Cube Volume Research Articles

Structure-Preserving Algorithms for the Lorenz System

Based on a splitting method and a composition method, we construct some structure-preserving algorithms with first-order, second-order and fourth-order accuracy for a Lorenz system. By using the Liouville's formula, it is proven that the structure-preserving algorithms exactly preserve the volume of infinitesimal cube for the Lorenz system. Numerical experimental results illustrate that for the conservative Lorenz system, the qualitative behaviour of the trajectories described by the classical explicit fourth-order Runge–Kutta (RK4) method and the fifth-order Runge–Kutta–Fehlberg (RKF45) method is wrong, while the qualitative behaviour derived from the structure-preserving algorithms with different orders of accuracy is correct. Moreover, for the small dissipative Lorenz system, the norm errors of the structure-preserving algorithms in phase space are less than those of the Runge–Kutta methods.

A Brief History of Impossibility

Sooner or later every student of geometry learns of three impossible problems: 1. Trisecting the angle: Given an arbitrary angle, construct an angle exactly one-third as great. 2. Duplicating the cube: Given a cube of arbitrary volume, find a cube with exactly twice the volume. 3. Squaring the circle: Given an arbitrary circle, find a square with the same area. These problems originated around 430 BC at a time when Greek geometry was advanc ing rapidly. We might add a fourth problem: inscribing a regular heptagon in a circle. Within two centuries, all these problems had been solved (see [3, Vol. I, p. 218-270] and [1] for some of these solutions). So if these problems were all solved, why are they said to be impossible? The im possibility stems from a restriction, allegedly imposed by Plato (427-347 BC), that geometers use no instruments besides the compass and straightedge. This restriction requires further explanation. For that, we turn to Euclid (fl. 300 BC), who collected and systematized much of the plane geometry of the Greeks in his Elements. Euclid's goal was to develop geometry in a deductive manner from as few basic assumptions as possible. The first three postulates in the Elements are (in modernized form): 1. Between any two points, there exists a unique straight line. 2. A straight line may be extended indefinitely. 3. Given any point and any length, a circle may be constructed centered at the point with radius equal to the given length. These three postulates correspond to the allowable uses of compass and straightedge: to draw a line that passes through two given points; to extend a given line segment indefinitely; and to draw a circle about any given point with any given radius. To solve a problem using compass and straightedge means to use only these operations, repeated a finite number of times. The construction's validity can then be proven using only the postulates of Euclidean geometry.

A hybrid-assembled MEMS Fabry-Pérot wavelength tunable filter

We have developed a wavelength tunable fiber optic filter by using a hybrid-assembly technique of glass reflector cubes (volume 1mm3) onto a bulk-micromachined electrostatic translation stage (4mm×5mm). More than 0.6-μm-change of the Fabry-Pérot interferometer gap could be electrostatically controlled with a tuning voltage upwards of 4V with the aid of a DC bias voltage of 64V. A wide tuning range from 1479 to 1590nm was achieved with a bandwidth of 1.3–1.2nm and insertion loss of 1.59–1.76dB.

Analysis of Pore Network in Three-dimensional (3D) Grain Bulks Using X-ray CT Images

The knowledge of distribution of pore space inside grain bulks is essential for determining the airflow resistance of grains. In this study, the internal pore structure and the 3D-distribution of air paths inside grain bulks were studied using X-ray computed tomography images. Image analysis methods were applied to the binary 3D X-ray CT images on the spatial distribution of voids to generate the connected, individualized pore objects of different size and shapes. Morphometric parameters, such as 3D air path volume distribution, structure separation factor, Euler number, fragmentation index, and structure model index were calculated based on hexahedral marching cubes volume model and marching cubes 3D surface construction algorithm. The quantified numerical measures of spatial integrity of air path networks were analyzed and compared with the airflow resistance of grain bulks. The results showed that the connectivity of airspace and the nonuniform distribution of air-path network inside grain bulks were responsible for the difference in airflow resistance between horizontal and vertical directions to the airflow of grain bulks.

