Power-law Approach Research Articles

ON A MICROSCOPIC MODEL OF VISCOUS FRICTION

We consider a body moving along the x-axis under the action of an external force E and immersed in an infinitely extended perfect gas. We assume the gas to be described by the mean-field approximation and interacting elastically with the body. In this setup, we discuss the following statement: "Let V0 be the initial velocity of the body and V∞ its asymptotic velocity, then for |V0 - V∞| small enough it results |V(t) - V∞| ≈ C t-d-2 for t large, where V(t) is the velocity of the body at time t, d the dimension of the space and C is a positive constant depending on the medium and on the shape of the body". The reason for the power law approach to the stationary state instead of the exponential one (usually assumed in viscous friction problems), is due to the long memory of the dynamical system. In a recent paper by Caprino, Marchioro and Pulvirenti,3 the case of E constant and positive, with 0 < V0 < V∞, for a disk orthogonal to the x-axis has been discussed. Here we complete the analysis in the cases E > 0 with V0 > V∞ and E = 0. We also approach the problem of an x-dependent external force, by choosing E of harmonic type. In this case we obtain the power-like asymptotic time behavior for the body position X(t). The investigation is done in detail for a disk orthogonal to the x-axis and then, by a sketched proof, extended to a body with a general convex shape.

Power Law Analysis Estimates of Analyte Concentration and Particle Size in Highly Scattering Granular Samples from Photon Time-of-Flight Measurements

Optical measurements of particle size and composition in granular samples are difficult to make due to complex light scattering from particles. These multiple scattering events bias absorption estimates and complicate the calculation of scattering and absorption coefficients used to estimate sample properties. Time series data, such as chromatograms and photon time-of-flight (TOF) profiles, contain self-repeating (fractal) characteristics. Power law analysis of photon TOF profiles allows the determination of absorption coefficients and particle sizes in a single experiment. A correlation dimension algorithm was used on photon TOF data from scattering samples. MLR models were then obtained from correlation dimension plots for the estimation of sample properties. Estimates of particle sizes and absorption coefficients were shown to agree well with theoretical values when compared using independent validation sets. Results show close to a 3-fold and up to a 5-fold decrease in the errors of estimation of dye concentration and particle size, respectively, as compared to steady-state measurements. The power law approach provides a useful means of determining sample properties in highly scattering media.

Universality of the power-law approach to the jamming limit in random sequential adsorption dynamics

Random sequential adsorption (RSA), on a two-dimensional continuum substrate, of different types of zero area objects that disallow domain formation and hence lead to jamming, is examined by simulation. In all the cases, in the asymptotic time regime, the approach of the number density rho(t) at instant t to jamming density rho(infinity) is found to exhibit power law rho(infinity)-rho(t) approximately t{-p} as that for RSA of finite area objects. These results suggest the possibility of the power law being universal for all jamming systems in RSA on a continuum substrate. A generalized analytical treatment is also proposed.

Kinetic Study on the Acid-Catalyzed Hydrolysis of Cellulose to Levulinic Acid

A variety of interesting bulk chemicals is accessible by the acid-catalyzed hydrolysis of cellulose. An interesting example is levulinic acid, a versatile precursor for fuel additives, polymers, and resins. A detailed kinetic study on the acid-catalyzed hydrolysis of cellulose to levulinic acid is reported in this paper. The kinetic experiments were performed in a temperature window of 150−200 °C, sulfuric acid concentrations between 0.05 and 1 M, and initial cellulose intakes between 1.7 and 14 wt %. The highest yield of levulinic was 60 mol %, obtained at a temperature of 150 °C, an initial cellulose intake of 1.7 wt %, and a sulfuric acid concentration of 1 M. A full kinetic model covering a broad range of reaction conditions was developed using the power-law approach. Agreement between the experimental data and the kinetic model is good. The kinetic expressions were used to gain insights into the optimum process conditions for the conversion of cellulose to levulinic acid in continuous-reactor configu...

