MULTIFUNCTIONAL COLLABORATIVEMODELING AND ANALYSIS METHODS IN

ENGINEERING SCIENCE

Jonathan B. RansomNASA Langley Research Center

Mail Stop 240, Hampton, VA 23681j.b.ransom@larc.nasa.gov

FEMCI Workshop 2001Innovative FEM Solutions to Challenging Problems

May 16-17, 2001

CONVENTIONAL MODELING AND ANALYSISOF COMPLEX SYSTEMS

• Multi-fidelity FE Modeling• Requires months to model• Changes expensive, time consuming and error prone• Model often tied to discipline

Multifunctional Collaborative Methodology

• Multi-fidelity• Multiple Methods• Multiple Disciplines

OBJECTIVES

• Present general methodology providingcapability for multifunctional modeling,analysis and solution

• Identify computational aspects and relatedalgorithmic issues for this methodology

• Demonstrate the formulation to scalar- andvector-field applications in engineeringscience

KEY TERMINOLOGY• Multifunctional - Computational methodologies for

rapid, robust solutions featuring multi-fidelity modelingand multiple methods

• Collaborative - Mechanism by which two or morephysical domains or methods are integrated/interfaced

• Engineering science - Broad spectrum of engineering(science, mathematics, numerical analysis)

• Homogeneous modeling - Same spatial modelingapproach among subdomains

• Heterogeneous modeling - Different spatial modelingapproaches among subdomains

Example - Previous interface technology demonstratedcollaborative method for homogeneous FE modelingapplied to solid mechanics

MULTIFUNCTIONAL METHODOLOGY

Multifunctional Methods Concept and General Formulation

Multiple MethodInterfacing

Non-FEM ModelInterfacing

FEM ModelInterfacing

Multiple DisciplineInterfacing

Homogeneous Modeling Heterogeneous Modeling

Structures

Materials

Electromagnetics

Acoustics

Aerodynamics Flight Systems

? ? ? ?

? ?? ? ? ?

Finite elementmodeling

Finite differencemodeling

? ?

? ?

Boundary elementmodeling

Capability which Enables the Synthesis of Diverse Models

Multi-fidelity

Multiple-domain

Multiple methods

Multiple disciplines

OUTLINE

• Multifunctional Formulation– Basic assumption– Method of weighted residuals for MFC approach– Collaborative interface treatment– General system of equations

• Selected Applications

• Concluding Remarks

? ? ? ?

? ?? ? ? ?

Pseudo-nodes

CollaborativeInterface

Finite elementmodeling

Finite differencemodeling

? ?

? ?

Boundary elementmodeling

MULTIFUNCTIONAL FORMULATION-BASIC ASSUMPTION-

Deformation, v, along the interface connecting thesubstructures, ? i, may be expressed as:

where:

• T is an interpolation matrix formed using cubic splines

• qI is a vector of interface degrees of freedom

v = TqI

MULTIFUNCTIONAL FORMULATION- METHOD OF WEIGHTED RESIDUALS -

Define: ? ? 0I

ps

1cc

1???? ??

??

N

i

N

mm

ii

ssRRRR

3t̂2t̂

? ?

? ?

? ?

? ?? ? ? ?

Vu1

u2

u3

1t̂ ? ?

where the orthogonalized residuals associated with:

0RFR ??? ??

mmmmm

dGoverning equation within the domain

? ? 0uv?R ???? ? ?? ?

I1

c dI

p

i

i

n

jjij

Compatibility constraint for primary variableon interface boundary

Compatibility constraint for secondary variableon interface boundary

0t?R ??? ? ?? ?

I1

c dˆˆI

s

i

i

n

jji

MULTIFUNCTIONAL FORMULATION- COLLABORATIVE INTERFACE TREATMENT -

substituted into:

0d ??? ??

mmmmm

RFR

? ? 0d I1 I

p ???? ? ?? ?

i

i

n

jjijc uv?R

0dˆˆI

1 I

s ??? ? ?? ?

i

i

n

jjic t?R

Assume for each interface i:

jjijjjijjj S?T?aStTqvqNu ????? and,ˆ ˆ,, I

? ?

? ?

? ?? ?

? ?? ? ? ?

V1q

Iq 3q

2q

3t̂

2t̂

1t̂

mm NF ?FE domains:

FD domains : ? ? ? ?llllm yxyyxx ,, ?? ????F

For weight functions:

MULTIFUNCTIONAL FORMULATION- GENERAL SYSTEM OF EQUATIONS -

General matrix form for multifunctional collaborative approach

For scalar-fields

??

??

?

??

??

??

??

??

?

??

??

?

???

?

?

???

?

?

00f

aqq

0KKK00K0K

ITIp

I

s

• Sparse

• Non-positive definite

• May be unsymmetric

Matrix characteristics:

? ?

? ?

? ? ? ?

? ?? ? ? ?