Ceramic components manufacturing by selective laser sintering

In the present paper, technology of selective laser sintering/melting is applied to manufacture net shaped objects from pure yttria–zirconia powders. Experiments are carried out on Phenix Systems PM100 machine with 50 W fibre laser. Powder is spread by a roller over the surface of 100 mm diameter alumina cylinder. Design of experiments is applied to identify influent process parameters (powder characteristics, powder layering and laser manufacturing strategy) to obtain high-quality ceramic components (density and micro-structure). The influence of the yttria–zirconia particle size and morphology onto powder layering process is analysed. The influence of the powder layer thickness on laser sintering/melting is studied for different laser beam velocity V ( V = 1250–2000 mm/s), defocalisation (−6 to 12 mm), distance between two neighbour melted lines (so-called “vectors”) (20–40 μm), vector length and temperature in the furnace. The powder bed density before laser sintering/melting also has significant influence on the manufactured samples density. Different manufacturing strategies are applied and compared: (a) different laser beam scanning paths to fill the sliced surfaces of the manufactured object, (b) variation of vector length (c) different strategies of powder layering, (d) temperature in the furnace and (e) post heat treatment in conventional furnace. Performance and limitations of different strategies are analysed applying the following criteria: geometrical accuracy of the manufactured samples, porosity. The process stability is proved by fabrication of 1 cm 3 volume cube.

Unsaturated flow and solute transport through the Chalk: Tracer test and dual permeability modelling

A tracer test was carried out in the unsaturated Chalk at the Fleam Dyke research site in Cambridgeshire, UK, to investigate the role of the Chalk fractures and matrix in unsaturated flow and solute transport. The experiment, under natural rainfall conditions, involved distributing deuterated water on a grass-covered lysimeter (a cube of volume 125 m3) and on an adjacent 4 m × 4 m field plot. Tracer migration was monitored through regular core sampling and collection of lysimeter drainage water. The presence of occasional secondary peaks in sampling of the vertical tracer profile suggested the occurrence of fracture flow, allowing some tracer to bypass the Chalk matrix. However, in the 15 months following application, none of the tracer was detected in the lysimeter drainage at 5 m depth. Modelling of the tracer results was undertaken with the 1-D numerical transient dual permeability model MACRO 5.0, initially developed for macroporous soils. Modelling results showed that MACRO 5.0 could reliably simulate transient recharge through the Chalk. The simulations suggested that fracture flow is important at the site, but that it is only initiated at 1 m depth or deeper. The extent of fracture flow appeared to be highly variable in different layers of the profile, varying between 40% and 85% of the cumulative flux, mainly depending on the saturated hydraulic conductivity of the matrix. Diffusion between the fractures and the matrix tended to equalize solute concentrations in both flow domains, although solute bypass through the fractures occurred in some Chalk strata. Besides diffusive exchange, the modelling stressed the importance of advective exchange of solutes. The results suggest that the Chalk aquifer at the Fleam Dyke site is only moderately vulnerable to pollution, even though for moderate rainfall conditions some bypass flow was possible.

Gaussian Limits for Multidimensional Random Sequential Packing at Saturation

Consider the random sequential packing model with infinite input and in any dimension. When the input consists of non-zero volume convex solids we show that the total number of solids accepted over cubes of volume $\lambda$ is asymptotically normal as $\lambda \to \infty$. We provide a rate of approximation to the normal and show that the finite dimensional distributions of the packing measures converge to those of a mean zero generalized Gaussian field. The method of proof involves showing that the collection of accepted solids satisfies the weak spatial dependence condition known as stabilization.

Effect of processing variables on cube texture formation in powder metallurgically prepared Ni and Ni–W alloy tapes for use as substrates for coated conductor applications

Development of textures have been studied in powder metallurgically prepared pure Ni and Ni–5 at.% W alloy after heavy cold rolling (∼95% reduction in thickness) and annealing for use as substrates for coated superconductor applications. The cold rolling textures of the two materials do not show any significant difference. However, Ni–5 at.% W alloy is found to develop a much sharper cube texture after annealing which indicate the beneficial role of W in increasing the cube volume fraction in Ni. W increases the volume fraction of the cube component by decreasing the volume fraction of the rolling direction (RD)-rotated cube grains. Partial recrystallization studies clearly indicate that in contrast to pure Ni sharp cube oriented grains form in Ni–5 at.% W alloy right from the beginning of recrystallization. A near 100% cube texture is obtained in Ni–5 at.% W alloy by an optimized two stage rolling (TSR) procedure followed by annealing at 1350 °C.

