[American Institute of Aeronautics and Astronautics 38th Structures, Structural Dynamics, and...

Copyright© 1997, American Institute of Aeronautics and Astronautics, Inc.

AIAA-97-1378

DESIGN TOOL FOR INFLATABLE SPACE STRUCTURES

Arthur L. Palisoc and Yuli Huang*L'Garde, Inc.

Tustin, California

Abstract

A finite element code was developed for theprediction of the on-orbit static and modal dynamicperformance of inflatable antennas and inflatable solarconcentrators. The computer program for the FiniteElement Analysis of Inflatable Membranes (FAIM) is ageometric nonlinear finite element solver with nonlinearmaterial capability. The code was interfaced to an RFantenna code, a ray-tracing code, and a commerciallyavailable graphical pre- and post-processor. The resultwas an integrated set of tools for the analysis and designof inflatable antennas and concentrators. This enables oneto design an inflatable antenna and predict its surfaceaccuracy and RF antenna parameters. For solarconcentrators, the companion code, RAYTRK, calculatesthe solar collected intensity and concentration ratios. Thecode calculates the deformations and stresses due to theapplied loads and outputs the stiffness and mass matricesfor use by a companion code that calculates the naturalfrequencies and mode shapes.

Introduction

Inflatable structures have been shown to possesstremendous potential as antennas and reflective devices inaerospace applications. They are very lightweight andpackaging volume reductions achievable are extremelyimpressive. Their low weight and small packaging volumepermit even larger diameter inflatable antennas to belaunched from only ATLAS-type boosters. NASA, JPL,and L'Garde, Inc. have teamed up and launched a14-meter diameter inflatable antenna on May 20, 1996.The antenna was launched from the Space ShuttleEndeavour (STS-77) and was called the InflatableAntenna Experiment (IAE).1'2

For these inflatable structures to be effective, theirgeometric shape must be accurately determined andcontrolled when subjected to pressure, thermal, anddynamic loads. Solar reflectors for example require themembrane to assume and maintain a parabolic shape of

Copyright © 1997 by the American Aeronautics andAstronautics, Inc. All rights reserved.

considerable accuracy. The most critical issues of concernare their long term survival, structural integrity, and theirsurface accuracy.

The success or failure of inflatable antennas andreflectors is directly tied to their ability to maintain asmooth parabolic shape. One of the major issues that mustbe solved is what is commonly referred to as the inverseproblem. This may be stated in the form of a question:what must be the initial shape of an inflatable shellstructure such that it attains a smooth parabolic surfaceafter experiencing large structural deformations?Specifically, a parabolic surface that maintains a surfaceslope on the order of 0.001 radian must be achieved afteran initially prescribed surface has undergone a largedeformation due to the inflation pressure. Furthermore,the fact that these structures are constructed fromspecially-shaped flat gores, lead to structural models thatdefy closed-form solutions. L'Garde, Inc. fabricates thesestructures from precision-cut, flat, pre-shaped gores thatare joined together by a doubler material at the seams. Inthe early 1980's L'Garde, Inc. developed the PLATE codeto calculate the precise gore shapes.3'4 It became obviousthat an analytical tool able to simulate the deformation ofinitially flat gores upon inflation is needed. Do they formthe desired parabolic shape?

It was in the mid-1980's that L'Garde, Inc. developeda finite element modeling program, FAIM (Finite ElementAnalysis of Inflatable Membranes), for the analysis of thedeformations of inflatable membranes. From 1994 to1996, through a Phase I and Phase II SBIR contracts fromNASA-JPL, L'Garde further improved FAIM by (a)adding nonlinear material capability, (b) adding a modaldynamic capability, (c) interfacing with a commerciallyavailable graphical pre- and post-processor, (d) inter-facing with an RF antenna code, and (e) interfacing witha ray-tracer.5'6

The FAIM Code

FAIM Description

FAIM is a specialized, problem-dependent, materialand geometrically nonlinear finite element program. It isiterative in nature; an initially prescribed, uninflated

