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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.
Principle of corresponding states for hard polyhedron fluids
Published in Molecular Physics, 2019
We next compare our simulated data with Equation (7). Additionally, we compare the derived equation of state (Equation (7)) with the predictions of both the virial (Equation (1)) and free volume (Equation (2)) equations where there are reported sets of fitted parameters. Relevant virial coefficients used for the evaluation of Equations (1) and (2) are reported in the Appendix and adapted from Irrgang et al. [35]. Figure 2(a–d) compare predictions using the tabulated virial, free volume, and analytical equation – shown in red, green and black lines, respectively – with computed equation of state data from simulation (scattered data). The moment of inertia and volume of each polyhedron are computed using an implementation of the algorithm proposed by Mirtich [41]. The inscribed sphere radius is determined by growing a sphere positioned at the centre of mass of the polyhedron until it first contacts the particle's surface. The sphere's volume and moment of inertia are then determined using the standard formulas.
Brownian dynamics simulations of a cubic haematite particle suspension with a more effective treatment of steric layer interactions
Published in Molecular Physics, 2020
Here, we describe the sphere-connected model, shown in Figure 1(c), for treating the repulsive interaction due to the overlap between the steric layers that uniformly coat the cubic particles. In this model, the cube-like particle with side length d employs three different size spherical particles with diameters denoted as dcenter, dedge and dcorner. An inscribed sphere of diameter dcenter(=d) is initially placed at the centre of the cube and then 12 spheres of diameter dedge are arranged so that each sphere is in contact with the inscribed sphere and also with the two faces which constitute the intersection line of the side of the cube. Finally, in order to represent the corners of the cube, 8 spheres with diameter dcorner are then arranged in contact with the inscribed sphere and also with the three faces which constitute the corner of the cube. As it is assumed each spherical particle is coated by a steric layer with thickness δ, then the steric layer model shown in Figure 1(c) can be employed for evaluating the repulsive interaction energy or force between the two cubic particles coated by a uniform steric layer in the manner shown in Figure 1(a). Each diameter dcenter, dedge and dcorner can be straightforwardly evaluated from a geometric calculation and are expressed as The above-mentioned steric layer model has a weakness in that the sides and the corners of the cube are not described with sufficient accuracy, although in this approximate model a face-to-face steric layer interaction is expressed with a relatively higher accuracy. However, for the suspension of magnetic cube-like particles addressed in the present study, it was clearly shown from Monte Carlo simulations [15] where solid particles without a steric layer were addressed, that particle aggregates with face-to-face contact are predominantly formed in the system. Hence, the above model treatment for the steric interaction is expected to give rise to physically reasonable results with sufficient accuracy.