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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
Membrane structures, in the form tensioned roofing forms, can be form-found using statistically prevalent load, such as pre-stress, and chosen boundaries/supports. When the chosen pre-stress is constant, the resulting structures take on natural forms, known as minimal surfaces (Otto & Rasch, 1995) that can be modelled using soap-film models (Hildebrand & Tromba, 1985).
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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
It is known that the uniform stress surface implies the least strain energy surface and corresponds to minimal area surface. The problem, which studies minimal surfaces spanned in boundaries, is called the Plateau problem (Hildebrant and Tromba, 1985). The problem is generally a multimodal nonlinear problem with some extremums. The sequential computation to minimize the surface area yields the coordinates of surfaces to the prescribed reducing area. This process leads to an optimization to minimize the surface area. Searching at any initial surfaces, the obtained minimal surface may be a local optimal solution and not always a global optimal solution. A numerical strategy is necessary to avoid capturing local minimal solutions and to obtain the least area surface.
Multivariate functionals
Published in Louis Komzsik, Applied Calculus of Variations for Engineers, 2019
From a differential geometry point-of-view, a minimal surface is a surface for which the mean curvature of the form κm=κ1+κ22 vanishes, where κ1 and κ2 are the principal curvatures. A subset of minimal surfaces are the surfaces of minimal area, and surfaces of minimal area passing through a closed space curve are minimal surfaces. Finding minimal surfaces is called the problem of Plateau.
Additively manufactured mechanical metamaterials based on triply periodic minimal surfaces: Performance, challenges, and application
Published in Mechanics of Advanced Materials and Structures, 2022
Deepak Sharma, Somashekhar S. Hiremath
TPMSs surfaces have zero mean curvature that means the sum of principal curvature at any point on the minimal surface is zero. A typical example of a minimal surface is the soap film, in which the surface tension tries to minimize its surface area at each point [34]. These minimal surfaces can be repeated with different periodicity and relative density to get the desirable mechanical properties. TPMS topologies are created using mathematical formulas, in (Eq. 1) when φ(r) ≤ 0 the region is solid, and φ(r) > 0 defines the empty region. The iso-surface is defined at φ(r) = 0, which means half volume solid and half volume void [35]. TPMS are further divided into balanced and unbalanced surfaces. The balanced surface divides the volume into similar regions. These regions may or may not mirror one another, whereas the unbalanced surface divides the space into labyrinths of unequal morphologies. The G and P are balanced surfaces, and I-WP (wrapped package) is the unbalanced surface. The example of balanced (G) and unbalanced (I-WP) are shown in Figure 2.
Flow and thermal transport characteristics of Triply-Periodic Minimal Surface (TPMS)-based gyroid and Schwarz-P cellular materials
Published in Numerical Heat Transfer, Part A: Applications, 2021
The strut-based regular cellular materials despite offering better structural and functional performance than stochastic metal foams may develop large stress concentrations at the strut junctions. An alternative to this issue can be a sheet-based cellular materials called triply-periodic minimal surfaces (TPMS) [18]. The TPMS are mathematically defined smooth and continuous surfaces that divide a given volume into two congruent regions. These are called minimal surfaces because they represent minimum possible area for any surface within specified boundaries. These surfaces also have zero-mean curvature i.e. sum of principal curvatures at each point is zero. TPMS is popular in biomedical sector for its usage in tissue engineering and orthopedic implants. TPMS closely mimic the structural and functional properties of bones. For tissue regeneration, the cellular scaffolds should avoid stress shielding, have enough porosity for nutrition and oxygen transport, have sufficient surface area and possess concave surfaces for promoting tissue growth. TPMS can be tailored to achieve the desired permeability and mechanical properties [21, 22]. The zero-mean curvature of TPMS is also similar to trabecular bone and has concave type surface tortuosity. The continuous sheet-like surface eliminates the high local stress concentration zones as well, making TPMS favorable for tissue engineering.
Performance-based design optimization for minimal surface based form
Published in Architectural Science Review, 2018
Minimal surfaces are able to depart from right angles being characteristic of Euclidean forms. Therefore, a minimal surface form was used because it can transform into curvilinear forms and thus create various spatial configurations, during the optimization process. Minimal surface is a branch of mathematics and exists in various forms (Figure 1). Minimal surfaces are surfaces that are formed by covering the smallest area within given boundaries in 3D space, and in mathematical terms, they are surfaces that minimize their areas on a local scale (Velimirović et al. 2008; Emmer 2013). In this work, while the curves basically create form, the minimal surface, which was connected with the curves, also shape the upper part of the form (roof). Thus, during the optimization process, the form occurs with curvilinear lines (i.e. more freely) instead of sharp lines. Thus, in this study, a model was created that allows the form to move more freely during the optimization process and can create more complex forms. The forms that were produced using the MSO model are referred to as complex forms in the inferences of this study.