Universal Meshes for Coupled Crack Propagation Problems and their Application to Hydraulic Fracturing

该奖项的研究目标是创建一类算法来模拟移动或演变的三维几何问题，特别是裂纹或断裂在物体中传播的问题。这类问题的模拟需要将几何体划分为规则的部分，如直四面体和弯曲四面体。这种分区称为网格。对于任何给定的几何体，这通常是一个需要人工干预的劳动密集型步骤。在几何体不断变化的问题中，这种干预是不可行的。这一奖项将研究一类算法，使唯一的网格变形，从而准确地划分整个几何类别。这种独特的网格被称为通用网格。目标是确定使这些算法自动化和健壮的条件，这意味着计算机不应该需要人工干预来执行计算。该项目的一个副产品将是研究在三维锐角四面体棱柱中构建自适应精细网格的算法。如果成功，这些研究将成为模拟石油和天然气开采以及增强地热系统工程的水力压裂策略的重要一步。特别是，它们将有助于在时间尺度很重要的瞬变场景中模拟裂缝的扩展，例如热诱导裂缝和多孔弹性材料中的裂缝。更广泛地说，这一奖项的结果将推动几何演化问题模拟的最先进水平。通用网格在流固耦合问题、形状优化问题、熔化或凝固问题等方面有着广泛的应用。这个项目将支持斯坦福大学的一名研究生，并将通过在正在进行的暑期实习计划的背景下，致力于与该项目相关的方面，让来自代表性不足的少数族裔机构的本科生接触计算力学和应用力学。

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.