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Related Concept Videos

Extended Versions of Green’s Theorem01:27

Extended Versions of Green’s Theorem

Green’s Theorem connects the circulation of a vector field around a closed curve with the behavior of the field across the region enclosed by that curve. It provides a way to replace a line integral around a boundary with a double integral over the interior region, making it especially useful in plane geometry, fluid flow, and vector calculus.Although Green’s Theorem is often introduced using simple regions without gaps, it can also be applied to regions made from several simple parts. This...
Green’s Theorem01:27

Green’s Theorem

Green’s Theorem establishes a relationship between a line integral around a closed plane curve and a double integral over the region enclosed by that curve. It applies to a vector field F(x, y) = 〈P(x, y), Q(x, y)〉, where P and Q have continuous first partial derivatives on an open set containing the region.Let C be a positively oriented, simple, closed, piecewise smooth curve, and let R be the plane region bounded by C. Green’s Theorem states that\begin{equation*}\oint_C P\,dx+Q\,dy =\iint_R...
Vector Forms of Green’s Theorem01:26

Vector Forms of Green’s Theorem

The study of fluid motion often involves understanding how local rotational behavior relates to global circulation. In the context of a pond with pollutants, direct measurement of water movement along an irregular shoreline can be impractical. Green’s Theorem in vector form provides an alternative by relating the circulation around a closed boundary to properties of the flow within the enclosed region.Measurements of water velocity at different points define a continuous vector field that...
Triple Integrals over General Regions01:28

Triple Integrals over General Regions

Triple integrals over general bounded regions extend the concept of double integrals from planar domains to three-dimensional solids. A solid region E in space is commonly enclosed within a rectangular box B, and a continuous function f(x, y, z) is integrated over the region by defining F such that it coincides with f on E and is zero outside the solid. The triple integral is therefore expressed as\begin{equation*}\iiint_E f(x,y,z) dV \end{equation*}The existence of the integral requires that f...
Surface Integrals of Vector Fields: Flux01:22

Surface Integrals of Vector Fields: Flux

Understanding the movement of air masses is fundamental to meteorological analysis and atmospheric modeling. A key component in this process is quantifying the total mass of air that flows into or out of a defined region over a specified period of time. This is achieved by evaluating the mass flux across a boundary surface, a conceptual tool that simplifies the complex dynamics of atmospheric systems.To begin, an imaginary boundary surface S is introduced, enclosing the region of interest. The...
Line, Surface, and Volume Integrals01:15

Line, Surface, and Volume Integrals

A line integral for a vector field is defined as the integral of the dot product of a vector function with an infinitesimal displacement vector along a prescribed path. If the prescribed path is closed, the integrals reduce to a closed-line integral. The closed-contour integral of the vector field is referred to in terms of the circulation of the vector field around the closed path. A vector with zero circulation around every closed path is called a conservative field, while one with non-zero...

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Related Experiment Video

Updated: Jun 16, 2026

Interfacial Molecular-level Structures of Polymers and Biomacromolecules Revealed via Sum Frequency Generation Vibrational Spectroscopy
09:43

Interfacial Molecular-level Structures of Polymers and Biomacromolecules Revealed via Sum Frequency Generation Vibrational Spectroscopy

Published on: August 13, 2019

Multilevel Green's function interpolation method for analysis of 3-D frequency selective structures using

Yan Shi1, Chi Hou Chan

  • 1School of Electronic Engineering, Xidian University, Xi'an, Shaanxi 710071, China.

Journal of the Optical Society of America. A, Optics, Image Science, and Vision
|February 4, 2010
PubMed
Summary

The multilevel Green's function interpolation method (MLGFIM) efficiently analyzes complex 3D periodic structures. This computational electromagnetics technique achieves linear O(N) complexity for dielectric and conducting objects.

Related Experiment Videos

Last Updated: Jun 16, 2026

Interfacial Molecular-level Structures of Polymers and Biomacromolecules Revealed via Sum Frequency Generation Vibrational Spectroscopy
09:43

Interfacial Molecular-level Structures of Polymers and Biomacromolecules Revealed via Sum Frequency Generation Vibrational Spectroscopy

Published on: August 13, 2019

Area of Science:

  • Computational Electromagnetics
  • Applied Physics
  • Materials Science

Background:

  • Analyzing three-dimensional doubly periodic structures with dielectric and conducting components is computationally intensive.
  • Existing methods often struggle with the complexity of simulating structures like photonic bandgap materials and split ring resonators.
  • Efficient numerical techniques are crucial for advancing the design and understanding of periodic electromagnetic systems.

Purpose of the Study:

  • To introduce the multilevel Green's function interpolation method (MLGFIM) for analyzing 3D doubly periodic structures.
  • To demonstrate the MLGFIM's capability in handling structures composed of dielectric media and conducting objects.
  • To validate the MLGFIM's efficiency and accuracy through simulations of various complex periodic structures.

Main Methods:

  • Utilized volume integral equation (VIE) for dielectric objects and surface integral equation (SIE) for conducting objects within a unit cell.
  • Employed conformal basis functions on curvilinear hexahedron and quadrilateral elements to solve the volume/surface integral equation (VSIE).
  • Implemented Ewald's transformation for accelerating Green's function computation and a periodic octree scheme for adaptive analysis.

Main Results:

  • The MLGFIM was validated against published data, showing accurate results for various periodic structures.
  • Simulations included dielectric coated conducting shells, folded dielectric structures, photonic bandgap structures, and split ring resonators (SRRs).
  • The proposed MLGFIM demonstrated a computational complexity of O(N) for analyzing doubly periodic structures.

Conclusions:

  • The MLGFIM provides an efficient and accurate method for the analysis of complex 3D doubly periodic electromagnetic structures.
  • The linear computational complexity makes MLGFIM suitable for large-scale simulations and design optimization.
  • This method advances the simulation capabilities for metamaterials, photonic crystals, and other periodic electromagnetic devices.