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 and XIII (2016, 2018), as well as Polygonal and Polyhedral Discretizations in Computational Mechanics at USNCCM (2015, 2017, 2019, 2021, 2023) and WCCM XIII (2018).

Publications

• E. B. Chin, F. Dassi, G. Manzini and N. Sukumar (2024), "Numerical Integration in the Virtual Element Method with the Scaled Boundary Cubature Scheme," International Journal for Numerical Methods in Engineering, DOI: https://doi.org/10.1002/nme.7549.
• A. Chen, J. E. Bishop and N. Sukumar (2024), "Stress-Hybrid Virtual Element Method on Six-Noded Triangular Meshes for Compressible and Nearly-Incompressible Linear Elasticity," Computer Methods in Applied Mechanics and Engineering, Vol. 426, Article 116971.
• J. Jaśkowiec and N. Sukumar (2024), "Penalty-free discontinuous Galerkin method," International Journal for Numerical Methods in Engineering, Vol. 125, Number 12, Article e7472.
• A. Russo and N. Sukumar (2024), "Quantitative Study of the Stabilizaion Parameter in the Virtual Element Method," in Beirao da Veiga, H. and F. Minhós and N. Van Goethem and L. Sanchez Rodrigues (Eds), Nonlinear Differential Equations and Applications (PICNDEA 2022), CIM Series in Mathematical Sciences, Vol. 7, Springer, Cham.
• L. Dieci and F. V. Difonzo and N. Sukumar (2024), "Nonnegative Moment Coordinates on Finite Element Geometries," Mathematics in Engineering, Vol. 6, Number 1, pp. 81–99.
• A. Chen and N. Sukumar (2024) "Stress-Hybrid Virtual Element Method on Quadrilateral Meshes for Compressible and Nearly-Incompressible Linear Elasticity," International Journal for Numerical Methods in Engineering, Vol. 125, Number 3, Article e7384.
• A. Chen and N. Sukumar (2023) "Stabilization-Free Serendipity Virtual Element Method for Plane Elasticity," Computer Methods in Applied Mechanics and Engineering, Vol. 404, Article 115784.
• A. Chen and N. Sukumar (2023) "Stabilization-Free Virtual Element Method for Plane Elasticity," Computers and Mathematics with Applications, Vol. 138, pp. 88–105.
• N. Sukumar and J. E. Bolander (2022) "Virtual Element Method for Modeling the Deformation of Multiphase Composites," Mechanics Research Communications, Vol. 124, Article 103907.
• N. Sukumar and M. R. Tupek (2022) "Virtual Elements on Agglomerated Finite Elements to Increase the Critical Time Step in Elastodynamic Simulations," International Journal for Numerical Methods in Engineering, Vol. 123, Number 19, pp. 4702–4725.
• E. Benvenuti, A. Chiozzi, G. Manzini and N. Sukumar (2022), "Extended Virtual Element Method for Two-Dimensional Linear Elastic Fracture," Computer Methods in Applied Mechanics and Engineering, Vol. 390, Article 114352. Available at arXiv: 2111.04150
• E. B. Chin and N. Sukumar (2021), "Scaled Boundary Cubature Scheme for Numerical Integration over Planar Regions with Affine and Curved Boundaries," Computer Methods in Applied Mechanics and Engineering, Vol. 380, Article 113796.
• J. Jaśkowiec and N. Sukumar (2021), "Addendum to the Paper "High-Order Symmetric Cubature Rules for Tetrahedra and Pyramids," International Journal for Numerical Methods in Engineering, Vol. 122, Number 7, pp. 1875–1883. [Cubature Rules]
• J. Jaśkowiec and N. Sukumar (2021), "High-Order Symmetric Cubature Rules for Tetrahedra and Pyramids," International Journal for Numerical Methods in Engineering, Vol. 122, Number 1, pp. 148–171. [Cubature Rules]
• E. B. Chin and N. Sukumar (2020), "An Efficient Method to Integrate Polynomials Over Polytopes and Curved Solids," Computer Aided Geometric Design, Vol. 82, 101914. [C++ Code]
• J. Jaśkowiec and N. Sukumar (2020), "High-Order Cubature Rules for Tetrahedra," International Journal for Numerical Methods in Engineering, Vol. 121, Number 11, pp. 2418–2436. [Cubature Rules]
• R. Silva-Valenzuela, A. Ortiz-Bernardin, N. Sukumar, E. Artioli and N. Hitschfeld-Kahler (2020), "A Nodal Integration Scheme for Meshfree Galerkin Methods using the Virtual Element Decomposition," International Journal for Numerical Methods in Engineering, Vol. 121, Number 10, pp. 2174–2205.
• J. E. Bishop and N. Sukumar (2020), "Polyhedral Finite Elements for Nonlinear Solid Mechanics using Tetrahedral Subdivisions and Dual-Cell Aggregation," Computer Aided Geometric Design, Vol. 77, Article 101812.
• E. Benvenuti, A. Chiozzi, G. Manzini and N. Sukumar (2019), "Extended Virtual Element Method for the Laplace Problem with Singularities and Discontinuities," Computer Methods in Applied Mechanics and Engineering, Vol. 356, pp. 571–597.
• E. B. Chin, J. E. Bishop, R. V. Garimella and N. Sukumar (2018), "Finite Deformation Cohesive Polygonal Finite Elements for Modeling Pervasive Fracture," International Journal of Fracture, Vol. 214, Number 2, pp. 139–165.
• K. Hormann and N. Sukumar (Eds.) (2017), "Generalized Barycentric Coordinates in Computer Graphics and Computational Mechanics," CRC Press, Boca Raton, FL. [AMAZON] [Taylor & Francis]
• 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, Vol. 112, Number 7, pp. 655–684.
• 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.
• 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.
• 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.
• 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. 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.
• 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

