Nonlinear and adaptive frame approximation schemes for elliptic PDEs: Theory and numerical experiments

This article is concerned with adaptive numerical frame methods for elliptic operator equations. We show how specific noncanonical frame expansions on domains can be constructed. Moreover, we study the approximation order of best n‐term frame approximation, which serves as the benchmark for the performance of adaptive schemes. We also discuss numerical experiments for second order elliptic boundary value problems in polygonal domains where the discretization is based on recent constructions of boundary adapted wavelet bases on the interval. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Approximate solutions and error estimates for a Stokes boundary value problem

The aim of this paper is the numerical treatment of a boundary value problem for the system of Stokes’ equations. For this we extend the method of approximate approximations to boundary value problems. This method was introduced by Maz’ya (DFG-Kolloquium des DFG-Forschungsschwerpunktes Randelementmethoden, 1991) and has been used until now for the approximation of smooth functions defined on the whole space and for the approximation of volume potentials. In the present paper we develop an approximation procedure for the solution of the interior Dirichlet problem for the system of Stokes’ equations in two dimensions. The procedure is based on potential theoretical considerations in connection with a boundary integral equations method and consists of three approximation steps as follows. In a first step the unknown source density in the potential representation of the solution is replaced by approximate approximations. In a second step the decay behavior of the generating functions is used to gain a suitable approximation for the potential kernel, and in a third step Nyström’s method leads to a linear algebraic system for the approximate source density. For every step a convergence analysis is established and corresponding error estimates are given.     

Advanced finite element formulations for modeling thin piezoelectric structures

This contribution is concerned with mixed finite element formulations for modeling piezoelectric beam and shell structures. Due to the electromechanical coupling, specific deformation modes are joined with electric field components. In bending dominated problems incompatible approximation functions of these fields cause incorrect results. These effects occur in standard finite element formulations, where interpolation functions of lowest order are used. A mixed variational approach is introduced to overcome these problems. The mixed formulation allows for a consistent approximation of the electromechanical coupled problem. It utilizes six independent fields and could be derived from a Hu-Washizu variational principle. Displacements, rotations and the electric potential are employed as nodal degrees of freedom. According to the Timoshenko theory (beam) and the Reissner-Mindlin theory (shell), the formulations account for constant transversal shear strains. To incorporate three dimensional constitutive relations all transversal components of the electric field and the strain field are enriched by mixed finite element interpolations. Thus the complete piezoelectric coupling is appropriately captured. The common assumption of vanishing transversal stress and dielectric displacement components is enforced in an integral sense. Some numerical examples will demonstrate the capability of the presented finite element formulation. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A finite element formulation of a viscoelastic model for dielectric elastomers

Smart materials by definition, are solids, fluids or gases which react independently on changing external conditions and modify one or more properties without external stimuli. Sensu lato an external energy can produce the reaction, such as stress, temperature, moisture, pH, magnetic or electric fields. The distinguishing characteristic for electroactive polymers (EAP) is, that they react with a deformation by the application of an electrical field. This contribution presents a nonlinear electro-viscoelastic model for dielectric elastomers and its finite element implementation. This type of smart materials belong to the group of EAP's and consists out of soft elastomer between compliant and conducting electrodes. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the complexity of hydromount-models in full vehicle simulation

The influence of the complexity of hydromount-models on the results in full vehicle simulation is studied. In particular, a Kelvin-Voigt model, a linear lever model and a non-linear model are assessed with respect to their predictive capabilities as compared to the necessary parameterization and computation efforts. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Gabor analysis over finite Abelian groups

Gabor frames for signals over finite Abelian groups, generated by an arbitrary lattice within the finite time–frequency plane, are the central topic of this paper. Our generic approach covers both multi-dimensional signals as well as non-separable lattices, and in fact the multi-window case as well. Our generic approach includes most of the fundamental facts about Gabor expansions of finite signals for the case of product lattices, as they have been given by Qiu, Wexler–Raz or Tolimieri–Orr, Bastiaans and Van-Leest and others. In our presentation the spreading representation of linear operators between finite-dimensional Hilbert space as well as a symplectic version of Poisson's summation formula over the finite time–frequency plane are essential ingredients. They bring us to the so-called Fundamental Identity of Gabor Analysis. In addition, we highlight projective representations of the time–frequency plane and its subgroups and explain the natural connection to twisted group algebras. In the finite-dimensional setting discussed in this paper these twisted group algebras are just matrix algebras and their structure provides the algebraic framework for the study of the deeper properties of finite-dimensional Gabor frames, independent of the structure theory theorem for finite Abelian groups.     

