Mixed time integration methods for transient thermal analysis of structures are studied. An efficient solution procedure for predicting the thermal behavior of aerospace vehicle structures was developed. A 2D finite element computer program incorporating these methodologies is being implemented. The performance of these mixed time finite element algorithms can then be evaluated employing the proposed example problem

Liu, W. K.

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https://ntrs.nasa.gov/search.jsp?R=19830012012 2020-03-21T04:04:55+00:00Z
Mixed Time Integration Methods for Transient*
Thermal Analysis of Structures
by	 MAR In83
RECEIVED
W. R. Liu	 Principal Investigator41
T. F. Zhang Visiting Scholar	 'im am
J. Lin	 Research Assistant
Northwestern University
Introduction
The purpose of this report is to give a brief summary of the progress
made in the study of mixed time integration method.- for transient thermal
analysis of structures during the period from October 1, 1981 to January 15,1982.
A simple and illustrative example problem is also proposed (as suggested by
Dr. Olsen) to demonstrate the practicability and usefulness of our proposed
approach. We believe that we have now developed an efficient solution
procedure for predicting the thermal behavior of aerospace vehicle
structures. Currently, a 2D finite element computer program incorporating
these methodologies is being implemented. The performance of these mixed time
:Finite element algorithms can then be evaluated employing the proposed example
problem. Related discussion is in later section.
Summary of Developments
A family of mixed time integration schemes in which different time
integration methods (implicit/explicit) with different time steps can be used
in each element group is developed. Its underlying theoretical framework for
*This research is sponsored by a NASA Grant No. NAG-1-210 and Dr. George Olsen
is the Program Manager.
(NASA-CR-1700F 1)	 GUYED TICS ItTEGEATICN	 N83-20283
FETHCDS FOE TRANSIENT THEPPAL ANAIISIS OF
STRUCTURES, IPPENDIX 5 (Northwcsterr Univ.)
8 p HC A02/Mf A01	 CSCL 20K	 Unclas
9 D^1 a	 ^ L:
ORIGINAL PAGE N
OF POOR QWlM
large scale multi-dimensional linear transient analysis is studied. The
s`	 effective computer implementation aspects of these proposed techniques are
also investigated. The "active column equation solver" is the key to success
of this technique. The equations system for each element group is conatructed
in such a way that they are uncoupled and hence each group can be integrated
at its own group time step with different integration methods. The stability
characteristic of these mixed time partition algorithms is also studied. An
energy balance technique is employed to carry out the stability analysis. It
is found that the critical time step restrictions are governed by each
individual explicit element group only. Of more significance, these mixed
time implicit-explicit methods provide a natural framework for the further
development of efficient, clean and modularized computer codes.
The above developments are summarized in the attached two papers.
Evaluation of the Mixed Time Implicit-Explicit Algorithms
A number of simple 1D numerical examples are presented in the two papers
to demonstrate the fessibilitiss (i.e. accuracy and stability behavior) of the
algorithms. We have formulated herein two finite element thermal models for
the Thermal Protection System (TPS) of the space shuttle to evaluate the
performance of our mixed time implicit-explicit, method. The first one is a 1D
model while the second one is a 2D model. They are depicted in figures 1 and
2 respectively. The initial phase of the research project is the development
of the mentioned methodologies in which linear assumption is employed.
Nonlinear effects such as internal and external radiation, and natural
convections are ignored. Further only the assumed mean temperature material
properties of the varicis components of the TPS are considered. Its material
properties are tabulated in table 1. The number of elements, the minimum
characteristic length Lin , the estimated explicit critical time step at min,
ithe proposed integration method and the proposed element group time step for
each group are included in table 2. As can be seen from table 2. due to the
various thermal time scales (e.g. ot min M 0.083 for AL, and
at 
min' 133.6 for RSI), a single integration method is definitely not
effective. For example, if an explicit method is employed, a time step of
0.083 has to be used; while if an implicit method is employed, there is no
stability-imposed limitation on of ; however, wide-banding and/or non-
convergence of the resulting matrix equations may decrease its advantage. The
family of mixed time integration schemes developed in best suited for this
type of problem. The attributes of the various time integration methods are
fully achieved using the proposed approach as can readily be seen from
table 2. It should also be observed that At mincan be se: as high as 33.42
in this mixed model. Subsequently of - 33.42 is employed for element groups
1 and 2 (042 Coating and RSI), At - 66.84 for element group 3 (RSI),