• "Virtual Element Method: A Stabilized Galerkin Method on Polytopal Meshes," Invited Seminar, Applied and Computational Math Seminar, Portland State University, Portland, OR, April 2023.

• "Virtual Element Method: A Stabilized Galerkin Method on Polytopal Meshes," Invited Seminar, Dassault Systemes Simulia Corporation, Johnston, RI, March 2023.

• "Scaled Boundary Cubature Scheme in Higher Dimensions: Integration over Polytopes and Curved Regions (with E. B. Chin)," Invited Lecture, POEMS Workshop, Milano, Italy, December 2022.

• "Virtual Element Method: A Stabilized Galerkin Method on Polytopes," Invited Seminar, Department of Aerospace Engineering, IIT Bombay, Mumbai, India, September 2022.

• "Introduction to Virtual Element Methods, Numerical Integration over Polytopes and Extended Finite Element Methods," Invited Lectures, Department of Mechanical Engineering, Indian Institute of Science, Bengaluru, India, July 2022.

• "Recent Advances in Polyhedral Virtual Element Methods and Physics-Informed Deep Neural Networks to Solve PDEs," Invited Seminar, Department of Mechanical Engineering, Indian Institute of Science, Bengaluru, India, July 2022.

• "Numerical Integration over Polytopes and Curved Regions and use of VEM to Advance Finite Element Technology," Invited Lecture, VEM Mini-Course (SDS 2022 Meeting), Rosa Marina, Italy, June 2022.

• "Recent Advances in Polyhedral Virtual Element Methods and Physics-Informed Deep Neural Networks to Solve PDEs," Invited Seminar, T-3 Division, Los Alamos National Laboratory, Los Alamos, NM, May 2022.

• "Recent Advances in Polyhedral Virtual Element Methods and Physics-Informed Deep Neural Networks to Solve PDEs," Invited Seminar, Engineering Sciences Center, Sandia National Laboratories, Albuquerque, NM, May 2022.

• "Stabilization-Free Virtual Element Method for Plane Elasticity (with A. Chen)," Invited Seminar, T-3 Division, Los Alamos National Laboratory, Los Alamos, NM, March 2022.

• "Virtual Elements on Agglomerated Finite Elements to Increase the Critical Time Step in Explicit Three-Dimensional Elastodynamic Simulations," US National Congress on Computational Mechanics, Chicago, IL, July 2021.

• "Scaled Boundary Cubature Scheme for Numerical Integration over Planar Regions with Affine and Curved Boundaries (with E. B. Chin)," Plenary Lecture, INdAM Conference on Polygonal Methods for PDEs: Theory and Applications, Rome, Italy, May 2021.

