Polygonal finite elements for topology optimization: A unifying paradigm

Abstract

SUMMARYIn topology optimization literature, the parameterization of design is commonly carried out on uniform grids consisting of Lagrangian-type finite elements (e.g. linear quads). These formulations, however, suffer from numerical anomalies such as checkerboard patterns and one-node connections, which has prompted extensive research on these topics. A problem less often noted is that the constrained geometry of these discretizations can cause bias in the orientation of members, leading to mesh-dependent sub-optimal designs. Thus, to address the geometric features of the spatial discretization, we examine the use of unstructured meshes in reducing the influence of mesh geometry on topology optimization solutions. More specifically, we consider polygonal meshes constructed from Voronoi tessellations, which in addition to possessing higher degree of geometric isotropy, allow for greater flexibility in discretizing complex domains without suffering from numerical instabilities.

Optimizing Voronoi Diagrams for Polygonal Finite Element Computations

“…As a consequence, generalizations of FE methods to arbitrary polygonal or polyhedral meshes have gained increasing attention, both in computational physics [34,36,37] and in computer graphics [43,25]. For instance, when cutting a tetrahedron into two pieces, polyhedral FEM can directly process the resulting two elements, while standard FEM has to remesh them into tetrahedra first.…”

“…For instance, when cutting a tetrahedron into two pieces, polyhedral FEM can directly process the resulting two elements, while standard FEM has to remesh them into tetrahedra first. Polygonal FEM was also shown to yield good results in topology optimization, since the resulting meshes are less biased to certain principle directions than the typically employed regular tessellations [37]. Independent of the employed element type, the accuracy and stability of numerical computations crucially depend on a high quality mesh consisting of well-shaped, non-degenerate elements only.…”

“…Most polygonal FE methods use Voronoi tessellations to discretize the simulation domain [35,32,37]. The corresponding mesh generation and optimization is then typically based on centroidal Voronoi tessellations (CVTs, see Figure 1).…”

Generalized Gaussian quadrature rules on arbitrary polygons

“…Quadrature rules on regular polygons can be presented as software libraries and readily used in codes where integration over polygons is needed [13][14][15][16]26]. The node elimination algorithm is very flexible, and to illustrate its benefits, quadrature rules on convex and concave polygons were also presented.…”

GOTICA - generation of optimal topologies by irregular cellular automata

“…One can find only isolated examples of implementation of triangular or hexagonal lattices, e.g. Saxena (2009), Talischi et al (2009), Jain and Saxena (2010, Talischi et al (2010), Sanaei andBabaei (2011), Talischi et al (2012), Christiansen et al (2014), Wang et al (2014), Jain et al (2015). However, in many cases a practical engineering analysis and design require using highly irregular meshes for complicated geometries and/or stress concentration regions.…”