Finite Integral Method

The finite integral method is based on the finite integration theory (FIT) and is a numerical simulation method for the approximate solution of the basic electromagnetic equations according to Maxwell . It forms the mathematical basis of simulation programs for electromagnetic problems such as B. MAFIA and CST MICROWAVE STUDIO® .

Basics

The finite integral method, first introduced in 1976 by Thomas Weiland , solves the basic electromagnetic equations named after Maxwell

${\displaystyle \oint _{C(A)}{\vec {E}}\cdot d{\vec {s}}\;=\;-\int _{A}{\frac {\partial }{\partial t}}{\vec {B}}\cdot d{\vec {A}}}$

${\displaystyle \oint _{C(A)}{\vec {H}}\cdot d{\vec {s}}\;=\;\int _{A}\left({\frac {\partial }{\partial t}}{\vec {D}}+{\vec {J}}\right)\cdot d{\vec {A}}}$

${\displaystyle \oint _{A(V)}{\vec {D}}\cdot d{\vec {A}}\;=\;\int _{V}\rho \cdot dV}$

${\displaystyle \oint _{A(V)}{\vec {B}}\cdot d{\vec {A}}\;=\;0}$

the material equations approximate in their integral form and by means of integral approximation

${\displaystyle {\vec {D}}\;=\;\varepsilon _{0}\varepsilon _{r}{\vec {E}}}$

${\displaystyle {\vec {B}}\;=\;\mu _{0}\mu _{r}{\vec {H}}}$

${\displaystyle {\vec {J}}\;=\;\kappa {\vec {E}}+{\vec {J_{s}}}}$

in discretized form.

method

The entire problem area is subdivided into a first (or primary) three-dimensional network of individual, as small as possible mesh cells with the material properties , each of which is calculated with regard to its electrical edge voltage and its magnetic flux through the edge surfaces . ${\displaystyle \varepsilon _{r},\mu _{r},\kappa }$ ${\displaystyle e_{i}=E_{i}\cdot a}$ ${\displaystyle b_{j}=B_{j}\cdot A_{j}}$

In addition, a second (dual) grid cell network placed orthogonally to the first grid is calculated with regard to the magnetic edge voltage and the electrical flux through the edge surfaces , taking into account the continuity conditions. ${\displaystyle h_{i}=H_{i}\cdot a}$ ${\displaystyle d_{j}=D_{j}\cdot A_{j}}$

Curve integral over the electric field strength in a grid cell

The cuboid shape of the grid cells simplifies the contour integral of the electric field strength to the sum of the edge stresses of a cuboid wall of the grid cell. ${\displaystyle \oint _{C(A)}{{\vec {E}}\cdot d{\vec {s}}}=\int _{{C}_{1}}{{\vec {E}}\cdot d{\vec {s}}}+\int _{{C}_{2}}{{\vec {E}}\cdot d{\vec {s}}}-\int _{{C}_{3}}{{\vec {E}}\cdot d{\vec {s}}}-\int _{{C}_{4}}{{\vec {E}}\cdot d{\vec {s}}}}$ ${\displaystyle \sum _{i=1}^{4}e_{i}}$

The time derivative of the magnetic flux through the edge surface of the grid cell is now set equal to this sum, so that the following equation results:

${\displaystyle \sum _{i=1}^{4}e_{i}\;=\;e_{1}+e_{2}-e_{3}-e_{4}\;=\;-{\frac {\partial }{\partial t}}b_{n}}$

This calculation must be repeated for all six edge areas of a grid cell. The system of equations results in matrix notation

${\displaystyle {\begin{pmatrix}\ldots &\ldots &\ldots &\ldots &\ldots &\ldots &\ldots \\1&\ldots &1&\ldots &-1&\ldots &-1\\\ldots &\ldots &\ldots &\ldots &\ldots &\ldots &\ldots \end{pmatrix}}\cdot {\begin{pmatrix}e_{i}\\\vdots \\e_{j}\\\vdots \\e_{k}\\\vdots \\e_{l}\end{pmatrix}}\;=\;-{\frac {\partial }{\partial t}}\;{\begin{pmatrix}\vdots \\b_{n}\\\vdots \end{pmatrix}}}$

${\displaystyle {\textbf {C}}\cdot {\vec {e}}\;=\;-{\frac {\partial }{\partial t}}\;{\vec {b}}}$

The descriptive matrix only has the values ​​1, 0, −1 as elements. ${\displaystyle {\textbf {C}}}$

The other Maxwell's equations are treated analogously. The grid Maxwell equation system results in matrix notation

${\displaystyle {\textbf {C}}\cdot {\vec {e}}\;=\;-{\frac {\partial }{\partial t}}\;{\vec {b}}}$

${\displaystyle {\textbf {C}}_{\text{Dual}}\cdot {\vec {h}}\;=\;{\frac {\partial }{\partial t}}\;{\vec {d}}+{\vec {j}}}$

${\displaystyle {\textbf {S}}_{\text{Dual}}\cdot {\vec {d}}\;=\;{\vec {q}}}$

${\displaystyle {\textbf {S}}\cdot {\vec {b}}\;=\;{\vec {0}}}$

The matrix corresponds to the analytical rotation operator , the matrix corresponds to the analytical divergence operator . The index indicates the calculation of the edge stresses and fluxes in the dual lattice. ${\displaystyle {\textbf {C}}}$ ${\displaystyle {\textbf {S}}}$ ${\displaystyle Dual}$

The material equations are discretized analogously to Maxwell's equations.

${\displaystyle {\vec {d}}\;=\;{\textbf {M}}_{\varepsilon }\;\cdot \;{\vec {e}}}$

${\displaystyle {\vec {b}}\;=\;{\textbf {M}}_{\mu }\;\cdot \;{\vec {h}}}$

${\displaystyle {\vec {j}}\;=\;{\textbf {M}}_{\kappa }\;\cdot \;{\vec {e}}\;+\;{\vec {j}}_{s},}$

whereby the material sizes can be location, frequency and direction dependent.

The FIT method is applicable to all electromagnetic problems in the time and frequency domain, both in electrostatics and in electrodynamics. Due to the special tailoring of the FIT method to Maxwell's equations and the resulting discrete analogue, the continuity conditions are met a priori and the analytical properties of the vector operations are retained.

For electrodynamic problems, all time derivatives are replaced by in the frequency domain . The result of a simulation in the frequency domain provides the impulse response to a mono-frequency input signal. ${\displaystyle j\omega }$

Broadband excitation with free signal curves is permitted in the time domain. In this case, the simulation calculation describes the frequency behavior over a previously defined frequency range.

literature

• T. Weiland: A method for solving Maxwell's equations for six-component fields on a discrete basis , AEÜ, Volume 31, Issue 3, pp. 116-120, 1977
• T. Weiland: A Discretization Method for the Solution of Maxwell's Equations for Six-Component Fields , Electronics and Communications AEUE, vol. 31, no. 3, pp. 116-120, 1977.