of the Lagrangian basis functions in the physical frame is strictly related to the number of nodes defining the geometry of the element.…”

Numerical integration of polynomials and discontinuous functions on irregular convex polygons and polyhedrons

“…Divergence theorem can also be applied to convert the domain integrals into boundary integrals that can be carried out in a more straightforward manner [8,9]. In two dimensions, when the goal is to produce a quadrature for the integration of polynomials of order d or lower, the set of basis functions includes all bivariate monomials up to order d, P d = {x i y j , i, j ∈ Z, i + j ≤ d}.…”

“…Conforming polygonal finite elements [1][2][3][4] and finite elements on convex polyhedra [5][6][7] require the integration of nonpolynomial basis functions. The integration of polynomials on irregular polytopes arises in the non-conforming variable-element-topology finite element method [8,9], discontinuous Galerkin finite elements [10], finite volume element method [11] and mimetic finite difference schemes [12][13][14]. In partition-of-unity methods such as the extended finite element method (X-FEM) [15,16], discontinuous functions are integrated to form the stiffness matrix of elements that are cut by a crack or an interface.…”

“…It bears mention that for monomials one can use divergence theorem to transform the surface integrals to line integrals [8], which can be handled efficiently. In three dimensions, successive application of the divergence theorem results in line integrals over the edges of the polyhedron [9,26,28]. Lasserre's method [17,18], which is a technique for the integration of homogeneous functions on convex polygons and polyhedra based on Euler's theorem, transforms the integration of a homogeneous function on a convex polygon to line integrations over its edges.…”

Overview and construction of meshfree basis functions: from moving least squares to entropy approximants

“…In Reference [4], Laplace basis functions [7] are constructed on regular polygons, and through an isoparametric mapping, the basis functions are defined on irregular polygons. The Wachspress basis functions and mean value co-ordinates are directly computed on irregular polygons, which is also the case in a recently proposed non-conforming finite element method on polyhedral meshes [43]. The interested reader can refer to Reference [40] and the references therein for further details on the construction and implementation of polygonal interpolants.…”