On a finite element method with variable element topology

Conforming polygonal finite elements

“…Recently, Dasgupta and Malsch [6][7][8] have used symbolic computations to compute the Wachspress basis functions and also explored the construction of shape functions for concave elements. In computational solid mechanics, Rashid and Gullett [39] proposed a variable element topology finite element method (VETFEM), in which shape functions for convex and non-convex elements are constructed in the physical space (x ∈ ) using a constrained minimization procedure. The VETFEM implementation falls within the class of non-conforming methods and hence as opposed to conforming methods, here consistency and the satisfaction of the patch test [40,41] need to be addressed [39].…”

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

“…Rashid and Gullet [19] proposed a variable element topology finite element method, in which shape functions for convex and nonconvex elements are computed in the physical space using constrained minimization procedure. Based on the assumed stress hybrid formulation, Ghosh et al [20] developed the Voronoï cell finite element method.…”

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.…”

“…[29] to three dimensions is unwieldy. 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].…”