Trapped Surfaces in Vacuum Spacetimes

An earlier construction by the authors of sequences of globally regular, asymptotically flat initial data for the Einstein vacuum equations containing trapped surfaces for large values of the parameter is extended, from the time symmetric case considered previously, to the case of maximal slices. The resulting theorem shows rigorously that there exists a large class of initial configurations for non-time symmetric pure gravitational waves satisfying the assumptions of the Penrose singularity theorem and so must have a singularity to the future.Comment: 14 page

Vacuum Spacetimes with Future Trapped Surfaces

In this article we show that one can construct initial data for the Einstein equations which satisfy the vacuum constraints. This initial data is defined on a manifold with topology $R^3$ with a regular center and is asymptotically flat. Further, this initial data will contain an annular region which is foliated by two-surfaces of topology $S^2$. These two-surfaces are future trapped in the language of Penrose. The Penrose singularity theorem guarantees that the vacuum spacetime which evolves from this initial data is future null incomplete.Comment: 19 page

Modeling seismic wave propagation and amplification in 1D/2D/3D linear and nonlinear unbounded media

To analyze seismic wave propagation in geological structures, it is possible to consider various numerical approaches: the finite difference method, the spectral element method, the boundary element method, the finite element method, the finite volume method, etc. All these methods have various advantages and drawbacks. The amplification of seismic waves in surface soil layers is mainly due to the velocity contrast between these layers and, possibly, to topographic effects around crests and hills. The influence of the geometry of alluvial basins on the amplification process is also know to be large. Nevertheless, strong heterogeneities and complex geometries are not easy to take into account with all numerical methods. 2D/3D models are needed in many situations and the efficiency/accuracy of the numerical methods in such cases is in question. Furthermore, the radiation conditions at infinity are not easy to handle with finite differences or finite/spectral elements whereas it is explicitely accounted in the Boundary Element Method. Various absorbing layer methods (e.g. F-PML, M-PML) were recently proposed to attenuate the spurious wave reflections especially in some difficult cases such as shallow numerical models or grazing incidences. Finally, strong earthquakes involve nonlinear effects in surficial soil layers. To model strong ground motion, it is thus necessary to consider the nonlinear dynamic behaviour of soils and simultaneously investigate seismic wave propagation in complex 2D/3D geological structures! Recent advances in numerical formulations and constitutive models in such complex situations are presented and discussed in this paper. A crucial issue is the availability of the field/laboratory data to feed and validate such models.Comment: of International Journal Geomechanics (2010) 1-1

An MBO scheme for minimizing the graph Ohta-Kawasaki functional

We study a graph based version of the Ohta-Kawasaki functional, which was originally introduced in a continuum setting to model pattern formation in diblock copolymer melts and has been studied extensively as a paradigmatic example of a variational model for pattern formation. Graph based problems inspired by partial differential equations (PDEs) and varational methods have been the subject of many recent papers in the mathematical literature, because of their applications in areas such as image processing and data classification. This paper extends the area of PDE inspired graph based problems to pattern forming models, while continuing in the tradition of recent papers in the field. We introduce a mass conserving Merriman-Bence-Osher (MBO) scheme for minimizing the graph Ohta-Kawasaki functional with a mass constraint. We present three main results: (1) the Lyapunov functionals associated with this MBO scheme Γ-converge to the Ohta-Kawasaki functional (which includes the standard graph based MBO scheme and total variation as a special case); (2) there is a class of graphs on which the Ohta-Kawasaki MBO scheme corresponds to a standard MBO scheme on a transformed graph and for which generalized comparison principles hold; (3) this MBO scheme allows for the numerical computation of (approximate) minimizers of the graph Ohta-Kawasaki functional with a mass constraint

Physics-Based Earthquake Ground Shaking Scenarios in Large Urban Areas

With the ongoing progress of computing power made available not only by large supercomputer facilities but also by relatively common workstations and desktops, physics-based source-to-site 3D numerical simulations of seismic ground motion will likely become the leading and most reliable tool to construct ground shaking scenarios from future earthquakes. This paper aims at providing an overview of recent progress on this subject, by taking advantage of the experience gained during a recent research contract between Politecnico di Milano, Italy, and Munich RE, Germany, with the objective to construct ground shaking scenarios from hypothetical earthquakes in large urban areas worldwide. Within this contract, the SPEED computer code was developed, based on a spectral element formulation enhanced by the Discontinuous Galerkin approach to treat non-conforming meshes. After illustrating the SPEED code, different case studies are overviewed, while the construction of shaking scenarios in the Po river Plain, Italy, is considered in more detail. Referring, in fact, to this case study, the comparison with strong motion records allows one to derive some interesting considerations on the pros and on the present limitations of such approach

