Convergence of a Vector Penalty Projection Scheme for the Navier-Stokes Equations with moving body

In this paper, we analyse a Vector Penalty Projection Scheme (see [1]) to treat the displacement of a moving body in incompressible viscous flows in the case where the interaction of the fluid on the body can be neglected. The presence of the obstacle inside the computational domain is treated with a penalization method introducing a parameter $\eta$. We show the stability of the scheme and that the pressure and velocity converge towards a limit when the penalty parameter $\epsilon$, which induces a small divergence and the time step $\delta$t tend to zero with a proportionality constraint $\epsilon$ = $\lambda$$\delta$t. Finally, when $\eta$ goes to 0, we show that the problem admits a weak limit which is a weak solution of the Navier-Stokes equations with no-sleep condition on the solid boundary. R{\'e}sum{\'e} Dans ce travail nous analysons un sch{\'e}ma de projection vectorielle (voir [1]) pour traiter le d{\'e}placement d'un corps solide dans un fluide visqueux incompressible dans le cas o` u l'interaction du fluide sur le solide est n{\'e}gligeable. La pr{\'e}sence de l'obstacle dans le domaine solide est mod{\'e}lis{\'e}e par une m{\'e}thode de p{\'e}nalisation. Nous montrons la stabilit{\'e} du sch{\'e}ma et la convergence des variables vitesse-pression vers une limite quand le param etre $\epsilon$ qui assure une faible divergence et le pas de temps $\delta$t tendent vers 0 avec une contrainte de proportionalit{\'e} $\epsilon$ = $\lambda$$\delta$t. Finalement nous montrons que leprob{\`i} eme converge au sens faible vers une solution des equations de Navier-Stokes avec une condition aux limites de non glissement sur lafront{\`i} ere immerg{\'e}e quand le param etre de p{\'e}nalisation $\eta$ tend vers 0

Towards nanoscale magnetic memory elements : fabrication and properties of sub - 100 nm magnetic tunnel junctions

