545 research outputs found
PyFrac: A planar 3D hydraulic fracture simulator
Fluid driven fractures propagate in the upper earth crust either naturally or
in response to engineered fluid injections. The quantitative prediction of
their evolution is critical in order to better understand their dynamics as
well as to optimize their creation. We present a Python implementation of an
open-source hydraulic fracture propagation simulator based on the implicit
level set algorithm originally developed by Peirce & Detournay (2008) -- "An
implicit level set method for modeling hydraulically driven fractures". Comp.
Meth. Appl. Mech. Engng, (33-40):2858--2885. This algorithm couples a finite
discretization of the fracture with the use of the near tip asymptotic
solutions of a steadily propagating semi-infinite hydraulic fracture. This
allows to resolve the multi-scale processes governing hydraulic fracture growth
accurately, even with relatively coarse meshes. We present an overview of the
mathematical formulation, the numerical scheme and the details of our
implementation. A series of problems including a radial hydraulic fracture
verification benchmark, the propagation of a height contained hydraulic
fracture, the lateral spreading of a magmatic dyke and the handling of fracture
closure are presented to demonstrate the capabilities, accuracy and robustness
of the implemented algorithm
A localized orthogonal decomposition method for semi-linear elliptic problems
In this paper we propose and analyze a new Multiscale Method for solving
semi-linear elliptic problems with heterogeneous and highly variable
coefficient functions. For this purpose we construct a generalized finite
element basis that spans a low dimensional multiscale space. The basis is
assembled by performing localized linear fine-scale computations in small
patches that have a diameter of order H |log H| where H is the coarse mesh
size. Without any assumptions on the type of the oscillations in the
coefficients, we give a rigorous proof for a linear convergence of the H1-error
with respect to the coarse mesh size. To solve the arising equations, we
propose an algorithm that is based on a damped Newton scheme in the multiscale
space
Reactive Flows in Deformable, Complex Media
Many processes of highest actuality in the real life are described through systems of equations posed in complex domains. Of particular interest is the situation when the domain is variable, undergoing deformations that depend on the unknown quantities of the model. Such kind of problems are encountered as mathematical models in the subsurface, or biological systems. Such models include various processes at different scales, and the key issue is to integrate the domain deformation in the multi-scale context. Having this as the background theme, this workshop focused on novel techniques and ideas in the analysis, the numerical discretization and the upscaling of such problems, as well as on applications of major societal relevance today
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