Rehydration of Apple Dried by Different Methods

Apple var. Idared cut into cubes was dried by convection and freeze-dried. Dried cubes were equilibrated to variable water activities by storage over saturated salt solutions (range 0–0.33). Rehydration was done for one hour in water at 20°C. Increase in both mass and volume of cubes was followed during rehydration. Changes in height of a cube were recorded using a minimal displacement gauge. Electron transmission microscopy was used to measure cell wall thickness in raw and rehydrated material. Modes of drying affected the behavior of apple cubes upon rehydration. Convective dried apple imbibed water and swelled during rehydration, and a collapse of structure was observed at the beginning of the wetting process. Analysis of both mass and volume increase showed that at the beginning of rehydration capillaries and pores were filled with water and thereafter swelling of biopolymers occurred. Cell walls imbibed water, but their thickness after one hour of rehydration was half of that of raw apple. Freeze-dried apple increased in mass much faster than these dried by convection. Rehydration caused collapse of structure, which was not rebuilt during further wetting. On the other hand, cell walls swelled significantly during rehydration and reached thickness almost that of raw apple. It was inferred that freezing injured tissue, which after drying contained numerous discontinuities. Hence, swelling of domains, which did not form a continuous structure, was not propagated and expressed by the increase in volume on a macroscopic scale. Initial water activity of both investigated materials affected collapse of structure during rehydration. The higher the water activity, the faster the collapse.

Visualisation of the numerical solution of partial differential equation systems in three space dimensions and its importance for mathematical models in biology

Numerical analysis and computational simulation of partial differential equation models in mathematical biology are now an integral part of the research in this field. Increasingly we are seeing the development of partial differential equation models in more than one space dimension, and it is therefore necessary to generate a clear and effective visualisation platform between the mathematicians and biologists to communicate the results. The mathematical extension of models to three spatial dimensions from one or two is often a trivial task, whereas the visualisation of the results is more complicated. The scope of this paper is to apply the established marching cubes volume rendering technique to the study of solid tumour growth and invasion, and present an adaptation of the algorithm to speed up the surface rendering from numerical simulation data. As a specific example, in this paper we examine the computational solutions arising from numerical simulation results of a mathematical model of malignant solid tumour growth and invasion in an irregular heterogeneous three-dimensional domain, i.e., the female breast. Due to the different variables that interact with each other, more than one data set may have to be displayed simultaneously, which can be realized through transparency blending. The usefulness of the proposed method for visualisation in a more general context will also be discussed.

Cube Texture Development in Pure Ni during Annealing in a High Strength Magnetic Field

A 10 Tesla magnetic field was applied during annealing of cold-rolled high purity Ni at 300°C for 2 hours, with samples aligned at various different angles to the magnetic field direction. The effect of the magnetic field annealing on the cube texture evolution and the microstructural development was investigated by use of electron backscatter pattern (EBSP) analysis in the scanning electron microscope. The results show that both the cube texture evolution and the grain size are affected by the application during annealing of a high strength magnetic field, and that the effect varies as a function of the angle between the sample normal and the magnetic field direction. The cube texture volume fraction resulting from 2 hours annealing increases with increasing angle between the field direction and the sample normal direction, though in nearly all cases lower cube volume fractions were obtained compared to annealing without a magnetic field. The average size for all grains and for just cube-oriented grains both decrease with decreasing cube texture fraction, suggesting that the presence of a magnetic field either leads to enhanced recrystallization nucleation, or to a retardation of grain growth after recrystallization.

Monte Carlo Simulation of Cube-Texture Evolution during Grain Growth of High-Purity Nickel

A Monte Carlo Potts model has been used to investigate cube-texture strengthening during grain growth in rolled high-purity Ni-tapes. The initial conditions for the simulations have been taken from electron back-scatter pattern (EBSP) orientation maps of already fully recrystallized samples. Experimentally, grain growth leads to an increase in the cube volume fraction to >95% , accompanied by an approximately ten-fold increase in the grain size. High cube volume fractions can be predicted under a number of conditions, though a small surface energy advantage of just 2% for cube-oriented grains is required to match the texture strengthening to the grain size change. An additional issue of interest is the influence on the grain growth of the large area-fraction of twin boundaries in the fully recrystallized condition. The presence of boundaries with low energy has a strong influence on the simulated microstructural evolution.

Reducing Cube Volume in Recrystallized Texture by Controlling the Finishing Temperature and Rolling Strain in Al-5%Mg Alloy

Reducing Cube Volume in Recrystallized Texture by Controlling the Finishing Temperature and Rolling Strain in Al-5%Mg Alloy

The Expected Volume of a Tetrahedron whose Vertices are Chosen at Random in the Interior of a Cube

We calculate E[V 4(C)], the expected volume of a tetrahedron whose vertices are chosen randomly (i.e. independently and uniformly) in the interior of C, a cube of unit volume. We find $$E[V_4 {\rm (cube)}]={3977\over 216000} - {\pi^2 \over 2160}=0.01384277574\ldots$$ The result is in convincing agreement with a simulation of 3000 · 106 trials.

Monte Carlo simulation of a 1,4,7,10-tetraazacyclododecane/lithium complex in aqueous solution.

