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POLYGONAL AND POLYHEDRAL FINITE ELEMENT METHODS

PHI PHI PHI
PHI PHI PHI

Polygonal and polyhedral elements: Shape functions and meshes

(Supported by NSF Grant CMMI-1334783, 09/2013–08/2017)


The development of Galerkin finite element methods on arbitrary polygonal and polyhedral elements to solve PDEs is pursued. The origins of this approach can be traced to Wachspress basis functions, which is a particular generalization of finite elements to planar convex polygons. Since then, many new shape functions on polygonal and polyhedral domains have emerged: mean value coordinates, metric coordinates, Laplace shape functions, maximum-entropy coordinates, harmonic coordinates, etc.

Current research focuses on contributions related to the virtual element method, a consistent and stable Galerkin method on arbitrary (polygonal and polyhedral) meshes with higher-order and smooth approximations; and on enabling fracture simulations on polygonal and polyhedral meshes. Coorganized minisymposium on Computational Fracture at SES 2014, SES 2015, USNCCM (2015, 2017) and WCCM XII (2016), as well as Polygonal and Polyhedral Discretizations in Computational Mechanics at USNCCM (2015, 2017).


Publications

  • A. Ortiz-Bernardin, A. Russo and N. Sukumar (2017), "Consistent and stable meshfree Galerkin methods using the virtual element decomposition," International Journal for Numerical Methods in Engineering, DOI: 10.1002/nme.5519. [PDF]
  • E. B. Chin, J. B. Lasserre and N. Sukumar (2015), "Numerical Integration of Homogeneous Functions on Convex and Nonconvex Polygons and Polyhedra," Computational Mechanics, Vol. 56, Number 6, pp. 967–981. [PDF]
  • A. Cangiani, G. Manzini, A. Russo and N. Sukumar (2015), "Hourglass Stabilization and the Virtual Element Method," International Journal for Numerical Methods in Engineering, Vol. 102, Number 3-4, pp. 404–436. [PDF]
  • G. Manzini, A. Russo and N. Sukumar (2014), "New Perspectives on Polygonal and Polyhedral Finite Element Methods," Math. Models Methods Appl. Sci., Vol. 24, Number 8, pp. 1665–1699. [PDF]
  • M. Floater, A. Gillette and N. Sukumar (2014), "Gradient Bounds for Wachspress Coordinates on Polytopes," SIAM Journal on Numerical Analysis, Vol. 52, Number 1, pp. 515–532. [PDF] Available at arXiv:1306.4385
  • N. Sukumar (2013), "Quadratic Maximum-Entropy Serendipity Shape Functions for Arbitrary Planar Polygons," Comput. Meth. Appl. Mech. Engrg., Vol. 263, pp. 27–41. [PDF]
  • 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. [PDF]
  • 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. [PDF]
  • 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. [PDF]
  • 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. [PDF]
  • 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. [PDF]
  • 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. [PDF]
  • 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. [PDF]
  • 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. [PDF] (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. [PDF]
  • N. Sukumar (September 2006), "Where Do We Stand on Meshfree Approximation Schemes?" in Online Blog on Meshfree Methods [HTML] or [HTML]
  • 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. [HTML]
  • 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 [PDF] 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 [PDF]
  • N. Sukumar and A. Tabarraei (2004), "Conforming Polygonal Finite Elements," International Journal for Numerical Methods in Engineering, Vol. 61, Number 12, pp. 2045–2066 [PDF]
  • 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 [PDF]
  • 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 [PDF]
  • 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] [PDF]
  • References [BibTeX Entries] [PDF]

Presentations

Online Resources

Software

  • Polygonal (Laplace) Shape Functions (in Fortran) [Tar] [ZIP] [README]

  • JAVA applets to visualize polygonal basis functions: [1D, 2D, 3D]

Disclaimer: Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation


© Copyright 2004–2015 N. Sukumar. All rights reserved.