Arthur L. Palisoc, Member; Yuli Huang, Senior Member

2922American Institute of Aeronautics and Astronautics


structural shape is input to be analyzed for stress anddeformations. An 8-node isoparametric quadrilateral, a6-node isoparametric triangle, and a 3-node tension-onlycable are the basic elements in FAIM. A consciousdecision was made early on to discard the computationaladvantage of the 4-node quadrilateral and the 3-nodetriangle shell elements since these elements were deemedtoo crude to provide the requisite deformationinformation. Also, because the need to include the effectof the (thicker) seams was recognized early on, a thirdelement was added to FAIM's library. It is a 3-nodetension-only cable element. It is true that the thickerseams could be modeled by the quad and triangleelements but the seam width is much smaller than the gorewidth. Modeling the seam this way would result in aprohibitively large number of elements. We have modeledthe seams as thicker quads and triangles for the muchsmaller F.E. model of an axisymmetric on-axis inflatableantenna by considering only half of the gore.7 The resultsare in very good agreement with a similar model but usingcable elements for the seams. The 3-node cable elementallows one to simulate inexpensively, the local stiffeningeffect produced by the seams.

FAIM's basic elements are shown in Fig. 1. Figure 2shows how the flat gores are joined together to form theinitial unstressed antenna profile. A finite element modelof the antenna in Fig. 2 is shown in Fig. 3. Because ofazimuthal symmetry, half of a gore may be used in themodeling.

'L

Fig. 1 (a) 8-node quadrilateral, (b) 6-node triangle

Isometric ViewTop View

Fig. 2. Flat gores joined at the seams to form the initialunstressed antenna profile

Fig. 3. Finite element model of an (axisymmetric) on-axis parabolic antenna

FAIM Analysis Capabilities

A number of enhancements have been made to FAIMover the years in order to increase its analysis capabilitiesas well as its computational efficiency. A "springboundary condition" capability was coded in anticipationof the need to characterize the outer radius mountingsurface of the parabolic shell to a flexible rather than aninfinitely rigid support. Another feature available in FAIM



is what is called a "skew" boundary condition. Referringto the axisymmetric finite element model depicted in Fig.3 we note that this model required the x-y-z displacementsat the outer radius to be zero as well as specifying that thecircumferential displacements along the radial edges bezero due to symmetry conditions. The symmetryrequirement on the circumferential displacementnecessitated an implementation of the "skew" boundarycondition of the finite element code. A skew boundarycondition means a node is restrained to move along aplane not parallel to the global x-y, y-z, and x-z planes.

The significant computational efficiencyenhancements to FAIM include a new solution strategyfor solving the large number of simultaneous equations aswell as a "restart capability". The iterative solution of thenonlinear equilibrium equations associated with largedeformations of shell membranes leads to a set ofsimultaneous, algebraic equations. There are threesignificant contributions to the coefficient matrix of theseequations: (1) elastic terms, (2) geometric terms, and (3)pressure terms. Whereas (1) and (2) lead to a symmetriccoefficient matrix, the third leads to a skew symmetric.The combination of these three effects leads to anunsymmetric matrix. FAIM has two simultaneousequation algorithms for solving the equilibrium equationof a shell membrane model: (1) a symmetric, bandedcoefficient matrix and (2) an unsymmetric, banded,coefficient matrix solution. Both solution strategies areimplemented in-core. This has an obvious advantage anddisadvantage at the same time. Because it is an in-coresolution, it is very fast. However, for the same reason, itcan analyze only a problem "small enough" that will fit inmemory.

The symmetric solution strategy takes somenumerical shortcuts to achieve efficiency. Although thecombination of the three effects mentioned above leads toan unsymmetric matrix, one may ignore the minorcontributions of the pressure terms by assuming that theassembled matrix is symmetric. The computationalshortcut significantly reduces the execution time of atypical analysis by a factor of 5 to 10 and requires onlyabout half as much computer storage. Once a satisfactoryequilibrium state has been achieved to a desired degree ofaccuracy with the symmetric albeit slightly inaccurate,coefficient matrix, one can repeatedly solve thesimultaneous equations with the proper unsymmetriccoefficient matrix. This is tantamount to using thesymmetric solution strategy to obtain a goodapproximation to the deformed shell membrane followedby the second strategy in conjunction with the restartfeature, to obtain the final deformed shell membrane.Experience to date has shown that in most cases, theskew-symmetric contribution from the pressure terms is

negligible so that the symmetric and unsymmetricequations solvers give nearly identical results.

