Time-Domain Analysis of Lightning Interaction to
Aeronautical Structures Composite Materials
Marcello D’Amore, IEEE Fellow
Maria Sabrina Sarto, IEEE Member
University of Rome “La Sapienza”, Department of Electrical Engineering
via Eudossiana18,00184 Rome, Italy
Abstract: An efficient procedure based on the three-dimensional
finite-difference time-domain approach is developed for the
electromagnetic simulation of lightning interaction to structures of
composite materials, realized by thin panels protected by conductive
paint. The procedure is applied to the analysis of the
electromagnetic field inside a cubic box with inner wiring, excited
by a transient low current injection.
covers and different bonding arrangement. The thin panel
formulation [5] is applied to the modeling of bilayered slabs,
realized by metal coated non-conducting composite material
(typically fiberglass, Kevlar, PEEK). Actually, standard FDTD
schemeswith regular grid cannot be applied to the analysis of these
structures because of computer limitation; moreover, equivalent
surface impedance approachesare limited to ‘the modeling of thick
layers or plane wave interaction to thin slabs [7].
INTRODUC-ITON
The susceptibility of aircraft to lightning is a critical issue due to the
widespreading use of advanced composite materials (ACM) in the
aeronautical industry and to the increasing automation of onboard
equipment. Direct lightning effects are critical from both the
thermo-mechanicaland electromagneticpoints of view [ 11.
An extensive report on the lightning phenomenologyand the related
risks in the aircraft environment is provided by [2]. The
electromagnetic characterization of lightning stricken aircraft is
described in [3], basing on a finite-difference time-domain (FDTD)
procedure;the metallic structure is consideredas perfect conducting,
whereas composite panels are simulated as lossless dielectric
homogeneousslabs. A 3D-FDTD procedure is also presented in [4]
for the analysis of the electromagnetic interaction of incident plane
wave to helicopters; the composite parts of the structure are
considered to be realized in fiberglass or Kevlar and described as
thin dielectric panels [5]. These studies are mainly focused on the
simulation of complex geometry rather than on the modeling of the
composite material; moreover, the effects of lightning protections
realized by metallic sheet or covers are not analyzed.
The aim of this paper is to develop efficient FDTD models of
structures realized by thin composite panels, including also the
lightning protections, for the electromagnetic analysis of the direct
lightning effects. The study is carried out assumingthat the thermal
and electromagnetic phenomenology related to the direct lightning
event can be studied independently, becausethey are characterized
by very different time-constants. Actually, when the lightning
attaches the structure, the injected current during the first stroke,
typically characterized by front time of a few microseconds and
emivalue time of several tens of microseconds, flows on surfaces
which are not damagedyet by the thermal effects, describedby timeconstant in the range of milliseconds. The same assumption can be
made for the following strokes, considering that the attachment
point moves along the hit surface.
In a preliminary study a 3D-FDTD procedure was developedfor the
numerical simulation of lightning tests on thin advanced composite
panels [6]. In the following, the approach is extended to the analysis
of more complex configurations, including wiring inside the stricken
structure, in order to verify the effectiveness of protective metallic
397
TIME-D•
MAINELECTROMAGNETICMODELING
Consider a three-dimensional structure of composite material; the
walls are realized by panels of thermoplastic resin or glassfiber,
covered by a thin conductive substrate representing the lightning
protection. A transient current is injected on the surface of the
structure, in order to simulate the direct lightning attachment. In the
time-domain the lightning current can be described by the following
function [S]:
i,(t)A- (f/da exP(-tb2)
h l+( t/$
(1)
in which c1= 10, I,& is the current maximum amplitude, with h=
0.93, and zl=90 ns, ~40 us are the time-constants. The maximum
frequency excited by the lightning current reaches about 20 MHz.
For typical configurations, the thickness of the composite slab is in
the range of l-2 cm, and can be consideredto be small compared to
the minimum wavelength (about 7 m for PEEK), as well as the
thickness of the metal coating (10 pm roughly) is small comparedto
the minimum skin-depth (about 360 pm for aluminum paint).
The analysis of the configuration is performed in the time-domain
basing on the integral formulation of the curl Maxwell’s equations,
which are expressed in finite-difference form by applying Yee’s
scheme [9]. Since the minimum geometrical dimensions of the
composite structure can be small or comparable with respect to the
minimum wavelength in air, a subgrid approach is developed in
order to reduce the computational efforts. The thin panels realizing
the walls of the enclosure are modeled by applying the procedure
described in [6]. Special care is paid to the simulation of the current
injection and grounding connections of the structure, realizing the
current return path.
