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Standard Variation Principle Formulation
Let
where
is a self-adjoint operator satisfying
and positive definite, that is
The inner product is defined by
Then, the solution of
given by
that is
Proof:
can be obtained by minimizing the functional
Let
For simplicity, assume all values are real. Then,
.
Enforce
for
Then,
where
is an
matrix, defined by
is an
column vector, defined by
This result is the same as Galerkin’s Method of Moments.
Generalized Variational Principles
where
Galerkin’s Method
From Maxwell’s Equations
Let
be a test function satisfying boundary condition on
the above in volume
. Integrate
,
By vector identity
we have
Since also satisfies the first boundary condition, the first term in the
right hand side becomes
Applying the second boundary, the above equation becomes
Finally,
Let
where
satisfying the boundary condition on
,
then
Boundary Conditions
1. Perfect Magnetic Conductor
.
2.
The meaning of this is no special treatment is needed for PMC
boundary, that is, the same as no boundary at all.
Perfect Electric Conductor
3.
Impedance Boundary Condition
For good conductor, then the thickness is much larger than skip
depth, we have
4.
Radiation Boundary Condition
For scattering problem
radiation boundary condition.
Port Excitation
Suppose
and
. But, only
satisfies
is mode pattern and propagation constant. Then
at the port surface, we have
Then,
If the mode pattern is normalized, that is,
the reflection coefficient can be computed by
.
Triangular Edge Basis
Interpolation Functions:
Defined by
,
Properties
1.
2.
, satisfy Maxwell Equation in sourceless region.
Tangential component of edge i-j only dependent on
constant. That also means
is perpendicular to the other two
edges.
Tetrahedral Edge Basis
Interpolation functions:
,
and is a
.
Properties similar to Triangular case.
Spurious Mode
When solving cavity or waveguide problem, due to the errors in the
numerical computation, not realistic modes are produced.