Condition number analysis and preconditioning of the finite cell method

The (Isogeometric) Finite Cell Method - in which a domain is immersed in a structured background mesh - suffers from conditioning problems when cells with small volume fractions occur. In this contribution, we establish a rigorous scaling relation between the condition number of (I)FCM system matrices and the smallest cell volume fraction. Ill-conditioning stems either from basis functions being small on cells with small volume fractions, or from basis functions being nearly linearly dependent on such cells. Based on these two sources of ill-conditioning, an algebraic preconditioning technique is developed, which is referred to as Symmetric Incomplete Permuted Inverse Cholesky (SIPIC). A detailed numerical investigation of the effectivity of the SIPIC preconditioner in improving (I)FCM condition numbers and in improving the convergence speed and accuracy of iterative solvers is presented for the Poisson problem and for two- and three-dimensional problems in linear elasticity, in which Nitche's method is applied in either the normal or tangential direction. The accuracy of the preconditioned iterative solver enables mesh convergence studies of the finite cell method

Discontinuities without discontinuity: The Weakly-enforced Slip Method

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

Inverting elastic dislocations using the Weakly-enforced Slip Method

Earthquakes cause lasting changes in static equilibrium, resulting in global deformation fields that can be observed. Consequently, deformation measurements such as those provided by satellite based InSAR monitoring can be used to infer an earthquake's faulting mechanism. This inverse problem requires a numerical forward model that is both accurate and fast, as typical inverse procedures require many evaluations. The Weakly-enforced Slip Method (WSM) was developed to meet these needs, but it was not before applied in an inverse problem setting. Consequently, it was unknown what effect particular properties of the WSM, notably its inherent continuity, have on the inversion process. Here we show that the WSM is able to accurately recover slip distributions in a Bayesian-inference setting, provided that data points in the vicinity of the fault are removed. In a representative scenario, an element size of 2 km was found to be sufficiently fine to generate a posterior probability distribution that is close to the theoretical optimum. For rupturing faults a masking zone of 20 km sufficed to avoid numerical disturbances that would otherwise be induced by the discretization error. These results demonstrate that the WSM is a viable forward method for earthquake inversion problems. While our synthesized scenario is basic for reasons of validation, our results are expected to generalize to the wider gamut of scenarios that finite element methods are able to capture. This has the potential to bring modeling flexibility to a field that if often forced to impose model restrictions in a concession to computability.Comment: The associated software implementation is openly available in zenodo at https://doi.org/10.5281/zenodo.507179

On the singular nature of the elastocapillary ridge

The functionality of soft interfaces is crucial to many applications in biology and surface science. Recent studies have used liquid drops to probe the surface mechanics of elastomeric networks. Experiments suggest an intricate surface elasticity, also known as the Shuttleworth effect, where surface tension is not constant but depends on substrate deformation. However, interpretations have remained controversial due to singular elastic deformations, induced exactly at the point where the droplet pulls the network. Here we reveal the nature of the elastocapillary singularity on a hyperelastic substrate with various constitutive relations for the interfacial energy. First, we finely resolve the vicinity of the singularity using goal-adaptive finite element simulations. This confirms the universal validity, also at large elastic deformations, of the previously disputed Neumann's law for the contact angles. Subsequently, we derive exact solutions of nonlinear elasticity that describe the singularity analytically. These solutions are in perfect agreement with numerics, and show that the stretch at the contact line, as previously measured experimentally, consistently points to a strong Shuttleworth effect. Finally, using Noether's theorem we provide a quantitative link between wetting hysteresis and Eshelby-like forces, and thereby offer a complete framework for soft wetting in the presence of the Shuttleworth effect.Comment: 17 Pages, 7 figure

Procedure-based assessment for laparoscopic cholecystectomy can replace global rating scales

