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Gauss's Law: Spherical Symmetry01:26

Gauss's Law: Spherical Symmetry

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A charge distribution has spherical symmetry if the density of charge depends only on the distance from a point in space and not on the direction. In other words, if the system is rotated, it doesn't look different. For instance, if a sphere of radius R is uniformly charged with charge density ρ0, then the distribution has spherical symmetry. On the other hand, if a sphere of radius R is charged so that the top half of the sphere has a uniform charge density ρ1 and the bottom half has a...
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Gauss's Law: Cylindrical Symmetry01:20

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A charge distribution has cylindrical symmetry if the charge density depends only upon the distance from the axis of the cylinder and does not vary along the axis or with the direction about the axis. In other words, if a system varies if it is rotated around the axis or shifted along the axis, it does not have cylindrical symmetry. In real systems, we do not have infinite cylinders; however, if the cylindrical object is considerably longer than the radius from it that we are interested in,...
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Area Computation by the Alternative Coordinate Method01:24

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The alternative coordinate method, also known as the Shoelace Formula, is a technique for determining the area of a traverse using Cartesian coordinates. This method relies on the sequential arrangement of x and y coordinates for each point of the shape, ensuring accuracy and ease of application.In this approach, each corner's x and y coordinates are listed as fractions, with the x-coordinate as the numerator and the y-coordinate as the denominator. These coordinates are arranged sequentially...
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Plastic Deformations of Members with a Single Plane of Symmetry01:21

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When a structural member undergoes plastic deformation due to bending, it is crucial to understand the position of the neutral axis and the stress distribution. This member, characterized by a single plane of symmetry, exhibits a uniform stress distribution, with negative stress above the neutral axis and positive stress below. Notably, the neutral axis does not align with the centroid of the cross-section. This misalignment is typical in cases where the cross-section is not rectangular or...
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Gauss's Law: Planar Symmetry01:27

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A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...
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The shear center of a channel section with uniform thickness, height, and width, is determined by computing the shear force in the member and calculating the moments of inertia of the sections.
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Related Experiment Video

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An Efficient and Flexible Cell Aggregation Method for 3D Spheroid Production
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Schwarz alternating methods for anisotropic problems with prolate spheroid boundaries.

Zhenlong Dai1, Qikui Du1, Baoqing Liu2

  • 1Jiangsu Key Laboratory for NSLSCS, School of Mathematics Sciences, Nanjing Normal University, No. 1 Wenyuan Road, Nanjing, 210023 People's Republic of China.

Springerplus
|September 15, 2016
PubMed
Summary

This study introduces an efficient Schwarz alternating algorithm for solving 3D anisotropic problems using the natural boundary element method. The algorithm is validated through numerical examples, demonstrating its effectiveness.

Keywords:
Artificial boundaryExterior anisotropic problemIteration methodProlate ellipsoidalSchwarz alternating algorithm

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Area of Science:

  • Computational Mathematics
  • Numerical Analysis
  • Boundary Element Methods

Background:

  • Solving exterior anisotropic problems in 3D domains presents significant computational challenges.
  • Existing methods may lack efficiency or require complex formulations.

Purpose of the Study:

  • To develop and analyze a Schwarz alternating algorithm for 3D exterior anisotropic problems.
  • To transform the anisotropic problem into a harmonic problem for simplified solution.

Main Methods:

  • The Schwarz alternating algorithm is adapted based on the natural boundary element method.
  • Coordinate transformation is employed to convert the anisotropic problem into a harmonic one.
  • Convergence analysis and error estimation are performed.

Main Results:

  • The modified algorithm's convergence properties are analyzed.
  • A contraction factor for the algorithm's convergence is determined.
  • Numerical examples confirm the algorithm's efficiency.

Conclusions:

  • The proposed Schwarz alternating algorithm provides an effective solution for 3D exterior anisotropic problems.
  • The coordinate transformation simplifies the problem, enhancing computational efficiency.
  • The algorithm's convergence and error estimates are rigorously analyzed.