The development of polygonal finite element approximations and their use to solve PDEs is pursued. The construction of different linearly complete approximations on polygonal domains is studied: Wachspress, mean value coordinates, and meshfree (Laplace shape functions) shape functions are used within a Galerkin method. Issues pertaining to numerical integration, imposition of essential boundary conditions, and mesh generation are of note. The use of Laplace shape functions leads to a particular generalization of finite elements to convex polygons of arbitrary order. As an application of the above advance, conforming approximations on quadtree meshes are developed for adaptive simulations. The data approximation problem is also studied using the principle of maximum entropy. Kai Hormann maintains the Barycentric Coordinates and Transfinite Interpolation web page that provides links to online resources.

Publications

• S. E. Mousavi and N. Sukumar (2011), "Numerical Integration of Polynomials and Discontinuous Functions on Irregular Convex Polygons and Polyhedrons," Computational Mechanics, Vol. 47, Number 5, pp. 535–554.
• S. E. Mousavi, H. Xiao and N. Sukumar (2010), "Generalized Gaussian Quadrature Rules on Arbitrary Polygons," International Journal for Numerical Methods in Engineering, Vol. 82, Number 1, pp. 99–113.
• N. Sukumar and J. E. Bolander (2009), "Voronoi-based Interpolants for Fracture Modelling," in Tessellations in the Sciences; Virtues, Techniques and Applications of Geometric Tilings, Springer Verlag, pp. xxx–xxx.
• K. Hormann and N. Sukumar (2008), "Maximum Entropy Coordinates for Arbitrary Polytopes," Computer Graphics Forum, Vol. 27, Number 5, pp. 1513–1520. Proceedings of SGP 2008.
• A. Tabarraei and N. Sukumar (2008), "Extended Finite Element Method on Polygonal and Quadtree Meshes," Computer Methods in Applied Mechanics and Engineering, Vol. 197, Number 5, pp. 425–438.
• N. Sukumar and R. W. Wright (2007), "Overview and Construction of Meshfree Basis Functions: From Moving Least Squares to Entropy Approximants," International Journal for Numerical Methods in Engineering, Vol. 70, Number 2, pp. 181–205.
• A. Tabarraei and N. Sukumar (2007), "Adaptive Computations Using Material Forces and Residual-Based Error Estimators on Quadtree Meshes," Computer Methods in Applied Mechanics and Engineering, Vol. 196, Number 25–28, pp. 2657–2680.
• N. Sukumar and E. A. Malsch (2006), "Recent Advances in the Construction of Polygonal Finite Element Interpolants," Archives of Computational Methods in Engineering, Vol. 13, Number 1, pp. 129–163. (proof)
• A. Tabarraei and N. Sukumar (2006), "Application of Polygonal Finite Elements in Linear Elasticity" International Journal of Computational Methods, Vol. 3, Number 4, pp. 503–520.
• N. Sukumar (September 2006), "Where Do We Stand on Meshfree Approximation Schemes?" in Online Blog on Meshfree Methods or
• N. Sukumar (2005), "Maximum Entropy Approximation," in Proceedings of the 25th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering, Eds. K. H. Knuth, A. E. Abbas, R. D. Morris and J. P. Castle, AIP Conference Proceedings, Vol. 803, Number 1, pp. 337–344.
• A. Tabarraei and N. Sukumar (2005), "Adaptive Computations on Conforming Quadtree Meshes," Finite Elements in Analysis and Design, Vol. 41, Number 7-8, pp. 686–702 or UC repository postprints
• N. Sukumar (2004), "Construction of Polygonal Interpolants: A Maximum Entropy Approach," International Journal for Numerical Methods in Engineering, Vol. 61, Number 12, pp. 2159–2181
• N. Sukumar and A. Tabarraei (2004), "Conforming Polygonal Finite Elements," International Journal for Numerical Methods in Engineering, Vol. 61, Number 12, pp. 2045–2066
• N. Sukumar and A. Tabarraei (2004), "Polygonal Interpolants: Construction and Adaptive Computations on Quadtree Meshes," in European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2004), Eds. P. Neittaanmäki, T. Rossi, S. Korotov, E. Onate, J. Periaux, and D. Knörzer, Jyväskylä, Finland
• N. Sukumar and A. Tabarraei (2004), "Numerical Formulation and Application of Polygonal Finite Elements," in Proceedings of the Seventh ESAFORM Conference on Metal Forming, Ed. S. Stören, Trondheim, Norway, pp. 73--76
• N. Sukumar (2003), ``Voronoi Cell Finite Difference Method for the Diffusion Operator on Arbitrary Unstructured Grids,'' International Journal for Numerical Methods in Engineering, Vol. 57, Number 1, pp. 1-34 [Abstract]
• References [BibTeX Entries]

Presentations

"Natural Neighbors and Voronoi Tessellations in Computational Solid Mechanics," Keynote Lecture, The World a Jigsaw: Tessellations in the Sciences, Lorent Center, Leiden University, The Netherlands, March 6-10, 2006