Manifold Techniques Research Articles

Classification of Global and Blow-up Sign-changing Solutions of a Semilinear Heat Equation in the Subcritical Fujita Range : Second-order Diffusion

AbstractIt is well known from the seminal paper by Fujita [22] for 1 < p < p0, and Hayakawa [36] for the critical case p = p0, that all the solutions u ≥ 0 of the semilinear heat equationut= Δu + |u|p−1u in ℝN× ℝ+, in the range, (0.1)with arbitrary initial data u0(x) ≥ 0, ≢ 0, blow-up in finite time, while for p > p0there exists a class of sufficiently “small” global in time solutions. This fundamental result from the 1960-70s (see also [39] for related contributions), was a cornerstone of further active blow-up research. Nowadays, similar Fujita-type critical exponents p0, as important characteristics of stability, unstability, and blow-up of solutions, have been calculated for various nonlinear PDEs. The above blow-up conclusion does not include solutions of changing sign, so some of them may remain global even for p ≤ p0. Our goal is a thorough description of blow-up and global in time oscillatory solutions in the subcritical range in (0.1) on the basis of various analytic methods including nonlinear capacity, variational, category, fibering, and invariant manifold techniques. Two countable sets of global solutions of changing sign are shown to exist. Most of them are not radially symmetric in any dimension N ≥ 2 (previously, only radial such solutions in ℝNor in the unit ball B1⊂ℝNwere mostly studied). A countable sequence of critical exponents, at which the whole set of global solutions changes its structure, is detected:, l = 0, 1, 2, ... .. See [47, 48] for earlier interesting contributions on sign changing solutions.

Large Eddy Simulation of the PRECCINSTA burner

For the first time the Reaction–Diffusion Manifold (REDIM) technique is implemented together with a presumed Filtered Density Function (FDF) to carry out Large Eddy Simulation (LES) of lean premixed swirling flames in the well-known PRECCINSTA burner. The REDIM approach reduces detailed chemical kinetics based on low-dimensional reaction–diffusion manifolds in composition space, so that many species can be accurately estimated at a relatively low computational cost via a look-up table. In the present work, only one reduced coordinate, the CO2 mass fraction, is considered in the REDIM look-up table. Implementation of the REDIM/FDF model is performed with two presumed FDF shapes, top-hat and clipped Gaussian. Two premixed flames are simulated, with equivalence ratios 0.75 and 0.83, respectively. Velocity, temperature and species concentration profiles are discussed and compared to the experimental data. The results are in good overall agreement with the corresponding experiment. It is also shown that the LES results obtained with the two presumed FDFs are very similar and the reason for this is analysed.

Lymphome B intravasculaire et dermatite granulomateuse interstitielle avec arthrites : association inédite de deux affections rares et trompeuses

A three-degree-of-freedom vibro-impact system with symmetric two-sided rigid constraints is considered. Since the symmetric period n−2 motion of the vibro-impact system corresponds to the symmetric fixed point of the Poincaré map of the vibro-impact system, we investigate bifurcations of the symmetric period n−2 motion by researching into bifurcations of the associated symmetric fixed point. The Poincaré map of the system has symmetry property, and can be expressed as the second iteration of another unsymmetric implicit map. Based on both the Poincaré map and the unsymmetric implicit map, the center manifold technique and the theory of normal forms are applied to deduce the normal form of the Neimark–Sacker-pitchfork bifurcation of the symmetric fixed point. By numerical analysis, we obtain the Neimark–Sacker-pitchfork bifurcation of the symmetric fixed point of the Poincaré map in the vibro-impact system.