Topology optimization of 2D continua for minimum compliance using parallel computing

Topology optimization is often used in the conceptual design stage as a preprocessing tool to obtain overall material distribution in the solution domain. The resulting topology is then used as an initial guess for shape optimization. It is always desirable to use fine computational grids to obtain high-resolution layouts that minimize the need for shape optimization and postprocessing (Bendsoe and Sigmund, Topology optimization theory, methods and applications. Springer, Berlin Heidelberg New York 2003), but this approach results in high computation cost and is prohibitive for large structures. In the present work, parallel computing in combination with domain decomposition is proposed to reduce the computation time of such problems. The power law approach is used as the material distribution method, and an optimality criteria-based optimizer is used for locating the optimum solution [Sigmund (2001)21:120–127; Rozvany and Olhoff, Topology optimization of structures and composites continua. Kluwer, Norwell 2000]. The equilibrium equations are solved using a preconditioned conjugate gradient algorithm. These calculations have been done using a master–slave programming paradigm on a coarse-grain, multiple instruction multiple data, shared-memory architecture. In this study, by avoiding the assembly of the global stiffness matrix, the memory requirement and computation time has been reduced. The results of the current study show that the parallel computing technique is a valuable tool for solving computationally intensive topology optimization problems.

Maximum fluid power condition in solar chimney power plants – An analytical approach

Main features of a solar chimney power plant are a circular greenhouse type collector and a tall chimney at its centre. Air flowing radially inwards under the collector roof heats up and enters the chimney after passing through a turbo-generator. The objective of the study was to investigate analytically the validity and applicability of the assumption that, for maximum fluid power, the optimum ratio of turbine pressure drop to pressure potential (available system pressure difference) is 2/3. An initial power law model assumes that pressure potential is proportional to volume flow to the power m, where m is typically a negative number between 0 and −1, and that the system pressure drop is proportional to the power n, where typically n = 2. The analysis shows that the optimum turbine pressure drop as fraction of the pressure potential is ( n − m)/( n + 1), which is equal to 2/3 only when m = 0, implying a constant pressure potential, independent of flow rate. Consideration of a basic collector model proposed by Schlaich leads to the conclusion that the value of m is equal to the negative of the collector floor-to-exit efficiency. A more comprehensive optimization scheme, incorporating the basic collector model of Schlaich in the analysis, shows that the power law approach is sound and conservative. It is shown that the constant pressure potential assumption ( m = 0) may lead to appreciable underestimation of the performance of a solar chimney power plant, when compared to the analyses presented in the paper. More important is that both these analyses predict that maximum fluid power is available at much lower flow rate and much higher turbine pressure drop than predicted by the constant pressure potential assumption. Thus, the constant pressure potential assumption may lead to overestimating the size of the flow passages in the plant, and designing a turbine with inadequate stall margin and excessive runaway speed margin. The derived equations may be useful in the initial estimation of plant performance, in plant performance analysis and in control algorithm design. The analyses may also serve to set up test cases for more comprehensive plant models.

Green Chemicals: A Kinetic Study on the Conversion of Glucose to Levulinic Acid

Levulinic acid has been identified as a promising green, biomass derived platform chemical. A kinetic study on one of the key steps in the conversion of biomass to levulinic acid, i.e., the acid catalysed decomposition of glucose to levulinic acid has been performed. The experiments were performed in a broad temperature window (140–200°C), using sulphuric acid as the catalyst (0.05–1 M) and a initial glucose concentration between 0.1 and 1 M. A kinetic model of the reaction sequence was developed including the kinetics for the intermediate 5-hydroxymethyl-2-furaldehyde (HMF) and humins byproducts using a power-law approach. The yield of levulinic acid is favoured in dilute glucose solution at high acid concentration. On the basis of the kinetic results, continuous reactor configurations with a high extent of back-mixing are preferred to achieve high levulinic acid yields.

Dimensional analysis of foam drainage

Dimensional analysis of the process of steady drainage of liquid from foam has shown that, if surface and inertial effects are neglected, the drainage rate non-dimensionalised as a Stokes-type number may be described by a function of the liquid volume fraction only. In addition it is demonstrated that a simple power law relationship between these two quantities is a good approximation to channel-dominated and node dominated foam drainage models and that the approach enables the prediction of the dependency of drainage upon bubble size. The power law approach requires two adjustable constants which is the minimum required since a priori estimates of both surface shear viscosity and viscous losses in the vertices of the foam are problematical. In addition, a simple dimensionless parameter has been introduced that can modify the power law model to take into account capillary suction so that transients in one-dimensional drainage problems can be modelled.