The rapidly growing field of spintronics has recently attracted much attention. Spintronics is electronics in which the spin degree of freedom has been added to conventional chargebased electronic devices. A magnetic tunnel junction (MTJ) is an example of a spintronic device. MTJs consist of two ferromagnetic layers separated by a thin insulating barrier. The tunnel current that flows through the barrier depends on the relative alignment of the magnetization in the ferromagnetic layers. As a consequence of this dependence, the MTJ exhibits two different resistance values that distinguish a logical ‘0’ and a logical ‘1’, corresponding to anti-parallel and parallel magnetization. Due to these two distinct states, the MTJs can be used as magnetic memory elements, and serve as bits for information storage in magnetic random access memories (MRAMs). MRAMs can lead to instant-on computers and longer battery lifetimes for mobile devices, which gives MRAMs the potential to replace the current RAM technologies. However, for the current RAM technologies to be replaced by MRAMs, the dimensions of the MTJs have to decrease to sub - 100 nm in order to achieve a high enough areal density and to match the semiconductor technology. Therefore, the research in this Thesis aims at fabricating sub - 100 nm MTJs and investigating the influence of the reduced dimensions on the modification of magnetic and electronic properties. The MTJ is incorporated in an engineered multilayer stack to promote stability and reproducibility of the magnetic and electric response of the MTJ. For the structuring of these layers, we have used top-down nanofabrication techniques to produce the sub - 100 nm MTJs. For MTJs with a surface area of less than 0.01 µm2, the Al2O3 barrier has an approximate thickness of a nanometer to ensure an appropriate tunnel current. Therefore, we have concentrated on the plasm oxidation of sub -nm thin Al layers to produce Al2O3 barriers. We have shown that over-oxidation of sub -nm thin Al2O3 barriers of MTJs can be observed in real-time using in situ differential ellipsometry measurements. The change in ellipsometry signal of Al layers grown on CoFe films, is proportional to the amount of oxidized metallic material. As a result, the derivative of this signal is a direct measure of the oxidation rate. Further analysis of this oxidation rate allowed us to determine the onset of the CoFe oxidation. We have found the onset to be proportional to the deposited Al layer thickness. The amount of CoO determined from in situ X-ray Photoelectron Spectroscopy data on identical samples is found to be proportional to that obtained from ellipsometry. In short, this means that the point in time on which over-oxidation starts can be precisely determined. This is a critical necessity in producing exact, well-functioning MTJs. We have used electron beam (EB) lithography to pattern the sub - 100 nm features. With EB lithography features of only a few nanometer in size can be defined, hence EB lithography enables the exploration of the fundamental boundaries of magnetic and electric scaling properties. For the structuring of large-area samples with ultra-dense arrays of sub - 100 nm MTJs the throughput is limited because of the sequential writing process. However, with the developed special high-speed EB writing strategy we could pattern a sample area of 16 mm2 with ultra-dense arrays of sub - 100 nm elliptical features in 10 minutes. The strategy is employed to define the pattern of hard Ta masks with sub - 100 nm features. The Ta masks are etched at -50 ¿C in a SF6/O2 plasma to an etch depth that can be controlled with nanometer precision. Ar+ ion beam milling is used to transfer the pattern and to produce dense arrays of sub - µm MTJs. Insight in the magnetic switching behavior of nanoscale MTJs as a function of the size, shape and thickness is vital for MRAM application. Especially, the collective properties of high areal density arrays of MTJs are of interest, because magnetostatic coupling mechanisms between elements can be a limiting factor for applications. In order to understand the effects of geometry and coupling mechanisms, the switching of 5 nm thick polycrystalline nanoscale Co dots is examined using SQUID measurements and the switching of sub - µm MTJs is studied with MOKE measurements. An array consists of approximately 108 elements with a width ranging from 50 to 300 nm, and a length to width aspect ratios of 1.5 to 2.5, arranged on a rectangular lattice. The measured switching fields of the Co dots and MTJs were low compared to predictions using the Stoner-Wohlfarth model. The deviations of the Stoner-Wohlfarth behavior could be explained in term of interdot coupling and edge roughness. Comparison with the outcome of OOMMF simulation of the switching of a single dot revealed that the interdot coupling has a major influence on the magnetic switching behavior of arrays of nanoscale magnetic elements. This implies that for the feasibility of ultra-high areal density arrays of nanoscale MTJs for information storage new strategies are needed. For example, a more complex toggle MTJ multilayer stacks can be used for the nanostructuring of sub - µm MTJs. Faceting of the etch mask due to physical sputtering of the mask material is a problem during deep etching of ultra-high bit density arrays of sub - µm MTJs for MRAMs. Besides that, chlorinated etch residues can reduce the magnetization of patterning magnetic materials substantially, and therefore constitutes a considerable concern. To get more insight into the magnetization losses, CoFeB dots were etched in a high ion density Cl2- based plasma with a width ranging from 0.3 to 6.4 µm. The magnetic properties of the CoFeB dots were measured by SQUID magnetometry. The sub-µm CoFeB dots showed significant magnetization reductions, despite H2O rinsing. Scanning electron microscopy (SEM) studies revealed that etching in a Cl2-based plasma caused faceting of the masks, leading to sloped sidewalls. SEM pictures were used to determine the geometric volume which was compared to the effective magnetic volume resulting from the magnetometry measurements. The SEM data are in good agreement with the magnetometry data, and a chloride penetration depth of only a few nanometer could be derived, indicating that the postetch rinsing is sufficient to prevent considerable corrosion of the CoFeB dots. This means that the chlorinated etch residues could be removed from the samples, without severely effecting the magnetic properties. The I-V characteristics of nanometer thin AlOx barriers are measured by applying a voltage over a 1×1 µm2 square MTJ pillar located at the cross-point of the bottom and top electrode. The I-V response of the MTJs showed in principle three different characteristics, that is an ohmic like response, a response resembling breakdown, and a tunneling response. Approximately 30 % of the measured 32 MTJs showed a tunneling response and had a resistance-area product between approximately 20 and 50 k µm2. Furthermore, the conductance showed roughly a parabolic behavior implying an asymmetrical barrier. As a consequence, we used the Brinkman formula to fit the experimental data. The fits yielded average barrier heights between 1.2 and 2.3 eV, and an asymmetry parameter ranging from 0.3 to 0.8 eV, which are close to reported observations in literature. Resistance measurements yielded no significant magnetoresistance for the 1 µm2 MTJs. Probably, as a consequence of interface roughnesses, a strong N´eel coupling originates, through which an independent switching of the two magnetic layers of the MTJ is hindered. As an alternative electrical characterization technique, the conductive atomic force microscopy (c-AFM) technique can used to measure the local electrical transport properties of nanoscale MTJs. We have explored this c-AFM technique and performed I-V measurements by applying a constant bias voltage to the bottom electrode and measuring the tunnel current through the barrier. However, due to resist remains no significant bias voltage dependency was observed