Experimental and Numerical Study of Various MT-SOFC Flow Manifold Techniques: Single MT-SOFC Analysis

Standard anode supported micro tubular-solid oxide fuel cell (MT-SOFC) stacks may provide the oxidant, in relation to the fuel, in three different manifold regimes. Firstly, “co-flow” involves oxidant outside the MT-SOFC flowing co-linearly in relation to the fuel inside. Secondly, “counter flow” involves oxidant outside the MT-SOFC flowing counter-linearly in relation to the fuel inside the MT-SOFC. Finally, “cross-flow” involves the oxidant outside the MT-SOFC flowing perpendicular to the fuel flow inside the MT-SOFC. In order to examine the effect of manifold technique on MT-SOFC performance, a combination of numerical simulation and experimental measurements was performed. Furthermore, the cathode current tap location, in relation to the fuel flow, was also studied. It was found that the oxidant manifold and the location of the cathode current collection point on the MT-SOFC tested and modeled had negligible effect on the MT-SOFC's electrical and thermal performance. In this study, a single MT-SOFC was studied in order to establish the measurement technique and numerical simulation implementation as a prerequisite before further test involving a 7 cell MT-SOFC stack.

2-D GEL ELECTROPHORESIS MAP OF METHICILLIN-RESISTANT <i>STAPHYLOCOCCUS AUREUS</i> TREATED WITH <i>QUERCUS INFECTORIA</i> GALL EXTRACT

The widespread outbreak of Methicillin-Resistant Staphylococcus Aureus (MRSA) has caused clinical and epidemiological concern in hospital environment. The emergence of Vancomycin-Intermediate S. Aureus (VISA) and, more recently, Vancomycin-Resistant S. Aureus (VRSA) has further alarmed clinician and scientist worlwide. The objective of this study is to determine the optimum concentration of sample protein from MRSA after treatment with acetone extract from Quercus infectoria gall. Comparison of the Protein Expression Profile (PEP) between the treated MRSA and untreated strain as control was obtained using 2-dimensional gel electrophoresis. The Minimum Inhibitory Concentration (MIC) value of acetone extract of galls from Q. infectoria against two strains of MRSA; ATCC 33591 and PPUKM clinical isolate was determined by broth microdilution method. The MIC value of acetone extracts against both strains of MRSA was 0.3125 mg mL-1 compared to vancomycin (0.00195 mg mL-1). The optimum concentration of MRSA protein that produced the best resolution was 100 µg. Manifold technique was observed to produce a gel with better resolution and greater number of spot compared with the strip holder technique. This study showed that there were 7 protein spots that represented the increased in the protein expression of more than 2-fold in the MRSA treated with acetone extract of galls Q. infectoria compared to the untreated group. This preliminary study on the PEP of Q. infectoria galls extract-treated MRSA may provide an insight of its antimicrobial mechanism which could lead to the identification of target protein in the future development of a new effective regimen for the treatment of MRSA infections.

Multiple Solutions for a Class of Concave-Convex Quasilinear Elliptic Systems with Nonlinear Boundary Condition

In this paper, a quasilinear elliptic system is investigated, which involves concave-convex nonlinearities and nonlinear boundary condition. By Nehari manifold, fibering method and analytic techniques, the existence of multiple nontrivial nonnegative solutions to this equation is verified.

Preparation of Mg2FeH6 Nanoparticles for Hydrogen Storage Properties

Magnesium (Mg) and iron (Fe) nanoparticles are prepared by thermal decomposition of bipyridyl complexes of metals. These prepared Mg‐Fe (2 : 1) nanoparticles are hydrogenated under 4 MPa hydrogen pressure and 673 K for 48 hours to achieve Mg2FeH6. Their structural analysis was assessed by applying manifold techniques. The hydrogen storage properties of prepared compound were measured by Sieverts type apparatus. The desorption kinetics were measured by high pressure thermal desorption spectrometer (HP‐TDS). More than 5 wt% hydrogen released was obtained by the Mg2FeH6 within 5 min, and during rehydrogenation very effective hydrogen absorption rate was observed by the compound.

Parameter identification of unknown biased sinusoid

An algorithm composed of a normalised frequency estimator and a first-order low-pass filter is introduced to identify online the unknown values of bias, amplitude, frequency and phase angles of a sinusoid signal. Slow integral manifold technique and averaging method are used to educe a non-linear autonomous system for frequency estimation. The global exponential stability of isolated equilibrium point and the ultimate boundedness under unknown bounded disturbance are investigated respectively. The identification and the estimation errors of bias, amplitude and phase angle under bounded disturbance are also formulated and exhibited by simulation.

