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Single Particle Motion
Published in Rob Appleby, Graeme Burt, James Clarke, Hywel Owen, The Science and Technology of Particle Accelerators, 2020
Rob Appleby, Graeme Burt, James Clarke, Hywel Owen
In our analysis we started with an arbitrary field and made an expansion, which we called the multipole expansion. When we used the expansion in our discussion of Hill's equation, we only kept the first two terms, equivalent to the constant and linear terms in a Taylor series. These correspond to dipole and quadrupole fields. The complete multipole expansion for the transverse fields looks like By+iBx=∑n=1∞Cnzn−1=∑n=1∞Cn(x+iy)n−1,
High-Accuracy Parasitic Extraction
Published in Louis Scheffer, Luciano Lavagno, Grant Martin, EDA for IC Implementation, Circuit Design, and Process Technology, 2018
One of the first sparsification techniques applied to capacitance extraction was the fast multipole method (FMM) [4–6] with O(n) complexity in time and memory. The idea, illustrated in Figure 26.2, is to approximate a group of distant charges within some radius R as a single charge, and then use that approximation for evaluation of potentials at points a distance r away, where r >> R. Using a single charge to represent this group is a monopole expansion, and using a series of higher-order representations, such as a monopole, dipole, quadrapole, etc., is a multipole expansion. Similarly, the potential due to many multipole expansions can be expressed with a local expansion centered at a point around a cluster of evaluation points. By using multipole and local expansions and by keeping the ratio r/R constant for all expansions, Px can be computed in O(n) time and memory for a given error tolerance. The details of applying an iterative solution algorithm and FMM sparsification to the capacitance problem can be found in [32], with a proof of O(n) given in [33].
Function Spaces
Published in Mattias Blennow, Mathematical Methods for Physics and Engineering, 2018
Since the eigenvalue does not depend on m and −ℓ≤m≤ℓ, there are 2ℓ +1 different spherical harmonics with eigenvalue ℓ(ℓ + 1), i.e., all of the eigenvalues except zero correspond to more than one eigenfunction. a selection of different spherical harmonics are shown in Fig. 5.14. The functions with different values of ℓ in an expansion into spherical harmonics are referred to as different multipoles, e.g., the ℓ = 0 spherical harmonic leads to a monopole, the ℓ =1 solutions dipoles, the ℓ =2 solutions quadrupoles, etc. An expansion in terms of the spherical harmonics is often referred to as a multipole expansion.
Time domain potential and source methods and their application to twin-hull high-speed crafts
Published in Ships and Offshore Structures, 2023
Hess and Smith (1964) method is used for the analytical integration of 1/r and 1/r′ in Equations (1–3) over each panel. Exact integration, multipole expansion and monopole expansion are used for the small, intermediate and large values of r and r′, respectively. The memory part of the transient wave Green function in Equation (4) is predicted analytically (Liapis 1986; King 1987; Kara 2000), while its integration over each panel is done numerically with 2 × 2 Gaussian quadrature after mapping the panels into unit squares. The integration on line elements is done with 16 Gaussian points, after subdividing interaction elements into straight lines. It is assumed that the line elements’ potential or source strength equals to those of the panels underneath them.
Chemical and physical aspects of recent bent-shaped liquid crystals exhibiting chiral and achiral mesophases
Published in Liquid Crystals, 2022
Supreet Kaur, Vidhika Punjani, Neelam Yadav, Abinash Barthakur, Anshika Baghla, Surajit Dhara, Santanu Kumar Pal
Bent-shaped mesogens may also possess complex orientational order apart from nematic and polar orders. Comprehensive and thorough theories of complex orientational order parameters exhibited by bent-shaped mesogens have been formulated [32]. They are explained in terms of multipole expansion theory and mass-moment tensors are used to construct the order parameters. The lowest orientational order is polar or dipolar denoted by a first-rank tensor or vector Pi. A second rank tensor Qij is used for the nematics or quadrupolar order. For octupolar order, the multipole expansion continues, and a third rank tensor Tijk represents it. A tetrahedratic phase is formed when the octupolar order exists by itself and Qij= Pi= 0. This phase has four directions that are equal to each other. The dielectric tensor eij is proportional to the identity matrix because Qij is zero. Due to this, tetrahedratic phase seems to be optically isotropic, whereas in reality, its orientational order is extremely difficult to be observed. The LC can become optically anisotropic on the application of an electric field which induces quadrupolar order to a tetrahedratic phase. Thus, optical anisotropy which is dependent on a bias electric field, can act as a signature for tetrahedratic phase [33,34]. Experimentally, the existence of this type of order in the nematic phase has been found by calorimetric and magneto-optical studies [35,36]. Further, more complex distorted phases are created if the octupolar order is present together with quadrupolar or dipolar order. This multipole expansion theory of mass moments which describes the symmetry breaking of nematic phase can give rise to numerous mesophases.
Investigation of the peeling and pull-off behavior of adhesive elastic fibers via a novel computational beam interaction model
Published in The Journal of Adhesion, 2021
Maximilian J. Grill, Christoph Meier, Wolfgang A. Wall
The following SSIP law aims to describe the electrostatic interaction of nonconducting, circular cross-sections with constant surface charge densities . It is based on the monopole expression, i.e., the first term of the multipole expansion of each cross-section’s surface charge distribution and reads