Imaging by sensitized oxygenations of photochromic anthracene films: examination of effects that improve performance and reversibility

The aliphatic anthracene compound 1 and the oligomeric anthracene 2 were synthesized. Thin films of 1 and 2 mixed with the sensitizers tetraphenylporphyrin (TPP) and methylene blue (MB) were irradiated with visible light in air. Upon formation of singlet oxygen, the anthracene units were converted quantitatively to the corresponding endoperoxides. Heating of the irradiated samples afforded the parent anthracenes with high yields. Here, we demonstrate that the kinetics and reversibility of this reaction strongly depend on the microenvironment of the anthracene groups in the two compounds. The photooxidation of thin films of 1 is accompanied by interesting changes in the morphology of the film and allows the first application of 1 as a nondestructive negative-tone photoresist for lithography and as an oxidizing ink. The morphology of 2 remained unchanged after photooxidation as a result of the stabilizing oligomer backbone. This stabilizing effect significantly improves the photochromic performance of 2. The reversibility of the photooxidation is very high (>90 %) for oligomeric films of 2 after several cycles of irradiation and heating. Decomposition of the anthracene and a loss of the activity of the sensitizer diminish slightly the performance of the monomeric species.     

Investigation of the ionisation and fragmentation behaviour of different nitroaromatic compounds occurring as polar metabolites of explosives using electrospray ionisation tandem mass spectrometry

In order to develop a liquid chromatography/electrospray ionisation tandem mass spectrometry (LC/ESI-MS/MS) method for identification and quantification of polar metabolites of explosives using a triple quadrupole system, the mass spectrometric ionisation and fragmentation behaviour of different nitrophenols, nitro- and aminonitrobenzoic acids, nitrotoluenesulfonic acids, and aminonitrotoluenes was investigated. Due to their different molecular structures, the substances concerned showed a very different ionisation efficiency in the ESI process. Interestingly, 2,4-dinitrobenzoic acid yielded no mass signals in the Q1 scan suggesting a thermal decarboxylation in the ion source, whereas the corresponding 3,5-isomer showed a high ionisation yield. Using negative ionisation polarity, carboxylic, phenolic, and sulfonic acid groups were deprotonated resulting in molecular anions, which could be fragmented in a collision cell. A pronounced dependency of the produced fragment ion series on the kind and position of substituents at the nitrobenzene ring (ortho effects) was observed and exploited for the development of substance-specific detection methods in the multiple reaction monitoring mode. In case of benzoic and sulfonic acids, decarboxylation and desulfonation, respectively, were observed as the most frequent fragmentation reactions. Furthermore, besides loss of NO(2), NO fragmentation occurred and preceded a decarbonylation of the benzene ring. The expulsion of the open-shell molecules NO and NO(2) led to a variety of distonic radical anions.     

Density-cumulant functional theory

Starting point is the energy expectation value as a functional of the one-particle density matrix gamma and the two-particle density cumulant lambda(2). We decompose gamma into a best idempotent approximation kappa and a correction tau, that is entirely expressible in terms of lambda(2). So we get the energy E as a functional of kappa and lambda(2), which can be varied independently. Approximate n-representability conditions, derived by perturbation theory are imposed on the variation of lambda(2). A nonlinear system of equations satisfied by lambda(2) is derived, the linearized version of which turns out to be equivalent to the coupled electron-pair approximation, variant zero. The start for kappa is Hartree-Fock, but kappa is then updated to become the best idempotent approximation of gamma. Relations to density matrix functional theory and Kohn-Sham type density functional theory are discussed.     

Phase-sensitive stimulated Raman adiabatic passage in dipolar extended lambda systems

The authors investigate the possible phase-sensitive behavior of (multiphoton) stimulated Raman adiabatic passage population transfer in extended lambda systems, if more than one state of an anharmonic progression of target levels is accessible in transitions of different photonicities. They use a minimal model four-level system (4LS) with one initial state separated from two target states by an apex state. The parameters of the 4LS are adapted from the bend states of the HCN-HNC system. Using a dressed-state analysis within the rotating wave approximation (RWA), the authors identify phase-dependent diabatic transitions between the two dressed states contributing to the state vector as the mechanism leading to phase-sensitive target populations. The essential features giving rise to the phase dependence are found to be different (non-zero-) diagonal elements of the dipole matrix, i.e., permanent dipole moments, and the presence of a direct two-photon overtone coupling between the apex state and the lower target state which formally enters the RWA Hamiltonian upon inclusion of permanent dipole moments. Among the parameters controlling the extent of the effect are the anharmonic properties of the target progression and the absolute values of the field frequencies, so that in view of the requirement to tune the driving fields into the vicinity of resonance, details of the level structure are of importance. A comparative numerical study executed without invoking RWA shows that qualitatively there are similar trends in the appearance of phase sensitivity, although the effects are considerably more pronounced in the full treatment. In the full treatment the authors also explore off-resonance conditions and discuss the signatures of phase sensitivity in the target populations.