Simulation of JD-ACM Structures
Consider the uniform space discretization described by the space
step As, which is related to the minimum wavelength in the medium
surrounding the structure. The time-discretization step At is related
to As by Courant’s condition [4]. In the standard Yee’s FDTD
scheme the space-grid for the calculation of the magnetic field
0-7803-4140-6/97/$10.00
components is shifted of As/2 along the x-, y-, and z-axes with
respectto the electric field spacegrid:
Assume that a rectangular (n&xn,A~xn,A~) enclosureis placed in
the electric-field grid, and is bounded by the planes X=&A& x=&As,
y= j&, J+h, z=k&, z=k&, with it = ib+&, jr = j$-n,,, kt = kb+n,.
The magnetic field is continuous across the non-magnetic thin
bilayered panels realizing the walls of the enclosure,as well as the
tangential componentsof the electric field. The compositeslabs are
then modeled in the FDTD grid consideringas additional unknowns
the normal componentsof the electric field in each layer [6].
magnetic field at y = ob-l/2)fh.
The additional unknowns E,,I, Ey2in
the two thin layers at y=jbhs are computed as functions of the
horizontal componentsof the magnetic field at y=(jb-L/2)b.
Special care must be paid to the modeling of the enclosure edges,
where thin compositepanels cross. For instance along the edge x=
i&r, y=j&, the updating equation for Hz at X=(ib-l/2)&,
y=
0;+1/2)As includes the contribution of both I342 and E,I,z which are
the electric field normal componentsto the boundary walls at X=i&
and y=j,A.s,respectively.At the same edge the updating equation for
E, is unchanged,whereasthe additional unknowns EXl,2and Ey1,2are
computed considering that the flux of the electric field takes place
Let’s then consider for instance the upper wall of the enclosure,
partially in air, partially in the medium with conductivity cl,2 and
located in the (x, z) plane at y = j&s. Fig.l(a) shows the (x, y)
external surface of the generic cell of the magnetic field grid permittivity sr,2.The resulting equationsare reported in Appendix.
crossing the thin panel. The z-componentof the electric field at y =
Simulation of Lightning Injection and Grounding Connections
j& is expressed in finite-difference form by applying Ampere’s
law, considering that the conductivity and electrical permittivity of Assume that the lightning current (1) is injected on the upper wall of
the two layers of the panel, having thicknessdr and d2, are ~1, ~1and the enclosure, at x=&As and z=k&s. Considering that the current
or, ~2,respectively. Similarly, the z-componentof the magnetic field lead is a perfect electrical conductor, the normal componentsof the
at y = (ir+l/2)A3 is obtained by applying the integral form of the electric field to the panel in the injection point are set to zero. The
electric field curl-Maxwell equation to the contour in Fig.l(b). lightning current is(t) is then supposedto share equally among the
Notice that HZ is expressedas function of the normal componentsto horizontal componentsof the magnetic field on the surface of the
the layers of the slab, E,,r, E,Q. These additional unknowns are still slab, H, and Hz, computed at x=z&As, z=(k&1/2)As
and
computed by using Ampere’s law as functions of the magnetic field x=(i&l/2)As, z=k&s, respectively.
componentsH,, HZ at y = (jt+l/2)dr.
The current return path is modeled by a delayed current source,
on the lower face of the enclosure, located at y = jb.&, similar which is applied in the groundedpoint of the structure, at x=i,,As,
expressionsare applied to upgradethe horizontal componentsof the y=jO&, z=k,,&. The outgoing current is characterizedby the same
electric field EX,E, at y =jbA.s, and the horizontal componentsof the waveform (1) of the injected one, but is opposite in amplitude. The
time-delay is computed with reference to the length of the shortest
path connecting the injection and outgoing points of the current,
along the metallic surfaceof the enclosure:
1
(i+ll2, jftll2, k+1/2)
(i-1/2, j&l/2, k+1/2)
74
I
I
Ip=npAs
(2)
I Hx
I
With:
01,
El
I
As
np =min(n~X,n~X)+min(n~,n~Y)+min(n~Z,n~)
(3)
in which:
(i-1/2,jr1/2, k+1/2)
(i+1/2,jr-l/2, k+1/2)
n*P = *imax+nx FiinFi 0”
(44
n*py = *i,,
(4b)
+ny T jhT j,,
n*pr =&k,,+nZTki,Tk
(a)
ou
(4c)
assuming that the enclosure is centered in the three-dimensional
(i,,Asxj,&x
km&s) grid.