Contact mechanics and tip shape in AFM-based nanomechanical measurements

Stiffness–load curves obtained in quantitative atomic force acoustic microscopy (AFAM) measurements depend on both the elastic properties of the sample and the geometry of the atomic force microscope (AFM) tip. The geometry of silicon AFM tips changes when used in contact mode, affecting measurement accuracy. To study the influence of tip geometry, we subjected ten AFM tips to the same series of AFAM measurements. Changes in tip shape were observed in the scanning electron microscope (SEM) between individual AFAM tests. Because all of the AFAM measurements were performed on the same sample, variations in AFAM stiffness–load curves were attributed to differences in tip geometry. Contact-mechanics models that assumed simple tip geometries were used to analyze the AFAM data, but the calculated values for tip dimensions did not agree with those provided by SEM images. Therefore, we used a power-law approach that allows for a nonspherical tip geometry. We found that after several AFAM measurements, the geometry of the tips at the very end is intermediate between those of a flat punch and a hemisphere. These results indicate that the nanoscale tip-sample contact cannot easily be described in terms of simple, ideal geometries.

A kinetic study on the decomposition of 5-hydroxymethylfurfural into levulinic acid

Levulinic acid (LA), accessible by the acid catalyzed degradation of biomass, is potentially a very versatile green intermediate chemical for the synthesis of various (bulk) chemicals for applications like fuel additives, polymers, and resin precursors. We report here a kinetic study on one of the key steps in the conversion of biomass to levulinic acid, i.e. the reaction of 5-hydroxymethylfurfural (HMF) to levulinic acid. The kinetic experiments were performed in a temperature window of 98–181 °C, acid concentrations between 0.05–1 M, and initial HMF concentrations between 0.1 and 1 M. The highest LA yield was 94% (mol/mol), obtained at an initial HMF concentration of 0.1 M and a sulfuric acid concentration of 1 M. The yield at full HMF conversion is independent of the temperature. An empirical rate expression for the main reaction as well as the side reaction to undesired humins was developed using the power law approach. Agreement between experimental and model data is good. The rate expressions were applied to gain insights into optimum process conditions for batch processing.

Roller extrusion of biscuit doughs

Biscuit doughs are dense solid–liquid pastes which exhibit complex rheological behaviour. The rolling behaviour of sheets of commercial short and hard biscuit doughs was investigated using an instrumented counter-rotating roll mill, which allowed the roll torque, separating force and surface pressure to be monitored. The rheological characteristics of the two doughs were quantified by analysing data from ram extrusion experiments in terms of a quasi-plastic model (following the Benbow–Bridgwater approach) and a power law fluid model. The results indicated that the doughs were not ideally suited to the quasi-plastic analysis. The power law parameters varied noticeably between the doughs but both were strongly shear-thinning (power law shear indices of 0.25 and 0.5), with large extensional viscosities. These rheological model parameters were subsequently used to generate predictions of dough behaviour during rolling. A standard sheet metal plasticity model was modified to include strain rate dependency but neither dough’s behaviour could be adequately described by this model. The power law fluid approach based on the lubrication assumption reported by Levine tended to underpredict the work requirement owing to the significant contribution from extensional deformation. Interestingly, the system could be modelled in terms of an apparent power law rheology by fitting data to Levine’s model. Better agreement was obtained for some parameters but not consistently so, indicating that the power law approach is not adequate for these soft-solid materials.

Bilateral filtering for structural topology optimization

Filtering has been a major approach used in the homogenization-based methods for structural topology optimization to suppress the checkerboard pattern and relieve the numerical instabilities. In this paper a bilateral filtering technique originally developed in image processing is presented as an efficient approach to regularizing the topology optimization problem. A non-linear bilateral filtering process leads to a suitable problem regularization to eliminate the checkerboard instability, pronounced edge preserving smoothing characteristics to favour the 0–1 convergence of the mass distribution, and computational efficiency due to its single pass and non-iterative nature. Thus, we show that the application of the bilateral filtering brings more desirable effects of checkerboard-free, mesh independence, crisp boundary, computational efficiency and conceptual simplicity. The proposed bilateral technique has a close relationship with the conventional domain filtering and range filtering. The proposed method is implemented in the framework of a power-law approach based on the optimality criteria and illustrated with 2D examples of minimum compliance design that has been extensively studied in the recent literature of topology optimization and its efficiency and accuracy are highlighted. Copyright © 2005 John Wiley & Sons, Ltd.

