Full%20Wave%20Analysis%20of%20the%20Contribution%20to%20the%20Radar%20Cross%20Section%20of%20the%20Jet%20Engine%20Air%20Intake%20of%20a%20Fighter%20Aircraft

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Full Wave Analysis of the Contribution to the
Radar Cross Section of the Jet Engine Air Intake
of a Fighter Aircraft
Pim Hooghiemstra

• Department of Flight
• Physics and Loads AVFP

10 April 2007
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About NLR

• NLR Nationaal Lucht- en Ruimtevaartlaboratoriu
  m
• (National Aerospace Laboratory NLR)
• Two residences (Amsterdam, NOP Marknesse)
• Independent Technological Institute
• 700 employes

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Overview

• RADAR
• Governing Equations
• Finite Element Method
• Dispersion Analysis
• Present Solution Process
• Future Research

OUTWASH
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Radar Detection
The platform is detected if the signal-to-noise
ratio exceeds the minimum level of received
signal
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Radar Equation
Received power by enemy radar
Power of enemy radar
Gain of enemy radar
Wavelength of enemy radar
Distance to enemy radar
If the RCS is decreased by a factor 16, the
maximum detection distance is decreased by a
factor 2 only.
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Radar Cross Section (RCS)

• Measure of detectability assuming isotropic
  scattering.
• Radar excited on the front -gt jet engine air
  intake (modeled as a cylinder) accounts for a
  great part of RCS for a large range of incident
  angles.
• RCS of total aircraft computed by subdividing the
  aircraft in simple geometries as flat planes and
  cones. For these simple geometries the RCS is
  known.
• RCS is computed in stead of measured since
  measurements are not always possible (expensive,
  not available or in development stage).

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Computing the RCS

• Approximate the electric field on the aperture
  numerically
• Compute the far-field components of the
  electric field and the RCS, proportional to

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Maxwells Equations
OUTWASH
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Vector Wave Equation

• RCS computed for one frequency only. Introduce
  phasor notation and derive the wave equation for
  the electric field.

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Vector Wave Equation (2)
To obtain a well-posed problem define a boundary
condition on the cavity wall and a integral
equation for the aperture.
To analyse the vector wave equation in detail it
is made dimensionless.
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Wave Number
Wave number important number for the
dimensionless vector wave equation. It is related
to the wavelength by The product is
characteristic for the RCS. An appropriate test
problem should have this ratio in the same order
of magnitude to obtain an equivalent problem.
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Finite Element Method The Elements

• Tetrahedral elements are chosen for two reasons
• They easily follow the shape of the object.
• They are compatible with the triangles used for
  the discretization of the aperture

OUTWASH
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Finite Element Method The Basis functions
Vector based basis functions are chosen to
prevent spurious solutions to occur.
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Higher Order Basis functions
Interpolation points on the edge and face for
construction of basis functions (higher order,
more points)
Higher order basis functions are chosen for
efficiency. This choice decreases the number of
DoF.
OUTWASH
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The System of Equations
Nonzero pattern for the discreti- zation matrix
A. The matrix has a sparse and a fully
populated part. The fully populated part
consist of half of the total number of nonzeros.
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Dispersion (1)

• Assume a wave front entering a cavity. After
  reflection
• through the cavity, phase differences occur
  between different rays of the front.
• The strength of the electric field (needed for
  computing the RCS) depends on this phase
  difference.
• Due to the discretization the dispersion error is
  introduced which influences the phase difference.
  For a deep cavity this error accumulates and
  dominates the problem.

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Dispersion (2)

• Exact phase difference
• Dispersion error
• Computed phase difference

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Dispersion (3)
Wave without error
Wave with dispersion error
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Complexity

• Purpose Minimize dispersion error and compute
  RCS accurately

• Dependence of matrix size on
• dispersion error
• element order

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Degrees of Freedom

• The total number of unknowns in the system is a
  function of the element order p, the error .
  It is observed that the total DoF decreases for
  higher order elements.

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Present Solution Method

• A direct method is used. Iterative methods with
  well known preconditioners (ILU, approximate
  inverse) were considered but not promising.
• Frontal solver
• Eliminate fully summed variables immediately
• Store active (front) variables only

Benefits Can be used to compute the RCS for
multiple right hand sides and an
answer is always obtained. Drawback
Takes a long time to compute
accurate solution
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Ideas for Future Research

• Use efficient iterative method (GCR or COCG)
• Construct an effective preconditioner.
  Observed simple preconditioners do not
  work.

• ILU and approximateinverse are implemented
  and they are not efficient
• Idea use the shifted Laplace operator as
  preconditioner. This works well for the Helmholtz
  equation, expected to work for vector wave
  equation.

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Questions and Suggestions

• Questions?
• Suggestions?
• Other Remarks?