C. Macaskill

Iterative approach for the numerical simulation of scattering from one- and two-dimensional rough surfaces

We describe the use of iterative techniques for the solution of the integral equation that arises in an exact treatment of scalar wave scattering from randomly rough surfaces. The surfaces vary in either one or two dimensions, and the special case of a Dirichlet boundary condition is treated. It is found that these techniques, particularly when preconditioning is applied, are much more efficient than direct inversion techniques. Moreover, convergence is obtained for rms roughness of the order of 1, so the techniques have applicability over a wide parameter regime. Convergence is always to the exact solution found by direct inversion, exceptly for cases of extremely large-scaled rms surface heights in which the iterative techniques fail. In addition, by monitoring the residuals in the iteration process, it is immediately clear if the iterative techniques are failing, or performing badly in any given case. Finally, numerical results are compared with existing data in the enhanced backscattering regime.

Wetting-induced ignition in cellulosic materials

Abstract A mathematical model of the ignition of cellulose is described which includes both the dry oxidation reaction and also the water-mediated exothermic reaction which is probably hydrolysis. It also takes account of endothermic evaporation of water inside the porous material, exothermic condensation of water vapour and transport of water vapour by diffusion. Appropriate boundary conditions are chosen and the results reported here are restricted to one spatial dimension, but are time dependent. Experimental data obtained recently for bagasse (ground extracted sugar cane) are used as input for the model. However, it is quite likely that the results are of general interest for other cellulose-based materials, many of which are of considerable commercial importance and many of which are believed to undergo spontaneous ignition induced by the presence of water or water vapour. A significant number of experimental observations and conjectures made over a long period are shown to be predictable and interpretable by use of the model, and good agreement with experiment occurs where results are available.

Simulation of a Chain of Collapsible Contracting Lymphangions With Progressive Valve Closure

The aim of this investigation was to achieve the first step toward a comprehensive model of the lymphatic system. A numerical model has been constructed of a lymphatic vessel, consisting of a short series chain of contractile segments (lymphangions) and of intersegmental valves. The changing diameter of a segment governs the difference between the flows through inlet and outlet valves and is itself governed by a balance between transmural pressure and passive and active wall properties. The compliance of segments is maximal at intermediate diameters and decreases when the segments are subject to greatly positive or negative transmural pressure. Fluid flow is the result of time-varying active contraction causing diameter to reduce and is limited by segmental viscous and valvular resistance. The valves effect a smooth transition from low forward-flow resistance to high backflow resistance. Contraction occurs sequentially in successive lymphangions in the forward-flow direction. The behavior of chains of one to five lymphangions was investigated by means of pump function curves, with variation of valve opening parameters, maximum contractility, lymphangion size gradation, number of lymphangions, and phase delay between adjacent lymphangion contractions. The model was reasonably robust numerically, with mean flow-rate generally reducing as adverse pressure was increased. Sequential contraction was found to be much more efficient than synchronized contraction. At the highest adverse pressures, pumping failed by one of two mechanisms, depending on parameter settings: either mean leakback flow exceeded forward pumping or contraction failed to open the lymphangion outlet valve. Maximum pressure and maximum flow-rate were both sensitive to the contractile state; maximum pressure was also determined by the number of lymphangions in series. Maximum flow-rate was highly sensitive to the transmural pressure experienced by the most upstream lymphangions, suggesting that many feeding lymphatics would be needed to supply one downstream lymphangion chain pumping at optimal transmural pressure.

Incorporating measured valve properties into a numerical model of a lymphatic vessel.

An existing lumped-parameter model of multiple lymphangions (lymphatic vascular segments) in series is adapted for the incorporation of recent physiological measurements of lymphatic vascular properties. The new data show very marked nonlinearity of the passive pressure–diameter relation during distension, relative to comparable blood vessels, and complex valve behaviour. Since lymph is transported as a result of either the active contraction or the passive squeezing of vascular segments situated between two one-way valves, the performance of these valves is of primary importance. The valves display hysteresis (the opening and closing pressure drop thresholds differ), a bias to staying open (both state changes occur when the trans-valve pressure drop is adverse) and pressure-drop threshold dependence on transmural pressure. These properties, in combination with the strong nonlinearity that valve operation represents, have in turn caused intriguing numerical problems in the model, and we describe numerical stratagems by which we have overcome the problems. The principal problem is also generalised into a relatively simple mathematical example, for which solution detail is provided using two different solvers.

Development of a model of a multi-lymphangion lymphatic vessel incorporating realistic and measured parameter values

Our published model of a lymphatic vessel consisting of multiple actively contracting segments between non-return valves has been further developed by the incorporation of properties derived from observations and measurements of rat mesenteric vessels. These included (1) a refractory period between contractions, (2) a highly nonlinear form for the passive part of the pressure–diameter relationship, (3) hysteretic and transmural-pressure-dependent valve opening and closing pressure thresholds and (4) dependence of active tension on muscle length as reflected in local diameter. Experimentally, lymphatic valves are known to be biased to stay open. In consequence, in the improved model, vessel pumping of fluid suffers losses by regurgitation, and valve closure is dependent on backflow first causing an adverse valve pressure drop sufficient to reach the closure threshold. The assumed resistance of an open valve therefore becomes a critical parameter, and experiments to measure this quantity are reported here. However, incorporating this parameter value, along with other parameter values based on existing measurements, led to ineffective pumping. It is argued that the published measurements of valve-closing pressure threshold overestimate this quantity owing to neglect of micro-pipette resistance. An estimate is made of the extent of the possible resulting error. Correcting by this amount, the pumping performance is improved, but still very inefficient unless the open-valve resistance is also increased beyond the measured level. Arguments are given as to why this is justified, and other areas where experimental data are lacking are identified. The model is capable of future adaptation as new experimental data appear.