Descent algorithms on oblique manifold for source-adaptive ICA contrast.

A Riemannian manifold optimization strategy is proposed to facilitate the relaxation of the orthonormality constraint in a more natural way in the course of performing independent component analysis (ICA) that employs a mutual information-based source-adaptive contrast function. Despite the extensive development of manifold techniques catering to the orthonormality constraint, only a limited number of works have been dedicated to oblique manifold (OB) algorithms to intrinsically handle the normality constraint, which has been empirically shown to be superior to other Riemannian and Euclidean approaches. Imposing the normality constraint implicitly, in line with the ICA definition, essentially guarantees a substantial improvement in the solution accuracy, by way of increased degrees of freedom while searching for an optimal unmixing ICA matrix, in contrast with the orthonormality constraint. Designs of the steepest descent, conjugate gradient with Hager-Zhang or a hybrid update parameter, quasi-Newton, and cost-effective quasi-Newton methods intended for OB are presented in this paper. Their performance is validated using natural images and systematically compared with the popular state-of-the-art approaches in order to assess the performance effects of the choice of algorithm and the use of a Riemannian rather than Euclidean framework. We surmount the computational challenge associated with the direct estimation of the source densities using the improved fast Gauss transform in the evaluation of the contrast function and its gradient. The proposed OB schemes may find applications in the offline image/signal analysis, wherein, on one hand, the computational overhead can be tolerated, and, on the other, the solution quality holds paramount interest.

Time–frequency manifold for nonlinear feature extraction in machinery fault diagnosis

Time–frequency feature is beneficial to representation of non-stationary signals for effective machinery fault diagnosis. The time–frequency distribution (TFD) is a major tool to reveal the synthetic time–frequency pattern. However, the TFD will also face noise corruption and dimensionality reduction issues in engineering applications. This paper proposes a novel nonlinear time–frequency feature based on a time–frequency manifold (TFM) technique. The new TFM feature is generated by mainly addressing manifold learning on the TFDs in a reconstructed phase space. It combines the non-stationary information and the nonlinear information of analyzed signals, and hence exhibits valuable properties. Specifically, the new feature is a quantitative low-dimensional representation, and reveals the intrinsic time–frequency pattern related to machinery health, which can effectively overcome the effects of noise and condition variance issues in sampling signals. The effectiveness and the merits of the proposed TFM feature are confirmed by case study on gear wear diagnosis, bearing defect identification and defect severity evaluation. Results show the value and potential of the new feature in machinery fault pattern representation and classification.

Trajectory Design Combining Invariant Manifolds with Discrete Mechanics and Optimal Control

A mission design technique that combines invariant manifold techniques, discrete mechanics, and optimal control produces locally optimal low-energy trajectories. Previously, invariant manifolds of the planar circular restricted three-body problem have been used to design trajectories with relatively small midcourse change in velocity ΔV. A different method of using invariant manifolds is explored to design trajectories directly in the four-body problem. Then, using the local optimal control method DMOC (Discrete Mechanics and Optimal Control), it is possible to reduce the midcourse ΔV to zero. The influence of different boundary conditions on the optimal trajectory is also demonstrated. These methods are tested on a trajectory that begins in Earth orbit and ends in ballistic capture at the moon. Impulsive DMOC trajectories require up to 19% less ΔV than trajectories using a Hohmann transfer. When applied to low-thrust trajectories, DMOC produces an improvement of up to 59% in the mass fraction and 22% in travel time when compared with results from shooting methods.

