Examples illustrating the use of renormalization techniques for singularly perturbed differential equations

With nine examples, we seek to illustrate the utility of the Renormalization Group approach as a unification of other asymptotic and perturbation methods.

Reduction of amplitude equations by the renormalization group approach

This article elucidates and analyzes the fundamental underlying structure of the renormalization group (RG) approach as it applies to the solution of any differential equation involving multiple scales. The amplitude equation derived through the elimination of secular terms arising from a naive perturbation expansion of the solution to these equations by the RG approach is reduced to an algebraic equation which is expressed in terms of the Thiele semi-invariants or cumulants of the eliminant sequence { Zi } i=1 . Its use is illustrated through the solution of both linear and nonlinear perturbation problems and certain results from the literature are recovered as special cases. The fundamental structure that emerges from the application of the RG approach is not the amplitude equation but the aforementioned algebraic equation. © 2008 The American Physical Society.

The renormalization group and the implicit function theorem for amplitude equations

This article lays down the foundations of the renormalization group (RG) approach for differential equations characterized by multiple scales. The renormalization of constants through an elimination process and the subsequent derivation of the amplitude equation [Chen, Phys. Rev. E 54, 376 (1996)] are given a rigorous but not abstract mathematical form whose justification is based on the implicit function theorem. Developing the theoretical framework that underlies the RG approach leads to a systematization of the renormalization process and to the derivation of explicit closed-form expressions for the amplitude equations that can be carried out with symbolic computation for both linear and nonlinear scalar differential equations and first order systems but independently of their particular forms. Certain nonlinear singular perturbation problems are considered that illustrate the formalism and recover well-known results from the literature as special cases. © 2008 American Institute of Physics.

Viability of Pseudomonas aeruginosa in cough aerosols generated by persons with cystic fibrosis

Background Person-to-person transmission of respiratory pathogens, including Pseudomonas aeruginosa, is a challenge facing many cystic fibrosis (CF) centres. Viable P aeruginosa are contained in aerosols produced during coughing, raising the possibility of airborne transmission. Methods Using purpose-built equipment, we measured viable P aeruginosa in cough aerosols at 1, 2 and 4 m from the subject (distance) and after allowing aerosols to age for 5, 15 and 45 min in a slowly rotating drum to minimise gravitational settling and inertial impaction (duration). Aerosol particles were captured and sized employing an Anderson Impactor and cultured using conventional microbiology. Sputum was also cultured and lung function and respiratory muscle strength measured. Results Nineteen patients with CF, mean age 25.8 (SD 9.2) years, chronically infected with P aeruginosa, and 10 healthy controls, 26.5 (8.7) years, participated. Viable P aeruginosa were detected in cough aerosols from all patients with CF, but not from controls; travelling 4 m in 17/18 (94%) and persisting for 45 min in 14/18 (78%) of the CF group. Marked inter-subject heterogeneity of P aeruginosa aerosol colony counts was seen and correlated strongly (r=0.73–0.90) with sputum bacterial loads. Modelling decay of viable P aeruginosa in a clinic room suggested that at the recommended ventilation rate of two air changes per hour almost 50 min were required for 90% to be removed after an infected patient left the room. Conclusions Viable P aeruginosa in cough aerosols travel further and last longer than recognised previously, providing additional evidence of airborne transmission between patients with CF.

Bistable regime of surface magnetoplasma waves excitation at a semiconductor-metal interface

The theoretical analysis of the bistability associated with the excitation of surface magnetoplasma waves (SWs) propagating across an external magnetic field at the semiconductor-metal interface by the attenuated total reflection (ATR) method is presented. The Kretschmann-Raether configuration of the ATR method is considered, i.e. a plane electromagnetic wave is incident onto a metal surface through a coupling prism. The third-order nonlinearity of the semiconductor medium is considered in the general form using the formalism of the third-order nonlinear susceptibilities and of the perturbation theory. The examples of the nonlinear mechanisms which influence the SW propagation are given. The analytical and numerical analyses show that the realization of bistable regimes of the SW excitation is possible. The SW amplitude values providing bistability in the structure are evaluated and are reasonably low to provide the experimental observation.

