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Related Concept Videos

Mesh Analysis01:20

Mesh Analysis

590
Mesh analysis is a valuable method for simplifying circuit analysis using mesh currents as key circuit variables. Unlike nodal analysis, which focuses on determining unknown voltages, mesh analysis applies Kirchhoff's voltage law (KVL) to find unknown currents within a circuit. This method is particularly convenient in reducing the number of simultaneous equations that need to be solved.
A fundamental concept in mesh analysis is the definition of meshes and mesh currents. A mesh is a closed...
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Electromagnetic Wave Equation01:24

Electromagnetic Wave Equation

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Maxwell's equations for electromagnetic fields are related to source charges, either static or moving. These fields act on a test charge, whose trajectory can thus be determined using suitable boundary conditions. The objective of electromagnetism is thus theoretically complete.
However, although electric and magnetic fields were first introduced as mathematical constructs to simplify the description of mutual forces between charges, a natural question emerges from Maxwell's equations:...
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Graphing the Wave Function01:13

Graphing the Wave Function

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Consider the wave equation for a sinusoidal wave moving in the positive x-direction. The wave equation is a function of both position and time. From the wave equation, two different graphs can be plotted.
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Equations of Wave Motion01:02

Equations of Wave Motion

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Mathematically, the motion of a wave can be studied using a wavefunction. Consider a string oscillating up and down in simple harmonic motion, having a period T. The wave on the string is sinusoidal and is translated in the positive x-direction as time progresses. Sine is a function of the angle θ, oscillating between +A and −A and repeating every 2π radians. To construct a wave model, the ratio of the angle θ and the position x is considered.
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Velocity and Acceleration of a Wave00:51

Velocity and Acceleration of a Wave

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A wave propagates through a medium with a constant speed, known as a wave velocity. It is different from the speed of the particles of the medium, which is not constant. In addition, the velocity of the medium is perpendicular to the velocity of the wave. The variable speed of the particles of the medium implies that there must be acceleration associated with it. 
The velocity of the particles can be obtained by taking the partial derivative of the position equation with respect to time....
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Mesh Analysis with Current Sources01:10

Mesh Analysis with Current Sources

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Mesh analysis becomes simpler when analyzing circuits with current sources, whether independent or dependent. The presence of current sources reduces the number of equations required for analysis. Two cases illustrate this:
Current Source in One Mesh: The analysis process is straightforward when a current source is found in only one mesh within the circuit. Mesh currents are assigned as usual, with the mesh containing the current source excluded from the analysis. Kirchhoff's voltage law...
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WaveGFD: Data and methods for numerically solving the wave equation using a meshless Generalized Finite Differences

Gerardo Tinoco-Guerrero1,2, Francisco J Domínguez-Mota1,2, José A Guzmán-Torres1

  • 1Civil Engineering Faculty, Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán, 58020, Mexico.

Data in Brief
|September 2, 2024
PubMed
Summary

WaveGFD introduces efficient meshless finite difference schemes for solving the wave equation in complex, irregular domains. This repository offers novel methods and data for precise and rapid simulations in various engineering fields.

Keywords:
Generalized finite differenceIrregular regionsMeshless methodNumerical solutionWave equation

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Area of Science:

  • Computational Mathematics
  • Numerical Analysis
  • Scientific Computing

Background:

  • Solving the wave equation in irregular domains presents significant computational challenges.
  • Conventional numerical methods often struggle with the complexity and precision required for these problems.
  • Meshless methods offer a promising alternative for handling complex geometries.

Purpose of the Study:

  • To present WaveGFD, a repository of meshless finite difference schemes for the wave equation.
  • To provide efficient and accurate methods for solving partial differential equations in highly irregular domains.
  • To facilitate the application of these methods across diverse engineering disciplines.

Main Methods:

  • Development and analysis of meshless finite difference schemes.
  • Utilizing polygonal approximations for geographical regions and complex domains.
  • Implementation of methods for approximating solutions to the wave equation.

Main Results:

  • Demonstration of precision and efficiency in solving the wave equation.
  • Validation with test data ranging from simple squares to complex geographical regions.
  • Successful application examples in civil, aerospace, and electrical engineering.

Conclusions:

  • WaveGFD provides effective and efficient solutions for wave equation problems in complex domains.
  • The developed methods overcome limitations of conventional techniques.
  • The repository supports broad applicability in engineering and physical sciences.