Computation of high order derivatives in optimal shape design

Summary. In shape optimization problems, each computation of the cost function by the finite element method leads to an expensive analysis. The use of the second order derivative can help to reduce the number of analyses. Fujii ([4], [10]) was the first to study this problem. J. Simon [19] gave the second order derivative for the Navier-Stokes problem, and the authors describe in [8], [11], a method which gives an intrinsic expression of the first and second order derivatives on the boundary of the involved domain. In this paper we study higher order derivatives. But one can ask the following questions: -- are they expensive to calculate? -- are they complicated to use? -- are they imprecise? -- are they useless? \medskip\noindent At first sight, the answer seems to be positive, but classical results of V. Strassen [20] and J. Morgenstern [13] tell us that the higher order derivatives are not expensive to calculate, and can be computed automatically. The purpose of this paper is to give an answer to the third question by proving that the higher order derivatives of a function can be computed with the same precision as the function itself. We prove also that the derivatives so computed are equal to the derivatives of the discrete problem (see Diagram 1). We call the discrete problem the finite dimensional problem processed by the computer. This result allows the use of automatic differentiation ([5], [6]), which works only on discrete problems. Furthermore, the computations of Taylor's expansions which are proposed at the end of this paper, could be a partial answer to the last question. Received January 27, 1993/Revised version received July 20, 1993     

Dual-mode registration of dynamic contrast-enhanced ultrasound combining tissue and contrast sequences

This study proposes a new method for automatic, iterative image registration in the context of dynamic contrast-enhanced ultrasound (DCE-US) imaging. By constructing a cost function of image registration using a combination of the tissue and contrast-microbubble responses, this new method, referred to as dual-mode registration, performs alignment based on both tissue and vascular structures. Data from five focal liver lesions (FLLs) were used for the evaluation. Automatic registration based on the dual-mode registration technique and tissue-mode registration obtained using the linear response image sequence alone were compared to manual alignment of the sequence by an expert. Comparison of the maximum distance between the transformations applied by the automatic registration techniques and those from expert manual registration reference showed that the dual-mode registration provided better precision than the tissue-mode registration for all cases. The reduction of maximum distance ranged from 0.25 to 9.3 mm. Dual-mode registration is also significantly better than tissue-mode registration for the five sequences with p  -values lower than 0.03$0.03$. The improved sequence alignment is also demonstrated visually by comparison of images from the sequences and the video playbacks of the motion-corrected sequences. This new registration technique better maintains a selected region of interest (ROI) within a fixed position of the image plane throughout the DCE-US sequence. This should reduce motion-related variability of the echo-power estimations and, thus, contribute to more robust perfusion quantification with DCE-US.     

On-line detection of fatigue cracks using an automatic mode tracking technique

Experimental fatigue tests usually require large testing times. In addition to the resulting increased time-to-market, the large fatigue test time also implies that any structural health monitoring technique that is used should be automatic. When using the modal parameters as damage indicators, an important amount of user interaction is still needed to separate physical poles from computational ones. In this paper, an experimental framework will be developed to automatically track the health of the structure on-line with the performance of fatigue tests. The modal parameters are tracked using a combination of the maximum likelihood estimator and an auto-regressive model. Since confidence levels on the modal parameter are available it is possible to detect if damage is present. In addition, the quasi-static stiffness with computed confidence levels is also used as a damage indicator. The proposed techniques are demonstrated on a steel beam with a propagating fatigue crack.     

On solutions of second and fourth order elliptic equations with power-type nonlinearities

We study second and fourth order semilinear elliptic equations with a power-type nonlinearity depending on a power p$p$ and a parameter λ>0$\lambda >0$. For both equations we consider Dirichlet boundary conditions in the unit ball B⊂Rⁿ$B\subset {\mathbb{R}}^n$. Regularity of solutions strictly depends on the power p$p$ and the parameter λ$\lambda$. We are particularly interested in the radial solutions of these two problems and many of our proofs are based on an ordinary differential equation approach.     

