Date February 5, 2016
Speaker Nathaniel Trask Brown University
Topic Compatible meshless discretization through $\ell_2$-optimization
Abstract Meshless methods provide an ideal framework for scalably simulating problems involving boundaries undergoing large deformation or interfaces between multiple materials. Discretization points may be moved in a Lagrangian fashion without the need for either costly mesh topology updates or diffuse Eulerian treatment of interfaces. Of the range of meshless discretizations available, there is a distinct lack of methods that maintain a sense of compatibility while simultaneously achieving high-order accuracy. In this talk, we present a new discretization that generalizes staggered primal/dual discretizations to unstructured point sets. Using only the epsilon-neighborhood graph of discretization points and solving inexpensive optimization problems, we construct divergence and gradient operators that mimic the algebraic structure of compatiblemesh-based discretizations. When applied to a model div-grad diffusion problem, we obtain high-order convergence for smooth solutions and observe monotone fluxes for problems with discontinuous material properties. We then present a new mixed meshless discretization for the Stokes equations, using a divergence-free moving least squares method for velocity and staggered moving least squares for the pressure. This approach achieves equal order convergence for both velocity and pressure, making it ideal for simulating problems in dense suspension flows dominated by lubrication forces. We finally assemble the Stokes solver, a Poisson-Boltzmann solver based on the staggered scheme, and a 6-DOF solver for colloid dynamics together into a monolithic, fully implicit scheme that we use to study electrophoretic suspensions. By using auxiliary space algebraic multigrid preconditioning to solve the resulting system, we obtain an efficient, robust, and highly accurate new tool for studying these problems in complex geometries.