# SC17-014: Virtual Element Method in Solid and Fluid Mechanics

N. Sukumar, University of California, Davis
Gianmarco Manzini, Los Alamos National Laboratory

In this short course, we will present the foundations and applications of the Virtual Element Method (VEM)  in solid and fluid mechanics. The VEM is a stabilized Galerkin formulation that permits robust and accurate computations on arbitrary (convex and nonconvex) polygonal and polyhedral meshes. It provides a variational framework for mimetic finite differences and also generalizes hourglass finite elements to arbitrary polytopal meshes.  Over the past decade it has become the subject of substantial research and new formulations of the method have appeared to solve initial/boundary-value problems in solid and fluid continua.  The VEM afford significant flexibility in element geometries that are permissible: for example, nonconvex elements, elements with short edges in 2D and small faces in 3D, and hanging nodes in a mesh to name a few.  In addition, it provides new opportunities to enable accurate and robust computations for finite elements on poor-quality finite element meshes. This short course will be beneficial to graduate student researchers, scientists and academic faculty to gain expertise in this emerging method in computational mechanics. The course will contain 5 lectures and a hands-on two-hour tutorial session.  For the hands-on tutorial session, participants should bring a laptop with an installation of Matlab.  The course outline follows.

Topics Covered in Lectures

1. Introduction to VEM: Introduction to polytopal computations and the conforming virtual element method, drawing on its connections to hourglass finite elements. Accurate and efficient Numerical integration of polynomials and nonpolynomial functions over polytopes. (Sukumar)
2. First-order (k = 1) VEM for the Poisson problem in 2D and 3D: Connections of mimetic finite difference schemes to the VEM. Introduction to conforming virtual element spaces and the element formulation in 2D and 3D. Numerical implementation of the method and solution of a few benchmark problems will be presented. (Manzini)
3. High-order (k > 1) VEM for the Poisson problem in 2D and 3D and Solution of Biharmonic Equation in 2D: Extension of the VEM to high-order C^0 formulations in 2D and 3D, and a C^1 VEM for the biharmonic equation in 2D. (Manzini)
4. VEM in Solid Mechanics:  First-order conforming VEM for isotropic, linear elasticity in 2D and 3D.  Stabilization-free first-order and serendipity (k = 2,3) VEM for plane elasticity will be presented. (Sukumar)
5. VEM in Fluid Mechanics: Formulation of VEM for the Stokes equation (exact discrete divergence-free element) in 2D and its extension for the time-independent Navier-Stokes equations in 2D. (Manzini)

Hands-on Tutorial

Matlab computer code to solve the 2D Poisson problem using low- and high-order VEM will be made available to participants.  Explanation of the code in light of the formulation will be presented, and verification tests of the method (accuracy and rates of convergence) will be assessed through the code. (Manzini and Sukumar)

Schedule

Registration: 8:00 am to 8:30 am

Lectures 1-4: 8:30 am to 12:30 am (15 min coffee break after the second lecture)

Lunch: 12:30 pm to 1:30 pm

Lecture 5: 1:30 pm to 2:20 pm

Tutorial: 2:30 pm to 4:30 pm

References

  L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, A. Russo, Basic principles of the virtual element method, Math. Models Methods Appl. Sci., 23, 119-214, 2013