Gravity-induced coronal plane joint moments in the adolescent scoliotic spine

INTRODUCTION Calculating segmental (vertebral level-by-level) torso masses in Adolescent Idiopathic Scoliosis (AIS) patients allows the gravitational loading on the scoliotic spine during relaxed standing to be estimated. METHODS Existing low dose CT scans were used to calculate vertebral level-by-level torso masses and joint moments occurring in the spine for a group of female AIS patients with right-sided thoracic curves. Image processing software, ImageJ (v1.45 NIH USA) was used to reconstruct the torso segments and subsequently measure the torso volume and mass corresponding to each vertebral level. Body segment masses for the head, neck and arms were taken from published anthropometric data. Intervertebral joint moments at each vertebral level were found by summing each of the torso segment masses above the required joint and multiplying it by the perpendicular distance to the centre of the disc. RESULTS AND DISCUSSION Twenty patients were included in this study with a mean age of 15.0±2.7 years and a mean Cobb angle 52±5.9°. The mean total trunk mass, as a percentage of total body mass, was 27.8 (SD 0.5) %. Mean segmental torso mass increased inferiorly from 0.6kg at T1 to 1.5kg at L5. The coronal plane joint moments during relaxed standing were typically 5-7Nm at the apex of the curve (Figure 1), with the highest apex joint of 7Nm. CT scans were performed in the supine position and curve magnitudes are known to be 7-10° smaller than those measured in standing [1]. Therefore joint moments produced by gravity will be greater than those calculated here. CONCLUSIONS Coronal plane joint moments as high as 7Nm can occur during relaxed standing in scoliosis patients, which may help to explain the mechanics of AIS progression. The body mass distributions calculated in this study can be used to estimate joint moments derived using other imaging modalities such as MRI and subsequently determine if a relationship exists between joint moments and progressive vertebral deformity.

Novel solutions for a model of wound healing angiogenesis

We prove the existence of novel, shock-fronted travelling wave solutions to a model of wound healing angiogenesis studied in Pettet et al (2000 IMA J. Math. App. Med. 17 395–413) assuming two conjectures hold. In the previous work, the authors showed that for certain parameter values, a heteroclinic orbit in the phase plane representing a smooth travelling wave solution exists. However, upon varying one of the parameters, the heteroclinic orbit was destroyed, or rather cut-off, by a wall of singularities in the phase plane. As a result, they concluded that under this parameter regime no travelling wave solutions existed. Using techniques from geometric singular perturbation theory and canard theory, we show that a travelling wave solution actually still exists for this parameter regime. We construct a heteroclinic orbit passing through the wall of singularities via a folded saddle canard point onto a repelling slow manifold. The orbit leaves this manifold via the fast dynamics and lands on the attracting slow manifold, finally connecting to its end state. This new travelling wave is no longer smooth but exhibits a sharp front or shock. Finally, we identify regions in parameter space where we expect that similar solutions exist. Moreover, we discuss the possibility of more exotic solutions.

Canards in advection-reaction-diffusion systems in one spatial dimension

This thesis contains a mathematical investigation of the existence of travelling wave solutions to singularly perturbed advection-reaction-diffusion models of biological processes. An enhanced mathematical understanding of these solutions and models is gained via the identification of canards (special solutions of fast/slow dynamical systems) and their role in the existence of the most biologically relevant, shock-like solutions. The analysis focuses on two existing models. A new proof of existence of a whole family of travelling waves is provided for a model describing malignant tumour invasion, while new solutions are identified for a model describing wound healing angiogenesis.

Butterfly catastrophe for fronts in a three-component reaction–diffusion system

We study the dynamics of front solutions in a three-component reaction–diffusion system via a combination of geometric singular perturbation theory, Evans function analysis, and center manifold reduction. The reduced system exhibits a surprisingly complicated bifurcation structure including a butterfly catastrophe. Our results shed light on numerically observed accelerations and oscillations and pave the way for the analysis of front interactions in a parameter regime where the essential spectrum of a single front approaches the imaginary axis asymptotically.