As
In practical test contigurations the current outgoing is not
concentrated in a single point of the composite structure, but
generally an external wall, or a part of it, is grounded.In this case a
distribution of delayed current sources is considered; the total
injected current is assumedto share equally among the grounded
cells of the structure.
Subgrid Procedure
(b)
Figure 1. Contours on the magnetic (a) and electric (b) field
grids crossing the upper wall of the enclosure at y=j&.
In order to reduce the computational efforts of the procedure the
calculation is performed by using the time-step A,T, considering a
coarsegrid having discretization step AS and (ImaxxJmawxKmax)
cells,
truncated by MIX’S absorbing boundary conditions [lo]. A finer
space-time-grid, having (imaxx&,axxkmax)
cells and discretization
398
steps As, At, is considered in a (n,ASxn&Sxn,AS) selected
subvolumeof the analysis domain, for the accuratemodeling of the
stricken structure. At this purpose each (ASxASxAS) cubic cell of
the primary uniform mesh is divided in nSUb(As~A.~~As)cubic
subcells; nSUbis assumed to be an even integer number. At the
boundary of the fine grid the tangential componentsof the electric
field at eachnode (ij,k) are computedby linear interpolation in time
and space of the computed values of the electric field tangential
componentsat the neighbouringnodes of the primary grid.
Consider for instance the boundaries of the fine grid located at
K=(K,,,=-nJ2 or K=(Kmax+nsr)/2.The x-component of the electric
field I?;, at x = (i+1/2)Aq J+A.s in the sub-meshat time step t=nAt,
With:
(I-n,)n,,bsi<(z+1/2-n,)n,,b
PO
(J-n=~)n,,~j<(j+l-n,)n,,b
(5b)
(N-l)n,,I
PC)
A,=(~-A+%,)
is given by the following expression,accordingto Fig.2(a):
+A3Eygw+U +A4 E;u;y.J+‘)
(6)
in which the x-componentsE, of the electric field in the primary grid
nodes,at time step nAt, are obtained by interpolation of the values at
times (N-1)AT and NAT:
(l-&hub)
A,=(Ai/n,"b)(Aj/n,b)
> A, =( A+‘&,) (I- Aj/n,,)
(8a)
2 A, =(I- Ai/n,)
W)
(Ahut,)
With:
Ai=i+1/2-(I-l/2-n,)n,,
W
Aj=j-
PI
t J-nsy 1nsub
Similar expression is obtained for the evaluation of the field
component.t?, at x = (i+l/2)dr,~jA~, with (Fig.2(b)):
(z+1/2-n,)YzS”~Ii<(z+1-n,)n,“b
(10)
Expression(6) is rewritten in the following form:
@$.O
n<Nn,,
jj(i+W) = A, @{;&J) +A, pp
x.(n)
and:
= A1E;I$sJ) +A, @ ‘$,“)
+A&;‘-$5’2.J+‘) +A4 @(+,‘)‘z.J+‘)
(11)
The parameters AM are still given by (8a,b), in which Ai is
expressedby:
Ai=i+1/2-(Z+1/2-nsX)n,,b
(12)
ELECTROMAGNETICANALYSISOFLIGHTNINGATTACHED
COMPOSITE
!hRUCTURES
The proposedprocedureis applied to the electromagneticanalysis of
non-disruptive lightning tests on a cubic box of 1.8 m side with
inner wires, having the arrangementof Figs3(a-f). The considered
(4
(b)
(4
Figure 2. Coarse and fine grids on (x& boundary surfaces.
Space-interpolation schemesfor &at (i+1/2&, with
(Z-1&5&(Z+l/2-n,)
(a) or (Z+1/2-n,)ri<(Z+l-n,)
(b).
399
(e>
Figure 3. Sketch of test configurations (distances in meters).
notice that the maximum amplitude reaches100 mA. It is interesting
to observethe consistentreduction of the amplitudes of the electric
and magnetic fields over the horizontal plane at 1.35 m below the
upper face. Fig.%(a) shows that the electric field assumes the
Figure 4. Composite panel with metallic paint.
maximum of about 93 V/cm correspondingto the attachmentpoint.
