Multiplicative Operator Splittings in Nonlinear Diffusion: From Spatial Splitting to Multiple Timesteps

Operator splitting is a powerful concept used in many diversed fields of applied mathematics for the design of effective numerical schemes. Following the success of the additive operator splitting (AOS) in performing an efficient nonlinear diffusion filtering on digital images, we analyze the possibility of using multiplicative operator splittings to process images from different perspectives.We start by examining the potential of using fractional step methods to design a multiplicative operator splitting as an alternative to AOS schemes. By means of a Strang splitting, we attempt to use numerical schemes that are known to be more accurate in linear diffusion processes and apply them on images. Initially we implement the Crank-Nicolson and DuFort-Frankel schemes to diffuse noisy signals in one dimension and devise a simple extrapolation that enables the Crank-Nicolson to be used with high accuracy on these signals. We then combine the Crank-Nicolson in 1D with various multiplicative operator splittings to process images. Based on these ideas we obtain some interesting results. However, from the practical standpoint, due to the computational expenses associated with these schemes and the questionable benefits in applying them to perform nonlinear diffusion filtering when using long timesteps, we conclude that AOS schemes are simple and efficient compared to these alternatives.We then examine the potential utility of using multiple timestep methods combined with AOS schemes, as means to expedite the diffusion process. These methods were developed for molecular dynamics applications and are used efficiently in biomolecular simulations. The idea is to split the forces exerted on atoms into different classes according to their behavior in time, and assign longer timesteps to nonlocal, slowly-varying forces such as the Coulomb and van der Waals interactions, whereas the local forces like bond and angle are treated with smaller timesteps. Multiple timestep integrators can be derived from the Trotter factorization, a decomposition that bears a strong resemblance to a Strang splitting. Both formulations decompose the time propagator into trilateral products to construct multiplicative operator splittings which are second order in time, with the possibility of extending the factorization to higher order expansions. While a Strang splitting is a decomposition across spatial dimensions, where each dimension is subsequently treated with a fractional step, the multiple timestep method is a decomposition across scales. Thus, multiple timestep methods are a realization of the multiplicative operator splitting idea. For certain nonlinear diffusion coefficients with favorable properties, we show that a simple multiple timestep method can improve the diffusion process.     

Measurement of Helium Leakage Rates through Closures of Dangerous Goods Packagings for the Assessment of Potentially Explosive Mixtures in Freight Containers

The objective was to find out whether an explosive atmosphere can be created in a freight container by gaseous leakage flow of vapour‐air‐mixture through leaks in the closures of dangerous goods packagings filled with hazardous liquids. Because of high temperatures during intercontinental carriage, there is a gauge pressure in the free vapour phase inside the packagings which can cause a gaseous leakage flow. Two different methods were applied: Helium limit leakage rates for 23 quantitatively important hazardous liquids concerning their lower explosion limit (LEL) were calculated for a worst case transport scenario (Method 1). Helium leakage rates of five closure types of dangerous goods packagings with volumes of approximately 6 l were measured using the pressure technique by accumulation (Method 2). All types of closures of steel packagings were uncritical. The maximum measured leakage was 33% of the limit leakage rate. The leakage rates of screw closures of plastic jerricans can exceed the LEL if there are production‐related patterns such as non‐concentricity of the closures and flashes on the neck. Especially for plastic packagings it is important to minimize gaseous leakage flow, because an explosive atmosphere can also be reached by permeation of the individual filling substance or by a combination of both effects. For the assessment of potentially explosive mixtures in freight containers, both mass transfer mechanisms have to be taken into account. Copyright © 2015 John Wiley & Sons, Ltd.     

Modeling segregation of bidisperse granular materials using physical control parameters in the quasi‐2D bounded heap

Quantitatively predicting segregation of size‐disperse granular materials is of potential value in many industrial applications. We consider granular segregation of size‐bidisperse particles in quasi‐2D bounded heaps, a canonical granular flow, using an advection‐diffusion transport equation with an additional term to account for particle segregation. The equation is characterized by two dimensionless parameters that are functions of control parameters (flow rate, system size, and particle sizes) and kinematic parameters (flowing layer depth, diffusion coefficient, and percolation length scale). As the kinematic parameters are usually difficult to measure in practice, their dependence on the control parameters is determined directly from discrete element method simulations. Using these relationships, it is possible to determine which values of the control parameters result in a mixed or segregated heap. The approach used here is broadly applicable to a wide range of other flow geometries and particle systems. © 2015 American Institute of Chemical Engineers AIChE J, 61: 1524–1534, 2015     

Volume-Surface Trees

Many algorithms in computer graphics improve their efficiency by using Hierarchical Space Subdivision Schemes (HS³), such as octrees, kD‐trees or BSP trees. Such HS³ usually provide an axis‐aligned subdivision of the 3D space embedding a scene or an object. However, the purely volume‐based behavior of these schemes often leads to strongly imbalanced surface clustering. In this article, we introduce the VS‐Tree, an alternative HS³ providing efficient and accurate surface‐based hierarchical clustering via a combination of a global 3D decomposition at coarse subdivision levels, and a local 2D decomposition at fine levels near the surface. First, we show how to efficiently construct VS‐Trees over meshes and point‐based surfaces, and analyze the improvement it offers for cluster‐based surface simplification methods. Then we propose a new surface reconstruction algorithm based on the volume‐surface classification of the VS‐Tree. This new algorithm is faster than state‐of‐the‐art reconstruction methods and provides a final semi‐regular mesh comparable to the output of remeshing algorithms.     

