A new locking-free polygonal plate element for thin and thick plates based on Reissner-Mindlin plate theory and assumed shear strain fields

A new $n-$ noded polygonal plate element is proposed for the analysis of plate structures comprising of thin and thick members. The formulation is based on the discrete Kirchhoff Mindlin theory. On each side of the polygonal element, discrete shear constraints are considered to relate the kinematical and the independent shear strains. The proposed element: (a) has proper rank; (b) passes patch test for both thin and thick plates; (c) is free from shear locking and (d) yields optimal convergence rates in $L^2-$norm and $H^1-$semi-norm. The accuracy and the convergence properties are demonstrated with a few benchmark examples

Analysis and new constructions of generalized barycentric coordinates in 2D

Different coordinate systems allow to uniquely determine the position of a geometric element in space. In this dissertation, we consider a coordinate system that lets us determine the position of a two-dimensional point in the plane with respect to an arbitrary simple polygon. Coordinates of this system are called generalized barycentric coordinates in 2D and are widely used in computer graphics and computational mechanics. There exist many coordinate functions that satisfy all the basic properties of barycentric coordinates, but they differ by a number of other properties. We start by providing an extensive comparison of all existing coordinate functions and pointing out which important properties of generalized barycentric coordinates are not satisfied by these functions. This comparison shows that not all of existing coordinates have fully investigated properties, and we complete such a theoretical analysis for a particular one-parameter family of generalized barycentric coordinates for strictly convex polygons. We also perform numerical analysis of this family and show how to avoid computational instabilities near the polygon’s boundary when computing these coordinates in practice. We conclude this analysis by implementing some members of this family in the Computational Geometry Algorithm Library. In the second half of this dissertation, we present a few novel constructions of non-negative and smooth generalized barycentric coordinates defined over any simple polygon. In this context, we show that new coordinates with improved properties can be obtained by taking convex combinations of already existing coordinate functions and we give two examples of how to use such convex combinations for polygons without and with interior points. These new constructions have many attractive properties and perform better than other coordinates in interpolation and image deformation applications