Discontinuities without discontinuity: The Weakly-enforced Slip Method

Abstract

Tectonic faults are commonly modelled as Volterra or Somigliana dislocations in an elastic medium. Various solution methods exist for this problem. However, the methods used in practice are often limiting, motivated by reasons of computational efficiency rather than geophysical accuracy. A typical geophysical application involves inverse problems for which many different fault configurations need to be examined, each adding to the computational load. In practice, this precludes conventional finite-element methods, which suffer a large computational overhead on account of geometric changes. This paper presents a new non-conforming finite-element method based on weak imposition of the displacement discontinuity. The weak imposition of the discontinuity enables the application of approximation spaces that are independent of the dislocation geometry, thus enabling optimal reuse of computational components. Such reuse of computational components renders finite-element modeling a viable option for inverse problems in geophysical applications. A detailed analysis of the approximation properties of the new formulation is provided. The analysis is supported by numerical experiments in 2D and 3D.Comment: Submitted for publication in CMAM

Finite element simulation of pressure-loaded phase-field fractures

“…Following the same arguments as in [28], the H 1 -norm is expected to diverge with Oðh À 1 2 Þ independent of the order and regularity of the discretization, which is in agreement with the results reported in Fig. 12b.…”

Computational vademecums for real‐time simulation of surgical cutting in haptic environments

“…Finally, both will be combined online as detailed in the last part of the present section. While several possibilities arise to treat the discontinuous part of the displacement, such a , for instance, we have preferred to employ the so‐called cracking node method . It has been preferred because it provides naturally a discontinuous approximation that can be used with almost no modification also for visualization purposes.…”

Inverting elastic dislocations using the Weakly‐enforced Slip Method

“…It bears repeating that the above considerations are speculative. While the mathematical soundness of the WSM was proved, 11 it is in the present paper that we set out to thoroughly test the utility of the WSM in the problem setting for which it was devised. To this end a number of synthetic but otherwise fully representative case studies are presented and analyzed using the WSM, as well as validated against the exact solutions.…”

“…This is why such analyses are often based on the assumption of a homogeneous halfspace, [8][9][10] for which cheaply evaluable analytical expressions are available. It is with this situation in mind that we proposed the Weakly-enforced Slip Method (WSM) 11 to strike a better balance between modeling complexity and computational cost.…”

“…7 The efficiency of WSM does come at a price: the displacement field it produces is continuous, and therefore unable to capture the jump at the fault. It was however shown 11 that the error thus incurred decays exponentially with distance to the fault. It was therefore hypothesized that this drawback is of little consequence in the context of satellite observations, as most surface measurements are sufficiently far removed from the dislocation.…”