The influence of the wire is strongly mitigated in this case; the
set up representsa canonical configuration, suitable to understand peaks appearingon the left of the E- and H- maps in FigsX(a,b) are
the electromagneticphenomenarelated to direct lightning stroke, due to the presenceof the inner conductor.
such as coupling effects in wires and sparks. Moreover, it should be
of interest to predict currents induced on the navigation-lampground If the 1.2 m long wire is open-endedat both sides (Fig.3(c)) the
wires inside the tail unit fin cap, which is classified as “zone 1B” maximum induced current reaches 40 mA. If it lays on the (x,z)
with high probability of lightning attachment[ 11.
plane in the middle of the enclosure, as shown in Fig.S(d), the
current waveform along the wire is characterized by the shape
The walls of the enclosure are realized by PEEK panels, having reported in Fig.9, where the peak value is reduced to about 20 mA.
thickness dr=l cm, conductivity 0i=10-‘~ S/m, relative permittivity In fact the excitation of the wire, represented by the tangential
E&S, protected on the external faces by a thin layer of aluminum component of the electric field, which assumes maximum value
paint, with dz=lO pm, oz=105S/m, ~a=2 (Fig.4).
along the current-injection direction, is decreased.
A current having the waveform described in (1) with Lo=32 A is
injected on the upper face of the box, in order to simulate lowcurrent injection test [2]. The current outgoing is assumedto be
distributed along an area of (1.2 mxl.2 m) centeredin the (1.8 mx
1,8 m) lower face of the box.
Figs.lO(a,b) show the distributions of the electric and magnetic
fields on the (XJ) plane located at 1.35 m below the hit surface of
the box in Fig.3(e), in which two vertical open-ended wire are
located. The two peaks appear correspondingto the positions of the
The analysis is performed applying the described time-domain
numerical procedure, considering the primary grid with AS=O.3m
and the fine mesh with n&=4. The inner conductors,having radius
of 1 mm, are modeled by applying the thin-wire formalism [I 11,
neglecting the effects of frequency-or temperature-dependinglosses.
The spatial distributions of the electric and magnetic fields are
computedon the (x,z) plane located at 1.35 m below the hit surface,
where the effects of the wiring is more evident, at time ~250 ns,
correspondingto the peak of the injected current.
In the first application the box is stricken in the middle of the upper
face (Fig.3(a)), a 1.2 m long wire is electrically connectedat the
attachment point and open-endedat the other side. The obtained
results show that the most of the injected current flows through the
thin wire, where the maximum reachesabout 27 A. The distribution
of the current density over the struck surface is reported in Fig.5 at
time t-250 ns. The maps of the electric and magnetic field
amplitudes over the considered(x,z) plane are shown in Figs.d(a,b).
The presenceof the wire modifies sensibly the field distribution; the
amplitude of the electric field reaches the maximum of about 2.5
kV/cm immediately below the lower end of the conductor.
Next, the calculation is performed assumingthat a 1.8 m-long wire
is connectedbetween the upper and lower faces of the box, and the
current is injected in proximity of a comer (Fig.3(b)). The time
waveform of the induced current along the wire is reported in Fig.7;
.8
@I
Figure 6. Distributions of the electric (a) and magnetic (b)
field amplitudes on the (x,z) plane at 1.35 m below
the hit surface of the box in Fig.3(a).
1.8
Figure 5. Current density distribution on the hit surface
of the box in Fig.S(a).
0
Figure 7. Time-waveform of the current distribution along
the 1.8 m long wire inside the box of Fig.3(b)
400
[V/cm]
kli
1.8
1.8 -0
(b)
Figure 8. Distributions of the electric (a) and magnetic (b)
field amplitudes on the (XJ) plane at 1.35 below the
hit surface of the box in Fig.3(b).
(b)
Figure 10. Distributions of the electric (a) and magnetic (b)
field amplitudes on the f&z) plane at 1.35 m below the
upper face of the box in Fig.3(e).
[V/cm]
150
75
0
x[m]
t
Ins1
1.8
0
Figure 9. Time-waveform of the current distribution along
the 1.2 m long wire inside the box of Fig.3(d)
two conductors. Notice that the closer is the wire to the injection
point, the higher is the maximum induced current: about 90 mA and
40 rn4 for the nearestand farthest wires, respectively.