V1q

Iq 3q

2q

For vector-fields

where are dependent on fluid mechanics formulation and is discipline-specific

aKKK ,,, TIpsu

??

??

?

??

??

??

??

??

?

??

??

?

???

?

?

???

?

??

??

??

?

??

??

?

???

?

?

???

?

?

00f

aqq

0KKK00K0K

aqq

00000000M

ITIp

I

s

I

..

• Patch Test Example Problems

• Torsion of Prismatic Bar

• Heat Conduction Problem

• Potential Flow Problem

• Plane Stress Problem

• Plane Flow Problem

• Boeing Crown Panel

• Douglas Stub-Box

APPLICATIONS

HEAT CONDUCTION PROBLEM

x

y

1.0

T=0

T=00?

??

xT

0???

yT

1.0

Square Plate

x

y Interface

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.0 0.2 0.4 0.6 0.8 1.0x

kT(x,0)Q

Analytical SolutionRitz Approx.MD/FE- C/CMD/FE -F/FMD/FE - C/FMD/FE - F/C

Interface

FE/FE Modeling

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.0 0.2 0.4 0.6 0.8 1.0x

kT(x,0)Q Analytical Solution

Ritz Approx.MD/FD- F/FMD/FD -C/FMD/FD - F/C

Interface

FD/FD Modeling

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.0 0.2 0.4 0.6 0.8 1.0x

kT(x,0)Q

Analytical SolutionRitz Approx.MD/HM- F/FMD/HM -C/FMD/HM - F/C

Interface

FD/FE Modeling

0.0 0.2 0.4 0.6 0.8 1.0x

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

kT(x,0)Q

0.0 0.2 0.4 0.6 0.8 1.0x

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

kT(x,0)Q

0.0 0.2 0.4 0.6 0.8 1.0x

0.35

0.30

0.25

0.20

0.15

0.10

0.05

0.00

kT(x,0)Q

PLANE STRESS PROBLEM

2a

2b? ?0xN ? ? ? ? hN xx 00 ??

x

yr

?

R0

0

0

?

?

t

n

T

T

a

b

0?? tn TT

0?u

x

y

0?v

? ?0

0

?

??

t

xn

T

T ?

r

A B

D

C

Two Configurations:Infinite plate:2a/R0 = 402b/R0 = 20

Finite-width plate:2a/R0 = 42b/R0 = 2

Plate with Central Cutout

E=10,000 ksi?=0.3h=0.1 in.

Geometric Configuration

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

r/R 0

N θ

(N x) nom

AnalyticalMultiple-Domain(FD/FE)Multiple-Domain(FE/FE)

Edge ofCutout

Interface

STRESS RESULTANT DISTRIBUTIONFOR FINITE-WIDTH PLATE

? ? ? ? hN nomxnomx ??? ? ???

??? ????

bR

bhhRbAnet0

0 1222 ; ? ?net

nomx AP

?? ;

Interface

Spatial Modeling

Alongline AB

Alongline CD

A B

C

D

A B

C

D

Stress Resultant Contours

COLLABORATIVE METHODOLOGY DEMONSTRATED ONBOEING CROWN PANEL

Hat StiffenerGapJ-Frame

Mouse Hole Area

Crown Panel Geometry

68 in.

64 in.

Radius = 122 in.

COLLABORATIVE METHODOLOGY DEMONSTRATED ONBOEING CROWN PANEL

Mouse Hole Area

Gap Combined Finite Element ModelNested Finite Element Models

Deformation Contour Hoop Stress Contour

Deformed Shape

Nonlinearly Deformed Unstiffened BayAccess Door Cutout

Tipload(kips)

x

Gages C/D

C

D

20000-2000-4000-6000-80000

40

80

120

160

Gage CGage D

Gage CGage D

Axial Strain (microstrains)

Test

Analysis

xx

Gage A

Tipload(kips)

20000-2000-4000-6000-80000

40

80

120

160

Gage AGage B

Gage AGage B

Circumferential Strain (microstrains)

Test

Analysis

Gage BGage A

xx

Gage A

xx

Failureline

Local Modelof NonlinearRegion

APPLICATION OF COLLABORATIVE METHODOLOGY IN NONLINEAR ANALYSIS OF WING STUB BOX

Tip load

Extension structure

Compositewing stub box

Hydraulicjack

Experimental Setup

SUMMARY

• Results presented for patch test, scalar-field, and vector-field problems

• Results for all problems and multifunctionalapproaches in overall good agreement

• Finite element solutions more accurate thanfinite difference solutions for discretizationsconsidered

• Results with heterogeneous modeling notas accurate as homogeneous modeling

CONCLUSIONS

• Multifunctional collaborative methodologiesand analysis procedures formulated and placedon solid mathematical foundation– Scalar-field and vector-field problems– Homogeneous and heterogeneous modeling

• Collaborative role of modeling approaches hasbeen illustrated

• Capabilities demonstrated on benchmarkproblems and large scale applications

• Computational challenges overcome

• Application of FD method limits general use