Establishing the impact of temporary tissue expanders on electron and photon beam dose distributions

Purpose: This study investigates the effects of temporary tissue expanders (TTEs) on the dose distributions in breast cancer radiotherapy treatments under a variety of conditions. Methods: Using EBT2 radiochromic film, both electron and photon beam dose distribution measurements were made for different phantoms, and beam geometries. This was done to establish a more comprehensive understanding of the implant’s perturbation effects under a wider variety of conditions. Results: The magnetic disk present in a tissue expander causes a dose reduction of approximately 20% in a photon tangent treatment and 56% in electron boost fields immediately downstream of the implant. The effects of the silicon elastomer are also much more apparent in an electron beam than a photon beam. Conclusions: Evidently, each component of the TTE attenuates the radiation beam to different degrees. This study has demonstrated that the accuracy of photon and electron treatments of post-mastectomy patients is influenced by the presence of a tissue expander for various beam orientations. The impact of TTEs on dose distributions establishes the importance of an accurately modelled high-density implant in the treatment planning system for post-mastectomy patients.

Accounting for management costs in sensitivity analyses of matrix population models

Traditional sensitivity and elasticity analyses of matrix population models have been used to inform management decisions, but they ignore the economic costs of manipulating vital rates. For example, the growth rate of a population is often most sensitive to changes in adult survival rate, but this does not mean that increasing that rate is the best option for managing the population because it may be much more expensive than other options. To explore how managers should optimize their manipulation of vital rates, we incorporated the cost of changing those rates into matrix population models. We derived analytic expressions for locations in parameter space where managers should shift between management of fecundity and survival, for the balance between fecundity and survival management at those boundaries, and for the allocation of management resources to sustain that optimal balance. For simple matrices, the optimal budget allocation can often be expressed as simple functions of vital rates and the relative costs of changing them. We applied our method to management of the Helmeted Honeyeater (Lichenostomus melanops cassidix; an endangered Australian bird) and the koala (Phascolarctos cinereus) as examples. Our method showed that cost-efficient management of the Helmeted Honeyeater should focus on increasing fecundity via nest protection, whereas optimal koala management should focus on manipulating both fecundity and survival simultaneously. These findings are contrary to the cost-negligent recommendations of elasticity analysis, which would suggest focusing on managing survival in both cases. A further investigation of Helmeted Honeyeater management options, based on an individual-based model incorporating density dependence, spatial structure, and environmental stochasticity, confirmed that fecundity management was the most cost-effective strategy. Our results demonstrate that decisions that ignore economic factors will reduce management efficiency. ©2006 Society for Conservation Biology.

Novel non-linear elastic structural analysis with generalised transverse element loads using a refined finite element

In the finite element modelling of structural frames, external loads such as wind loads, dead loads and imposed loads usually act along the elements rather than at the nodes only. Conventionally, when an element is subjected to these general transverse element loads, they are usually converted to nodal forces acting at the ends of the elements by either lumping or consistent load approaches. In addition, it is especially important for an element subjected to the first- and second-order elastic behaviour, to which the steel structure is critically prone to; in particular the thin-walled steel structures, when the stocky element section may be generally critical to the inelastic behaviour. In this sense, the accurate first- and second-order elastic displacement solutions of element load effect along an element is vitally crucial, but cannot be simulated using neither numerical nodal nor consistent load methods alone, as long as no equilibrium condition is enforced in the finite element formulation, which can inevitably impair the structural safety of the steel structure particularly. It can be therefore regarded as a unique element load method to account for the element load nonlinearly. If accurate displacement solution is targeted for simulating the first- and second-order elastic behaviour on an element on the basis of sophisticated non-linear element stiffness formulation, the numerous prescribed stiffness matrices must indispensably be used for the plethora of specific transverse element loading patterns encountered. In order to circumvent this shortcoming, the present paper proposes a numerical technique to include the transverse element loading in the non-linear stiffness formulation without numerous prescribed stiffness matrices, and which is able to predict structural responses involving the effect of first-order element loads as well as the second-order coupling effect between the transverse load and axial force in the element. This paper shows that the principle of superposition can be applied to derive the generalized stiffness formulation for element load effect, so that the form of the stiffness matrix remains unchanged with respect to the specific loading patterns, but with only the magnitude of the loading (element load coefficients) being needed to be adjusted in the stiffness formulation, and subsequently the non-linear effect on element loadings can be commensurate by updating the magnitude of element load coefficients through the non-linear solution procedures. In principle, the element loading distribution is converted into a single loading magnitude at mid-span in order to provide the initial perturbation for triggering the member bowing effect due to its transverse element loads. This approach in turn sacrifices the effect of element loading distribution except at mid-span. Therefore, it can be foreseen that the load-deflection behaviour may not be as accurate as those at mid-span, but its discrepancy is still trivial as proved. This novelty allows for a very useful generalised stiffness formulation for a single higher-order element with arbitrary transverse loading patterns to be formulated. Moreover, another significance of this paper is placed on shifting the nodal response (system analysis) to both nodal and element response (sophisticated element formulation). For the conventional finite element method, such as the cubic element, all accurate solutions can be only found at node. It means no accurate and reliable structural safety can be ensured within an element, and as a result, it hinders the engineering applications. The results of the paper are verified using analytical stability function studies, as well as with numerical results reported by independent researchers on several simple frames.

