An Optimization-Based Approach to Membrane Problems Arising in the

Analysis of Gossamer Structures

Frank Baginski, Michael Barg & Stacey Vezina, Department of Mathematics,

The George Washington University, Washington, DC 20052, baginski@gwu.edu

Laura Cadonati, MIT, LIGO Project, NW17-161, 175 Albany St.,

Cambridge, MA 02139, cadonati@ligo.mit.edu

William Collier, Flight Software Maintenance Group, Computer Sciences Corporation,

7700 Hubble Drive, Lanham-Seabrook, Md. 20706, wcollier@csc.com

Willi W. Schur, Physical Sciences Laboratory, NMSU, NASA-GSFC-WFF,

Wallops Island, VA 23337, willi.w.schur.1@gsfc.nasa.gov

Supported with funding from NASA’s Balloon Program Office, a mathematical model is being developed for theanalysis of large scientific balloons. Initially focusing on fully inflated and partially inflated zero-pressure natural-shape balloons, our approach has proven to be adaptable to a variety of problems that lend themselves to avariational formulation with an optimization-based solution process. Our model is implemented into a programcalled EMsolver.

We present a collection of membrane problems for gossamer structures, including natural-shape balloons,pumpkin balloons, a solar neutrino balloon, and an NGST-like sunshield. In each of these applications, thestructure consists of membrane elements and reinforcing tendons. Wrinkling in the membrane is modeled viaA. C. Pipkin’s energy relaxation approach.

Nomenclature

S - complete balloon shape.Ω - complete reference configuration.FI - discretized configuration in IR3

FI - flat unstrained reference configuration in IR2.T - a triangle in FI

T - a triangle in FI.ng - number of gores in a complete shape.rB - bulge radius for the ideal pumpkin gore.ET - total energy of the balloon system.EP

R

Ω

12bz2 p0zk dS - hydrostatic pressure potential energy.

E f R

S wf zdA - gravitational potential energy of the film.

Et ngwt

R Lt

0 τs k ds - gravitational potential energy of the load tendons.Etop wtopztop - gravitational potential of top fitting.St ng

R

Γ W

t εds - relaxed tendon strain energy.Sf

R

ΩW

f dA - relaxed film strain energy.

dS NdS, N is the unit outward normal to S .dS is surface area measure on S .dA - area measure in the reference configuration,ds - arc length measure along Γ,τs IR3 is the position of the load tendon.

Variational Formulation and Optimization-Based SolutionSee Baginski & Collier (2001)

Total potential energy:

E

T S EPE f Et EtopSt Sf

EP - hydrostatic pressure potential (lifting gas)E f - gravitational potential energy of film Sf - relaxed strain energy of film

Et - gravitational potential energy of load tendons St - relaxed strain energy of load tendonsEtop - gravitational potential energy of top fitting S - membrane

Optimization Problem

For S C ,minimize: E

T S

subject to: GS 0

EMsolver uses Matlab’s fmincon to solve a discretization of Problem .

C - class of shapes; G - constraints

Design parameters and related constants for a ULDB Phase IV balloon. Comparing a

standard gore design and a supergore design.

Description ULDBTop fitting weight (N) wtop 831

Cap weight density (N/m2) wc 0.18387Cap thickness (µm) hc 10.8

Film weight density (N/m2) wf 0.3440Film weight with caps (N) 16,562

Film thickness (µm) hf 38.1Youngs modulus (MPa) E 404.2

Poisson ratio ν 0.825Tendon weight density (N/m2) wt 0.094

Tendon weight (N) 4,144Tendon stiffness (MN) Kt 0.651

Payload (N) Lbot 20393Specific buoyancy (N/m3) b 0.0763

Number of gores ng 290Cap length (m) lc 49.45

Constant pressure (Pa) p0 135Target volume (mcm) 0.515

Curved edge length (m) Ld 152.03Center gore length (m) Lc 152.02

Pumpkin generator length (m) Lp 150.02Length of tendon (m) Lt 147.57

Wrinkling and film strain energy relaxation

F. Baginski & W. CollierSee Collier (2000) & Baginski & Collier (2001)

Relaxed membrane strain energy density:

W

f T

0 δ1 0 and δ2 0 (slack)

12tEδ2

2 µ1 0 and δ2 0 (wrinkled)

12tEδ2

1 µ2 0 and δ1 0 ( wrinkled)

tE21ν2

δ21δ2

22νδ1δ2

µ1 0 and µ2 0 (taut)

(1)

Relaxed membrane energy:

Sf ilm Z

ΩW

f dA

Λ1-taut, Λ2-slack, other - wrinkled

δ1δ2-plane W

f - quasi-convexification of Wf

•

•

•• •

↑δ

2=−0.5

δ1=−0.5 →

← δ2 = −δ

1/νδ

2 = −ν δ

1 →

Λ2

Λ1

−2

−1

0

1

2

−2

−1

0

1

20

5

10

15

20

δ1

δ2

Wf*

Reference Configuration Natural state of single gore

.LcL

d . . . . .