Neimark–Sacker-pitchfork bifurcation of the symmetric period fixed point of the Poincaré map in a three-degree-of-freedom vibro-impact system

A three-degree-of-freedom vibro-impact system with symmetric two-sided rigid constraints is considered. Since the symmetric period n−2 motion of the vibro-impact system corresponds to the symmetric fixed point of the Poincaré map of the vibro-impact system, we investigate bifurcations of the symmetric period n−2 motion by researching into bifurcations of the associated symmetric fixed point. The Poincaré map of the system has symmetry property, and can be expressed as the second iteration of another unsymmetric implicit map. Based on both the Poincaré map and the unsymmetric implicit map, the center manifold technique and the theory of normal forms are applied to deduce the normal form of the Neimark–Sacker-pitchfork bifurcation of the symmetric fixed point. By numerical analysis, we obtain the Neimark–Sacker-pitchfork bifurcation of the symmetric fixed point of the Poincaré map in the vibro-impact system.

Large Eddy Simulations and experimental studies of turbulent premixed combustion near extinction

In this paper, lean premixed flames featuring local extinction are simulated numerically by Large Eddy Simulations (LES) and investigated experimentally by one-dimensional, spatially resolved Raman scattering. Two unconfined piloted lean premixed natural gas/air flames with equivalence ratios ϕ=0.71 and 0.65 are studied. The Reaction–Diffusion Manifold (REDIM) technique is employed together with the presumed joint Filtered Density Function (FDF) to carry out the LES. In the present work, two reduced coordinates, the CO2 and N2 mass fractions, are considered in the REDIM look-up table, and the joint FDF of them is modelled in terms of two statistically independent FDFs. Good overall agreement between the experimental data and the LES results is obtained. Scatter plots from measurements and instantaneous data from LES show states with local extinction, mainly in the leaner flame. The results demonstrate the capability of the proposed REDIM-presumed joint FDF method to deal with the phenomenon of local extinction in lean premixed flames.

Applying the manifold theory to Milky Way models: First steps on morphology and kinematics

We present recent results obtained by applying invariant manifold techniques to analytical models of the Milky Way. It has been shown that invariant manifolds can reproduce successfully the spiral arms and rings in external barred galaxies. Here, for the first time, we apply this theory to Milky Way models. We select five different models from the literature and, using the parameters chosen by the authors of the papers, and three different cases, namely Case 1, where only the COBE/DIRBE bar is included in the potential; Case 2, when the COBE/DIRBE and the Long bar are aligned, and Case 3, when the COBE/DIRBE bar and the Long bar are misaligned. We compute in each case and for each model the orbits trapped by the manifolds. In general, the global morphology of the manifolds can account for the 3-kpc arms and for the Galactic Molecular Ring.

Reduced order modeling of nonlinear time periodic systems subjected to external periodic excitations

In this work, new methodologies for order reduction of nonlinear systems with periodic coefficients subjected to external periodic excitations are presented. The periodicity of the linear terms is assumed to be non-commensurate with the periodicity of forcing vector. The dynamical equations of motion are transformed using the Lyapunov–Floquet (L–F) transformation such that the linear parts of the resulting equations become time-invariant while the forcing and nonlinearity takes the form of quasiperiodic functions. The techniques proposed here construct a reduced order equivalent system by expressing the non-dominant states as time-varying functions of the dominant (master) states. This reduced order model preserves stability properties and is easier to analyze, simulate and control since it consists of relatively small number of states in comparison with the large scale system. Specifically, two methods are discussed to obtain the reduced order model. First approach is a straightforward application of linear method similar to the ‘Guyan reduction’. The second novel technique proposed here extends the concept of ‘ invariant manifolds’ for the forced problem to construct the fundamental solution. Order reduction approach based on this extended invariant manifold technique yields unique ‘ reducibility conditions’. If these ‘ reducibility conditions’ are satisfied only then an accurate order reduction via extended invariant manifold approach is possible. This approach not only yields accurate reduced order models using the fundamental solution but also explains the consequences of various ‘ primary’ and ‘ secondary resonances’ present in the system. One can also recover ‘ resonance conditions’ associated with the fundamental solution which could be obtained via perturbation techniques by assuming weak parametric excitation. This technique is capable of handling systems with strong parametric excitations subjected to periodic and quasi-periodic forcing. It is anticipated that these order reduction techniques will provide a useful tool in the analysis and control system design of large-scale parametrically excited nonlinear systems subjected to external periodic excitations.

