This thesis presents novel approaches to improve the accuracy and efficiency of scientific simulations, particularly those involving complex geometries, intrinsic physical modeling, and demanding computational costs.
The first contribution extends the MPM to unstructured meshes, addressing the challenges of the transfer kernel's gradient continuity and stability issue on any general mesh tesselation. The Unstructured Moving Least Squares MPM (UMLS-MPM) incorporates a diminishing function into the MLS kernel's sample weights, ensuring an analytically continuous function and gradient reconstruction. It is the first-of-its-kind framework in this field. Several numerical test cases demonstrate the method's stability and accuracy.
The second contribution is a hybrid scheme for modeling the interaction between compressible flow, shock waves, and deformable structures. By combining recent advancements in time-splitting compressible flow and Material Point Methods (MPMs), this approach seamlessly integrates Eulerian and Lagrangian/Eulerian methods for monolithic flow-structure interactions. Reflective and penetrable boundary conditions handle deforming boundaries with sub-cell particles, while a mixed-order finite element formulation utilizing B-spline shape functions discretizes the coupled velocity-pressure system. This comprehensive framework accurately captures shock wave propagation, temperature/density-induced buoyancy effects, and topology changes in solids.
The third contribution addresses challenges in learning physical simulations on large-scale meshes using Graph Neural Networks (GNNs). Existing state-of-the-art methods often encounter issues related to over-smoothing and incorrect edge construction during multi-scale adaptation. To overcome these limitations, a novel pooling strategy, termed \textit{bi-stride}, is introduced. This approach, inspired by bipartite graph structures, involves pooling nodes on alternate frontiers of the breadth-first search (BFS), eliminating the need for labor-intensive manual creation of coarser meshes and mitigating incorrect edge problems. The proposed \textit{BSMS-GNN} framework employs non-parametrized pooling and unpooling through interpolations, resulting in a substantial reduction of computational costs and improved efficiency. Experimental results demonstrate the superiority of the \textit{BSMS-GNN} framework in terms of both accuracy and computational efficiency in representative physical simulations on large-scale meshes.