Monte Carlo simulations have been carried out for the system consisting of a 1,4,7,10-tetraazacyclododecane (cyclen)-lithium complex in 201 water molecules. The volume of the periodic cube was calculated using the experimental density of pure water at 298 K and 1 atm of 1 g.cm(-)(3), plus additional space occupied by the complex. The geometry of the complex is the alternated form, where the ion is located at the center of the cyclen. The complex-water interaction was represented by the cyclen-water and lithium-water pair potentials, both of which were developed on the basis of ab initio calculations. The results show two layers of solvation shells consisting of 2 and 6.9 water molecules. Two water molecules in the first solvation shell (O(1) and O(2)) bind directly to the ion in which the ion-oxygen distance is 2.38 A, the dipole vector points to the ion, and rotation takes place around the ion-oxygen axis. In the next layer, 4 water molecules coordinate simultaneously to the first 2 water molecules in the first shell and the NH functional groups of cyclen. The remaining 2.9 water molecules in the second layer are also coordinated to be in the first half-hydration shell of O(1) and O(2).

Modelling of the Recrystallization Behaviour of AA5xxx Aluminum Alloys after Hot Deformation

The recrystallization behaviour (recrystallization kinetics, recrystallized grain size and cube texture development) of three commercially significant aluminum alloys (AA5182, AA5052 and AA5005) was modelled using a recrystallization model developed by Vatne et al. The Vatne recrystallization model was originally developed based on thorough experimental investigations of an AA1050 alloy and an AA3104 alloy and has been applied to model the behaviour of AA5xxx alloys in order to assess its applicability to other non-heat treatable aluminum alloy systems. In general, good agreement between model predictions and experimental data was obtained for the AA5xxx alloys studied. The Vatne microstructure model appears to capture some of the important physical mechanisms of recrystallization in hot deformed aluminum as it can reasonably predict the recrystallization kinetics, recrystallized grain size and cube volume fraction as a function of different hot deformation conditions for a variety of aluminum alloys. However, the Vatne model still contains some "tuning" constants that are alloy specific and the model predictions need to be further validated against experimental data especially for multistand scenarios.

Development and stability during high temperature annealing of the cube texture in rolled Ni substrate materials

The development and stability of the cube texture formed during annealing has been investigated, using rolled nickel sheets of 99.95% and 99.999% purity. Microstructure and texture were characterized using electron back-scatter pattern analysis in the scanning electron microscope. The texture following primary recrystallization in both materials showed cube volume fractions of up to 50% and 80% (99.95% and 99.999% Ni respectively). The average grain size of the cube textured grains during recrystallization was larger than for grains of other orientations. For both materials high temperature annealing resulted only in normal grain growth, leading to a significant strengthening of the cube texture. The results are discussed in terms of the relative effects of size, boundary mobility and boundary energy on the process of grain growth. For the 99.999% Ni material an annealing process using a continuous heating ramp to 1200 °C gave a further improvement in the cube texture.

Cube recrystallization textures in a hot deformed Al–Mg–Si alloy

The formation of cube recrystallization textures has been studied on Al–Mg–Si polycrystals deformed at 400 °C by channel die compression tests and annealed at 510 °C. A cube recrystallization texture was observed on samples presenting a very low fraction of cube volumes before and after deformation.

CHANGES OF RHEOLOGICAL PROPERTIES OF APPLE TISSUE UNDERGOING CONVECTIVE DRYING

ABSTRACT The aim of this work was to investigate rheological properties of apple cubes undergoing convective drying. Compression-relaxation test was used to follow changes of properties under investigation. Raw apple appeared to be very heterogeneous material from the rheological point of view. The relationship between stress and strain was concave downwards and the concavity increased until water content of 2·5 g/g d.m. was reached. Then the relationship begun to straighten and for dry material developed stress was linearly dependent on strain. The resistance of deformation decreased with decreasing water content. Analysis of relaxation showed that the rate of relaxation increased with decreasing water content and the unrelaxed stress was smaller the lower was the water content. Apple cubes undergoing drying are pictured as material composed of three compartments with different rheological properties. The outer layer is dry, inelastic and difficult to deform. Underneath is moist material with no turgor and relatively easy to compress, and the core with properties of raw apples. During drying the second layer grows in expense of the core and porous structure with many voids filled with air is formed. Share of each compartment in the volume of apple cube undergoing drying affects average rheological properties of the material.

Neutron diffraction study of the cube texture development in FeNi53% alloy

Crystallographic texture evolution during thermal annealing of the Fe-Ni53% alloy (94% rolled) has been characterized by “in situ” neutron diffraction. The texture components analysis, has allowed to follow the cube volume fraction component formation and its expansion to the wreckage of the deformation ones.