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Algebraic Geometry
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
Let L/K be a field extension and α∈L. If there is a polynomial f(x)∈K[x] such that f(α)=0, then α is called an algebraic element over K. An algebraic extension is a field extension L/K such that every element of L is algebraic over K.
Evaluation of the downhole hydraulic fracturing using single-well shear wave scattering imaging -modeling and field applications
Published in Waves in Random and Complex Media, 2023
Assume that the acoustic source to the scattering point and back to the receiver’s propagation path is a straight ray, as shown in Figure 3, where S is the source point, R is the receiving point. For the case of vertical or weak lateral changes in velocity, the prestack time migration based on the Kirchhoff method uses the following equation to calculate the travel time [19], where represents a one-way travel time; represents a half offset; represents the midpoint position; and represents the root mean square velocity. Wave-field extension and imaging can be carried out by using the following Kirchhoff integral formula, where represents the tilt factor indicating the amplitude changes with the exit angle; represents the bin size; represents the imaging point position; represents the surface receiving point position; represents the delay time; and will be taken while performing the migration imaging and to ignore the far-field term in Equation (4).
Effects of carbon fibers with different particle sizes on the physical properties of MoS2-filled PTFE composites
Published in Philosophical Magazine Letters, 2021
Ke Zhang, Xin Ji, Yiming Mi, Lei Gao, Tianyu Wang
The CF/MoS2/PTFE-B surface shown in Figure 4b is smooth without excessive abrasive debris, whereas a large amount of abrasive debris is distributed on the surface of the CF/MoS2/PTFE-B in Figure 4d. The wear debris is in the form of small flakes or even powder without wire drawing. In the circumferential tip of an interfacial peeling, the non-reinforced board made of fibers/fillers and matrix materials bears far-field extension-bending, in-plane shear-twisting, and torsional loads [27–29]. Also, during the wear process, when the particle size of the filled CF becomes large, part of the wear debris will re-embed into defects under pressure, reducing direct wear [30]. When the friction reaches equilibrium, external conditions such as temperature and pressure are stable. The friction coefficient and wear of the composites tend to be stable.
Sharp interface immersed boundary method for simulating three-dimensional swimming fish
Published in Engineering Applications of Computational Fluid Mechanics, 2020
Zuo Cui, Zixuan Yang, Hongzhou Jiang
In the Cartesian-IB method, IB points are specified as Cartesian grid points located in the fluid but in the immediate vicinity of the solid body. In comparison, curvilinear grids are used in the curvilinear-IB method. The cut-cell methods use the interface of the solid body to divide the grid cells into two subcells, i.e. the solid phase and fluid phase. Combined with the Cartesian-IB method, Gilmanov and Sotiropoulos (2005) used the ray-tracing algorithm to identify the fluid, solid and IB points. Yang and Balaras (2006) applied a field-extension procedure to extrapolate the velocity and pressure fields of pseudo-fluid points after each deformation. Maitri, Das, Kuipers, Padding, and Peters (2018) used the sharp interface IB method to simulate the particle-laden flows, in which the ghost-cell approach was used to calculate the direct force. Zhu, Seo, and Mittal (2019) proposed a graph-partitioning framework for a sharp interface IB method, and it was suitable to simulate internal flows on large-scale parallel computers. In the study by Wang, Du, and Sun (2019), adaptive mesh refinement (AMR) was applied along with the sharp interface IB method, and the results showed that the AMR approach can increase the computational efficiency.