Because most materials of interest to inflatablestructures technology exhibit nonlinear material behavior,it became imperative that the analytical tools used mustalso be able to model the same behavior. Furthermore, thenatural frequency of inflatable systems must be known inorder to determine whether or not it could interactundesirably with closed-loop control systems. These arethe two most recent capabilities added to FAIM: (a)nonlinear material and (b) modal dynamic capability. Thetwo significant material models in FAIM are (a)orthotropic material properties and (b) orthotropic plasticbehavior.

The companion code to FAIM, SME (ShellMembrane Eigenvalue) uses the subspace iterationalgorithm to calculate the eigenvalues (naturalfrequencies) and eigenvectors (mode shapes). Input isobtained from the initial FAIM run where the membraneundergoes loading. The input to SME consists of thestiffness and mass matrices.

The loadings available in FAIM are: (a) followerpressure, (b) concentrated nodal forces, (c) 3D arbitrarybody force accelerations, and (d) nodal or elementtemperatures.

Bandwidth and Wavefront Solution Methods

The most efficient, in-core procedure for solving a setof simultaneous linear equations is the direct Gaussianelimination algorithm. The banded stiffness matrix ofmost finite element formulations, such as the one used inFAIM provides economy in both storage as well asequation solution (bandwidth). There are times, however,in which alternate methods must be considered when thebanded stiffness matrix is too large to fit within corestorage. The wavefront technique is one such method.8This technique has earned the reputation of being easy andinexpensive to use. This procedure, which uses the frontalmethod of equation assembly and reduction, is similar inscope to the Gaussian elimination algorithm. Theinterested reader may consult the references for a moredetailed description of the method.

Bandwidth and Wavefront Minimization

When either the bandwidth or the wavefront methodof solution is used, the user must construct the inputgeometry configuration in such a way that the bandwidthor the wavefront size is minimized in order to conservedisk storage, main memory usage, and reduce CPUexecution time. In some representative membrane



problems, the execution time has been reduced by 1 to 2orders of magnitude by simply finding the optimum nodenumbering to minimize the bandwidth. One problemanalyzed by the wavefront method was observed to beoscillating and not converging up to about 1000 iterations.When the input geometry was optimized for the wavefrontsolution, convergence was achieved in only 5 iterations!FAIM is equipped with a bandwidth and wavefrontminimization routine. It uses the GPS algorithm (Gibbs,Poole, and Stockmeyer) for bandwidth minimization andthe GK (Gibbs, King) algorithm for wavefront sizeminimization.9

A Pre- and Post-processor for FAIM

Individuals familiar with computer-aided finiteelement analysis will appreciate the fact that the datapreparation (i.e., the modeling part) can take up to 60 to80 percent of the entire finite element analysis task. Andthis is with the use of a computer graphical pre-processor.The input phase of a CAD-type data preparation for finiteelement analysis involves defining the geometry,determining the nodal coordinates, and forming theelement connectivities. This is usually followed bydefining the material and physical properties. Theloadings and boundary conditions must likewise bespecified. In a computer environment the whole processis usually done in a "windows-type" of environment withpull-down menus and a "mouse-driven" cursor for"clicking" on the desired parameter(s). The pre- andpost-processor by Engineering Mechanics and ResearchCorporation (EMRC), called DISPLAY III was selectedas the pre- and post-processor to use for FAIM.10'" Theonly reason for the choice is that we already haveDISPLAY III in-house for use with the NISA finiteelement package by the same company.

While executing, FAIM displays the program statusin real time on the computer monitor. Currently, thisgraphics capability is only for the PC, but it is not adifficult task to do the same for other CPU platforms. Onthe screen, the user sees (in graphic and numeric format)the convergence status at the current iteration, themaximum number of iterations allowed, and theconvergence criterion, among others. Furthermore, theuser may increase the number of iterations if theconvergence is slow or tighten or relax the convergencecriterion in real time.

