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。这些问题的模拟需要将几何体划分为规则的部分，例如直的和弯曲的四面体。这种划分称为网格。对于任何给定的几何形状，这通常是需要人工干预的劳动密集型步骤。这种干预在几何形状不断变化的问题中是不可行的。该奖项将研究一类算法，以变形一个独特的网格，以便准确地划分整个类的几何形状。这种独特的网格称为通用网格。我们的目标是确定条件，使这些算法自动化和鲁棒性，这意味着计算机不需要人为干预来执行计算。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细化网格的算法，如果成功，这些研究将构成石油和天然气开采水力压裂策略模拟和增强地热系统工程的重要一步。特别是，他们将促进在瞬态情况下，时间尺度是重要的，如热致断裂和多孔弹性材料断裂的裂缝传播的模拟。更一般地说，这个奖项的成果将推进最先进的模拟问题与不断变化的几何形状。通用网格在流体-结构相互作用问题、形状优化问题、熔化或凝固问题等方面有着广泛的应用。该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方案中研究与该项目有关的方面，使来自代表性不足的少数民族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.