Photoinduced magnetization in copper octacyanomolybdate

This article describes the studies of a photomagnetic cyanide-bridged Cu-Mo bimetallic assembly, Cu(II)(2)[Mo(IV)(CN)(8)].8H(2)O (Cu(II), S = (1)/(2); Mo(IV), S = 0) (1), which has an intervalence transfer (IT) band from Mo(IV)-CN-Cu(II) to Mo(V)-CN-Cu(I) around 480 nm. Wide-angle X-ray scattering and X-ray spectroscopic studies provide precise information about the 3D connectivity and the local environment of the transition metal ions. Irradiating with blue light causes solid 1 to exhibit a spontaneous magnetization (Curie temperature = 25 K). The thermal reversibility is carefully studied and shows the long-time stability of the photoinduced state up to 100 K. Photoreversibility is also observed; i.e., the magnetization is induced by irradiation with light below 520 nm, while the magnetization is reduced by irradiation with light above 520 nm. The UV-vis absorption spectrum after irradiation shows a decrease of the IT band and the appearance of the reverse-IT band in the region of 600-900 nm (lambda(max) = 710 nm). This UV-vis absorption spectrum is recovered to the original spectrum by irradiation with 658-, 785-, and 840-nm light. In this photomagnetic effect, the excitation of the IT band causes an electron transfer from Mo(IV) to Cu(II), producing a ferromagnetic mixed-valence isomer of Cu(I)Cu(II)[Mo(V)(CN)(8)].8H(2)O (Cu(I), S = 0; Cu(II), S = (1)/(2); Mo(V), S = (1)/(2)) (1'). 1' returns to 1 by irradiation of the reverse-IT band, which obeys the scheme for the potential energy surface in mixed-valence class II compounds.     

Testimony of the Correlation Between the Reactive Histidine Residue and the Arginase Catalytic Mechanism Using a Biochromatographic Concept and Mutagenesis Experiments

In a previous paper (Bagnost et al., J Chromatogr B Analyt Technol Biomed Life Sci 853:38–46, 2007), an arginase chromatographic support was developed to study the association mechanism of arginase (an enzyme which can reduce endothelial dysfunction and blood pressure rising in spontaneously hypertensive rats) with nor-NOHA a potential inhibitor of its activity. In this report mutagenesis experiments associated with this biochromatographic approach confirmed that the active-site residue Hist 141 is protonated as imidazolium cation. Hist 141 could function as a general acid to protonate the leaving amino group of l-ornithine during catalysis.     

Photophysical properties of ruthenium(II) tris(2,2'-bipyridine) complexes bearing conjugated thiophene appendages

A small series of ruthenium(II) tris(2,2'-bipyridine) complexes has been synthesized in which ethynylated thiophene residues are attached to one of the 2,2'-bipyridine ligands. The photophysical properties depend on the conjugation length of the thiophene-based ligand, and in each case, dual emission is observed. The two emitting states reside in thermal equilibrium at ambient temperature and can be resolved by emission spectral curve-fitting routines. This allows the properties of the two states to be evaluated in both fluid butyronitrile solution and a transparent KBr disk. It is concluded that both emitting states are of metal-to-ligand charge-transfer (MLCT) character, and despite the presence of conjugated thiophene residues, there is no indication for a low-lying pi,pi*-triplet state that promotes nonradiative decay of the excited-state manifold. A key feature of these systems is that the conjugation length imposed by the thiophene-based ligand helps to control the rate constants for both radiative and nonradiative decay from the two MLCT triplet states.     

Structural and luminescent properties of micro- and nanosized particles of lanthanide terephthalate coordination polymers

Reaction in water between rare earth ions (Ln = Y, La-Tm, except Pm) and the sodium salt of terephthalic acid leads to a family of lanthanide-based coordination polymers of general formula [Ln2(C8H4O4)3(H2O)4] n with Ln = La-Tm or Y. The isostructurality of the compounds with the previously reported Tb-containing polymer is ascertained on the basis of their X-ray powder diffraction diagrams. The coordination water molecules can be reversibly removed without destroying the crystal structure for compounds involving one of the lighter lanthanide ions (La-Eu). For compounds involving one of the heavier lanthanide ions (Tb-Tm) or yttrium, a structural change occurs during the drying process. X-ray diffraction data show this new anhydrous phase corresponding to the linking of pairs of Er(III) ions through mu-carboxylate bridges. Porosity profiles calculated for the anhydrous phases of Tb(III) and Er(III) show the presence of channels with very small sections. The luminescent properties of all the compounds have been recorded and the two most luminescent polymers, namely, the europium- and the terbium-containing ones, have been studied in more detail. Tb(III)-containing compounds display large quantum yields, up to 43%. Polyvinylpyrrolidone nanoparticles doped with [Ln2(C8H4O4)3(H2O)4] n (Ln = Eu, Tb, Er) have also been synthesized and characterized. The encapsulation of the coordination polymers results in somewhat reduced luminescence intensities and lifetime, but the nanoparticles can be dispersed in water and remain unchanged in this medium for more than 20 h.     