Ω1

Ω2

Ωn

g

Lc

Ld

The Pumpkin BalloonF. Baginski & W. Schur

Section of pumpkin gore Pumpkin Balloon

t(s)

b(s)

α(s)

j

← z = 0, −P(0) = p0

←z = ztop

, −P(ztop

) = bztop

+ p0

Averaged principal stresses of natural and pumpkin shapes

(a) σ1s meridional stress (b) σ1s

0 50 100 150

0

10

20

30

40

50

60

70

80

s

MP

a

σ1 − short tendon

σ1 − slack tendon

σ1 − film/tendon match

0 50 100 150

0

10

20

30

40

50

60

70

80

s

MP

a

σ1 − short tendon

σ1 − slack tendon

σ1 − film/tendon match

(c) σ2s hoop stress (d) σ2s

0 50 100 150

0

10

20

30

40

50

60

70

80

s

MP

a

σ2 − short tendon

σ2 − slack tendon

σ2 − film/tendon match

0 50 100 150

0

10

20

30

40

50

60

70

80

s

MP

a

σ2 − short tendon

σ2 − slack tendon

σ2 − film/tendon match

(a)-(b) slack tendons; (c)-(d) short tendons (structural lack-of-fit)

Max. principal stress µ2 vs. bulge radius rB for ng-gores.

F. Baginski & M. Barg

0.7 0.8 0.9 1 1.1 1.2 1.3 1.44

5

6

7

8

9

10

11

12

13

14

rB

MP

an

g=200

ng=220

ng=255

ng=290

ng=350

ng=375

ng=400

“Molded” Pumpkin Supergore

F. Baginski & W. Schur

Molded Pumpkin Supergore Flat unassembled configuration

Containment vessel for a solar neutrino detectorL. Cadonati & F. Baginski

Stainless Steel Water Tank18m ∅

Stainless SteelSphere 13.7m ∅

2200 8" Thorn EMI PMTs

WaterBuffer

100 ton fiducial volume

Borexino Design

PseudocumeneBuffer

Steel Shielding Plates8m x 8m x 10cm and 4m x 4m x 4cm

Scintillator

Nylon Sphere8.5m ∅

Holding Strings

200 outward-pointing PMTs

Muon veto:

Nylon filmRn barrier

Ø13700 mmØ10500 mm

OUTER VESSEL SSS

SUPPORTSTRAP

POLARSUPPORTROD

PIPES

Ø8500 mm

INNER VESSEL

Tendon/gore configuration in Borexino

Top mooring rope tobottom endcap

Bottom mooring rope totop endcap

Top endcap

Bottomendcap

Gore seam

−6 −4 −2 0 2 4 6

−1

0

1

2

3

4

5

6

7

8

9

10

vbot,ec

vbot,m

vtop,m

vi*

vn−i

*

vtop,ec

Mooring ropeto top endcap

Mooring ropeto bottom endcap

Polar support

Pipes

EMsolver and ABACUS Results Comparison

Constant Pressure Buoyant Force + Constant Pressure

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1minimum principal stress sp1 (MPa) ABA at seam

ABA midgoreEM at seamEM midgore

0 2 4 6 8 10 120

0.2

0.4

0.6

0.8

1maximum principal stress sp2 (MPa)

arc length (m)

0 2 4 6 8 10 120

1

2

3minimum principal stress sp1 (MPa)

ABA at seamABA midgoreEM at seam EM midgore

0 2 4 6 8 10 120

2

4

6

8

10maximum principal stress sp2 (MPa)

arc length (m)

NGST-like sunshieldF. Baginski & S. Vezina

Principal strains

−0.5 0 0.5 1 1.5 2 2.5

−1.5

−1

−0.5

0

0.5

1

(ST 0.2028) (S

F 4.488)

(EG

0)

(E

F −

0.06

3565

) (E

T

0

)

(f−cnt: 1886) (ETOT

0.01596)strn2/1 L:(m),F:(N),E:(J)

−2

−1

0

1

2

3

x 10−3

Initially flat membrane. Corners displaced 2.5 cm out-of-plane; Fixed center box.

Wrinkle Pattern

−0.5 0 0.5

−1.5

−1

−0.5

0

0.5

1

(ST 0.2028) (S

F 4.488)

(EG

0)

(E

F −

0.06

3565

) (E

T

0

)

(f−cnt: 1886) (ETOT

0.01596)Tension Field

Tension Field − blueTaut −red

Corners: 2.5 cm out−of−plane 0.2 mm in−plane Center square fixed

Nonuniqueness of Ascent ShapesBaginski (2002)

References F. Baginski and W. Collier, “Modeling the shapes of constrained partially inflated high altitude balloons,”

AIAA J., Vol. 39, No. 9, September 2001, pp. 1662-1672. Errata: AIAA J., Vol. 40, No. 9, September2002, pp. 1253.

F. Baginski and W. Schur, “Structural Analysis of Pneumatic Envelopes: A Variational Formulation andOptimization-Based Solution Process,” inflated high altitude balloons,” AIAA J., Vol. 41, No. 2, February2003, pp. 304-311.

F. Baginski, “Nonuniqueness of strained ascent shapes of high altitude balloons,” COSPAR02-A-01591, TheSecond World Space Congress/34th COSPAR Scientific Assembly, Houston, Texas 10-19 October 2002.

F. Baginski and J. Winker, “The natural shape balloon and related models,” 34th COSPAR Scientific As-sembly, PSB1-0009-02, Houston, Texas, 10-19 October 2003.

W. Collier, Applications of Variational Principles to Modeling a Partially Inflated Scientific Research Balloon,Ph.D. Thesis, Department of Mathematics, The George Washington University, January 2000.

W. W. Schur, “Analysis of load tape constrained pneumatic envelopes”, 40th AIAA/ASME/ASCE/AHS/ASCStructures, Structural Dynamics, and Materials Conference, AIAA-99-1526, St. Louis, MO, April 1999.

A. C. Pipkin, “Relaxed energy densities for large deformations of membranes,” IMA Journal of AppliedMathematics, Vol. 52, 1994, pp. 297-308.

Acknowledgement

F. Baginski, M. Barg, W. Collier and S. Vezina were supported in part by

NASA Awards NAG5-5292 and NAG5-5353.