FAIM Family of Programs

Figure 4 shows how a typical membrane analysisproceeds. Table 1 describes the functions of each of theseutility programs. In the table, NECREF is an RF antennacode by Ohio State University.12

'*'VnttfilrMm*iu&i'nn~''tiTSWP III1JIM1WW .

SHEStMl Membmrw

Efexnb.

'rSMENS

SME-TO4USA

\

**ii

—— *

FM2NSFMU-TO-MU

is?* iCMltlO

NUADWUTdB

£i«idfS$R\t>;^<^

Fig. 4. FAIM Family of Programs - FAIM AnalysisProcedure

Table 1. The FAIM Family of Utility Programs

FAIM_DA

FAIM_SA

NISA-DISPLAYIII

NS2FM

FM2NS

ONGEN

OFFGEN

REN

RAYTRK

NECREF

SME

FAIM code that uses the bandwidthtechnique.FAIM code that uses wavefrontstrategy.Pre- and post-processor for FEM byEMRC10-""NISA-to-FAIM". Code thattranslates DISPLAY III output toFAIM format.Code that converts FAIM output intoDISPLAY format for post-processingCode that generates FAIM model ofon-axis membrane antenna/reflector.Code that generates FAIM model ofoff-axis membrane antenna/reflector.Nodal/element renumbering programused to minimize bandwidth orwavefront. Input to REN must be inFAIM input format.A ray tracing code interface toFAIM.Antenna code by Ohio StateUniversity."Shell Membrane Eigensolver".Calculates eigenvalues andeigenvectors of shell membraneproblems.



AnalyticalAntennas

FAIM Code Verification

Model for On-Axis Seamless Parabolic

An exact solution has been derived for the "inverseproblem" described above.3 This is depicted schematicallyin Fig. 5. Given the desired inflation pressure p0, and finalparabolic shape, what must be the initial uninflated shapeso that on inflation to the pressure p0, the desiredparabolic shape is obtained?

IWaflitodStap. Flml (Daired) Shtpe

Fig. 5. Inverse problem solved by PLATE code.3'4

This analytical model was used to calculate theuninflated shape for a seamless paraboloid. The results areexpressed in terms of the dimensionless parameters (Z/2F)and (r/2F) where F is the paraboloid focal length, z is thevertical displacement, and r is the radial distance from thevertex. The result for three different values of thePoisson's ratio is shown in Fig. 6.

Each of the three different uninflated membranes hasbeen specified to FAIM as a seamless shell consisting ofa given number of doubly-curved gores. The bottomperimeter of each of the uninflated membranes has beendisplaced prior to the run, to r/2F=2.5 to simulate thestretching of the membrane on the frame prior to inflation.It is evident from Fig. 6 that there is excellent agreementbetween the FAIM results and the analytical solution. Inthe figure, p is the inflation pressure, E is Young'sModulus and t is the membrane thickness.

Large. Deflection of Inflatable Membrane

Inflated Polj»on'f r»Uo=0.1, 0.3, 0.5

O.a

WZFJ

Fig. 6. Comparison of exact paraboloid with FAIM-calculated inflated shapes for (pF/Et) = 1

Pressurized Circular Membrane

A classic series-solution for the stress anddeformation of a pressurized circular membrane may bederived.13 The finite element approximation to the largedeformation of the circular membrane was obtained bydiscretizing a 15° wedge of the membrane with 42quadrilateral and 3 triangle elements as shown in Fig. 7.Figure 8 shows the results for the circular membrane.

Fig. 7. FEM model of pressurized circular membrane



100

LEGEND

a Tluonttcot we FWta Ehnunt W

JJ t.0 7.SNond!men«tonal Radiut r/o

Pressurized Circular MembraneFig. 8a. Nondimensional deflections of circularmembrane

LEGENDo Flnlt* Bwmnt Sr< Th««»«MlSh* nnlU B«m»nt Sh

10 4.0 «,0 IJ

Nondtmonstonot Radlu* r/a

Pressurized Circular Membrane

104 |

(.10-) I

As a means of checking the modal analysis capability,the frequencies and mode shapes of a flat "drumhead"subjected to a uniform tension were determined forvarious finite element models. The theoretical results aredetermined from the zeros of the Bessel Functions J0, themembrane tension, and the mass density per unit area.14

The SME code computed the first 5 frequencies toless than 0.25 percent of the theoretical values whereasthe FAIM and exact mode shapes were indistinguishablewhen plotted to a common scale. The first 5 radial modeshapes are shown in Fig. 9. The parameters of the circularmembrane are given in Table 2.