New insights into the polymerization of methyl methacrylate initiated by rare-earth borohydride complexes: a combined experimental and computational approach

Polymerization of methyl methacrylate (MMA) initiated by the rare-earth borohydride complexes [Ln(BH(4))(3)(thf)(3)] (Ln=Nd, Sm) or [Sm(BH(4))(Cp*)(2)(thf)] (Cp*=eta-C(5)Me(5)) proceeds at ambient temperature to give rather syndiotactic poly(methyl methacrylate) (PMMA) with molar masses M(n) higher than expected and quite broad molar mass distributions, which is consistent with a poor initiation efficiency. The polymerization of MMA was investigated by performing density functional theory (DFT) calculations on an eta-C(5)H(5) model metallocene and showed that in the reaction of [Eu(BH(4))(Cp)(2)] with MMA the borate [Eu(Cp)(2){(OBH(3))(OMe)C=C(Me)(2)}] (e-2) complex, which forms via the enolate [Eu(Cp)(2){O(OMe)C=C(Me)(2)}] (e), is calculated to be exergonic and is the most likely of all of the possible products. This product is favored because the reaction that leads to the formation of carboxylate [Eu(Cp)(2){OOC-C(Me)(=CH(2))}] (f) is thermodynamically favorable, but kinetically disfavored, and both of the potential products from a Markovnikov [Eu(Cp)(2){O(OMe)C-CH(Me)(CH(2)BH(3))}] (g) or anti-Markovnikov [Eu(Cp)(2){O(OMe)C-C(Me(2))(BH(3))}] (h) hydroboration reaction are also kinetically inaccessible. Similar computational results were obtained for the reaction of [Eu(BH(4))(3)] and MMA with all of the products showing extra stabilization. The DFT calculations performed by using [Eu(Cp)(2)(H)] to model the mechanism previously reported for the polymerization of MMA initiated by [Sm(Cp*)(2)(H)](2) confirmed the favorable exergonic formation of the intermediate [Eu(Cp)(2){O(OMe)C=C(Me)(2)}] (e') as the kinetic product, this enolate species ultimately leads to the formation of PMMA as experimentally observed. Replacing H by BH(4) thus prevents the 1,4-addition of the [Eu(BH(4))(Cp)(2)] borohydride ligand to the first incoming MMA molecule and instead favors the formation of the borate complex e-2. This intermediate is the somewhat active species in the polymerization of MMA initiated by the borohydride precursors [Ln(BH(4))(3)(thf)(3)] or [Sm(BH(4))(Cp*)(2)(thf)].     

Investigating the interaction of crystal violet probe molecules on sodium dodecyl sulfate micelles with hyper-Rayleigh scattering

We report the use of the nonlinear optical technique of hyper-Rayleigh scattering to investigate the interaction of the cationic probe molecule crystal violet with micelles of sodium dodecyl sulfate. An absolute value of (847 +/- 80) x 10(-30) esu is measured at the fundamental wavelength of 870 nm for the molecular hyperpolarizability of crystal violet free in pure aqueous solutions. In aqueous solutions of sodium dodecyl sulfate, above and below the critical micelle concentration, the measured hyperpolarizability of crystal violet is weaker than in the solution free of sodium dodecyl sulfate. From the comparison with linear optical photoabsorption spectroscopy data, this difference is attributed to electrostatic interactions between the cationic crystal violet molecules and the negatively charged sodium dodecyl sulfate surfactant molecules present in excess. Polarization resolved hyper-Rayleigh scattering measurements are then performed to show that, below and above the critical micelle concentration, crystal violet molecules also undergo symmetry changes upon interaction with sodium dodecyl sulfate. Above the critical micelle concentration, the minimum fraction of micelles interacting with at least one CV molecule is estimated. For instance, for a crystal violet aqueous concentration of 150 microM, this fraction is larger than 7%.