The research objective of this award is to create a class of algorithms to simulate problems with moving or evolving three-dimensional geometries; in particular, problems in which a crack or fracture propagates in an object. Simulations of such problems require partitioning the geometry into regular parts, such as straight and curved tetrahedrons. This partition is called a mesh. For any given geometry, this is generally a labor-intensive step requiring human intervention. Such intervention is unfeasible in problems in which the geometry is continuously changing. This award will investigate a class of algorithms to deform a unique mesh so as to exactly partition an entire class of geometries. Such unique mesh is called a Universal Mesh. The goal is to identify conditions to make these algorithms automatic and robust, meaning that a computer should not require human intervention to perform the calculation. A byproduct of this project will be the investigation of algorithms to construct adaptively refined meshes in a prism of acute-angled tetrahedrons in three-dimensions.
If successful, these studies would constitute an important step towards the simulation of hydraulic fracture strategies for oil and gas extraction and for enhanced geothermal system engineering. In particular, they would facilitate the simulation of the propagation of fractures in transient scenarios in which time-scales are important, such as thermally-induced fracture and fracture in poroelastic materials. More generally, the outcomes of this award would advance the state-of-the-art in the simulation of problems with evolving geometries. Universal meshes have extensive applications to fluid-structure interaction problems, shape optimization problems, and melting or solidification problems, among others. This project will support a graduate student at Stanford, and will expose undergraduate students from underrepresented minorities institutions to computational mechanics and applied mechanics, by working on aspects related to the project in the context of an ongoing summer internship program.