Coordinate-independent sparse sufficient dimension reduction and variable selection

Sufficient dimension reduction (SDR) in regression, which reduces the dimension by replacing original predictors with a minimal set of their linear combinations without loss of information, is very helpful when the number of predictors is large. The standard SDR methods suffer because the estimated linear combinations usually consist of all original predictors, making it difficult to interpret. In this paper, we propose a unified method - coordinate-independent sparse estimation (CISE) - that can simultaneously achieve sparse sufficient dimension reduction and screen out irrelevant and redundant variables efficiently. CISE is subspace oriented in the sense that it incorporates a coordinate-independent penalty term with a broad series of model-based and model-free SDR approaches. This results in a Grassmann manifold optimization problem and a fast algorithm is suggested. Under mild conditions, based on manifold theories and techniques, it can be shown that CISE would perform asymptotically as well as if the true irrelevant predictors were known, which is referred to as the oracle property. Simulation studies and a real-data example demonstrate the effectiveness and efficiency of the proposed approach.

Modeling turbulent flow from dam break using slow manifolds

We present a novel approach based on centre manifold techniques to describe the large scale dynamics of the mean turbulent dam-break flow. We avoid empirical assumptions about the cross-stream profile of the velocity; instead a solution is obtained using free surface and bed boundary conditions that accommodate constant turbulent shear as an approximately neutral mode. We describe the turbulent dynamics across the flow, and identify important factors affecting the turbulent dissipation in the lateral direction. Available experimental data verify the results. References J. Carr. Applications of centre manifold theory, volume 35 of Applied Mathematical Sciences. Springer--Verlag, 1981. D. J. Georgiev, A. J. Roberts, and D. V. Strunin. The dynamics of the vertical structure of turbulence in flood flows. ANZIAM Journal, 48(CTAC-2006):C573--C590, 2007. http://anziamj.austms.org.au/ojs/index.php/ANZIAMJ/article/view/124. D. J. Georgiev, A. J. Roberts, and D. V. Strunin. Nonlinear dynamics on centre manifolds describing turbulent floods: $k$-$\omega $ model. Discrete and Continuous Dynamical Systems Supplements, (Special):419--428, 2007. http://aimsciences.org/journals/pdfs.jsp?paperID=2848&mode=full. A. J. Hogg and D. Pritchard. The effects of hydraulic resistance on dam-break and other shallow inertial flows. Journal of Fluid Mechanics, 501:179--212, 2004. I. M. Janosi, D. Jan, K. G. Szabo, and T. Tel. Turbulent drag reduction in dam-break flows. Experiments in Fluids, 37:219--229, 2004. P. Keast and P. H. Muir. {EPDCOL}: A more efficient {PDECOL} code. ACM Transactions on Mathematical Software, 17(2):153--166, 1991. Y. A. Kuznetsov. Elements of applied bifurcation theory, volume 112 of Applied Mathematical Sciences. Springer--Verlag, 1995. N. K. Madsen and R. F. Sincovec. {PDECOL}: General collocation software for partial differential equations. ACM Transactions on Mathematical Software, 5(3):326--351, 1979. I. Nezu. Open-channel flow turbulence and its research prospects in the 21st century. Journal of Hydraulic Engineering, 131(4):229--246, April 2005. T. M. Oezgoekmen, T. Iliescu, P. F. Fischer, A. Srinivasan, and J. Duan. Large eddy simulation of stratified mixing and two-dimensional dam-break problem in a rectagular enclosed domain. Ocean Modelling, 16:106--140, 2007. A. J. Roberts. Low-dimensional modelling of dynamics via computer algebra. Computer Physics Communications, 100:215--230, 1997. A. J. Roberts. Computer algebra describes flow of turbulent floods via the Smagorinsky large eddy closure. Technical report, University of Southern Queensland, 2008. http://eprints.usq.edu.au/4008/. A. J. Roberts, D. J. Georgiev, and D. V. Strunin. Model turbulent floods with the Smagorinski large eddy closure. http://arxiv.org/abs/0805.3192/. P. K. Stansby, A. Chegini, and T. C. D. Barnes. The initial stages of dam-break flow. Journal of Fluid Mechanics, 374:407--424, 1998.