Measured Amplitude Distribution of Automotive Ignition Noise

This paper describes a series of tests conducted 1) to statistically characterize the amplitude distribution of automotive ignition noise, and 2) determine whether such test results can be reproduced. Some tests showed the amplitude distribution to be Weibull and others indicated a log normal distribution. However, under controlled conditions the results obtained from an individual vehicle were quite repeatable.     

Implicit Brushes for Stylized Line‐based Rendering

We introduce a new technique called Implicit Brushes to render animated 3D scenes with stylized lines in realtime with temporal coherence. An Implicit Brush is defined at a given pixel by the convolution of a brush footprint along a feature skeleton; the skeleton itself is obtained by locating surface features in the pixel neighborhood. Features are identified via image‐space fitting techniques that not only extract their location, but also their profile, which permits to distinguish between sharp and smooth features. Profile parameters are then mapped to stylistic parameters such as brush orientation, size or opacity to give rise to a wide range of line‐based styles.     

An Inexpensive BRDF Model for Physically-based Rendering

A new BRDF model is presented which can be viewed as an kind of intermediary model between empirism and theory. Main results of physics are observed (energy conservation, reciprocity rule, microfacet theory) and numerous phenomena involved in light reflection are accounted for, in a physically plausible way (incoherent and coherent reflection, spectrum modifications, anisotropy, self-shadowing, multiple surface and subsurface reflection, differences between homogeneous and heterogeneous materials). The model has been especially intended for computer graphics applications and therefore includes two main features: simplicity (a small number of intuitively understandable parameters controls the model) and efficiency (the formulation provides adequation to Monte-Carlo rendering techniques and/or hardware implementations).     

Monitoring pH-dependent conformational changes in aqueous solutions of poly(methacrylic acid)-b-polydimethylsiloxane copolymer based on fluorescence spectra of pyrene and 1,3-bis(1-pyrenyl)propane

Aqueous solutions of poly(methacrylic acid)-b-polydimethylsiloxane copolymer (PMA-b-PDMS), and reference solutions of PMA homopolymer, were studied as a function of pH using fluorescent probes: pyrene (P) and 1,3-bis(1-pyrenyl)propane (P3P). In both polymers the conformational transition of PMA chains at pH>5 was reflected in the sharp increase of the intensity ratio of the first to the third vibronic band in emission spectra of P; an excimer emission appeared for P3P, as well as for P if the probe concentration was higher than its solubility limit in water. The excimer to monomer intensity ratio of both probes is a sensitive indicator of the conformational transitions of polymer chains, but P3P is significantly more sensitive compared to pyrene. The data clearly indicated that the presence of PDMS blocks markedly facilitated the transition, even though the ratio of dimethylsiloxane to methacrylic acid units in the copolymer was low, 1:34. The absence of the P3P excimer emission at pH≈3 was rationalized by assuming that clustering of sections of PMA chains around pyrene moieties imposes an extended conformation of the probe molecule.     

Direct ESR and spin trapping methods for the detection and identification of radical fragments in Nafion membranes and model compounds exposed to oxygen radicals

Direct electron spin resonance (ESR) and spin trapping methods are used in our laboratory in order to identify radical fragments in fluorinated membranes (Nafion) used in fuel cells and in model compounds when exposed to reactive oxygen species. Oxygen radicals were generated by the Fenton reaction, by the photo-Fenton method, and by UV-irradiation of aqueous H₂O₂ solutions. Direct ESR detection led to the identification of fluorinated radical fragments in UV-irradiated Nafion neutralized by Cu(II), Fe(II), and Fe(III) cations and exposed to H₂O₂. In Nafion exposed to the Fenton reagent in the presence of DMPO (5,5-dimethyl-1-pyrroline) as the spin trap, radical adducts of hydroxyl and carbon-centered radicals (CCRs) were detected. A fluorinated model compound, perfluro-(2-ethoxyethane) sulfonic acid (CF₃CF₂OCF₂CF₂SO₃H, PFEESA), that mimics the side chain of Nafion was chosen for study, with the goal to shed light on the more complicated degradation processes in the polymeric membranes. Analysis of the spin adducts obtained from UV-irradiated PFEESA in the presence of H₂O₂ and MNP (2-methyl-2-nitrosopropane) as the spin trap suggested that the side chain of Nafion is a possible site of attack by oxygen radicals.