Chapter 75: Architectural fabrics

Thin summer fabrics without anti-UV finishing were used as experimental samples and the influences of fabric thickness, tightness, structure and fiber on the anti-UV properties were observed carefully. The measured performance parameters were normalized, to make UV protection properties of different thickness, tightness, fabric structures and fiber materials are comparable. This investigation revealed that the dominant factors affecting fabric anti-ultraviolet property were fabric structure, thickness and fiber type. The anti-UV performance from strong to weak is satin, twill and plain weave in terms of fabric structure; and polyester, silk, hemp and cotton in terms of fiber material. In addition, the anti-UV performances of fabrics positively increase with their thickness, when the other parameters were kept the same.

This paper is an exploration of the numeric and visual properties of a square numerical grid containing 81 elements (“J9 Space”). J9 Space has been derived from the standard multiplication grid through the calculation of digital roots. Nested within J9 Space are eight smaller square grids. A three-dimensional representation of J9 Space has been created from the two-dimensional numerical grid, displaying the highly symmetrical properties contained within it. The three-dimensional representation is fundamentally a collection of two-dimensional triangular polygons that create a surface in three-dimensional space. J9 Space appears to be a fundamental part of the framework of the number system and has some interesting mathematical features. This paper also summarises the recording of a three-dimensional printed model of J9 Space as a denisyuk reflection hologram. This is an artistic impression of the model using a volume holographic technique. The author who has contributed the mathematical concept (Prashant Jadav) suggests that the use of digital roots to develop three-dimensional models from well-known mathematical tools is a new method by which to create and visualise structures within the number system.

We present the Operator design pattern which can be used for the design of both sequential and data parallel applications. To reach this goal, we show how the participants of this pattern can be implemented either for a sequential or a parallel execution. Besides, reusing the sequential design for a parallel application decreases the cost of parallelization by allowing the maintenance of a unique application for the two execution environments. The proposed approach for parallel programming does not require any dedicated compiler or code pre-processing. Nothing but object oriented features, such as inheritance and polymorphism, is used to provide the distributed behaviour of the parallel participants of the pattern. The Operator pattern can help to solve many design issues in relation with the development of reusable software components for data collection processing. Moreover, we are confident that many programmers who want to migrate their applications towards parallelism can find it helpful.

Lower envelopes are fundamental structures in computational geometry, which have many applications, such as computing general Voronoi diagrams and performing hidden surface removal in computer graphics. We present a generic, robust and efficient implementation of the divide-and-conquer algorithm for computing the envelopes of surfaces in IR3. To the best of our knowledge, this is the first exact implementation that computes envelopes in three-dimensional space. Our implementation is based on CGAL (the Computational Geometry Algorithms Library) and is designated as a CGAL package. The separation of topology and geometry in our solution allows for the reuse of the algorithm with different families of surfaces, provided that a small set of geometric objects and operations on them is supplied. We used our algorithm to compute the lower and upper envelope for several types of surfaces. Exact arithmetic is typically slower than floating-point arithmetic, especially when higher order surfaces are involved. Since our implementation follows the exact geometric computation paradigm, we minimize the number of geometric operations, and by that significantly improve the performance of the algorithm in practice. Our experiments show interesting phenomena in the behavior of the divide-and-conquer algorithm and the combinatorics of lower envelopes of random surfaces.

This paper focuses on three-dimensional grasping and manipulation by robotic fingers with soft fingertips. The authors have proposed two-dimensional parallel-distributed model of a soft fingertip to describe the dependency of its potential energy to the relative orientation between the fingertip and the object. Here we extend the previous two-dimensional model to three-dimensional model, incorporating the rotation in three-dimensional space. We formulate the elastic potential energy stored in a soft fingertip due to the contact with the planar surface in three-dimensional space. We sketch the Lagrangian of the system consisting of a rigid prism grasped by three fingers with soft fingertips.