Large-scale inference of the transcriptional regulation of Bacillus subtilis

This paper addresses the inference of the transcriptional regulatory network of Bacillus subtilis. Two inference approaches, a linear, additive model and a non-linear power-law model, are used to analyze the expression of 747 genes from B. subtilis obtained using Affymetrix GeneChip ® arrays under three different experimental conditions. A robustness analysis is introduced for identifying confidence levels for all inferred regulatory connections. Both the linear and non-linear methods produce candidate networks that share a scale-free or a “hub-and-spoke” topology with a small number of global regulator genes influencing the expression of a large number of target genes. The two computational approaches in tandem are able to identify known global regulators with a high level of confidence. The linear model is able to identify the interactions of highly expressed genes, particularly those involved in genetic information processing, energy metabolism and signal transduction. Conversely, the non-linear power-law approach tends to capture development regulation and specific carbon and nitrogen regulatory interactions.

On optimization of bio‐probes

AbstractThe present paper deals with the modelling and optimization of small bio‐probes that can be used for biological sensing; the bio‐probes can be classified as MicroElectroMechnical Systems (MEMS). The objective is to optimize the structure of the bio‐probes in order to maximize the sensing sensitivity. A biological coating results in a prestress on the sensing cantilever when certain molecules are present in the surrounding medium. The mechanical deformation due to the biological material is modelled by applying a prestress in the top layer of the bio‐probes. Topology optimization is used to improve the design. In the present work it is necessary to use an interpolation scheme different from the SIMP (power law) approach which is usually used in topology optimization. In calculating the sensitivities, needed for the optimization, complications due to the prestress occur, but also due to the coupling between the elastic field and the electric field which both must be used in an integrated model. These complications are dealt with and analytically obtained sensitivities are presented. Copyright © 2004 John Wiley & Sons, Ltd.

Mechanistic aspects of in vitro fatigue-crack growth in dentin

Although the propagation of fatigue cracks has been recognized as a problem of clinical significance in dentin, there have been few fracture mechanics-based studies that have investigated this issue. In the present study, in vitro cyclic fatigue experiments were conducted over a range of cyclic frequencies (1–50Hz) on elephant dentin in order to quantify fatigue-crack growth behavior from the perspective of understanding the mechanism of fatigue in dentin. Specifically, results obtained for crack extension rates along a direction parallel to the dentinal tubules were found to be well described by the stress-intensity range, ΔK, using a simple Paris power-law approach with exponents ranging from 12 to 32. Furthermore, a frequency dependence was observed for the crack-growth rates, with higher growth rates associated with lower frequencies. By using crack-growth experiments involving alternate cyclic and static loading, such fatigue-crack propagation was mechanistically determined to be the result of a “true” cyclic fatigue mechanism, and not simply a succession of static fracture events. Furthermore, based on the observed frequency dependence of fatigue-crack growth in dentin and observations of time-dependent crack blunting, a cyclic fatigue mechanism involving crack-tip blunting and re-sharpening is proposed. These results are deemed to be of importance for an improved understanding of fatigue-related failures in teeth.

Graph representation for structural topology optimization using genetic algorithms

This paper proposes a graph representation for evolutionary structural topology design. Based on graph theory, a valid topology in the design domain is represented by a connected simple graph and each edge of the graph is defined by a cubic Bézier curve with varying thickness. The graph is defined to have connecting paths between loading regions and support regions of the structure, to ensure a physically meaningful connected structure. Each path is formed by one or more Bézier curves in order to allow more local control of the shape. A real-valued chromosome encoding and decoding scheme and a partition crossover method are also developed based on graph theory. The graph representation GA is applied to structural topology optimization problems and its performance is compared with those of other methods. Compared with the power-law approach, the present graph representation GA can generate clearly defined and distinct geometries and perform a global search, but it requires more computational cost. The numerical results also demonstrate the improved performance of the present graph representation over that of the Voronoi-bar representation in terms of connectivity of the geometry and convergence speed. It is suggested that this graph representation method is both physically meaningful and computationally effective in the framework of topological optimum design using GAs.

