Universal Meshes for the Simulation of Brittle Fracture and Moving Boundary Problems

Abstract

Universal meshes have recently appeared in the literature as a computationally efficient and robust paradigm for the generation of conforming simplicial meshes for domains with evolving boundaries. The main idea behind a universal mesh is to immerse the moving boundary in a background mesh (the universal mesh), and to produce a mesh that conforms to the moving boundary at any given time by adjusting a few of elements of the background mesh. In this manuscript we present the application of universal meshes to the simulation of brittle fracturing. To this extent, we provide a high level description of a crack propagation algorithm and showcase its capabilities. Alongside universal meshes for the simulation of brittle fracture, we provide other examples for which universal meshes prove to be a powerful tool, namely fluid flow past moving obstacles. Lastly, we conclude the manuscript with some remarks on the current state of universal meshes and future directions.Dedicated to Michael Ortiz on the occasion of his 60 th birthday.

The shifted fracture method

“…This approach avoids mesh dependencies by mesh adaptation, and has been implemented also in the context of higher-order discretizations. 7,8 Mesh dependencies in the behavior of the fracture are also observed when considering element-deletion techniques, 4 and more generally fracture damage modeling techniques. 9,10 There, a damage model is introduced to reduce the stiffness of elements that lie in proximity of a fracture.…”

“…In the case in which unloading occurs in the cohesive zone, a history variable w ⋆ tracks the largest (equivalent) crack opening achieved at a point on the crack surface, and the unloading/reloading occurs along the line (t ⋆ ∕w ⋆ )w eq , which is also called the unloading/reloading branch (see again Figure 2). The boundary value problem associated with crack propagation involves the solution of (1) under the elastic constitutive law (2), the (outer) boundary conditions (6) and the fracture interface conditions (7). We also define the fracture energy…”

“…In order to maintain an optimal accuracy and avoid mesh dependencies, the cohesive zone model ( 7) also needs to be adapted appropriately. In particular, (7) can be used to express t in terms of ȗ as…”

“…More recently, Lew and collaborators developed the Universal Meshes algorithm, 5,6 in which a base computational grid is locally adapted as the fracture propagates through the domain. This approach avoids mesh dependencies by mesh adaptation, and has been implemented also in the context of higher‐order discretizations 7,8 …”

Improved algorithms for generalized finite element simulations of three‐dimensional hydraulic fracture propagation

“…However, this can lead to increased computational costs of data transfer between meshes. Another way to address this problem is to use the so‐called universal meshes, where a mesh is deformed to fit the fracture geometry at each step . To our knowledge, this methodology has not yet been extended to 3‐D fracture problems.…”