Method for computing self-contamination processes of a spacecraft
A method for computing self-contamination processes of a spacecraft by means of a data processing device comprising the following steps: receiving a first set of input parameters comprising general definitions of the spacecraft, receiving a second set of input parameters comprising control parameters for the spacecraft orbital data, physics, numeric, and a predetermined accuracy requirement of the computation, computing a self-contamination process of the spacecraft based on the received first and second sets of input data by either evaluating the analytical solution of a basic equation of emission or numerically solving the basic equation of emission for calculating a deposit of molecules outgassed from surfaces of the spacecraft with a numerical solver with the data processing device, wherein the numerical solver applies an adaptive stepsize control based on the preset accuracy requirement of the computation, and outputting the calculated deposit.
1. A method for computing self-contamination processes of a spacecraft by means of a data processing device comprising the following steps:
receiving a first set of input parameters comprising general definitions of the spacecraft,
receiving a second set of input parameters comprising control parameters for the spacecraft orbital data, physics, numeric, and a predetermined accuracy requirement of the computation,
computing a self-contamination process of the spacecraft based on the received first and second sets of input data by either evaluating the analytical solution of an equation of emission or numerically solving the equation of emission
W
.
dep
,
i
=
(
F
ji
-
δ
ij
τ
r
,
j
)
W
dep
,
j
+
F
ji
W
.
emitted
,
j
for calculating a deposit of molecules outgassed from surfaces of the spacecraft with a numerical solver with the data processing device, wherein the numerical solver applies an adaptive stepsize control based on the predetermined accuracy requirement of the computation, wherein F ji denotes a view factor from node j, first surface mesh element, to node i, second surface mesh element, δ ij denotes the Kronecker delta from node i to node j, τ rj denotes the temperature-dependent reemission time of node j, τ ej denotes the temperature-dependent emission time of node j, and wherein calculation of a deposit of molecules outgassed from surfaces of the spacecraft with a numerical solver comprises a stepwise analytical integration of the emission term W emittable of the equation of emission according to the following equation:
Δ
W
emittable
,
j
=
W
emittable
,
j
(
t
n
)
(
1
-
(
τ
e
,
j
(
t
n
)
τ
e
,
j
(
t
n
)
+
α
j
h
)
1
α
j
)
,
with α j being the secant slope defined by
α
j
=
τ
e
,
j
(
t
n
+
1
)
-
τ
e
,
j
(
t
n
)
t
n
+
1
-
t
n
,
τ e designating time dependent material parameters, and h designating the time interval h=t n+1 −t n , and
outputting the calculated deposit;
wherein the numerical solver applies a theta-method for calculating an analytical solution of the equation of emission for calculating a deposit of molecules outgassed from surfaces of the spacecraft and adapts the stepsize of the theta-method by performing the following steps:
a) choosing an initial stepsize h lte and setting it as actual stepsize,
b) calculating the local truncation error LTE of the theta-method for one step with the actual stepsize,
c) calculating the stepsize for the next step depending on the calculated local truncation error LTE and setting the calculated stepsize for the next step as actual stepsize,
d) repeating steps b) and c); and
wherein calculating of the stepsize h tol for the next step depending on the calculated local truncation error is performed according to the following equation:
h
tol
=
min
(
h
lte
TOL
m
LTE
m
(
Θ
prop
)
)
with h lte designating the chosen initial stepsize, Θ prop being either Θ 1 or Θ 2 depending on which one is selected as a propagator, LTE m (Θ prop ) designating the local truncation error calculated for a pair of nodes W m , V m for the selected theta-method Θ prop , wherein TOL m designates the predetermined accuracy requirements or tolerances that are applied at time t n+1 in the calculation of the next timestep and are selected from two predetermined values TOL m,abs and TOL rel according to the following rule:
if U m (t n+1 ,Θ prop )>TOL m,abs
TOL m =TOL rel u m (t n+1 ,Θ prop )
else
TOL m =TOL m,abs
where U m (t n+1 ,Θ prop ) is either W m (t n+1 ,Θ 1 ) or V m (t n+1 ,Θ 2 ) depending on which one is selected as propagator, and wherein
w
(
t
n
+
1
,
Θ
1
)
=
y
(
t
n
)
+
h
[
Θ
1
f
(
t
n
+
1
,
y
(
t
n
+
1
)
)
+
(
1
-
Θ
1
)
f
(
t
n
,
y
(
t
n
)
)
]
;
v
(
t
n
+
1
,
Θ
2
)
=
y
(
t
n
)
+
h
[
Θ
2
f
(
t
n
+
1
,
y
(
t
n
+
1
)
)
+
(
1
-
Θ
2
)
f
(
t
n
,
y
(
t
n
)
)
]
;
and
LTE
m
(
Θ
prop
)
=
(
1
2
-
Θ
prop
)
w
m
(
t
n
,
Θ
1
)
-
v
m
(
t
n
,
Θ
2
)
Θ
1
-
Θ
2
,
wherein y(t n ) denotes the exact solution of the LTE at a time t n , wherein y(t n+1 ) denotes the exact solution of the LTE at a next time t n+1 and wherein
f(t n+1 ,y(t n+1 ))={dot over (y)}(t n+1 ).
2. The method of claim 1 , wherein the calculating of the local truncation error LTE of the theta-method comprises
calculating a first local truncation error LTE explicit using the explicit Euler method,
calculating a second local truncation error LTE implicit using the implicit Euler method, and
selecting as the local truncation error LTE the one of the calculated first and second local truncation error LTE explicit and LTE implicit , with the smaller magnitude.
3. The method of claim 1 , wherein the calculating of the local truncation error LTE of the theta-method is performed according to the following equation:
LTE
m
(
Θ
prop
)
=
(
1
2
-
Θ
prop
)
w
m
(
t
n
+
1
,
Θ
1
)
-
v
m
(
t
n
+
1
,
Θ
2
)
Θ
1
-
Θ
2
with Θ 1 , Θ 2 designating different theta-methods, Θ prop being either Θ 1 or Θ 2 depending on which one is selected as a propagator, W m , V m designating the deposit on a pair of nodes or mesh elements of the surface of the spacecraft, which is represented as a collection of surfaces, which are subdivided into mesh elements with outgassing properties.
4. The method of claim 1 , wherein Θ 1 , Θ 2 are set to the explicit and implicit Euler method with Θ 1 =0, Θ 2 =1 the implicit Euler method Θ 2 is selected as propagator Θ prop .
5. The method of claim 2 , wherein the theta-method comprises a higher order linear multistep method, which is used to calculate the results and/or propagate the solution.
6. The method of claim 1 , wherein the calculating of the stepsize for the next step depending on the calculated local truncation error LTE comprises the following steps:
if the local truncation error LTE is smaller than or equal to the predetermined accuracy requirement TOL performing the following steps: accepting the step, calculating an increased stepsize for the next step depending on the calculated local truncation error LTE and setting the increased stepsize as actual stepsize, and
if the local truncation error LTE is larger than the predetermined accuracy requirement TOL performing the following steps: discarding the step, calculating a decreased stepsize for this step depending on the calculated local truncation error LTE and setting the decreased stepsize as actual stepsize.
7. The method of claim 1 , wherein the equation used for the stepwise analytical integration of the emission term W emittable of the equation of emission is expanded for small values of α j to
Δ
W
emittable
,
j
≅
W
emittable
,
j
(
t
n
)
(
1
-
exp
(
-
Δ
t
τ
e
,
j
(
t
n
)
)
(
1
+
(
Δ
t
)
2
α
j
2
τ
e
,
j
2
(
t
n
)
)
)
.