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Vector and Tensor Calculus
Published in Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou, ViscousFluid Flow, 2021
Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou
A coordinate system in the three-dimensional space is defined by choosing a set of three linearly independent vectors, B={e1, e2, e3}, representing the three fundamental directions of the space. The set B is a basis of the three-dimensional space, i.e., each vector v of this space is uniquely written as a linear combination of e1, e2, and e3: v=υ1e1+υ2e2+υ3e3.
Topological relations between spherical spatial regions with holes
Published in International Journal of Digital Earth, 2020
Jingwei Shen, Lan Zhang, Min Chen
Many geographical objects and phenomena are presented as surface distributions, such as lakes, forests, and cities. The spatial region in the two-dimensional space, three-dimensional space and spherical space are different. The region in the two-dimensional space is a plane (Open Geospatial Consortium 1999) and the region in the three-dimensional space is normally a free surface (Zlatanova 2000a; Lee and Kwan 2005; Open Geospatial Consortium 2012), whereas, the region in the spherical space is a surface that is strictly attached to the surface of the sphere (Otoo and Zhu 1993; Gold and Mostafavi 2000). The two-dimensional space, three-dimensional space and spheric space are also different. The two-dimensional space (Egenhofer and Herring 1991; Schneider and Behr 2006; Long and Li 2013; Shen, Chen, and Liu 2018) and three-dimensional space (Egenhofer 1995; Zlatanova 2000b; Kurata 2009) are usually defined as infinite space, whereas the spherical space (Egenhofer 2005) is usually defined as finite space. Moreover, many geographical objects and phenomena have holes, such as lakes with islands and a state within a state. As a type of spatial region, a spatial region with holes has the characteristics of the separated boundaries and separated holes (Egenhofer, Clementini, and Di Felice 1994). Topological relations between spatial regions with holes are more complex than topological relations between simple spatial regions.
Additive manufactured gyroid-based cell structures under compression: design, testing and simulation for biomedical applications
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2023
Juan Carlos Castro-Sandoval, Alfredo Chavez, Jorge Corona-Castuera, John Henao, Marco Antonio Rivera-Gil, Carlos A. Poblano-Salas
An implicit representation method, from which a 3D model of a TPMS-based cellular structure is generated, was employed in this work. The 3D model is generated from the evaluation of a set of mathematical functions that describe a physical object in a three-dimensional space. When the functions are evaluated in each x, y and z coordinate, the resulting value dictates whether such function lies inside the space occupied by the object or not. The coordinates of the three-dimensional space are represented by voxels, essentially in a Boolean scalar field. By employing this approach, 3D models of TPMS-based CS were constructed; volumetric meshing was employed to perform FE analyses of the structures under static compressive load conditions.
Time-variant feedback controller based on capture point and maximal output admissible set of a humanoid
Published in Advanced Robotics, 2019
In the above discussion, the author considered the three-dimensional space: x,y and z (where motion in the z direction was considered implicitly). This subsection makes the property of the MOA set clearer by limiting the discussion to the two-dimensional space, in the x and z direction.