Meshless methods provide an ideal framework for scalably simulating Lagrangian hydrodynamics in domains undergoing large deformation. For these schemes, interfaces can easily be treated without introducing artificial diffusion, and boundary deformation is handled without costly topological updates to the mesh. Unfortunately, abandoning a mesh also means abandoning a natural framework for performing analysis and as a result meshless methods have historically lacked the stability and conservation properties of their finite element/volume counterparts. In this work, we present a number of new schemes that use a combination of $\ell_2$-optimization and graph theory to achieve highly accurate and robust meshless discretizations. These schemes mimic the algebraic structure of compatible finite-element methods, and as a result inherit many of their favorable properties. We use these methods to develop monolithic schemes for suspension flows driven by electrokinetic effects. While these flows are challenging due to the presence of singular pressure forces and a range of relevant length scales of several orders of magnitude, we demonstrate that these new techniques easily resolve analytic benchmarks without the need for sub-grid scale lubrication models.
Trask, Nathaniel A.,
"Compatible high-order meshless schemes for viscous flows through l2 optimization"
Applied Mathematics Theses and Dissertations.
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