Generalized barycentric coordinates (e.g., Wachspress, mean value coordinates, maximum-entropy coordinates, and harmonic coordinates) have been widely adopted for applications in computer graphics and as basis functions in polytopal finite element methods. Over the past decade many new formulations on polygonal and polyhedral meshes have been developed. A few examples of such technologies are: continuous and discontinuous Galerkin methods, structure-preserving mimetic schemes, and virtual element methods. These novel discretization techniques purport to extend the regime of standard finite element approaches for the solution of partial differential equations — for instance, the use of polygonal and polyhedral meshes with convex and concave elements provide greater flexibility in mesh generation, and discretizations on such meshes afford robustness in material design simulations, enable large deformation simulations of Lagrangian solids, capturing flow in heterogeneous subsurface porous media, and reducing mesh-sensitivity to model complex dynamic fracture processes. In this talk, I will first present an overview of generalized barycentric coordinates and their use in polyhedral finite element methods. Then, I will focus on a recently developed stabilized Galerkin approach 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 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. The VEM provides flexibility in element technology to design arbitrary-order conforming and non-conforming methods, and trial spaces with Ck (k > 0) regularity on standard finite element as well as polytopal meshes. Applications of the VEM so far have been diverse: convection-diffusion-reaction equation, fluid flow (Stokes and Navier-Stokes equations), Maxwell equations (H-div and H-curl elements) and deformation of nonlinear solid continua to name a few. Numerical examples will be presented to affirm the accuracy and rate of convergence of the method on finite element and polytopal meshes, and I will draw connections of the VEM to hourglass control techniques in FEM.
Sukumar holds a B.Tech. in Metallurgical Engineering from IIT Bombay in 1989, a M.S. from Oregon Graduate Institute in 1992, and a Ph.D. in Theoretical and Applied Mechanics from Northwestern University in 1998. He held post-doctoral appointments at Northwestern and 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 Finite Elements in Analysis and Design. He has spent sabbatical visits at Cornell University (2007), SLAC National Accelerator Laboratory (2011) and LANL (2018). Sukumar’s research focuses on smooth maximum-entropy approximation schemes, novel discretizations on polytopal meshes, fracture modeling with extended finite elements, and new methods development (orbital-enriched partition-of-unity methods) for large-scale ab initio materials calculations.