EEE 431 Computational Methods in Electrodynamics

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EEE 431Computational Methods in Electrodynamics

• Lecture 18
• By
• Dr. Rasime Uyguroglu
• Rasime.uyguroglu_at_emu.edu.tr

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Variational Methods
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Variational Methods/Weighted Residual Method

• The name Method of Moments is derived from the
  original terminology that
• Is the nth moment of f. When is replaced by
  an arbitrary , we continue to call the
  integral a moment of f.

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Variational Methods/Weighted Residual Method

• The name method of weighted residuals is derived
  from the following interpretation

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Variational Methods/Weighted Residual Method

• Consider again the operator equation
• Linear Operator.
• Known function, source.
• Unknown function.
• The problem is to find g from f.

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Variational Methods/Weighted Residual Method

• Let f be represented by a set of functions
• scalar to be determined (unknown expansion
  coefficients.
• expansion functions or basis functions.

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Variational Methods/Weighted Residual Method

• Now, substitute (2) into (1)
• Since L is linear

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Variational Methods/Weighted Residual Method

• Now define a set of testing functions or
  weighting functions
• Define the inner product (usually an integral).
  Then take the inner product of (3) with each
  and use the linearity of the inner product

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Variational Methods/Weighted Residual Method

• If (3) represents an approximate equality, then
  the difference between the exact and approximate
  is
• R, is the error in the equation.

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Variational Methods/Weighted Residual Method

• The inner products are called the
  weighted residuals.
• In the weighted residual method, the weighting
  functions are chosen such that the integral
  of a weighted residual of the approximation is
  zero.

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Variational Methods/Weighted Residual Method

• Equation (4) can be obtained by setting all
  weighted residuals to zero.
• Which is equation (4).

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Variational Methods/Weighted Residual Method

• A system of linear equations can be written in
  matrix form as

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Variational Methods/Weighted Residual Method

• Where

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Variational Methods/Weighted Residual Method

• Solving for and substituting for
  in Eq. 2, gives an approximate solution to Eq. 1.
  However, there are different ways of choosing the
  weighting functions

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Variational Methods/Weighted Residual Method

• Selection of basis and weighting functions
• There are infinitely many possible sets of basis
  and weighing functions. Although the choice of
  these is specific to each problem, we can state
  rules that can be applied generally to optimize
  the change of success of obtaining accurate
  results in a minimum time and computer memory
  storage.

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Variational Methods/Weighted Residual Method

• Selection of basis and weighting functions
• They should form a set of linearly independent
  functions.
• should approximate the (expected)
  function
• reasonably well.

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Variational Methods/Weighted Residual Method

• Keep the following in mind in the selection of
  basis and weighting functions
• The desired accuracy of the solution,
• The size of the matrix A to be inverted,
• The realization of a well-behaved matrix A,
• The easy of evaluation of the inner products.

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Variational Methods/Weighted Residual Method

• Methods used for choosing the weighting
  functions
• Collocation (or point matching) method,
• Subdomain method,
• Galerkin Method,
• Least squares method.

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Variational Methods/Weighted Residual Method

• Let us discuss the point matching Method
• Collocation (or point matching) method
• It is the simplest method for choosing the
  weighting functions
• It basically involves satisfying the approximate
  representation
• at discrete points in the region of interest.

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Variational Methods/Weighted Residual Method

• Collocation (or point matching) method
• In terms of the MoM this is equivalent to
  choosing the testing functions to be Dirac delta
  functions. i.e.,

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Variational Methods/Weighted Residual Method

• Substituting Eq. 9 into
• Results
• We select as many matching points in the interval
  as there are unknown coefficients and
  make the residual zero at those points.

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Variational Methods/Weighted Residual Method

• The integrations represented by the inner
  products now become trivial, i.e..

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Variational Methods/Weighted Residual Method

• Although the point matching method is the
  simplest specialization for the computation, it
  is not possible to determine in advance for the
  particular operator equation what weighting
  functions would be suitable.

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Variational Methods/Weighted Residual Method

• Example Find an approximate solution to
• Using the method of weighted residuals.

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Variational Methods/Weighted Residual Method

• Let the approximate solution be
• Select to satisfy . So a
  reasonable choice is

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Variational Methods/Weighted Residual Method

• Now select
• If i2 the approximate solution is
• Where the expansion functions are to be
  determined.

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Variational Methods/Weighted Residual Method

• To find the residual R

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Variational Methods/Weighted Residual Method

• Point Matching Method
• Since we have two unknowns
• We select
• And set the residual equal to zero at those
  points. i.e

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Variational Methods/Weighted Residual Method

• Point Matching Method
• Solving these equations,
• And substituting

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Variational Methods/Weighted Residual Method

• Point Matching Method
• Select
• As the match points. Then

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Variational Methods/Weighted Residual Method

• Point Matching Method
• Solving these equations
• With the approximate solution