Table 2. Parameters of Circular Membrane

Radius, aTensionAreal density

100 inches1151b/inch7.5 x 10-4 Ib-secVin3

Circular Membrane Mode Shapes1 Deq Segment - Radial Modes Only

t.oo

0.1 0.2 &3 0.4 0.5 0.8 0.7 0.8 0.9Nondlmenstanal Radial Coordinate

Fig. 9. First five radial modes of circular membrane

The Rectangular Membrane

The parameters of the rectangular membrane studiedare listed in Table 3. Table 4 compares selected modeshapes calculated by FAIM-SME against theory.

Table 3. Parameters of Rectangular Membrane

Fig. 8b. Nondimensional radial and hoop stress

Length, aWidth, bTensionDensity

24 inches12 inches1 Ib/inch1.29x lO^lb-secVin4



Table 4. Comparison to Theory

ModeNo.

136

1520

m

12338

n

111

31

Theory

258.84 Hz

328.13 Hz4 18.28 Hz

778.22 Hz956.64 Hz

FAIM

258.75 Hz327.30 Hz417.22 Hz

778.77 Hz955.92 Hz

% Diff

0.004%0.25%0.25%0.07%0.08%

FAIM Interface to NECREF and RAYTRK

Interface to NECREF

We have coded an interface computer programbetween FAIM and an RF antenna code. Specifically, weused the Ohio State University's NECREF antenna code.12

The current version of the OSU Reflector Antenna codehas the capability of analyzing reflector antennas withcertain types of surface distortion from an ideal parabolicsurface. However, the code does not have the capability ofaccepting an actual antenna surface profile which isobtained either from measurement of the surface or froma numerically generated surface such as that generated byFAIM. We have modified NECREF so that one can nowinput the nodal coordinates of the surface as it is outputfrom FAIM. Figure 10 shows the interpolated surface fora 3m antenna formed from 36 flat gores. TheFAIM-calculated profile has a surface accuracy of 0.3 mmrms. The NECREF-calculated H-Plane patterns for 8 and25 GHz are shown in Figs. 11 and 12.

contour L«WI; Mn.-o in jMax-0.2 m |tncram«nt»0.01 m

-1.6-1.5 -1.2-0.9 -0.6 -0.3 0 0.3 0.6 0.9 1.2 1.5

____ ____________X (METER) ____________

Fig. 10. Interpolated surface for 3-meter antenna

PHI-O o«g.FREO-aQHzPEAK-4635 dBDEM. SURFACE

PHI-0 D«a.FHECUeQHzPEAK-48.74dBTRUE SURFACE

-10 0 10THETA(DEQ)

Fig. 11. Calculated H-plane patterns of 3m antenna at8 GHz

PHM)D»g.FRECU25GHZPEAK.66.75dBIDEAL SURFACE

PHM3D«g.FREQ.-25 GHzPEAK-55.71 dBREAL SURFACE

-10 0 10THETA(DEQ)

Fig. 12. Calculated H-plane patterns of 3m antenna at25 GHz



Interface to RAYTRK

RAYTRK, a ray-tracing computer code wasdeveloped to assess the performance of parabolic solarconcentrators. It is designed to take as input, the updatedsurface generated by code FAIM. This updated surfaceprofile is fed into RAYTRK in order to generate auniform grid distribution over the reflector surface. Then,a ray of light originating from a point on the solar disk istraced towards a grid point (and every point on thereflector thereafter) and towards a focus plane. Theintersection of this reflected ray with the focus plane iscalculated. The results of one study using FAIM andRAYTRK are shown in Fig. 13

Fig. 13. Concentration ratios

Summary and Conclusions

We have developed a finite element analytical toolfor the analysis and design of inflatable membranestructures. Its capabilities include (1) nonlinear materialmodel and (2) modal dynamics capability. A pre- andpost-processing interface program was coded and testedwhich renders the whole analysis/design process verystreamlined. The loadings available in FAIM are (1) nodalor element thermal loads, (2) general 3D G-loads, (3)concentrated point loads, and (4) follower pressure.Specified nodal displacements as well as skew boundaryconditions may be used. Furthermore, a nodal and elementreordering capability was also coded to reduce computerstorage and increase execution speed. This capability is avery important feature in that it significantly reducescomputer run time.