Accurate prediction of ground motion using an hp-adaptive discontinuous Galerkin finite-element method

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Coupling the spectral element method with a modal solution for elastic wave propagation in global earth models

International audienceWe present a new method for wave propagation in global earth models based upon the coupling between the spectral element method and a modal solution method. The Earth is decomposed into two parts, an outer shell with 3-D lateral heterogeneities and an inner sphere with only spherically symmetric heterogeneities. Depending on the problem, the outer heterogeneous shell can be mapped as the whole mantle or restricted only to the upper mantle or the crust. In the outer shell, the solution is sought in terms of the spectral element method, which stem from a high order variational formulation in space and a second-order explicit scheme in time. In the inner sphere, the solution is sought in terms of a modal solution in frequency after expansion on the spherical harmonics basis. The spectral element method combines the geometrical flexibility of finite element methods with the exponential convergence rate of spectral methods. It avoids the pole problems and allows for local mesh refinement, using a non-conforming discretization, for the resolution of sharp variations and topography along interfaces. The modal solution allows for an accurate isotropic representation in the inner sphere. The coupling is introduced within the spectral element method via a Dirichlet-to-Neumann (DtN) operator. The operator is explicitly constructed in frequency and in generalized spherical harmonics. The inverse transform in space and time requires special attention and an asymptotic regularization. The coupled method allows a significant speed-up in the simulation of the wave propagation in earth models. For spherically symmetric earth model, the method is shown to have the accuracy of spectral transform methods and allow the resolution of wavefield propagation, in 3-D laterally heterogeneous models, without any perturbation hypothesis

Post-emplacement dynamics of andesitic lava flows at Volcan de Colima, Mexico, revealed by radar and optical remote sensing data

We used optical and radar remote sensing datasets to map, estimate the volume, and measure the surface displacements of lava flows emplaced on the flanks of Volcan de Colima, Mexico by extrusion of lava dome material from the end of 2014 to early 2016. Our main result is that the flow motion of the lava contributes significantly to the recorded displacements several months after its emplacement. First, we mapped the deposits and estimated their volumes using two Digital Elevation Models (DEM), one derived from radar data acquired before the peak of activity and one derived from optical images acquired just after this peak of activity. Coherence information derived from the radar dataset added some temporal constraints on the timing of emplacement of various deposits. We thus estimated a mean extrusion rate of 1-2 m(3)s(-1) between November 2014 and February 2015. We then used a new approach to reconstruct the 3D displacement field, taking advantage of images acquired by the same satellite, on both ascending and descending tracks, and using a physical a priori on the direction of horizontal displacements. Our results show that about 2 cm yr(-1) of horizontal motion is still recorded a few months after the emplacement on the SW lava flow, which is the only one covered by the two-acquisition geometries. In order to differentiate the potential causes of the observed displacements, we modeled the thermal contraction of the lava flow using a finite element numerical method. Removing the contribution of thermoelastic contraction from the measured displacements enable to infer both the viscoelastic loading and flow motion effects from the residuals. Results show that, thermal contraction, flow motion and viscoelastic loading contribute significantly to the displacements recorded

EXPERIMENTAL INVESTIGATION OF SPATIAL VARIABILITY OF GROUND MOTIONS – MONITORING OF AN ARCH DAM

International audienceThe term " spatial variability of seismic ground motions " denotes the differences in amplitude and phase content of seismic motions recorded over extended areas or within the dimensions of a structure. The effect of such spatial variability on the response of civil infrastructure systems such as dams is still an open issue. In-situ experiments may be helpful in order to answer the questions regarding both the quantification of the spatial variability of the ground motion within the dimensions of a structure as well as the effect on its dynamic response. For this purpose, the 69-m-high double curvature Saint Guérin arch dam located nearby the village of Beaufort, Savoie, France as well as the surrounding area is instrumented with a seismological network with a few meters of inter-station distance. This very dense network consists of nineteen velocimeters which have been deployed for one year in total (June 2015-June 2016). The configuration of the network is such that the spatial variability of the ground motions can be captured on the dam-foundation rock interface (left and right side of the valley) and at the surrounding area. Coherency functions are computed and analyzed providing information about the effect of the on-site topography and the interaction with the dam on the ground motion. Besides, the measurements along the crest provide informations on the structure's response that might be useful for the interpretation of the results