Outflow boundary conditions for the incompressible non-homogeneous Navier-Stokes equations

International audienceIn this paper we propose the analysis of the incompressible non-homogeneous Navier-Stokes equations with nonlinear outflow boundary condition. This kind of boundary condition appears to be, in some situations, a useful way to perform numerical computations of the solution to the unsteady Navier-Stokes equations when the Dirichlet data are not given explicitly by the physical context on a part of the boundary of the computational domain. The boundary condition we propose, following previous works in the homogeneous case, is a relationship between the normal component of the stress and the outflow momentum flux taking into account inertial effects. We prove the global existence of a weak solution to this model both in 2D and 3D. In particular, we show that the nonlinear boundary condition under study holds for such a solution in a weak sense, even though the normal component of the stress and the density may not have traces in the usual sense

Analyse mathématique d'un systéme de transport-diffusion-réaction modélisant la restauration biologique d'un milieu poreux

Sin resume

Atom lithography without laser cooling

Using direct-write atom lithography, Fe nanolines are deposited with a pitch of 186 nm, a full width at half maximum (FWHM) of 50 nm, and a height of up to 6 nm. These values are achieved by relying on geometrical collimation of the atomic beam, thus without using laser collimation techniques. This opens the way for applying direct-write atom lithography to a wide variety of elements.Comment: 7 pages, 11 figure

Incompressible flow in porous media with fractional diffusion

In this paper we study the heat transfer with a general fractional diffusion term of an incompressible fluid in a porous medium governed by Darcy's law. We show formation of singularities with infinite energy and for finite energy we obtain existence and uniqueness results of strong solutions for the sub-critical and critical cases. We prove global existence of weak solutions for different cases. Moreover, we obtain the decay of the solution in $L^p$, for any $p\geq2$, and the asymptotic behavior is shown. Finally, we prove the existence of an attractor in a weak sense and, for the sub-critical dissipative case with $\alpha\in (1,2]$, we obtain the existence of the global attractor for the solutions in the space $H^s$ for any $s > (N/2)+1-\alpha$

A new fast method to compute saddle-points in constrained optimization and applications

International audienceThe solution of the augmented Lagrangian related system (A+r\,B^TB)\,\rv=f is a key ingredient of many iterative algorithms for the solution of saddle-point problems in constrained optimization with quasi-Newton methods. However, such problems are ill-conditioned when the penalty parameter \eps=1/r>0 tends to zero, whereas the error vanishes as \cO(\eps). We present a new fast method based on a {\em splitting penalty scheme} to solve such problems with a judicious prediction-correction. We prove that, due to the {\em adapted right-hand side}, the solution of the correction step only requires the approximation of operators independent on \eps, when \eps is taken sufficiently small. Hence, the proposed method is all the cheaper as \eps tends to zero. We apply the two-step scheme to efficiently solve the saddle-point problem with a penalty method. Indeed, that fully justifies the interest of the {\em vector penalty-projection methods} recently proposed in \cite{ACF08} to solve the unsteady incompressible Navier-Stokes equations, for which we give the stability result and some quasi-optimal error estimates. Moreover, the numerical experiments confirm both the theoretical analysis and the efficiency of the proposed method which produces a fast splitting solution to augmented Lagrangian or penalty problems, possibly used as a suitable preconditioner to the fully coupled system

A Sucrose Solution Application to the Study of Model Biological Membranes

The small-angle X-ray and neutron scattering, time resolved X-ray small-angle and wide-angle diffraction coupled with differential scanning calorimetry have been applied to the investigation of unilamellar and multilamellar dimyristoylphosphatidylcholine (DMPC) vesicles in sucrose buffers with sucrose concentrations from 0 to 60%. Sucrose buffer decreased vesicle size and polydispersity and increased an X-ray contrast between phospholipid membrane and bulk solvent sufficiently. No influence of sucrose on the membrane thickness or mutual packing of hydrocarbon chains has been detected. The region of sucrose concentrations 30%-40% created the best experimental conditions for X-ray small-angle experiments with phospholipid vesicles.Comment: PDF: 10 pages, 6 figures. MS Word sours