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Published in Kok Keong Choong, Mustafasanie M. Yussof, Jat Yuen Richard Liew, Recent Advances in Analysis, Design and Construction of Shell and Spatial Structures in the Asia-Pacific Region, 2019
Catenoid is a minimal surface which is spanned between parallel circles as the fixed boundaries. When the distance between boundaries is less than 1.325487 times the radius of the boundary circle, there are minimal and maximal surfaces. Confirming that the minimal area surface can be obtained even if the initial shape is far apart, the tunnelling step starts from the neighbourhood of the maximum solution. The boundary and the initial shape are shown in Figure 11.14. The radius of the boundary circle is 5 and the length between the boundaries is 6. The initial area is 177.8. The initial shape is divided by 24 circumferentially and by 10 vertically using triangular mesh. The control parameters are α = 0.97, λ = 1.0, r = 1.0.
Principle of constant stress in analytical form-finding for durable structural design
Published in Alphose Zingoni, Current Perspectives and New Directions in Mechanics, Modelling and Design of Structural Systems, 2022
It is known that constant stress (minimal) surface structures do not exist between any boundaries; in the case of a catenoid surface, its attainable height depends on the ratio of the larger to smaller radii of the boundaries. This has been demonstrated analytically by Alexander & Macho, 2020, and earlier found by Frei Otto through experimentation with soap-film models, helping him to determine the maximum attainable height of the IL Tentg (Figure 1). The structure is now a listed building.
Kinetic model for competitive condensation of vapor between concave and convex surfaces in a soot aggregate
Published in Aerosol Science and Technology, 2020
E. V. Ivanova, A. F. Khalizov, G. Y. Gor
Figure 3 shows the three model geometries for the meniscus at the vapor–liquid interface. The simplest representation is the cylindrical meniscus, which can be easily described in analytical form, and thus further referred to as analytical approximation. However, when we assume that the droplet has a simple rectangular cylinder shape (Figure 3a), this geometry does not take into account the nearly zero contact angle at the solid-liquid interface, which should correspond to the perfect wetting typical of organic liquids on soot (Chen et al. 2018). A more realistic meniscus geometry with the correct wetting angle can be achieved assuming that the droplet is a globoid, a body formed by the rotation of an arc of a circle around the line connecting the centers of the monomer spheres, as shown in Figure 3b (Rose 1958). However, the curvature of a globoid is not constant throughout its surface, so it does not represent an equilibrium configuration with the minimal surface energy. Therefore, we also introduce the meniscus in the shape of a catenoid, a body arising by rotating a catenary curve about an axis, as a more physical representation which has a constant curvature κ at any point (Figure 3c). For the catenoid, κ is a combination of two varying curvatures related to the two radii: