DISTANCE DIFFERENCE REPRESENTATIONS OF SUBSETS OF COMPLETE RIEMANNIAN MANIFOLDS

Volume: 2017.4 Host publication title: RIMS KôkyûrokuPeer reviewe

A numerical study of the influence from pre-existing cracks on granite rock fragmentation at percussive drilling

International audienceThe aim of this study is to investigate the effect of pre-existing, or structural, cracks on dynamic fragmentation of granite. Because of the complex behavior of rock materials, a continuum approach is employed relying upon a plasticity model with yield surface locus as a quadratic function of the mean pressure in the principal stress space coupled with an anisotropic damage model. In particular, Bohus granite rock is investigated, and the material parameters are chosen based on previous experiments. The equation of motion is discretized using a finite element approach, and the explicit time integration method is employed. The pre-existing cracks are introduced in the model by considering sets of elements with negligible tensile strength that leads to their immediate failure when loaded in tension even though they still carry compressive loads as crack closure occurs because of compressive stresses. Previously performed edge-on impact tests are reconsidered here to validate the numerical model. Percussive drilling is simulated, and the influence of the presence of pre-existing cracks is studied. The results from the analysis with different crack lengths and orientations are compared in terms of penetration stiffness and fracture pattern. It is shown that pre-existing cracks in all investigated cases facilitate the drilling process

Partial data inverse problem for hyperbolic equation with time-dependent damping coefficient and potential

We study an inverse problem of determining time-dependent damping coefficient and potential appearing in the wave equation in a compact Riemannian manifold of dimension three or higher. More specifically, we are concerned with the case of conformally transversally anisotropic manifolds, or in other words, compact Riemannian manifolds with boundary conformally embedded in a product of the Euclidean line and a transversal manifold. With an additional assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove that the knowledge of a certain partial Cauchy data set determines time-dependent damping coefficient and potential uniquely.Comment: arXiv admin note: text overlap with arXiv:1702.07974 by other author

Numerical modelling of fracture processes in thermal shock weakened rock

This paper presents some preliminary results of a research project aiming at the simulation of thermal shock assisted percussive drilling. In the present study, a numerical model for transient thermal shock induced damage in rock is presented. This model includes a rock mesostructure description accounting for different mineral properties and a thermo-mechanical constitutive model based on embedded discontinuity finite elements. In the numerical simulations, the thermal shock induced damge process is first simulated. Then the uniaxial compression test on thermally affected numerical rock samples is carried out. The effect of thermal shock is demonstrated by comparison to uniaxial compression test simulation on intact rock. The results show that the thermal-shock assisted rock breakage is a feasible idea to be extended to percussive drilling as well.Peer reviewe

Reconstruction of Riemannian manifold from boundary and interior data

This thesis focuses on geometric inverse problems. By this, we mean that the mathematical framework is the Riemannian geometry and the objects of interest are smooth Riemannian manifolds with or without boundaries. Electric impedance tomography, sonography, and seismic imaging are examples of geometric inverse problems that have been studied extensively. In inverse problems, one tries to obtain more information about the object of interest by doing indirect measurements and combining this additional information with some type of a priori information. The a priori information together with the measurements are called “the data”. We want to show that Riemannian manifolds with the same data have also some other geometric properties in common. For instance, two Riemannian manifolds admit the same data if and only if they are Riemannian isometric. In this thesis, we focus, on the uniqueness questions of the geometric inverse problems. In the first article, we study an inverse problem related to the obtaining information about the deep structures of the Earth from the travel time differences of seismic waves produced by earthquakes. We show that, under certain assumptions about the measurement area, the travel time difference functions determine the Riemannian manifold up to an isometry. In the second article, we show that, if in an open set of Euclidean space we have been given a wave that is produced by a single realization of a white noise source, then we can determine the Riemannian metric tensor, provided that the metric tensor is non-trapping and coincides with the Euclidean metric outside some compact set. In addition we also show that, if the solution mapping of the Riemannian wave equation with interior source is given in some open set, then we can determine the Riemannian structure up to an isometry. In the third article, we study an inverse problem of a reconstruction of a compact Riemannian manifold with a smooth boundary from the scattering data of internal sources. This data consists of the exit directions of geodesics that emanate from the interior points of the manifold. We show, that under certain generic assumptions on the metric, one can reconstruct an isometric copy of the manifold from such scattering data measured on the boundary. In the fourth article, we consider a generalization of the first article.Tämä työ sijoittuu soveltavan ja puhtaan matematiikan välimaastoon. Tutkimuksen tärkeimpinä motivaation lähteinä on toiminut seismologia ja maaperän rakenteen selvittäminen tekemällä seismisiä mittauksia maan pinnalla. Esimerkiksi jos pystymme selvittämään kuinka maanjäristyksen synnyttämä seisminen aalto etenee maan sisällä, saamme paljon tietoa maaperän koostumuksesta. Tämä johtuu siitä, että maaperän rakenne vaikuttaa siihen, miten seismiset aallot liikkuvat. Siksi tutkimalla matemaattisia malleja, jotka liittyvät aaltojen etenemiseen, voimme selvittää epäsuorasti myös maaperän rakennetta. Matemaattinen malli, joka on tämän työn taustalla, soveltuu varsin hyvin esimerkiksi seismisten aaltojen mallintamiseen. Väitöskirjassani olen tutkinut geometrisiä inversio-ongelmia. Tämän tyyppisessä matemaattisessa tutkimuksessa tavoitteena on todistaa, että ainoastaan geometrisesti samanlaiset kappaleet tuottavat samanlaisen teoreettisen mittausdatan. Työssä on tutkittu neljää erilaista mittausdataa. Kussakin tutkimuksessa on osoitettu edellä mainittu geometrinen yksikäsitteisyys. Olen käsitellyt datoja, jotka on määritelty sekä reunattoman Riemannin moniston avoimessa joukossa että reunallisen moniston reunalla. Työn ensimmäisessä osajulkaisussa on osoitettu, että seismisten aaltojen kulkuaikojen erotuksien avulla voimme selvittää, kuinka aallot etenevät mittausalueella ja sen ulkopuolella. Työn toisessa osassa on osoitettu, että mikäli tunnemme valkoisen kohinan synnyttämän aallon avoimessa mittausjoukossa, voimme selvittää minkä tahansa aallon etenemisen mittausjoukossa ja sen ulkopuolella. Kolmannessa osajulkaisussa on osoitettu, että mikäli tiedämme missä pisteessä ja missä suunnassa kappaleen sisältä pistelähteen synnyttämä aalto osuu kappaleen reunaan, niin voimme selvittää aaltojen etenemisen kappaleen sisällä. Työn viimeisessä osassa on laajennettu ensimmäisen osan tulosta koskemaan yleisempää mittaustilannetta