We present a cut cell method in R2 for enforcing Dirichlet and Neumann boundary conditions with nearly incompressible linear elastic materials in irregular domains. Virtual nodes on cut uniform grid cells are used to provide geometric flexibility in the domain boundary shape without sacrificing accuracy. We use a mixed formulation utilizing a MACtype staggered grid with piecewise bilinear displacements centered at cell faces and piecewise constant pressures at cell centers. These discretization choices provide the necessary stability in the incompressible limit and the necessary accuracy in cut cells. Numerical experiments suggest second order accuracy in L1. We target high-resolution problems and present a class of geometric multigrid methods for solving the discrete equations for displacements and pressures that achieves nearly optimal convergence rates independent of grid resolution.

@Article{ZWHCST12,
  author       = "Zhu, Yongning and Wang, Yuting and Hellrung, Jeffrey and Cantarero, Alejandro and Sifakis, Eftychios and Teran, Joseph",
  title        = "A Second-Order Virtual Node Algorithm for Nearly Incompressible Linear Elasticity in Irregular Domains",
  journal      = "Journal of Computational Physics",
  number       = "to appear",
  volume       = "accepted",
  year         = "2012",
  url          = "http://graphics.cs.wisc.edu/Papers/2012/ZWHCST12"
}