The Optimal Transportation Meshfree Method for General Fluid Flows and Strongly Coupled Fluid-Structure Interaction Problems

Abstract

This thesis develops a novel meshfree numerical method for simulating general fluid flows. Drawing from concepts in optimal mass transport theory and in combination with the notion of material point sampling and meshfree interpolation, the optimal transport meshfree (OTM) method provides a rigorous mathematical framework for numerically simulating three-dimensional general fluid flows with general, and possibly moving boundaries (as in fluid-structure interaction simulations). Specifically, the proposed OTM method generalizes the Benamou-Brenier differential formulation of optimal mass transportation problems which leads to a multi-field variational characterization of general fluid flows including viscosity, equations of state and general geometries and boundary conditions. With the use of material point sampling in conjunction with local max-entropy shape functions, the OTM method leads to a meshfree formulation bearing a number of salient features. Compared with other meshfree methods that face significant challenges to enforce essential boundary conditions as well as couple to other methods, such as the finite element method, the OTM method provides a new paradigm in meshfree methods. The OTM method is numerically validated by simulating the classical Riemann benchmark example for Euler flow. Furthermore, in order to highlight the ability of the OTM to simulate Navier-Stokes flows within general, moving three-dimensional domains, and naturally couple with finite elements, an illustrative strongly coupled FSI example is simulated. This illustrative FSI example, consisting of a gas-inflated sphere impacting the ground, is simulated as a toy model of the final phase of NASA's landing scheme devised for Mars missions, where a network of airbags are deployed to dissipate the energy of impact.