Finally, the configuration shown in Fig.3(f) is considered,in which
the lower face of the enclosure is completely electrically connected
to a perfect conducting plane. The current outgoing is located in a
single point outside the box, at x=-3 m, y=O, z=4.8 m on the ground
plane. The electric-field spatial distribution on the horizontal plane
at y=O.45 m inside the box is computed considering at first only the
two vertical wires; the obtained map is reported in Fig.1 1. The
comparison between Fig.lO(a) and Fig.1 1 shows that in the new
configuration the electric field map spread out and the amplitudes
are reduced, due to the increased area of the grounded part of the
structure. However, if the presence of the third horizontal wire is
considered in Fig.3(f), a consistent increment of the electric and
magnetic field on the same plane is observed(Figs.l2(a,b)), due to
the contribution of the current induced in the third wire, which
reachesthe maximum amplitude of about 350 mA.
401
Figure 11. Distribution of the electric field amplitude on the
(x,z) plane at FO.45 m inside the box of Fig.30
with only the two vertical wires.
CONCLUSIONS
An efficient procedure,basedon a 3D finite-difference time-domain
approach,is developedfor the electromagneticanalysis of lightning
interaction to composite structures realized by thin panels protected
by aluminum paint, with inner wiring.
The obtained results show that the electric field inside a lightning
striken cubic enclosure reaches maximum amplitude below the
attachmentpoint or in proximity of inner wires. It is also observed
that the highest currents are induced in conductors which are
parallel to the direction of the current injection. Moreover the peak
values of the electromagnetic field in the box are strongly affected
by the presence of the wires, and increase considerably if the
conductoris directly hit by the current injection.
APPENDIX
Along the edge x = i&s, y=j& of the enclosure, the updating
equations for the field componentsE, and Hz, at (ia j,, k+1/2) and
(ib-1/2,jt+1/2, k), respectively, are expressedin the following form:
E&i,,k+lt2)
(i,,j,-llZ,k+lIZ)
+bEc
2) [HX,(,,,,)
=a EC E G)(ibJ’tl,k+1’
.e+‘)
+H(ibtl/2,jt,k+l12)
H(ib-l/2,j,+1/2,k)
z,(n+l/2)
y,(n+ll2)
=H(j,-%it+1&k)
z,(n-112)
+bHo
t-4.1)
1
_ H(~b-112,jt,k+lLl)
y,(n+ll2)
_ H(ib,jt+l/2L+1/2)
x,(n+ll2)
{El’e(+;kJ’+“k)
+E$-;.6+1/2,k)
-[~-(dl+d,)/~][E~~~~‘.k)+E~~+1’2’k)]
(~.2)
in which:
1.8
2Ec -o&it
aEc =
(b)
Figure 12. Distributions of the electric (a) and magnetic (b)
field amplitudes on the (x,z) plane at-l.35 below the
hit surface of the box in Fig.30.
(A.3a)
2&, +g&
Ec=Ea[l-(d,
’
+d2)/As]+s1
(A.3b)
dl/As+EZdJAs
cc =a, d, /As+c2 d, /As
(A.3c)
hm = At/i JAI4
(A. 3d)
REFERENCES
PI
J. A. Plumer, J. D. Robb, “The direct effects of lightning on
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141 P. A. Tirkas, C. A. Balanis, M. P. Purchine, G. C. Barber,
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and E$;;,, E;;;“c;, are the normal componentsto the two layers of
the composite walls parallel to the b,z) and (x,z) planes, computed
along the edgeX=ib.&, y=j& by the following expressions:
E (j, A
xl,2(n+l)
;,,2
Ez;;;,
+b;1,2
[ #d;;i;
.k-112)
_ &k-k-;;,
,k+1’2)
=u
+H(ib-112,jt+l/2,k)
r,(n+l/2)
_ H(ib-112,jt-112,k)
z,(n+l/Z)
1
(A.4a)
The constantsa’sI,& b’sI,Zare given by:
(A.5a)
412=
’
2(%2
+E,, ) +a,,At
,
1
b’ -f!!!
(A.5b)
“” - As 2( q2 +so) +o,,,At
At the comers X=ib& y=j,&s and Fkb,& the updating equations
(A.4a,b) still apply, in which constantsa’E1,2,b’E1,2are replacedby:
2(%,2
+%)-01,2at
a’L’2 = 2 ( E,,~ +34
b” -- 8At
(A.6a)
+q,At
1
E1s2- As 2 (s,,, +34
(A.6b)
+o,,,At
402