A single saddle model for the -relaxation in supercooled liquids

We study the Langevin equation for a single harmonic saddle as an elementary model for the beta-relaxation in supercooled liquids close to Tc. The input of the theory is the spectrum of the eigenvalues of the dominant stationary points at a given temperature. We prove in general the existence of a time-scale t_eps, which is uniquely determined by the spectrum, but is not simply related to the fraction of negative eigenvalues. The mean square displacement develops a plateau of length t_eps, such that a two-step relaxation is obtained if t_eps diverges at Tc. We analyze the specific case of a spectrum with bounded left tail, and show that in this case the mean square displacement has a scaling dependence on time identical to the beta-relaxation regime of Mode Coupling Theory, with power law approach to the plateau and power law divergence of t_eps at Tc.

A double-blind, randomized, incomplete crossover trial to assess the dose proportionality of rosuvastatin in healthy volunteers

Background: Rosuvastatin, a 3-hydroxy-3-methylglutaryl coenzyme A reductase inhibitor, has been developed for the treatment of patients with dyslipidemia. Objective: This study assessed the dose proportionality and pharmacokinetics of single oral doses of rosuvastatin in healthy volunteers. Methods: This was a double-blind, randomized, incomplete crossover trial consisting of 3 trial days separated by ≥7-day washout periods. Healthy men were allocated to 1 of 2 treatment regimens: rosuvastatin 10, 20, and 80 mg, or rosuvastatin 10, 40, and 80 mg, administered as single doses on separate trial days in random order. Pharmacokinetic and tolerability assessments were made up to 96 hours after administration. Dose proportionality was tested using the power-law approach. Results: Eighteen healthy white men participated in the trial (mean age, 41.2 years; mean height, 178.4 cm; mean body weight, 81.6 kg). Geometric mean rosuvastatin maximum plasma concentration (C max) values of 3.75, 6.79, 10.3, and 30.1 ng/mL were achieved at a median time to C max of 5.0 hours after doses of 10, 20, 40, and 80 mg, respectively. The corresponding geometric mean values for rosuvastatin area under the plasma concentration-time curve from time 0 to time of the last measurable concentration (AUC 0−t) were 31.6, 56.8, 98.2, and 268 ng·h/mL. C max and AUC 0−t were both linearly related to dose. The estimates of the proportionality coefficient (90% CI) for CmaX and AUC o−t were 0.999 (0.898–1.099) and 1.024 (0.941–1.107), respectively; all values fell within the prespecified range of 0.847 to 1.153. Rosuvastatin was well tolerated in this group of healthy men when administered orally at doses of 10 to 80 mg. Conclusion: Rosuvastatin systemic exposure was dose proportional over the dose range of 10 to 80 mg.

Spatial and temporal scaling of population density and animal movement: A power law approach

Many ecological patterns are sensitive to spatial and temporal scale, but no general protocol has emerged for dealing with scale-dependence. We suggest that power laws, regularly used in the study of body size, can be applied to many such features. We used these techniques to describe two scale-sensitive aspects of caribou (Rangifer tarandus): population densities of North American herds when measured at varying spatial extents, and speeds of radiotracked individuals when observed at varying temporal resolutions. Log-log regression was applied, based on the power formula, Y= cX z. Non-isometric relationships existed for abundance versus area of the population range, but the pattern differed between groups. For sedentary caribou, animal numbers increased more slowly than spatial extent (z = 0.503). For migratory populations, this relationship was hyper-allometric (z > 1), and densities were generally lower for continental herds compared to insular Newfoundland herds living without wolves (Canis lupus). We surmise that defining population bounds may be more subjective for sedentary herds; migratory herd densities may increase with increasing population size. Distance moved versus the interval between radiolocations was also non-isometric (z = 0.551). We standardized these movements to eliminate the confounding effects of scale, enabling comparisons across seasons and populations. The power formula helps to clarify the debate over the use of ratios, where variables may change non-isometrically. Power laws may enable the synthesis of ecological patterns sensitive to scales of space and time.

Coloured noise influence on system evolution

We present a picture of phase transitions of the system with colored multiplicative noise. Considering the noise amplitude as the power-law dependence of the stochastic variable $x^a$ we show the way to phase transitions disorder-order and order-disorder. The governed equations for the order parameter and one-time correlator are obtained and investigated in details. The long-time asymptotes in the disordered and ordered domains on the phase portrait of the system are defined.