and of - 133.68 for element groups 4 and 5 (RTV, FELT, AL and AIR). However,
judging from the heat loading versus time curve (see figure 3) it is advisable
to pick at min 0 51.4/4 - 12.85 during the heat loading period (i.e. time - 0
to 1200) and at min" 33.42 in later time. Of course, an automatic selection
of the step size based on truncation error and accuracy considerations is
desirable. It is not being considered here because of the limited time frame
and budget of this investigation.
The importance of this proposed mixed time implicit-explicit finite
element concept can further be visualized if a nonlinear analysis of the above
finite element models is assumed. The thermal responses of the various
components of the TPS may be divided into regions of slowly and rapidly
varying temperatures and they are:
_, z- F M
s	 ^
ORIQiMAL PAGE 8
1. 042 Coating	 oF POM QUAD
The thermal diffusivity is almost independent of temperature and the
estimate at is so small (0.25) that explicit calculation is not cost
effective at all. Implicit calculation is best suited since only
limited arrtti_al reformation and refactorization of this element
group's effective stiffness will be needed. This is one of the major
advantages of the mixed time implicit-explicit concept.
2. RSI
The thermal diffusivity changes rapidly with temperature, therefore
reformation and refac.torization of this element group's effectiva
stiffness are frequently required if implicit method is employed.
Non-convergence and/or wide-banding of the resulting element group
effect i ve stiffness may decrease the advantage of applying the
implicit method here. The estimated at (based on mean temperature
value) is 33.42; therefore explicit is best suited (no matrices
operations).
3. RTV
Its thermal diffusivity is independent of temperature and the
estimated at is too small (0.213) that implicit method is
recommende3.
4. AL
Its thermal diffusivity is fairly independent of temperature (at
least in the operational temperature range of the space shuttle) and
its estimated at is so stringent (0.083) that implicit method is
ags'n recommended.
S.	 FELT
Its thermal diffusivity is independent of temperature and its
i
ORIGINAL pAQE R
r OF PWR QUALMY
tstals t-ed At is large (51.4), therefore implicit method or explicit
method can be used. For this example, we propose to employ implicit
method. However for 3D calculation, in order to reduce the band-
width and the unnecessary nonlinear calculations (due to the fact
that the adjacent groups might be implicit groups too), explicit
method will be recosmsncied.
6.	 AIR	 .
The variations of the thermal diffusivity with temperature are small
in our range of interest. Also the estimated at ie small, therefore
implicit method is again well suited.
Remarks:
(1) In the above At calculations, 
'lain 0 min Itx , ty I where t  and I 
are defined as follows:
T
ty
tx
(2) The estimated at is defined to be 12 
min /a. It is a conservative
critical time step calculation. Note that a is the thermal
diffusivity.
(3) By virtue of the fact that each element group's effective stiffness
is uncoupled to the global assemblied matrix equations system, any
element group can be reformed and refactorized at any instant if
required without affecting the global equations system. This arp tial
factorization procedure can further be enhanced if it is to be
combined with an iterative update procedure.
f f
M ^	 t
(4)
ORIGINAL PAGE !a
n _\j
N
OF OR QUALM
In the case of short time duration transients, mixed time explicit-
explicit is most effective. It is even more attractive if the mixed
time explicit-explicit procedures are unconditionally stable.
Conclusions and Future Directions
The work summarized here is not complete. The most important work in
progress is the development of mixed time integration methods for transient
thermal analysis of structures suitable for incorporation into most finite
element computer codes. These mixed time procedures are currently being
integrated into a pilot finite element computer program. Detailed studies of
solutions of the proposed examples are planned.
r+
OR OKU PAGE 4
OF POOR QUALITY
Table 1. Mean Temperature Material Properties
NATKKIAL
042
COATING RSI RTV FELT Al. AIR
PROPERTIES
A w/m-k 1.4338 0.1354 0.3113 0.0363 131.3573 0.0271
conductivity
p kg/w3 1665.92 144.17 L409.62 96.11 2851.28 1.126
density
C J/kg-k 1317.96 1255.20 1213.36 1213.36 962.32 1009.0
specific heat
a•	 m2 /sec 6.53x10-7 7'.48x10-7 1.82x10-7 3.11x10-7 4.78x10-5 2.39x10 5
PC
thermal
diffusivity
I%e mean temperature material properties are computed from the average of those at
1200 k. 950 k. 500 k. 477 k, 333 k and 300 k.
ELEMENT MA?ERIAL NUMBER OF ESTIMATED INTEGRATION EVXENT GROUP
GROUP TYPE ELEMENTS Imin(cws) At METHOD TIME STEP
UMBER
1 042 20 0.04 0.25 I 33.42
coating (-At)
2 RSI 50 0.5 33.42 E 33.42
3 RSI 80 1.04 133.6 E 66.84
(-2At)
4 RTV 20 0.02 0.213 133.68
FELT 10 0.40 51.40 I (-4At)
AL 28 0.20 0.083
5 AIR 32 1.0 4.18 I 133.68
total number r;f elements - 240	 I - implicit integration
9 - explicit integration


Mixed time integration methods for transient thermal analysis of structures, appendix 5

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