The finite element codes were improved using Hermitian elements and numerical integration of element relations. The question of real variable versus complex variable formulation was resolved and an integrated civil engineering system was implemented on the Georgia Tech Cyber 70/74. Efforts are underway to restructure the program to obtain the most efficient use of array storage

Horowitz, S. J.

Sigman, R. K.

Zinn, B. T.

English

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s(NASA—CR-164118) ACCUSTiC PROPERTIES CF	 N81-1 834
TUH80FAN INLETS Semiannual HePorte 1 Sep.
1980 — 28 Feb. 1981 (veoryia IuSt. of Teca.)
9 p HC A02/NF A01	 CSCL 20A	 UAC1dSG3/71 42118
Research Conducted Under
NASA GRANT NO. NSG 3036
ACOUSTIC PROPERTIES OF
TURBOFAN INLETS
SEMI-ANNUAL REPORT COVERING PERIOD
SEPTEMBER 1, 1980 - FEBRUARY 28, 1981
Prepared by
Br--n T. Zinn, Regents' Profess.-.)r
Robert K. Sigman, Research Engineer II
Scott J. Horowitz, Graduate Research Assistant
GEORGIA INSTITUTE OF TECHNOLOGY
SCHOOL OF AEROSPACE ENGINEERING
ATLANTA, GEORGIA
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INTRODUCTION
This report is a summary of the work completed under NASA Grant
NSG 3036, entitled "Acoustic Properties of Turbofan Inlets" during the
period September 1, 1980 to February 28, 1981. The work performed during
this period is divided into the following four categories: 1) improvements to
the new finite element codes including the use of Hermitian elements and
numerical integration of element relations, 2) resolution of the question of
efficiency of real variable vs. complex variable formulation,
3) implementation of GTICES, 4) preparation of an AIAA abstract.
FINITE ELEMENT CODES
Throughout the course of this investigation, the finite element
method (FEM) has been used as a technique for solving the acoustic
equations within the inlet duct. As described in previous reports the
procedure for application of the finite element method consists of 6 steps:
1), subdivision of the domain into finite elements 2) selection of
interpolation functicns, 3) calculation of element equations, 4) assembly of
element equation to form a global (rnatrix) equation. 5) solution of global
rnatrix equation, and 6) calculation of additional variables. Items 1,2, and 3
are usually performed as separate operations. Items 5, and 6 are usually
subroutines of the finite element computer codes and are improved regularly
without disturbing the essential structure of the codes. Item number 4 is the
backbone of any FEM, and any change in the approach to the assembly
routine requires the development of a new set of computer codes.
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t
During the course of this investigation, three separate approaches to
the problem of assembling the individual elements into a global matrix have
been developed. The original approach (FEM 1), developed in 1977, used only
linear elements (item 2) and were very efficient. Unfortunately, these codes
lacKed versatility and could not assemble higher order elements without
complete reprogramming. Thus, a more versatile set of codes (FEM 2) was
prepared in 1978 which could assemble either linear or quadratic elements.
These codes were not very efficient, however, resulting in longer run times
and larger mass storage requi rements. In an effort to combine the best
qualities of FF.M 1 and FEM 2, preparation of a new set of codes began in
1979. These new codes are now substantially complete. They are more
versatile than FEM 1 in that they can assemble linear, quadratic, or
Hermitian elements and can be easily modified to consider other higher
order element relations. They are more efficient than FEM 2 in that the
execution times for these more versatile codes are somewhat less than the
execution lines for FEM 1. The new codes have combined the advantages of
FEM I and FEM 2 without the disadvantages of either.
Some additional improvements are listed below.
Numerica l Integration of Element Relation s
When linear elements are employed, (Item 2) the integration of the
weighted governing equations over a typical element (Ite • rn 3) are
straightforward, resulting in simple algebraic expressions. For quadratic
elements. the algebra is more tedious, but the resulting equations are still
reasonable. These algebraic expressions are presented in Reference 1. For
6higher order elements, the algebra becomes overpowering, the chance for
error increases and the computation times increase enormously. For this
reason, it is standard practice among structural finite element programmers
to employ numerical quadrature techniques.
Since the approximating functions (linear, quadratic or Hermitian) are
polynomials, the correct Gauss-Legendre numerical quadrature scheme will
give .ae exact value for each element without resorting to tedious algebra
and with a reduction in computation time. A computer subroutine using a
4th order Gauss-Legendre quadrature scheme is included in the FEM codes.
This quadrature scheme will calculate exact element relations for linear,
quadratic, and Hermitian elements. This subroutine has been debugged and
extensively tested.
Lined Walls
The computer codes have provisions for acoustic liners on the
boundaries using the principle of continuity of particle displacement. In the
pr,&sence of a steady flow, either quadratic or Hermitian elements can be
used. This subroutine has not been verified at this time.
REAL. - VS - COMPLEX ARITHMETIC
It has been claimed by others (2) that real variable programming is
inefficient, and substantial improvements with regard to mass storage and
computation times can be achieved by using complex notation. It has been
our opinion, based on the advice of our computer advisors, that on the CDC
computers, real variable notation was more efficient. While some
computers are "hard wired" for complex arithmetic, CDC (and certain other)
computers perform complex arithmetic using external subroutines. Since
the use of external subroutines is time consuming, it is preferable to use real
variable arithmetic.
In our last report (3) we noted that the new finite element codes
(FEM3) using complex variables were slightly (about 10%) faster than FEM 1
codes using real variable notation. Because the codes were different, no
exact comparison could be made.
For reasons to be described in the next section, the new codes have
been prepared in two forms: one uses complex notation and arithmetic,
wbile the second form uses only real variable arithmetic. The computation
times were found to be:
Complex arithmetic:
	