Particle modeling of liquid drop formation on a solid surface in 3-D

Bathymetric mapping is an important tool for reservoir management, typically completed before reservoir construction. Historically, bathymetric maps were produced by interpolating between points measured at a relatively large spacing throughout a reservoir, typically on the order of a few, up to 10, meters or more depending on the size of the reservoir. These measurements were made using traditional survey methods before the reservoir was filled, or using sonar surveys after filling. Post-construction issues such as sedimentation and erosion can change a reservoir, but generating updated bathymetric maps is difficult as the areas of interest are typically in the sediment deltas and other difficult-to-access areas that are often above water or exposed for part of the year. We present a method to create complete reservoir bathymetric maps, including areas above the water line, using small unmanned aerial vehicle (sUAV) photogrammetry combined with multi-beam sonar data—both established methods for producing topographic models. This is a unique problem because the shoreline topographic models generated by the photogrammetry are long and thin, not an optimal geometry for model creation, and most images contain water, which causes issues with image-matching algorithms. This paper presents methods to create accurate above-water shoreline models using images from sUAVs, processed using a commercial software package and a method to accurately knit sonar and Structure from Motion (SfM) data sets by matching slopes. The models generated by both approaches are point clouds, which consist of points representing the ground surface in three-dimensional space. Generating models from sUAV-captured images requires ground control points (GCPs), i.e., points with a known location, to anchor model creation. For this study, we explored issues with ground control spacing, masking water regions (or omitting water regions) in the images, using no GCPs, and incorrectly tagging a GCP. To quantify the effect these issues had on model accuracy, we computed the difference between generated clouds and a reference point cloud to determine the point cloud error. We found that the time required to place GCPs was significantly more than the time required to capture images, so optimizing GCP density is important. To generate long, thin shoreline models, we found that GCPs with a ~1.5-km (~1-mile) spacing along a shoreline are sufficient to generate useful data. This spacing resulted in an average error of 5.5 cm compared to a reference cloud that was generated using ~0.5-km (~1/4-mile) GCP spacing. We found that we needed to mask water and areas related to distant regions and sky in images used for model creation. This is because water, objects in the far oblique distance, and sky confuse the algorithms that match points among images. If we did not mask the images, the resulting models had errors of more than 20 m. Our sonar point clouds, while self-consistent, were not accurately georeferenced, which is typical for most reservoir surveys. We demonstrate a method using cross-sections of the transition between the above-water clouds and sonar clouds to geo-locate the sonar data and accurately knit the two data sets. Shore line topography models (long and thin) and integration of sonar and drone data is a niche area that leverages current advances in data collection and processing. Our work will help researchers and practitioners use these advances to generate accurate post-construction reservoir bathometry maps to assist with reservoir management.

In this paper we describe a fast strain-limiting method that allows stiff, incompliant materials to be simulated efficiently. Unlike prior approaches, which act on springs or individual strain components, this method acts on the strain tensors in a coordinate-invariant fashion allowing isotropic behavior. Our method applies to both two-and three-dimensional strains, and only requires computing the singular value decomposition of the deformation gradient, either a small 2x2 or 3x3 matrix, for each element. We demonstrate its use with triangular and tetrahedral linear-basis elements. For triangulated surfaces in three-dimensional space, we also describe a complementary edge-angle-limiting method to limit out-of-plane bending. All of the limits are enforced through an iterative, non-linear, Gauss-Seidel-like constraint procedure. To accelerate convergence, we propose a novel multi-resolution algorithm that enforces fitted limits at each level of a non-conforming hierarchy. Compared with other constraint-based techniques, our isotropic multi-resolution strain-limiting method is straightforward to implement, efficient to use, and applicable to a wide range of shell and solid materials.

The main subject of this chapter is surfaces in three-dimensional space. In preparation for this, we begin with a study of some special curves in the plane —the conic sections. In the last two sections, we will do some calculus with curves in space. Applications of calculus to surfaces are given in Chapters 15 and 16.

Разработана методика расчета и анализа статистических характеристик боковых пиков двумерных, авто- и взаимных корреляционных функций шумоподобных сложных сигналов, определенных как поверхности в трехмерном пространстве с координатами, соответствующими частоте и времени. Суть и новизна методики состоит в установлении для любой двумерной корреляционной функции сигнала взаимосвязи значений ее боковых пиков, а также параметров их функции распределения со спектральной плотностью мощности исходного сигнала, рассчитанной по его реализации конечной длительности, и, в конечном итоге, с характеристиками боковых пиков различных одномерных автокорреляционных функций псевдослучайной последовательности, на основе которой он сформирован. Результатом данного подхода являются установленные количественные взаимосвязи статистических характеристик боковых пиков двумерной корреляционной функции сложного сигнала с длиной псевдослучайной последовательности, соответствующей ему в зависимости от ее типа. A technique has been developed for calculating and analyzing the statistical characteristics of the side peaks of two-dimensional and auto- and cross-correlation functions of spread spectrum signals, defined as surfaces in threedimensional space with coordinates corresponding to frequency and time. The essence and novelty of the technique is to establish for any two-dimensional correlation function of a signal the relationship between the values of its side peaks, as well as the parameters of their distribution function, with the power spectral density of the original signal, calculated from its implementation of finite duration, and, ultimately, with the characteristics of the side peaks of various one-dimensional autocorrelation functions of the pseudo-random sequence on the basis of which it is formed. The result of this approach is the established quantitative relationships between the statistical characteristics of the side peaks of the two-dimensional correlation function of a complex signal with the length of the pseudo-random sequence corresponding to it, depending on its type.

