A new approach to hyperbolic inverse problems

We present a modification of the BC-method in the inverse hyperbolic problems. The main novelty is the study of the restrictions of the solutions to the characteristic surfaces instead of the fixed time hyperplanes. The main result is that the time-dependent Dirichlet-to-Neumann operator prescribed on a part of the boundary uniquely determines the coefficients of the self-adjoint hyperbolic operator up to a diffeomorphism and a gauge transformation. In this paper we prove the crucial local step. The global step of the proof will be presented in the forthcoming paper.Comment: We corrected the proof of the main Lemma 2.1 by assuming that potentials A(x),V(x) are real value

Influence of melt feeding scheme and casting parameters during direct-chill casting on microstructure of an AA7050 billet

© The Minerals, Metals & Materials Society and ASM International 2012Direct-chill (DC) casting billets of an AA7050 alloy produced with different melt feeding schemes and casting speeds were examined in order to reveal the effect of these factors on the evolution of microstructure. Experimental results show that grain size is strongly influenced by the casting speed. In addition, the distribution of grain sizes across the billet diameter is mostly determined by melt feeding scheme. Grains tend to coarsen towards the center of a billet cast with the semi-horizontal melt feeding, while upon vertical melt feeding the minimum grain size was observed in the center of the billet. Computer simulations were preformed to reveal sump profiles and flow patterns during casting under different melt feeding schemes and casting speeds. The results show that solidification front and velocity distribution of the melt in the liquid and slurry zones are very different under different melt feeding scheme. The final grain structure and the grain size distribution in a DC casting billet is a result of a combination of fragmentation effects in the slurry zone and the cooling rate in the solidification range

Inverse Scattering for Gratings and Wave Guides

We consider the problem of unique identification of dielectric coefficients for gratings and sound speeds for wave guides from scattering data. We prove that the "propagating modes" given for all frequencies uniquely determine these coefficients. The gratings may contain conductors as well as dielectrics and the boundaries of the conductors are also determined by the propagating modes.Comment: 12 page

Effect of V and N on the microstructure evolution during continuous casting of steel

Low Carbon (LC) steel is not expected to be sensitive to hot tearing and/or cracking while microalloyed steels are known for their high cracking sensitivity during continuous casting. Experience of the Direct Sheet Plant caster at Tata Steel in Ijmuiden (the Netherlands), seems to contradict this statement. It is observed that a LC steel grade has a high risk of cracking alias hot tearing, while a High Strength Low Alloyed (HSLA) steel has a very low cracking occurrence. Another HSLA steel grade, with a similar composition but less N and V is however very sensitive to hot tearing. An extreme crack results in a breakout. A previous statistical analysis of the breakout occurrence reveals a one and a half times higher possibility of a breakout for the HSLA grade compared to the LC grade. HSLA with extra N, V shows a four times smaller possibility of breakout than LC. This study assigns the unexpected effect of the chemical composition on the hot tearing sensitivity to the role of some alloying elements such as V and N as structure refiners.This research was carried out under project number M41.5.08320 within the framework of the Research Program of the Materials innovation institute M2i (www.m2i.nl)

Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow

We compute the sum of the positive Lyapunov exponents of the Hodge bundle with respect to the Teichmuller geodesic flow. The computation is based on the analytic Riemann-Roch Theorem and uses a comparison of determinants of flat and hyperbolic Laplacians when the underlying Riemann surface degenerates.Comment: Minor corrections. To appear in Publications mathematiques de l'IHE

Formation of hot tear under controlled solidification conditions

Aluminum alloy 7050 is known for its superior mechanical properties, and thus finds its application in aerospace industry. Vertical direct-chill (DC) casting process is typically employed for producing such an alloy. Despite its advantages, AA7050 is considered as a "hard-to-cast" alloy because of its propensity to cold cracking. This type of cracks occurs catastrophically and is difficult to predict. Previous research suggested that such a crack could be initiated by undeveloped hot tears (microscopic hot tear) formed during the DC casting process if they reach a certain critical size. However, validation of such a hypothesis has not been done yet. Therefore, a method to produce a hot tear with a controlled size is needed as part of the verification studies. In the current study, we demonstrate a method that has a potential to control the size of the created hot tear in a small-scale solidification process. We found that by changing two variables, cooling rate and displacement compensation rate, the size of the hot tear during solidification can be modified in a controlled way. An X-ray microtomography characterization technique is utilized to quantify the created hot tear. We suggest that feeding and strain rate during DC casting are more important compared with the exerted force on the sample for the formation of a hot tear. In addition, we show that there are four different domains of hot-tear development in the explored experimental window-compression, microscopic hot tear, macroscopic hot tear, and failure. The samples produced in the current study will be used for subsequent experiments that simulate cold-cracking conditions to confirm the earlier proposed model.This research was carried out within the Materials innovation institute (www.m2i.nl) research framework, project no. M42.5.09340

Application of the "Full Cavitation Model" to the fundamental study of cavitation in liquid metal processing

Ultrasonic cavitation treatment of melt significantly improves the downstream properties and quality of conventional and advanced metallic materials. However, the transfer of this technology has been hindered by difficulties in treating large volumes of liquid metal. To improve the understanding of cavitation processing efficiency, the Full Cavitation Model, which is derived from a reduced form of the Rayleigh-Plesset equation, is modified and applied to the two-phase problem of bubble propagation in liquid melt. Numerical simulations of the sound propagation are performed in the microsecond time scale to predict the maximum and minimum acoustic pressure amplitude fields in the domain. This field is applied to the source term of the bubble transport equation to predict the generation and destruction of cavitation bubbles in a time scale relevant to the fluid flow. The use of baffles to limit flow speed in a launder conduit is studied numerically, to determine the optimum configuration that maximizes the residence time of the liquid in high cavitation activity regions. With this configuration, it is then possible to convert the batch processing of liquid metal into a continuous process. The numerical simulations will be validated against water and aluminium alloy experiments, carried out at Brunel University

The role of ultrasonic treatment in refining the as-cast grain structure during the solidification of an Al-2Cu alloy

The effect of Ultrasonic Treatment (UT) over selected temperature ranges during cooling and solidification of an Al-2Cu alloy melt on the grain structure and cooling behaviour of the alloy has been investigated using a molybdenum sonotrode introduced without preheating. UT was applied over various temperature ranges before, during and after the nucleation of primary aluminium grains. It was found that ultrasonic grain refinement was achieved only when UT was applied from more than 20 °C above the liquidus temperature until below the liquidus temperature after nucleation has occurred. Applying UT from 40 °C or 60 °C above the liquidus to just above the liquidus brings the melt to a condition that favours nucleation, survival of the nucleated grains and their subsequent transport throughout the melt. Continuing to apply UT beyond the liquidus for a short time enhances both nucleation and convection thereby ensuring the formation of a fine, uniform equiaxed grain size throughout the casting. The lack of grain refinement when UT was applied from 20 °C above the liquidus temperature or from temperatures below the liquidus temperature is attributed to the formation of a strong solidified layer on the sonotrode which hinders the effective transmission of ultrasonic irradiation into the liquid metal. The application of a preheated sonotrode showed that formation of a solid layer can be prevented by preheating the sonotrode to 285 °C. Thus, an appropriate amount of superheat of the liquid metal or sufficient preheating of the sonotrode is necessary for ultrasonic grain refinement when a sonotrode is introduced into the melt

Khinchin theorem for integral points on quadratic varieties

We prove an analogue the Khinchin theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that in this system, generic trajectories visit a family of shrinking subsets infinitely often.Comment: 19 page