Flight-training effect on the cervical muscle isometric strength of trainee pilots

External stimulus/loading initiates adaptations within skeletal muscle. It has been previously found that the cervical area has the highest loading while performing flying maneuvers under +Gz. The first purpose of this study was to examine the neck muscle response to the physical environment associated with flight training, incorporating limited exposure to +Gz force, in a Pilatus PC-9 aircraft. The second purpose was to examine the short-term range of movement (ROM) response to flight training. Isometric cervical muscle strength and ROM was monitored in 9 RAAF pilots completing an 8-mo flight-training course at Pearce Airbase in Western Australia, and in 10 controls matched for gender, age, height, and weight. Isometric cervical muscle strength and ROM were measured at baseline and at 8 mo using the multi-cervical rehabilitation unit (Hanoun Medical, Downsview, Ontario, Canada). Results indicated that an increase in pilot neck strength was limited to flexion while in a neutral position. No strength changes were recorded in any other site in the pilots or for the controls. These findings suggest that short-term exposure to the physical environment associated with flight training had a limited significant effect on increasing isometric cervical muscle strength. No significant changes were observed in pilot ROM, indicating that short-term exposure to flight does not effect ROM.

Non-destructive structural integrity assessment of a decommissioned rail wagon system

This paper presents a multi-criteria based approach for nondestructive diagnostic structural integrity assessment of a decommissioned flatbed rail wagon (FBRW) used for road bridge superstructure rehabilitation and replacement applications. First, full-scale vibration and static test data sets are employed in a FE model of the FBRW to obtain the best ‘initial’ estimate of the model parameters. Second, the ‘final’ model parameters are predicted using sensitivity-based perturbation analysis without significant difficulties encountered. Consequently, the updated FBRW model is validated using the independent sets of full-scale laboratory static test data. Finally, the updated and validated FE model of the FBRW is used for structural integrity assessment of a single lane FBRW bridge subjected to the Australian bridge design traffic load.

Characterization of alluvial formation by stochastic modelling of paleo-fluvial processes: The concept and method

Modelling fluvial processes is an effective way to reproduce basin evolution and to recreate riverbed morphology. However, due to the complexity of alluvial environments, deterministic modelling of fluvial processes is often impossible. To address the related uncertainties, we derive a stochastic fluvial process model on the basis of the convective Exner equation that uses the statistics (mean and variance) of river velocity as input parameters. These statistics allow for quantifying the uncertainty in riverbed topography, river discharge and position of the river channel. In order to couple the velocity statistics and the fluvial process model, the perturbation method is employed with a non-stationary spectral approach to develop the Exner equation as two separate equations: the first one is the mean equation, which yields the mean sediment thickness, and the second one is the perturbation equation, which yields the variance of sediment thickness. The resulting solutions offer an effective tool to characterize alluvial aquifers resulting from fluvial processes, which allows incorporating the stochasticity of the paleoflow velocity.