Self-heating and drying in two-dimensional bagasse piles

This paper describes a two-dimensional model for self-heating and changes in water levels in bagasse piles of constant rectangular or triangular cross section. (Bagasse is the residue, mainly cellulose, that remains after sugar has been extracted from sugar-cane.) After milling, the bagasse has almost 50% water by weight, as hot water is used to remove the last of the sugar. The bagasse can be used as fuel in electrical power stations, but needs to be dried out before use. This paper discusses the way in which the drying out of a pile depends on the ambient conditions, and the shape and size of the pile. Accordingly, the energy equation, and equations for liquid water, water vapour and oxygen are solved numerically using the method of lines. The equations include terms describing heat conduction, diffusion of water vapour and oxygen, condensation and evaporation and an Arrhenius self-heating term. In addition, recent measurements show that there is also self-heating due to the presence of water in the bagasse, with a maximum effect near 60 °C, which is modelled by a modified Arrhenius expression. The local maximum in the heat release curve for the problem leads to approximate steady-state behaviour on short time scales that eventually is lost as the pile dries out. This interesting physical behaviour motivates an approximate analytical model for the rate at which liquid water is reduced in the pile. Analytical and numerical results are presented for a variety of pile configurations and some fairly general conclusions are drawn.

ITERATIVE TECHNIQUES FOR ROUGH SURFACE SCATTERING PROBLEMS

Abstract The problem of wave scattering at a randomly rough surface gives rise to a non-Hermitian linear system, which corresponds to the discretization of a linear Fredholm integral equation of the first or second kind. In this paper, we describe the use of various iterative methods for the solution of the real linear non-symmetric system, corresponding to the complex system of the scattering problem, and the regions of convergence of the methods in terms of the basic parameters of the problem. It has been found that Generalized Conjugate Gradient method is the only approach valid for very rough surfaces, h l . 1.0 say (here h l is the ratio of the RMS surface height to surface correlation length). The regions of validity for the iterative methods in terms of convergence and energy conservation are also found and it is shown that this depends not only on h l and kl (here k is the wavenumber of the incident field) but also on the density of mesh points in the discretization. Some convergence properties are discussed by analyzing the spectral radius of the iterative matrix. It is found for small h l that the spectral radius is linearly proportional to h l and approximately proportional to √kl for large kl. It is also found numerically that when h l · √kl is greater than one, the iterative matrix for the linear integral equation of the second kind treated here has spectral radius greater than one, and therefore, the Generalized Conjugate Gradient method is not guaranteed to converge since the corresponding linear real system of the second kind may not have positive definite symmetric part. Finally, we present a comparison of iterative techniques for the linear integral equations of the first and second kinds, and find that when such iterative methods are to be employed, it is more efficient to use the second kind integral equation formulation than the first kind formulation.

Intensity fluctuations. Part II: Comparison with the Cobb experiment

The intensity fluctuations measured at 4 and 8 kHz in the Cobb experiment have been available for nearly 10 years and in that time have not successfully been predicted theoretically. We show that multiple scatter effects must be considered, and that neither the Born nor the Rytov approximation to the scattering formulation is appropriate. A companion paper [J. Acoust. Soc. Am. 74, 1474–14832  (1983)] provides the theoretical background for this work by presenting a general form of the analytical solution to the fourth moment equation in two dimensions for a point source transmission. We use the parameters of the Garrett–Munk model of the internal wave field appropriate to the Cobb experiment oceanographic regime to obtain the correlation functions of the acoustic refractive index field, and then predict the intensity fluctuations. We discuss corrections to the predicted spectrum that are due to fine structure effects and tidal motions. We include a discussion of the scattering parameters Γ and X, which ar...

The Shortley-Weller embedded finite-difference method for the 3D Poisson equation with mixed boundary conditions

This paper describes a method for the solution of the 3D Poisson equation, subject to mixed boundary conditions, on an irregularly shaped domain. A finite difference method is used, with the domain embedded in a rectangular grid. Quadratic treatment of the boundary conditions is shown to be necessary to obtain uniform error of O(@D^2). This contrasts with the Dirichlet case where both quadratic and linear treatments give O(@D^2) error, although the coefficient of error may be much larger for the linear case. Explicit error estimates demonstrating this behaviour are found for the 1D case with similar behaviour found in 2D and 3D numerical examples. Finally, the extension of this approach to the N-dimensional case is given, where N>3.

Acoustic intensity for a long vessel with noncircular cross section

The acoustic intensity distribution around and within long vessels of noncircular cross section was investigated for parameters typical of biomedical ultrasound and blood vessels. We have developed a collocation method for finding the acoustic field when a uniform plane wave is obliquely incident on a long, not necessarily cylindrical, impedance interface. Results are presented for vessels of noncircular cross section and for vessels with thick walls of nonuniform thickness. The intensity in the vicinity of the vessel, throughout the lumen, and in the wall, is calculated for intermediate length scales, i.e., vessel radius and wall thickness in the range 1 to 10 wavelengths. The intensity distribution is an interference pattern, with complicated regions of increased and decreased intensity. These results are compared with approximate intensity obtained using ray theory. Effects not predicted by ray theory and intensity variations that will be significant in any close ultrasonic investigation of these vessels are revealed.