Date and time: 14th September, 11.00 am.
Venue: PSB Seminar hall
Speaker: Prof. N. Sukumar (Uni California Davis)
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 (harmonic generalized barycentric coordinates). 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. This approach provides a means to extend hourglass finite element methods to polygonal and polyhedral meshes. On linear Delaunay meshes, VEM is identical to the FEM. The 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, 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.