49 cpu secs
Real arithmetic:
	
27 cpu secs
Thus, on a CDC computer the use of complex arithmetic uses twice the time
of the identical program using real arithmetic. Therefore, all calculations
will be carried out using real variable arithmetic.
^7
INTRODUCTION OF ICES
Roth finite elements and finite differences have been used
successfully to describe sound propagation in ducts. Both procedures
approximate the exact potential (or pressure) by a large number of discrete
values of the potential (or pressure) within the duct. Since it takes
approximately 6 to 10 discrete values to describe one cycle of a sine wave,
an equal number of points must be used for each wavelength of sound. Thus,
as the frequency of the sound propagating in the duct increases, the number
of discrete points or subdivisions must increase. This in turn increases the
size of the matrix which must be inverted, the amount of data which must
be stored in the computers memory, and the computation times. Limitations
on the amount of mass storage available, limit computations to relatively
low dimensionless frequencies (W — 15). it is this limitation on mass storage
which has severely restricted the usefulness of the steady state finite
element and finite difference formulations.
Since computers can store data on disks or magnetic tapes, the actual
mass storage capability of a modern high speed computer is virtually
unlimited. The limitation arises through the operating system which only
permits access to data store y on the activc disk at the time of execution.
Thus if the operating system is modified to permit transfer of additional
mass storage to and from magnetic tapes, the available mass storage will
increase and calculations can be carried out at higher frequencies. 	 i
This problem of assembling and storing large f inite element programs
has been previously addressed b} civil engineers. In fact, the majority of the
development of the finite element method as an engineering tool has been
carried out by structural engineers. The basic concept of a finite element
originated from a simple structural problem. This concept was developed
into a general method for structural analysis and later into a universal
technique applicable to fluid mechanics, acoustics, and other engineering
disciplines.
The motivation for the development of the FEM was the need to
analyze very large structures. Due to the size and complex geometries of
these structures, the strews matrices contained more nodal points than the
computer cou,d maintain in active memory. To overcome these difficulties,
the Civil Engineering Department at MIT developed an operating system
called ICES (Integrated Civ i l Engineering System) which can access memory
stored on peripheral equipment such as magnetic tape or discs. An improved
version of this system, known as GTICES, is available on the Georgia Tech
Cyber 70/74. The Civil Engineering Department employs a fulltime research
engineer, David Green. who is responsible for maintaining and updating this
system. During the past three months, we have been working closely with
Mr. Green in adapting the FEM aciustic codes to the GTICES system.
In order to modify the codes to make them compatible with the
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GTICES system several detail changes are being made. Since the GTICES
system can only access real variables the codes which were written using
complex variables have been rewritten using only real variables. As
previously reported, this change brought about a 406 reduction in execution
time. A necessary change from FORTRAN V to FORTRAN IV language
should be fairl y
 routine. With these changes the codes will execute under
the GTICES system. However, the execution times for programs executed
under GTICES are very dependent on the ordering and structure of the
W-
i	 3
k
program. Thus, considerable time must be spent in re-ordering the arrays
and order of operations. As an example, under the normal operating system,
the coordinates of node 100 are located in two arrays as follows:
X = X A(! 00)
R = RA(100)
Under the GTICES system it is considerably more efficient to store
the location in a single two dimensional array as follows:
X = XR(100,1)
R = XR(100,2)
Efforts are now underway to re-structure the program to obtain the
most efficient use of array storage.
AIAA ABSTRACT
In tfe previous progress report we described the results of a
successful iterative scheme for combining the FEM with an integral
technique. An abstract describing this procedure has been submitted for
presentation at the AIAA Aeroacoustics Conference.
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REFERENCES
l: Majjigi, R. K., R. K. Sigman, and B. T. Zinn, "Application of Finite
Element Techniques in Predicting the Acoustic Properties of
Turbofan Inlets," Interim Technical Report, NASA Lewis Research
Center, 1979.
2. Tag, 1. A. and E. Lumsdaine, "An Efficient Finite Element Technique
for Sound Propagation in Axisymmetric Hard Wall Ducts Carrying
High Subsonic Mach Number Flows," AIAA Paper No. 78-1154,
July 1973.
3. Zinn, B. T., R. K. Sigman, and S. J. Horowitz, "Acoustic Properties of
Turbofan Inlets," Semi-Annual Report Covering Period March 1, 1980
- August 31, 1980.


Acoustic properties of turbofan inlets

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