Advancing the shifted boundary method: scalable multi-physics simulations on octree meshes for complex geometries

Abstract

This dissertation advances the field of computational mechanics by developing, optimizing, and applying the Shifted Boundary Method (SBM) integrated with octree-based meshes for accurate and efficient simulation of large-scale problems involving complex geometries. The research is structured into four key contributions: First, the SBM framework is extended and optimized to improve the simulation of Partial Differential Equations (PDEs) on irregular domains. By shifting boundary conditions to surrogate boundaries on Cartesian meshes and correcting them with Taylor expansions, we achieve high accuracy without requiring boundary-fitted meshes. We demonstrate that the numerical error of SBM can be significantly reduced by an optimal choice of surrogate boundaries, with mathematical proofs validating optimal convergence. Second, the dissertation introduces a robust Octree-SBM framework for simulating incompressible Navier-Stokes equations. Leveraging efficient surrogate boundary construction on incomplete and adaptive octree meshes, we address the computational challenges of fluid dynamics in complex geometries. Benchmark simulations confirm the framework's scalability, accuracy, and efficiency across various flow regimes. Third, the SBM is applied to multiphysics thermal-flow simulations, coupling incompressible flow with heat transfer. Using a linearized Navier-Stokes RB-VMS formulation, the framework captures diverse thermal-flow phenomena with precision, enabling accurate enforcement of Dirichlet and Neumann boundary conditions on non-conformal meshes. Benchmark studies over a wide range of Rayleigh and Reynolds numbers validate the approach for laminar to turbulent regimes. Fourthly, the dissertation integrates SBM with adaptive mesh refinement (AMR) for incompressible flow and coupled thermal-flow problems. By employing vorticity-based adaptivity on hierarchical octree meshes, the framework dynamically resolves fine-scale features such as complex vorticity structures and steep thermal gradients while reducing computational costs. This integration enhances the capability to handle non-trivial geometries and evolving flow patterns in distributed-memory environments. Finally, the SBM framework is extended to simulate flow past open-interface geometries. By integrating SBM with octree meshes and leveraging Nitsche's method for weak enforcement of boundary conditions, this extension achieves numerical stability and accurate flow blockage without requiring boundary-fitted or excessively refined meshes. Applications to engineering scenarios confirm the framework's scalability and computational efficiency in addressing large-scale problems involving open-interface geometries. The methodologies and applications presented in this dissertation establish the Octree-SBM framework as a practical and effective tool for solving computational fluid dynamics and multiphysics problems. The framework is shown to be robust, scalable, and adaptable, addressing specific challenges such as complex geometries, diverse flow regimes, and efficient handling of large-scale simulations with accuracy and computational efficiency