Displacement based formulation over arbitrary polygon and polyhedra

Core Legato-team member: S Natarajan, Indian Institute of Technology-Madras, India.

Traditional FEM simulations rely strictly on tetrahedral or hexahedral meshes in 3D (or triangular, quadrilateral meshes in 2D). In generating a FE mesh, a balance is required between the accuracy and the flexibility in the mesh generation. For example, triangulation of a domain is relatively easy when compared with quadrangulation, whilst the quadrilateral mesh is more accurate than the triangular mesh. It is relatively easy to construct interpolants over standard shapes, viz., triangles and tetrahedrals. The use of standard shapes, viz., triangles (or quadrilaterals) and tetrahedrals (or hexahedral) simplifies the approach, however, allowing only a few element shapes can be too restrictive, because

  • it may require sophisticated meshing algorithm to generate high-quality meshes, especially with quadrilaterals, for meshing complex geometries;
  •  it may require complex remeshing to capture topological changes, for instance due to discontinuous surface growth.

In polygonal finite elements, the use of elements with more than four sides can provide the flexibility in meshing and the solution accuracy.  In polygonal finite element methods (PFEM) have been receiving increasing attention. In PFEM, the domain can be discretized without needing to maintain a particular element topology. This is advantageous in adaptive mesh refinement, where a straightforward subdivision of individual elements usually results in hanging nodes. Conventionally, this is eliminated by introducing additional edges/faces to retain conformity. This can be avoided if we can directly compute the stiffness matrices on polyhedral meshes with hanging nodes. Polygonal/polyhedral elements allow all elements to be treated using a quad-tree/oct-tree mesh within a single paradigm. For example, elements of class (1) quadrilaterals, (2) pentagons and (3) hexagons can be assembled within a single routine.

Domain discretized with polyhedra elements
Domain discretized with polyhedra elements


It was observed in that the strain smoothing technique over arbitrary polytopes yield less accurate solution when compared to other techniques, such as the conventional polygonal finite element method. The objective is to extend the strain smoothing technique to arbitrary polygons and polyhedra and to device a stable numerical technique within the framework of strain smoothing technique for arbitrary polygons and polyhedra.


S Natarajan, S Bordas, D Roy Mahapatra, Numerical integration over arbitrary polygonal domains based on Schwarz-Christoffel conformal mapping, International Journal for Numerical Methods in Engineering, v80, 103-134, 2009. Preprint

S Natarajan, D Roy Mahapatra, S Bordas, Integrating strong and weak discontinuities without integration subcells and example applications in an XFEM/GFEM framework, International Journal for Numerical Methods in Engineering, v83, 269-294, 2010. Preprint

S Natarajan, ET Ooi, I Chiong, C Song, Convergence and accuracy of displacement based finite element formulations over arbitrary polygons: Laplace interpolants, strain smoothing and scaled boundary polygon formulation, Finite Elements in Analysis and Design, v85, 101-122, 2014.

S Natarajan, S Bordas, ET Ooi, On the connection between the cell-based smoothed finite element method and the virtual element method. Preprint

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