• "Numerical Integration of Homogeneous Functions on Curved Geometries," US National Congress on Computational Mechanics, Austin, TX, July 2019.

• "Numerical Integration of Homogeneous Functions on Curved Geometries," Invited Lecture, SIAM Conference on Geometric Design, Vancouver, British Columbia, Canada, June 2019.

• "Virtual Element Method in Computational Mechanics," Invited Seminar, Department of Mechanical, Materials and Aerospace Engineering, Illinois Institute of Technology, Chicago, IL, December 2018.

• "Virtual Element Method in Computational Mechanics," Invited Seminar, CSRI Seminar, Sandia National Laboratories, Albuquerque, NM, July 2018.

• "Virtual Element Method in Computational Mechanics," Invited Seminar, T-3 Fluid Dynamics and Solid Mechanics Seminar, Los Alamos National Laboratory, Los Alamos, NM, July 2018.

• "Polygonal and Polyhedral Finite Element Technology in Computational Mechanics," Invited Seminar, Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, PA, November 2017.

• "Polygonal and Polyhedral Finite Element Technology in Computational Mechanics," Invited Seminar, ANSYS Inc., Canonsburg, PA, November 2017.

• "Numerical Integration of Homogeneous Functions on Convex and Nonconvex Polytopes: Applications in the Extended Finite Element Method," Invited Seminar, Department of Mechanical Engineering, UM - Shanghai Jiao Tong University Joint Institute, Shanghai, China, August 9, 2017.

• "Numerical Integration of Homogeneous Functions on Convex and Nonconvex Polytopes: Applications in the Extended Finite Element Method," Invited Seminar, State Key Laboratory of Scientific and Engineering Computing, Chinese Academy of Sciences, Beijing, China, July 31, 2017.

• "Numerical Integration of Homogeneous Functions on Convex and Nonconvex Polytopes: Applications in the Extended Finite Element Method," Invited Seminar, School of Mathematical Sciences, Peking University, Beijing, China, July 28, 2017.

• "Modeling Crack Discontinuities and Holes without Element-Partitioning in the Extended Finite Element Method," (with Eric B. Chin), 14th U.S. National Congress on Computational Mechanics, Montreal, Canada, July 2017.

• "Modeling Crack Discontinuities and Holes without Element-Partitioning in the Extended Finite Element Method," Invited Lecture, POEMS 2017, Milano, Italy, July 2017.

• "Modeling Crack Discontinuities without Element Partitioning in the Extended Finite Element Method (with Eric B. Chin and Jean B. Lasserre),", WCCM XII and APCOM VI, Seoul, South Korea, July 2016.

• "Numerical Integration of Homogeneous Functions and Polynomials on Polytopes (with Eric B. Chin and Jean B. Lasserre),", Polytopal Element Methods in Mathematics and Engineering (POEMs), Atlanta, GA, October 2015.

• "Modeling Crack Discontinuities without Element Partitioning in the Extended Finite Element Method (with Eric B. Chin and Jean B. Lasserre),", SES 52nd Annual Technical Meeting, College Station, TX, October 2015.

• "Quadrature Methods for Discontinuities and Singularities," Invited Lectures, X-DMS Short Course on Extended Discretization Methods, Ferrara, Italy, September 2015.

• "Cohesive Fracture Simulations on Polygonal Finite Element Meshes (with J. E. Bishop)," Invited Lecture, Symposium on Computational Fracture Mechanics, 51st Annual Technical Conference of Society of Engineering Sciences, Purdue University, West Lafayette, IN, October 2014.

• "Computations on Polygonal and Polyhedral Meshes with Entropy Approximants," Invited Seminar, Los Alamos National Laboratory, Los Alamos, NM, August 2014.

• "Computations on Polygonal and Polyhedral Meshes with Entropy Approximants," Invited Seminar, Sandia National Laboratories, Albuquerque, NM, August 2014.

• "Polygonal and Polyhedral Finite Element Methods in Computational Mechanics," Plenary Lecture, PACAM XIV: Fourteenth Pan-American Congress of Applied Mechanics, Santiago, Chile, March 2014.

• "Polygonal and Polyhedral Finite Element Methods in Computational Mechanics," Invited Seminar, Department of Civil and Environmental Engineering, Stanford University, CA, February 2014.

