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Three-Dimensional Finite Element Analysis
Published in Özlem Özgün, Mustafa Kuzuoğlu, ®-based Finite Element Programming in Electromagnetic Modeling, 2018
The concept of mesh quality was discussed in Chapter 5 for triangular elements, and it was observed that if the elements are close to equilateral triangles, the generated mesh is of high quality. There are also various quality measures for tetrahedral elements in the literature [4–6]. The ideal tetrahedron is the one whose four faces are all equilateral triangles, so a high-quality mesh is obtained if the generated tetrahedral elements do not deviate much from the ideal tetrahedron. One of the quality measure approaches is to compare the radii of circumscribed and inscribed spheres of a given tetrahedron. The circumscribed sphere (or circumsphere) of a tetrahedron is the smallest sphere that passes through all four vertices of the tetrahedron. The inscribed sphere (or insphere) is the largest sphere lying inside the tetrahedron and tangent to the faces of the tetrahedron. The radii of circumsphere and insphere (denoted by rcs and ris, respectively) can be computed by using the following formulas.
Construction of rhombic triacontahedron discrete global grid systems
Published in International Journal of Digital Earth, 2022
Xiaoyu Liang, Jin Ben, Rui Wang, Qishuang Liang, Xinhai Huang, Junjie Ding
The points belonging to the sphere can be determined by constructing the unit vector of each side using Snyder Polyhedral Projection because platonic solids are equipped with a circumscribed sphere that helps the vertices of the platonic solid translate directly to the sphere. Because the RT has no circumscribed sphere, the translation of polyhedral vertices directly to the sphere cannot be realized as in platonic solids. According to the spatial geometric relationship analysis, the distance between the vertex of the RT and its geometric center satisfies the multiple of or . Through these two parameters, the vertex of the spherical RT can be transformed into the corresponding vertex of the RT. Then, the rhombus to which one sphere point belongs can be determined by constructing the direction vector of RT on each side. The RTEA projection can be achieved through the above operations.
Computational Study of Hypersonic Rarefied Gas Flow over Re-Entry Vehicles Using the Second-Order Boltzmann-Curtiss Constitutive Model
Published in International Journal of Computational Fluid Dynamics, 2021
Tushar Chourushi, Satyvir Singh, Vishnu Asokakumar Sreekala, Rho Shin Myong
where is the speed of sound at an elemental interface, and the superscripts (+) and denote the left and right states of the element interface. Then, the substitution of Eq. (29) in Eq. (21) generates the weak formulation of the mixed form, Assembling all the elemental contributions yields a system of semi-discrete ordinary differential equations in time for each element, where M and refer to the elemental orthogonal mass matrix and the residual vector of the system of equations, respectively. These explicit systems of equations (31) are then solved using the three-stage, third-order accurate, strong stability preserving Runge–Kutta method (TVD-RK) proposed by Shu and Osher (Cockburn and Shu 1998), Here the term refers to the local time step for each element and is determined using the following relation, where CFL, h, and are the Courant–Friedrichs-Lewy number, the radius of the circumscribed sphere in the tetrahedral element , and the dimension of the element, respectively (Singh 2018; Singh and Battiato 2020, 2021b).
Optimal junction localization minimizing maximum miners’ evacuation distance in underground mining network
Published in Mining Technology, 2023
Zhixuan Shao, Maximilien Meyrieux, Mustafa Kumral
In accordance with Lemma, a function for the sphere having four given points on its boundary (i.e. the four-point circumscribed sphere) is required. This function is trivial to obtain, which implies the fact that the recursion is terminated. Effectively, the function is built with the set of formulas presented in Weisstein (1999).