Abstract
Abstract Spline-based meshes allow for a higher inter-element continuity. For coupled problems, e.g. poroelasticity, different meshes with different orders of interpolation are normally used for the various fields in order to avoid spurious oscillations. When including discontinuities in these meshes, there exist several options for the discretisation. Herein we will discuss two options which use T-splines, one aiming at a minimum number of degrees of freedom around the crack tip, the other trying to maximise this number. Both meshes retain a higher-order continuity along the fracture, but the mesh which maximises the number of degrees of freedom mesh introduces two additional degrees of freedom around the crack tip to allow for a sharper crack. The two discretisations are used to simulate a pressurised fracture inside a poroelastic material and the results are compared to results obtained using a Non-Uniform Rational B-Spline (NURBS) mesh. A comparison between the two discretisations shows the effect of including additional degrees of freedom close to the crack tip. However, both meshes yield similar results further away from the crack tip. It is shown that both T-spline meshes capture a fully closed discontinuity at the fracture tip, whereas the NURBS mesh retains a small opening due to the discontinuity which exists for the cracked as well as the intact elements. A fully closed fracture aperture results in T-splines with a lower discontinuity pressure compared to NURBS, making T-splines more suitable for simulations in which the fracture propagation is limited by the fluid transport within the fracture.
Highlights
An advantage of using spline-based interpolation functions compared to standard Lagrangian polynomials is their increased inter-element continuity
The two T-spline meshes result in the same fracture length, while the Non-Uniform Rational B-Splines (NURBS) mesh matches for the high permeability, but is a few elements ahead for the medium and lower permeabilities
The NURBS mesh, yields a small displacement jump ahead of the fracture tip in the still intact material. This is caused by the interface elements having been inserted for the fractured and the not yet fractured elements, the used dummy stiffness not being able to completely enforce a closed fracture at the fracture tip
Summary
An advantage of using spline-based interpolation functions compared to standard Lagrangian polynomials is their increased inter-element continuity. While it is possible that these oscillations do not occur when using equal order meshes, they tend to dominate the solution when they occur This necessitates the use of complex stabilisation schemes to retain usable results when using the same interpolants for the displacements and pressure [9,10,11]. Use of T-splines allows for several options with regard to the representation of discontinuities, of which two possible discretisation options are explored These two meshes are used to simulate a typical poroelasticity case, and their results are compared to a discretisation using NURBS, showing the impact of the different discretisation choices, and highlighting the advantages of T-splines
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