• "Polygonal and Polyhedral Finite Element Methods in Computational Mechanics," Keynote Lecture, 23rd International Workshop on Computational Mechanics of Materials, Singapore, October 2013.

• "Efficient Polygonal and Polyhedral Finite Element Methods (with G. Manzini and A. Russo)," Invited Lecture, 7th International Workshop on Meshfree Methods for Partial Differential Equations, Bonn, Germany, September 2013.

• "New Perspectives on Polygonal and Polyhedral Finite Element Methods (with G. Manzini and A. Russo)," Minisymposium on General Polyhedral Discretizations: Meshing, Formulations, and Applications, 12th U.S. National Congress on Computational Mechanics, Raleigh, NC, July 2013.

• "Meshfree Methods: From the Element-Free Galerkin Method to Maximum-Entropy Approximation Schemes," Invited Lecture, Advances in Computational Mechanics with Emphasis on Fracture and Multiscale Phenomena: Workshop Honoring Professor Ted Belytschko's 70th Birthday, Evanston, IL, April 2013.

• "Quadratic Maximum-Entropy Serendipity Shape Functions on Arbitrary Planar Polygons," Invited Lecture, Minisymposium on Meshfree and Generalized Finite Element Methods, Advances in Computational Mechanics: A Conference Celebrating the 70th Birthday of Thomas J. R. Hughes, San Diego, February 2013.

• "Generalized Barycentric Coordinates on Polygons and Polyhedra," Invited Lecture, Combined Applied Math and PDEs Seminar, Department of Mathematics, University of California at Davis, October 2012.

• "Quadratic Polygonal Finite Element Methods," Invited Lecture, Faculty of Informatics, Universita della Svizzera italiana, Lugano, Switzerland, September 21, 2012.

• "Polygonal Finite Element Methods," Invited Lecture, Workshop on Discretization Methods for Polygonal and Polyhedral Meshes, University of Milano-Bicocca, Milan, Italy, September 17-19, 2012.

• "Barycentric Finite Element Methods," Invited Lecture, Workshop on Generalized Barycentric Coordinates in Geometry Processing and Finite/Boundary Element Methods, Columbia University, New York, July 26, 2012.

• "Barycentric Finite Element Methods," Invited Lecture, Symposium on Barycentric Coordinates and Transfinite Interpolation, Tenth SIAM Conference on Geometric Design & Computing, San Antonio, TX, November 5, 2007.

• "Recent Advances in Polygonal Finite Element Methods," Invited Lecture, Mathematical Modeling and Analysis Group, Los Alamos National Laboratory, Los Alamos, New Mexico, April 2006.

• "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

• "Adaptive Computations using Conforming Quadtree Meshes," Invited Lecture, Mechanics Colloquim, Technical University of Kaiserslautern, Germany, September 2005

• "Adaptive Computations using Residual-Based Error Estimators on Conforming Quadtree Meshes," (with A. Tabarraei), Third International Workshop on Meshfree Methods for Partial Differential Equations, Bonn, Germany, September 2005

• "Construction and Application of Polygonal Finite Element Approximations," (with A. Tabarraei), Sixth World Congress on Computational Mechanics, Symposium on Meshfree and Extended/Fictitious FEM, Beijing, China, September 2004.

• "Polygonal Interpolants: Construction and Adaptive Computations on Quadtree Meshes," (with A. Tabarraei), European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2004), Symposium on Computational Fracture Mechanics, Jyvskyla, Finland, July 2004.

• "Polygonal Finite Elements: Construction and Applications to PDEs," Invited Lecture, Ecole Centrale de Nantes, Nantes, France, July 2004

• "Numerical Formulation and Application of Polygonal Finite Elements" (with A. Tabarraei), 7th International ESAFORM Conference on Material Forming, Trondheim, Norway, April 2004

• "Conforming Polygonal Finite Elements," Invited Lecture, Laboratoire de Mecanique des Systemes et des Procedes, ENSAM, Paris, France, April 2004

Online Resources

• Barycentric Coordinates and Transfinite Interpolation
• NSF-Sponsored Workshop on Generalized Barycentric Coordinates
• Workshop on Discretization Methods for Polygonal and Polyhedral Meshes
• POEMS 2015
• POEMS 2017

Software

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

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

• 2D VEM in 50 lines of Matlab (written by Oliver Sutton)

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