Title: Virtual Element Method: A Stabilized Galerkin Method on Polytopal Meshes
Abstract: In this presentation, I will introduce a stabilized Galerkin method on polytopal meshes by Brezzi and coworkers (2013) coined as the virtual element method (VEM), wherein the basis functions are virtual---they are not known nor do they need to be computed within the domain. In the VEM, the trial function in an element is defined to be harmonic in its interior and piecewise affine on its boundary. The degrees of freedom of the trial function in an element are selected so that suitable polynomial projection operators can be computed, which enable the decomposition of the bilinear form into two parts: a consistent term that reproduces a given polynomial space and a correction term that ensures stability. To compute the stiffness matrix, numerical integration of only monomials over the polytope is required. On linear Delaunay meshes, VEM is identical to the FEM, but VEM provides flexibility in element technology (convex and nonconvex elements are allowed) to design higher order conforming and non-conforming methods, simplifies conformity, contact enforcement and adaptivity on nonmatching meshes with hanging nodes, and spaces with high-order regularity and structure-preserving properties are readily constructed on standard finite element as well as polytopal meshes. Over the past five years, many novel formulations have been proposed for simulations in solid and fluid continua. I will begin with a brief overview of FEM, meshfree methods and generalized barycentric coordinates. This will be followed by the derivation, implementation and numerical results of the VEM for the Poisson problem. Finally, if time remains, I will show two recent applications of virtual elements as an enabling technology for finite elements: dramatic increase in the critical time step in elastic wave propagation (explicit dynamics simulations) on poor-quality tetrahedral meshes and a new approach to treat near-incompressibility in linear elasticity.
Bio: Sukumar holds a B.Tech. in Metallurgical Engineering from IIT Bombay (1989), a M.S. in Materials Science from Oregon Graduate Institute (1992), and a Ph.D. in Theoretical and Applied Mechanics from Northwestern University (1998). He was a post-doc at Northwestern University and a research associate in MAE at Princeton University, before joining UC Davis in 2001, where he is currently a Professor in Civil and Environmental Engineering. Sukumar is a Regional Editor of International Journal of Fracture and a member of the Editorial Board of Computer Methods in Applied Mechanics and Engineering and Finite Elements in Analysis and Design. Sukumar's research focuses on maximum-entropy and deep learning based PDE solvers, novel discretization methods on polytopal meshes for deformation of solid continua, cubature schemes over curved geometries, extended FEM, and orbital-enriched partition-of-unity methods for quantum-mechanical materials calculations. Sukumar is currently on a sabbatical visit to George Karniadakis's Group in Applied Mathematics at Brown University.