1. Department of Flight Physics and Loads AVFP
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

2. About NLR NLR = Nationaal Lucht- en Ruimtevaartlaboratorium
(National Aerospace Laboratory NLR)
Two residences (Amsterdam, NOP: Marknesse)
Independent Technological Institute
700 employes
12/5/2018

3. Overview RADAR Governing Equations Finite Element Method
Dispersion Analysis
Present Solution Process
Future Research
OUTWASH
Important feature of radar signature of helicopter is:
HERM = Helicopter Rotor Modulation
Similar to jet engine modulation ( reflection by rotating compressor fan)
RCS has Doppler spectrum caused by rotating rotor blades and hub.
Can be used for target detection, classification and recognition
Spectrum determined by
Shape of hub
Shape and material of blades
Number of blades
Rotational speed of rotor
Radar cross section computations based on asymptotic high frequency techniques: Physical Optics/ray-tracing and the Method of Equivalent Currents. Incorporation of landing gear, weapons and other protuberances would require (hybrid) full-wave modelling.
Here quasi-steady analysis of seperated rotor and fuselage.
Higher order effects to be included
Electromagnetic blade-blade and rotor-fuselage interaction
Deflection and/or pitch changes of rotor blades
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4. Radar Detection The platform is detected if
the signal-to-noise ratio exceeds
the minimum level of received
signal
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5. Received power by enemy radar
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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6. Radar Cross Section (RCS)
Measure of detectability assuming isotropic scattering.
Radar excited on the front -> 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).
Important feature of radar signature of helicopter is:
HERM = Helicopter Rotor Modulation
Similar to jet engine modulation ( reflection by rotating compressor fan)
RCS has Doppler spectrum caused by rotating rotor blades and hub.
Can be used for target detection, classification and recognition
Spectrum determined by
Shape of hub
Shape and material of blades
Number of blades
Rotational speed of rotor
Radar cross section computations based on asymptotic high frequency techniques: Physical Optics/ray-tracing and the Method of Equivalent Currents. Incorporation of landing gear, weapons and other protuberances would require (hybrid) full-wave modelling.
Here quasi-steady analysis of seperated rotor and fuselage.
Higher order effects to be included
Electromagnetic blade-blade and rotor-fuselage interaction
Deflection and/or pitch changes of rotor blades
12/5/2018

7. 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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8. Maxwell’s Equations OUTWASH
Important feature of radar signature of helicopter is:
HERM = Helicopter Rotor Modulation
Similar to jet engine modulation ( reflection by rotating compressor fan)
RCS has Doppler spectrum caused by rotating rotor blades and hub.
Can be used for target detection, classification and recognition
Spectrum determined by
Shape of hub
Shape and material of blades
Number of blades
Rotational speed of rotor
Radar cross section computations based on asymptotic high frequency techniques: Physical Optics/ray-tracing and the Method of Equivalent Currents. Incorporation of landing gear, weapons and other protuberances would require (hybrid) full-wave modelling.
Here quasi-steady analysis of seperated rotor and fuselage.
Higher order effects to be included
Electromagnetic blade-blade and rotor-fuselage interaction
Deflection and/or pitch changes of rotor blades
12/5/2018

9. 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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10. 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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11. 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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12. 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
Important feature of radar signature of helicopter is:
HERM = Helicopter Rotor Modulation
Similar to jet engine modulation ( reflection by rotating compressor fan)
RCS has Doppler spectrum caused by rotating rotor blades and hub.
Can be used for target detection, classification and recognition
Spectrum determined by
Shape of hub
Shape and material of blades
Number of blades
Rotational speed of rotor
Radar cross section computations based on asymptotic high frequency techniques: Physical Optics/ray-tracing and the Method of Equivalent Currents. Incorporation of landing gear, weapons and other protuberances would require (hybrid) full-wave modelling.
Here quasi-steady analysis of seperated rotor and fuselage.
Higher order effects to be included
Electromagnetic blade-blade and rotor-fuselage interaction
Deflection and/or pitch changes of rotor blades
12/5/2018

13. Finite Element Method – The Basis functions
Vector based basis functions are chosen to prevent spurious
solutions to occur.
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14. 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
Important feature of radar signature of helicopter is:
HERM = Helicopter Rotor Modulation
Similar to jet engine modulation ( reflection by rotating compressor fan)
RCS has Doppler spectrum caused by rotating rotor blades and hub.
Can be used for target detection, classification and recognition
Spectrum determined by
Shape of hub
Shape and material of blades
Number of blades
Rotational speed of rotor
Radar cross section computations based on asymptotic high frequency techniques: Physical Optics/ray-tracing and the Method of Equivalent Currents. Incorporation of landing gear, weapons and other protuberances would require (hybrid) full-wave modelling.
Here quasi-steady analysis of seperated rotor and fuselage.
Higher order effects to be included
Electromagnetic blade-blade and rotor-fuselage interaction
Deflection and/or pitch changes of rotor blades
12/5/2018

15. 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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16. 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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17. Dispersion (2) Exact phase difference Dispersion error
Computed phase difference
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18. Dispersion (3)
Wave without error
Wave with
dispersion error
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19. Complexity
Purpose: Minimize dispersion error and compute RCS accurately
Dependence of matrix size on
dispersion error
element order
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20. 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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21. 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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22. Ideas for Future Research
Use efficient iterative method (GCR or COCG)
Construct an effective preconditioner. Observed: ‘simple’ preconditioners do not work.
ILU and approximate inverse 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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23. Questions and Suggestions
Other Remarks?
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