A Robust Immersed Interface Method for Viscous Incompressible Fluids and Discrete Surfaces

The Immersed Interface Method (IIM) for models of fluid flow and fluid-structure interactionimposes jump conditions that capture stress discontinuities generated by forces that are concentratedalong immersed boundaries. Imposing these discontinuities requires decomposing the boundaryforce into its normal and tangential components, which respectively determine jump conditionsfor the pressure and velocity gradient. Our present work builds upon prior work to integrate finiteelement-type representations of the interface geometries with the IIM. First, we address the challengeswith geometries with sharp features by introducing a discontinuous Galerkin formulation for the jumpconditions. Then, we present a solution to the numerical leaking found in highly pressure-loadedmodels by introducing a smoothed representation of the discrete interfacial normal vector. Finally,we present a new modified bilinear interpolation operator for simulating incompressible fluidflows in thin gaps between closely spaced immersed boundaries. We also extend our near contactmethodology to provide an alternative approach to treating geometries with sharp features. For eachof these challenges, our implemented approaches demonstrate substantial improvements in accuracywhen compared to analytical benchmarks and prior studies.