Direct numerical simulation of hydrogen addition in turbulent premixed Bunsen flames using flamelet-generated manifold reduction

Direct numerical simulations of a lean premixed turbulent Bunsen flame with hydrogen addition have been performed. We show the results for a case with equivalence ratio of 0.7 and a molar fractional distribution of 40% H2 and 60% CH4. The flamelet-generated manifold technique is used to reduce the chemistry; flamelets with different equivalence ratios and inflow temperature are used to account for stretch effects that are enhanced by preferential diffusion. The three-dimensional simulation clearly shows enhanced burning velocity in regions convex toward the reactants and reduced burning velocity with possible extinction in regions concave toward the reactants. To obtain these effects it was found to be necessary to include two three-dimensional transport equations with essentially different diffusivities. This point is illustrated by comparison of the results with cases in which either a single transport equation was used or two transport equations with minor differences in diffusivities were used. These latter cases incorporated preferential diffusion in the 1D flamelets (and thus in the manifold), but not in the three-dimensional transport. Thus the three-dimensional preferential diffusion effects are shown to enhance curvature and thereby to increase the turbulent burning velocity and reduce the mean flame height. In addition the turbulent burning velocity increases because hydrogen addition leads to a larger laminar flamelet consumption speed. To demonstrate this second effect, results of the cases mentioned above are compared to the results of simulations of the Bunsen flame with 0% hydrogen added to the fuel.

Elimination of Fast Modes in the Coupled Process of Chemistry and Diffusion in Turbulent Nonpremixed Flames: An Application of the REDIM Approach

A computational study has been made of bluff-body stabilized turbulent jet flames with strong turbulence-chemistry interaction (Sydney Flames HM1 and HM3). The wide range of scales in the problem is described using a combination of a standard second moment turbulence closure, a joint scalar transported probability density function (PDF) method and the Reaction-Diffusion Manifold (REDIM) technique. The latter provides a reduction of a detailed chemistry mechanism, taking into account effects of laminar diffusion. In an a priori test it is evaluated to what extent the single shot experimental data are located on the reaction-diffusion manifold. Next, computed spatial profiles of mean and variance of independent and dependent scalar variables and profiles of conditional averages and variances (conditional on mixture fraction) are compared to the experimental results. The quality of these predictions is interpreted in relation to the a priori analysis. In general, simulations using the REDIM approach for reduction of detailed C2-chemistry confirm earlier findings for micro-mixing model behavior, obtained with a skeletal Cl-mechanism. Nevertheless it is concluded that the experiments show important features that are not described by the currently used REDIM.

Analysis of a strong mass-based flame stretch model for turbulent premixed combustion

In the present paper a theory describing effects of strong flame stretch on turbulent flame propagation [L. P. H. de Goey and J. H. M. ten Thije Boonkkamp, “A flamelet description of premixed laminar flames and the relation with flame stretch,” Combust. Flame 119, 253 (1999)] is extended to volume averaged quantities and validated with direct numerical simulation (DNS). The extended theory describes the fuel consumption rate in terms of subgrid scale contributions connected to propagation effects including strong flame stretch. In case there is no preferential diffusion present, it is predicted that the total consumption rate is not affected by local stretch at all. Then the total consumption is described by the unstretched mass burning rate multiplied with the flame surface density. DNSs of turbulent flame kernels have been carried out in order to support the results from the theory. The chemistry is described by application of the flamelet generated manifold technique. The strong stretch theory is shown to be valid up to realizations in the thin reaction zone regime by three independent methods. The local effects of stretch are described, evaluated, and interpreted. Locally the mass burning rate changes by fuel leakage tangential to the flame, but this has no integral effect. The method can be used for subgrid scale modeling of turbulent flame propagation.