The study of geometric flows for smoothing, multiscale representation, and analysis of two- and three-dimensional objects has received much attention in the past few years. In this paper, we first survey the geometric smoothing of curves and surfaces via geometric heat-type flows, which are invariant under the groups of Euclidean and affine motions. Second, using the general theory of differential invariants, we determine the general formula for a geometric hypersurface evolution which is invariant under a prescribed symmetry group. As an application, we present the simplest affine invariant flow for (convex) surfaces in three-dimensional space, which, like the affine-invariant curve shortening flow, will be of fundamental importance in the processing of three-dimensional images.

<p>A new simple approach inspired by MacAyeal (1979) to explain the time-asymmetric ‘saw-toothed’ shape and 100,000-year quasi-period of glacial-interglacial cycles since the Middle Pleistocene Transition, is presented. Using a simple model with fast-slow dynamics, the global ice volume is taken to be a function of two independently varying parameters, the solar insolation and ‘alpha’, a secondary control parameter, the study of which is the focus this research. The steady state of the model is a partially folded surface in three-dimensional space where insolation, ‘alpha’, and global ice volume are orthogonal axes. The pleated surface allows for the gradual increase and sudden decrease in ice volume that is observed in the paleoclimate record. To derive a time series of global ice volume, the Euler integration method is used, producing a time series which replicates the ‘saw-toothed’ pattern of glacial cycles in the late Pleistocene. The second control parameter, ‘alpha’, is proposed to be related to internal dynamics of the climate system, such as ice sheet dynamics.</p><p> </p><p>Reference</p><p>D. R.  MacAyeal, ‘A Catastrophe Model of the Paleoclimate Record’ , Journal of Glaciology , Volume 24 , Issue 90 , 1979 , pp. 245 – 257.</p>

Photonic crystals are artificial structures with periodic geometries that manipulate light at the wavelength scale. Recently, they have garnered significant attention in the field of topological photonics, where advanced band manipulation is explored to achieve breakthroughs in topological device applications. Conventional photonic crystals predominantly rely on a single spatial domain, characterized by spatially distributed dielectric modulation or metallic resonance. However, the degrees of freedom offered by spatial domains are often overlooked. Here, we theoretically propose and experimentally demonstrate photonic metacrystals that manipulate electromagnetic waves via minimal surfaces in three-dimensional space. These minimal surfaces partition the three-dimensional space into two complementary yet dual subspaces, each supporting a chiral topological semimetal. This spatial duality induces double-band degeneracy, analogous to the well-known time-reversal Kramers degeneracy, which has long been sought in classical systems. Previous proposals either relied on unrealizable strict symmetries or achieved double degeneracy only along high-symmetry lines. In contrast, we demonstrate that our double degeneracy exists globally, akin to the intrinsic spin degree of freedom of electrons. Furthermore, we extend the zero-frequency sixfold degeneracy of Maxwell's equations to a nonzero frequency range, which significantly facilitates the characterization of the intrinsic topological properties. Our study provides new insights into the geometric foundations of topological physics.

Displacement and pressure transfer between fluid and structure meshes in fluidstructure interaction analysis is complicated substantially when the two meshes on the domain interface are different and therefore form two distinct surfaces in three-dimensional space. A new, approximately energy-conserving interfacing strategy with two distinct components - mapping and interpolation - is proposed. The mapping between meshes is based on the assumption of a common parametric-based description of the wetted surface. The interpolation strategy is based on the smoothing element analysis method developed to recover stresses in finite element analysis, with an additional term to impose an energy-conserving constraint. The method is evaluated numerically with several examples from linear hydrodynamics/hydroelasticity. The advantage of these examples is that the components can be evaluated separately, and the performance of the method can be examined more clearly. Numerical results show the method is quite promising.

In this chapter we will study the simplest examples of curved surfaces in three-dimensional space. Just as a curve can be defined by a locus equation f(x,y) = 0 relating two variables, so a surface can be defined by a locus equation f(x, y, z) = 0 relating three variables.