In order to verify code performance, we compared theFAIM calculations to an analytical solution for asymmetric paraboloid. FAIM calculation results have alsobeen compared to other verification cases that include thefollowing, some of which were not presented in thispaper:

* classical solution of the deflection and vibration offlat circular membranes.

* classical solution of the vibration of rectangularmembranes

* analytical solution to the vibration of sphericalmembranes.

* experimental results on the natural frequencies ofright circular and conical cylindrical tubes.

Finally, in order to increase the utility of the code, wehave written interface programs between FAIM and an RFantenna code and between FAIM and a ray-tracer.Because of the FAIM code and the capabilities coded intoit, we now have the capability of predicting the overallperformance of membrane antennas and reflectors. TheFAIM family of codes may be considered as a completeset of tools for the analysis of the surface precision andstresses within all types of precision inflatable spaceantennas and support structures. It makes the analysis anddesign of inflatable antennas and reflectors a verystreamlined process.

7.0 Acknowledgments

The addition of nonlinear material and modaldynamics capability as well as interfacing FAIM toNECREF and RAYTRK was supported by a Phase I andII SBIR contract from NASA-JPL under contractNAS7-1219 and NAS7-1313, respectively. We also thankour consultant, Prof. Gerard C. Pardoen of the Universityof California at Irvine for his significant contributions.

References

1. Freeland, R. and Bilyeu, G., "In-Step InflatableAntenna Experiment," paper presented at the 43rdCongress of the International AstronomicalFederation, Aug. 28 - Sept 5, 1992, Washington,D.C.

2. Veal, G. and Freeland, R., "In-Step InflatableAntenna Description," AIAA 1995 Space Programsand Technologies Conference, Huntsville, Alabama,Sept. 26-28 ,1995.

3. Thomas, M., "Effect of Inflation Pressure onParaboloid Structure," L'Garde Memorandum,LM-79-MT-054, Aug. 10. 1979.

4. Thomas, M., "Changes in PLATE Code toDetermine Flat Patterns," L'Garde Memorandum,LM-80-MT-082, Dec. 15, 1980.



5. Palisoc, A. and Huang, Y., "Finite Element Analysisof Inflatable Membranes (FAIM)," SBIR Phase IIFinal Report, L'Garde Technical Report,LTR-96-AP-012, May 1996.

6. Palisoc, A., "A User Interface and Finite ElementAnalysis of Off-Axis Inflatable Antennas," SBIRPhase I final Report, L'Garde Technical Report,LTR-93-AP-043, August 1993.

7. Huang, Y., "Test of Cable Elements in FAIM,"L'Garde Technical Report, LTR-96-YH-023,October 1996.

8. Hinton, E. and Owen, D.R.J., "Finite ElementProgramming," Academic Press, London, 1977.

9. Lewis, J.G., "Implementation of the Gibbs-Poole-Stockmeyer and Gibbs-King Algorithms forReordering Sparse Matrices," ACM Transactions onMathematical Software, Vol. 8, No. 2, June 1982, pp.190-194.

10. DISPLAY-III - A pre- and post-processor byEngineering Mechanics Research Corporation(EMRC) for its Finite Element Computer Package -NISA, Troy, Michigan.

11. NISA- Numerically Integrated Elements for SystemAnalysis, A Finite Element Computer Package fromEngineering Mechanics Research Corporation, Troy,Michigan.

12. NECREF - Ohio State University Reflector AntennaCode - Version 3.0, Ohio State University ElectroScience Laboratory, Columbus, Ohio, Feb. 1994.

13. Hencky, H., "Uber den Spannunszustand inkreisrunden Flatten mil vershwindenderBiegungssteifigkeit," Zeits. Math. Phys., Vol. 63, pp.311-316, 1915.

14. Meirovitch, L., "Analytical Methods in Vibration,"McMillan Co., New York, 1971.


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