Introduction Global rating scales (GRSs) such as the Objective Structured Assessment of Technical Skills (OSATS) and Global Operative Assessment of Laparoscopic Surgery (GOALS) are assessment methods for surgical procedures. The aim of this study was to establish construct validity of Procedure-Based Assessment (PBA) and to compare PBA with GRSs for laparoscopic cholecystectomy. Material and methods OSATS and GOALS GRSs were compared with PBA in their ability to discriminate between levels of performance between trainees who can perform the procedure independently and those who cannot. Three groups were formed based on the number of procedures performed by the trainee: novice (1-10), intermediate (11-20) and experienced (>20). Differences between groups were assessed using the Kruskal-Wallis and Mann-Whitney U tests. Results Increasing experience correlated significantly with higher GRSs and PBA scores (all p < .001). Scores of novice and intermediate groups overlapped substantially on the OSATS (p = .1) and GOALS (p = .1), while the PBA discriminated between these groups (p = .03). The median score in the experienced group was higher with less dispersion for PBA (97.2[85.3-100]) compared to OSATS (82.1[60.7-100]) and GOALS (80[60-100]). Conclusion For assessing skill level or the capability of performing a laparoscopic cholecystectomy independently, PBA has a higher discriminative ability compared to the GRSs

Progress with the LOFAR Imaging Pipeline

One of the science drivers of the new Low Frequency Array (LOFAR) is large-area surveys of the low-frequency radio sky. Realizing this goal requires automated processing of the interferometric data, such that fully calibrated images are produced by the system during survey operations. The LOFAR Imaging Pipeline is the tool intended for this purpose, and is now undergoing significant commissioning work. The pipeline is now functional as an automated processing chain. Here we present several recent LOFAR images that have been produced during the still ongoing commissioning period. These early LOFAR images are representative of some of the science goals of the commissioning team members.Comment: 11 pages, 6 figures. Accepted for publication in proceedings of "ISKAF2010 Science Meeting", PoS(ISKAF2010)05

First LOFAR results on galaxy clusters

Deep radio observations of galaxy clusters have revealed the existence of diffuse radio sources related to the presence of relativistic electrons and weak magnetic fields in the intracluster volume. The role played by this non-thermal intracluster component on the thermodynamical evolution of galaxy clusters is debated, with important implications for cosmological and astrophysical studies of the largest gravitationally bound structures of the Universe. The low surface brightness and steep spectra of diffuse cluster radio sources make them more easily detectable at low-frequencies. LOFAR is the first instrument able to detect diffuse radio emission in hundreds of massive galaxy clusters up to their formation epoch. We present the first observations of clusters imaged by LOFAR and the huge perspectives opened by this instrument for non-thermal cluster studies.Comment: Proceedings of the 2012 week of the French Society of Astronomy and Astrophysics (SF2A) held in Nice, June 5th-8t

First LOFAR observations at very low frequencies of cluster-scale non-thermal emission: the case of Abell 2256

Abell 2256 is one of the best known examples of a galaxy cluster hosting large-scale diffuse radio emission that is unrelated to individual galaxies. It contains both a giant radio halo and a relic, as well as a number of head-tail sources and smaller diffuse steep-spectrum radio sources. The origin of radio halos and relics is still being debated, but over the last years it has become clear that the presence of these radio sources is closely related to galaxy cluster merger events. Here we present the results from the first LOFAR Low band antenna (LBA) observations of Abell 2256 between 18 and 67 MHz. To our knowledge, the image presented in this paper at 63 MHz is the deepest ever obtained at frequencies below 100 MHz in general. Both the radio halo and the giant relic are detected in the image at 63 MHz, and the diffuse radio emission remains visible at frequencies as low as 20 MHz. The observations confirm the presence of a previously claimed ultra-steep spectrum source to the west of the cluster center with a spectral index of -2.3 \pm 0.4 between 63 and 153 MHz. The steep spectrum suggests that this source is an old part of a head-tail radio source in the cluster. For the radio relic we find an integrated spectral index of -0.81 \pm 0.03, after removing the flux contribution from the other sources. This is relatively flat which could indicate that the efficiency of particle acceleration at the shock substantially changed in the last \sim 0.1 Gyr due to an increase of the shock Mach number. In an alternative scenario, particles are re-accelerated by some mechanism in the downstream region of the shock, resulting in the relatively flat integrated radio spectrum. In the radio halo region we find indications of low-frequency spectral steepening which may suggest that relativistic particles are accelerated in a rather inhomogeneous turbulent region.Comment: 13 pages, 13 figures, accepted for publication in A\&A on April 12, 201